Rethinking Elementary Mathematics Series Elementary... · TEXTEAMS Rethinking Elementary School...

Rethinking ElementaryMathematics SeriesRethinking ElementaryMathematics Series

Mathematics InstituteMathematics Institutehttp://www.texteams.org

Part IPart IPart I

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page ii

Permission is given to any person, group, or organization to copy and distribute Texas Teachers Empowered for Achievement inMathematics and Science (TEXTEAMS) materials for noncommercial educational purposes only, so long as the appropriate credit is given.This permission is granted by The Charles A. Dana Center, a unit of the College of Natural Sciences at The University of Texas at Austin.

Dwight D. Eisenhower Professional Development Program, Title II, Part BTexas Education AgencyTexas Statewide Systemic Initiative in Mathematics, Science, and Technology EducationCharles A. Dana Center, The University of Texas at Austin

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page iii

Acknowledgements

The Rethinking Elementary Mathematics institute was developed under the direction andassistance of the following:

Advisors/ReviewersJanie Schielack Texas A & M UniversityBonnie McNemar Independent ConsultantBrenda DeBorde Grand Prairie ISD

WritersDinah Chancellor Carroll ISDGeorge Christ Region X ESCWayne Gable Austin ISDPam Littleton Tarleton State UniversityAnn May Independent Consultant

Advisory CommitteeDonna Ashby Longview ISDCatherine Banks Texas Woman's UniversityJanice Bradley University of Texas at AustinErnestina Cano Edinburg CISDDinah Chancellor Carroll ISDGeorge Christ Region X ESCBrenda DeBorde Grand Prairie ISDSusan Empson University of Texas at AustinPaula Gustafson Texas Education AgencyPam Harris Mathematics ConsultantJulia Haun Plano ISDSusan Hudson Hull Charles A. Dana CenterVanessa Jones Austin ISDPam Littleton Tarleton State UniversityMary Longoria Laredo ISDWinifred Mallam Texas Woman's UniversityJose Alberto Marquez Ysleta ISDMerrie Lyn Martinez Edgewood ISDLaurie Mathis Charles A. Dana CenterAnn May Independent ConsultantDiane McGowan Charles A. Dana CenterBonnie McNemar Independent ConsultantBarbara Montalto Texas Education AgencyErika Pierce Charles A. Dana CenterJoAnn Reyes Charles A. Dana CenterJanie Schielack Texas A & M UniversityNita Copley University of HoustonChristi Taff Pearland ISDCarmen Whitman Charles A. Dana Center

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page iv

Table of Contents

About TEXTEAMS Institutes viiInstitute Introduction viiiTEXTEAMS Elementary Institute Organizational Chart for Week 1 ixDaily Materials Summaries xDirections for the Reflection Journal xv

Day One: The Lesson Planning Process

Preparing for Day One 2Section A Opening: Why do teachers need to be clear about what is expected? 4Section B Making Evidence of Understanding Clear to Students 11Section C The Process of Identifying Evidence of Understanding 22Section D Working Through the Lesson Planning Process 31Section E Applying the Lesson Planning Process 40

Day Two: Whole Number Concepts

Preparing for Day Two 2Section A Place Value Puzzles 5Section B Investigating Whole Number Relationships 12Section C Whole Number Sampler 31

Button, Button, Where’s the Button? 40Expanding Numbers 42In and Out 51It’s the Place that Counts 60Make It Zero (Base-ten Block Version) 65Make It Zero (Calculator Version) 71More 76Number, Number, Where’s the Number? 79See Saw 82

Section D Closure for Day Two: Shaping Up a Summary 89

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page v

Day Three: Fraction and Decimal Concepts

Preparing for Day Three 2Section A Fraction Meanings with Cuisenaire Rods 7Section B Understanding Fractions 17Section C Fraction/Decimal Menu Sampler 37

Cookie Sharing 56Fraction Rectangles 57Same Name 58More Same Name 59Shake and Spill 60Fraction Riddles 63Geoboard Fractions 65Tenths 67NOT Tenths 72Hundredths 74Measuring with Decimals 76Real-world Decimals 77Wipe Out ONE! 78Show Me! Tell Me! 79

Day Four: Addition and Subtraction

Preparing for Day Four 2Section A It’s Simply Addition 5Section B Investigating Addition and Subtraction 9Section C Addition and Subtraction Sampler 34

Decimal Addition 40Diffyboxes 46Dollar Addition and Subtraction 49Double More 52Doubles, No Trouble! 55Fill in the Blanks 60In and Out Revisited 77Magic Squares 84

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page vi

Day Five: Multiplication and Division

Preparing for Day Five 2Section A Fair Share 6Section B Investigating Multiplication and Division 18Section C Multiplication and Division Sampler 39

Colored Pencils 48My Flowers 50A Remainder of One 52Leftovers 55Let’s Paint! 58How Long? How Many? 604 In a Row 63Go Figure! 66The Greatest Product Wins 68Going Bananas! 70Multiple Towers 72Fresh Produce 76Fresh Produce Challenge 78What’s In Each Box? 80Tiffany’s Beanie Babies™ 83Marissa’s Garden 85The Greatest Product, Part 2 93Marissa’s Garden Again 97

Section D Closure for Day Five: Conceptual Models of Computational Fluency 102

Appendix

Related Books A-1Quotes A-3Deca… Spinner A-4

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page vii

About TEXTEAMS Institutes

TEXTEAMS Philosophy

� Teachers at all levels benefit from extending their own mathematical knowledge andunderstanding to include new content and new ways of conceptualizing the contentthey already possess.

� Professional development experiences, much like the school mathematics curriculumitself, should focus on few activities in great depth.

� Professional development experiences must provide opportunities for teachers toconnect and apply what they have learned to their day-to-day teaching.

Features of TEXTEAMS Institute Materials

Multiple representations (verbal, concrete, pictorial, tabular, symbolic, graphical)Mathematical ideas will be represented in many different formats. This helps both teachersand students understand mathematical relationships in different ways.

Integration of manipulative materials and graphing technologyThe emphasis of TEXTEAMS Institutes is on mathematics, not on learning about particularmanipulative materials or calculator keystrokes. However, such tools are used in variousways throughout the institutes.

Rich Connections within and outside mathematicsInstitutes focus on using important mathematical ideas to connect various mathematicaltopics and on making connections to content areas and applications outside of mathematics.

Questioning strategiesA variety of questions are developed within each activity that help elicit deep levels ofmathematical understanding and proficiency.

Hands-on approach with “get-up-and-move” activitiesInstitutes are designed to balance intense thinking with hands-on experiences.

The Charles A. Dana Center is approved by the State Board for Educator Certification as a registered Continuing Professional Education(CPE) provider. Hours received in TEXTEAMS institutes may be applied toward the required training for gifted and talented in the area ofcurriculum and instruction. Individual district/ campus acceptance of these hours for gifted and talented certification is a local decision.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page viii

Institute Introduction

The TEXTEAMS Professional Development Institute, Rethinking Elementary SchoolMathematics Series: Part I provides a five-day staff development experience formathematics educators interested in the improvement of number sense and computationalfluency in the elementary grades through the implementation of effective instructionaldesign. The institute provides participants with opportunities (a) to increase their ownunderstanding of the underlying principles of elementary mathematics; (b) to explore criticalcomponents of effective instructional decision-making such as selection of appropriatemodels, developmental sequencing, and conceptual connections; and (c) to practice theseinstructional decision-making skills through the analysis of and reflection on a variety ofactivities addressing number and operation sense for students in the elementary grades.

The mathematical topics addressed in this institute include the meaning of and relationshipsamong whole numbers, fractions and decimals and the meanings of, properties of, andcomputational methods for addition, subtraction, multiplication, and division of wholenumbers. Effective instructional design is addressed through a Lesson Planning Process thatincludes

� identifying the Big Idea connected to the mathematics in the TEKS that the lessoncontributes to,

� using the TEKS to identify evidences of understanding and making those clear tostudents so that they can begin to take ownership of their own learning,

� orchestrating the instructional environment so that students are engaged in rigorouslearning, and

� providing opportunities for students to talk about their learning.

Part I of this series of institutes has been developed to provide an opportunity for teachers toenhance their own mathematical knowledge while honing their instructional decision-makingskills in order to guide their students toward a deeper understanding of numbers andoperations in order to develop the flexibility, efficiency, and accuracy required forcomputational fluency.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page ix

TEXTEAMS Elementary Institute Organizational Chart for Week 1

Day 1:Lesson Planning

Process

Day 2:Whole Numbers

Day 3:Fractions and

Decimals

Day 4:Addition andSubtraction

Day 5:Multiplication and

Division1A Why setexpectations?

2 A Math Concepts

Place Value Puzzles

3 A Math Concepts

Fraction MeaningsWith Cuisenaire Rods

4 A Math Concepts

It's Simply Addition

5 A Math Concepts

Fair Share

1B Evidence ofUnderstanding

Mor

ning

1C "Big Ideas"

2 B InstructionalDecision Making –Models

Investigating NumberRelationships

3 B InstructionalDecision Making–DevelopmentalSequence

UnderstandingFractions

4 B InstructionalDecision Making–Models Revisited

Investigating Additionand Subtraction

5 B InstructionalDecision Making–Levels ofSophistication

InvestigatingMultiplication andDivision

1D Lesson PlanningProcess Introduction to Sampler of Learning Experiences

Lesson Planning Process: Big Ideas, Evidence of Understanding,Questioning examined by participants in each sampler

2 C Sampler ofLearning Experiences



5 C Sampler ofLearning ExperiencesA

fter

noon

1E Using the LessonPlanning Process

2D Closure for Day 2 3D Closure for Day 3 4D Closure for Day 4 5D Closure for Day 5

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page x

Day One Materials Summary

Reusable Materials Consumable Materials� Charlie’s Checklist by Rory Lerman,

Orchard Books, 1997� A variety of everyday containers of

various sizes and shapes (examples:boxes or containers that held pizza,cereal, oatmeal, milk (quart), toothpaste,stapes, salt, margarine, animal crackers,yogurt, dry cat food, toothpicks)

� Copies of mathematics TEKS for Grades1 to 5

� The Great Divide, by Dayle Ann Dodds,Candlewick Press, 1999

� Linking Cubes� Counters� Calculators

� 4 sheets of sturdy construction paper ofvarious colors per group

� 1 roll of transparent tape per group� 5” X 7” note cards� Chart paper� Markers� Paper for creating books� For the leader: Chart paper and markers

or overhead projector and pens

For a list of transparencies and handouts, see the materials list at the beginning of Day 1.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page xi

Day Two Materials Summary

Reusable Materials Consumable Materials� Twelve Ways to Make Eleven by Eve

Merriam, Aladdin Paperbacks, 1996� Teddy bear counters� Two-color counters� Linking cubes� Base-ten blocks� Money� Calculators� Decks of cards with face cards removed� Small paper sacks numbered 1-10� Paper clip and pencil or clear plastic

spinner per pair of participants� Small paper cups labeled with numerals

1-10� Buttons� Scissors� Counters� Small cups� Bags of small beans or centimeter cubes� Pinto beans

� Chart paper� Markers� 5" X 7" note cards� Square, Circle, Triangle Charts made

from chart paper� Sticky notes� Manila paper 11"X 17"


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page xii

Day Three Materials Summary

Reusable Materials Consumable Materials� Cuisenaire Rods� Pattern blocks� Linking cubes� Two-color counters� Tiles or cubes of different colors� Paper or plastic cup� Geoboards� Geobands� Scissors� Linear measuring tools such as cm

measuring tape or meter stick� Fraction die

� Chart paper� Crayons (16 count)� Centimeter graph paper� Graph paper (to match the size of the tiles

or cubes� Construction paper� Glue sticks� Participants bring in examples of

decimals from the newspaper, magazines,catalogs, etc.

� Cards (with fractions and decimals suchas 5/10, 0.5, 0.50, 8/10, 0.8, 0.80, 35/100,0.35, 2 75/100, 2.75)


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page xiii

Day Four Materials Summary

Reusable Materials Consumable Materials� Teddy bear counters� Two-color counters� Straws and rubber bands; linking cubes� Base-ten blocks� Money� Calculators� Scissors� Play money in $10, $1, $0.10, $0.01� Decahedral (10-sided) dice or “0-9”

spinner� Dice number cubes� Decks of cards with face cards removed� Small paper sacks, numbered 1-20� Counters� Small cups

� Pencil� Paper� 5” X 7” note cards� Chart paper� Paper for poster or book� Markers


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page xiv

Day Five Materials Summary

Reusable Materials Consumable Materials� Base-ten blocks� Color tiles� Pattern blocks� Counters or color tiles� Bare Bear's New Clothes by Peter S.

Seymour, 1986� A Remainder of One by Elinor J. Pinczes� Dice, at least one per participant� 6 small paper plates per pair of players� Cuisenaire rods� Paper clips, 2 per pair of players� Linking cubes� Counters, a different color for each player� Calculators� Scissors

� Chart paper� Markers� Blank transparencies� Overhead pens� Centimeter grid paper� Chart paper or construction paper� Tape


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Page xv

Directions for the Reflection Journal

Why: We suggest that you provide a journal that will be a specified location to collect theirreflections about their learning across the week.

What: The directions below help participants construct their own journals. This is atechnique that is just as useful with elementary students as it is with teachers.

Where: There are reflection prompts with transparencies at the end of each section that areintended to inspire thoughtful responses to each of the learning experiences.

How: Have participants use a notebook (or have them make a booklet per directions below)in which to keep their journal of reflections throughout the week.

Directions for making a booklet:

1. Fold 5-10 sheets of 8.5 x 11 paper in half.

2. Set aside one sheet of paper. Take the rest of the sheets of paper, keeping themstacked together and cut a one-inch slit along the fold from either end toward themiddle.

3. Take the page you set aside and cut it along the fold EXCEPT for one inch on eitherend.

4. Roll the stack lengthwise and slip the stack through the cut in the middle of the singlepage and fit the slits together to make a booklet.

cut

cut

cut

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 1 Page 1

Day OneThe Lesson Planning Process

Table of Contents

Preparing for Day One 2

Section A Opening: Why do teachers need to be clear about what is expected? 4

Section B Making Evidence of Understanding Clear to Students 11

Section C The Process of Identifying Evidence of Understanding 22

Section D Working Through the Lesson Planning Process 31

Section E Applying the Lesson Planning Process 40

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 1Preparing for Day One Page 2

Day One Materials

Reusable Materials Consumable MaterialsSection AOpening: Why do teachers need to be clear about what is expected?Transparencies:� Benefits of Reflection (1A-1)� Day 1A Reflection Prompt (1A-2)

� 4 sheets of sturdy construction paper ofvarious colors per table group

� 1 roll of transparent tape per table group� 5” X 7” note cards� For the leader: Chart paper and markers

or overhead projector and pensSection BMaking Evidence of Understanding Clear to Students� Charlie’s Checklist by Rory Lerman� A variety of everyday containers of

various sizes and shapes (examples:boxes or containers that held pizza,cereal, oatmeal, milk (quart), toothpaste,stapes, salt, margarine, animal crackers,yogurt, dry cat food, toothpicks)

Transparencies:� Charlie’s Checklist (1B-1)� Mathematics TEKS for Contain It, Grade

3 (1B-2)� Constructing Understandings (1B-3)� Day 1B Reflection Prompt (1B-4)

� 5" X 7" note cards

Section CThe Process of Identifying Evidence of Understanding� Copies of mathematics TEKS for Grades

1 to 5

Transparencies:� Filters for Selecting Big Ideas (1C-1)� Lesson Planning Process Chart (1C-2)� Grade 3 Mathematics TEKS (1C-3)� Examples of “Big Ideas” (1C-4)� Day 1C Reflection Prompt (1C-5)

� Chart paper� Markers

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 1Preparing for Day One Page 3

Section DWorking Through the Lesson Planning Process� The Great Divide, by Dayle Ann Dodds,

Candlewick Press, 1999� Linking cubes� Counters� Calculators

Transparencies:� Problem Solving Model (1D-1)� Lesson Planning Process Chart – The

Great Divide (1D-2)� Benefits of Discussion (1D-3)� Day 1D Reflection Prompt (1D-4)

� Paper for creating books� Markers� Chart paper

Section EApplying the Lesson Planning Process� Copies of the Mathematics TEKS for

Grades 1 to 5

Transparencies:� Why is Rigorous Learning Important?

(1E-2)� Lesson Planning Process Chart (1E-3)� Day 1E Reflection Prompt (1E-4)

� Chart Paper� Markers

Handouts:� What Should I Look for in a Math

Classroom? (1E-1)� Lesson Planning Process Chart (1E-3)

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 1: Section AOpening: Why do teachers need to be clear about what is expected? Page 4

Opening:Why do teachers need to be clear about what is expected?

Key Questions:� Why do teachers need to be clear about student learning expectations?� What learning experiences have we had when we did not know what was expected?

Frame:We begin with an activity that leads participants to be actively engaged in learning with eachother. The task is designed to set the stage for thinking about why it is important to clearlyestablish learning expectations.

Materials:� 4 sheets of sturdy construction paper of various colors per group� 1 roll of transparent tape per group� 5" X 7" note cards� chart paper or overhead projector for facilitator

Transparencies/Handouts:� Benefits of Reflection (1A-1)� Day 1A Reflection Prompt (1A-2)

Procedures Notes

1. Participants work together in smallgroups of no more than 4 at each table.Each group will use only the followingmaterials: four sheets of constructionpaper and transparent tape. Each groupwill use these materials to constructsomething so that there is oneconstruction per group. Participantsshould be given approximately sevenminutes permitted for construction.

This activity is to initiate our thinking about“Lesson Design.” Participants should be cuedto reflect as

� A learner

� A teacher

� A participant in the institute

� A presenter of the institute

Ask them to think about how models provideexamples of clear learning expectations towhich they can add their creativity and newinsights.


2. Display these four characteristics on apiece of chart paper or on the overheadprojector.

tallsturdyuseful

attractiveAsk participants to attend to each of thecharacteristics as they plan and buildtheir construction. Tell participants thatwhen the constructions are complete,each group will examine all of theconstructions to rate them based on thesecharacteristics.

Note: The characteristics are listed in thisfashion to provide discussion points later inthe activity.

3. Ask participants to show "thumbs up" ifthey understand the instructions. Clarifyany questions about the directions and geta "thumbs up" from each person in theaudience before proceeding.

Note: It is important for the discussion thatfollows that all participants haveacknowledged that they understand thedirections.

4. When all participants have agreed thatthey understand the instructions, directthem to begin. Allow approximatelyseven minutes (the time of two well-chosen songs if you care to use music) tocomplete the constructions in theirgroups.

5. As groups are working, place a note cardwith an identifying letter (A, B, C, etc.)next to each construction on the tables.

6. At the end of the construction period,distribute one note card to each group.Ask for one person in each group to holdup the card. That person will be therecorder. Determine the number ofconstructions. Have the recorder write avertical column of numbers on the leftside of the card to match the number ofconstructions. (If there are 8constructions labeled A to H, then writethe numbers 1 to 8.)

12345678


7. Explain the ranking process. Each groupwill examine each of the constructions inthe room. After viewing all theconstructions, each group will determinewhich construction best meets all four ofthe characteristics. The construction thatbest matches is ranked number one andthe letter labeling that construction iswritten next to number one on the notecard. The letter for the construction thatis the next best at matching thecharacteristics is written next to numbertwo. Continue rank ordering until theletter labeling the construction that leastmatches the four characteristics is writtennext to the last number. Members of thegroup must agree upon the ranking order.Answer any questions about theprocedure for rank ordering beforedirecting the groups to stand and movearound the room to examine theconstructions.

Participants may have questions about theranking process. For each participant’squestion, ask for one or two restatements ofthe question by other participants to assureunderstanding by the group. Questions maylead to additional tasks, such as allowinggroups to tell about the name and purpose oftheir construction or adding a title to theconstruction. If participants ask to explaintheir constructions before the rankingprocess, you may wish to allow each group15 seconds to tell about its construction.

8. Allow several minutes for the groups toexamine and rank each of theconstructions before you direct them toreturn to their seats. Have the personholding the ranking note card hand it toanother person in the group. That personwill be the reporter. Write the rankingnumbers in a vertical column on chartpaper or on an overhead transparency. Asthe reporter calls out the ranking orderstarting with number one and continuingdown the list, you write letters next to thecorresponding numbers.

1. H B H H C2. B C C B H3. E H A C A4. C G B D B5. A A D A D6. D D G F F7. F E E E G8. G F F G E

9. Examine the patterns. You may wish toidentify the letter(s) that appears mostoften in each row.

Allow the participants’ comments to lead thedirection of your discussion.

Debriefing:Open the discussion by asking participants to tell how they felt about the task. By followingthe lead of the participants' comments and by asking leading questions, you will want tobring out the following points:


� The instructions were clear. (They showed thumbs up.) Despite clear instructions,participants may have felt some uncertainty about the expectations. Many times wefeel we are clear when students know what to do and can complete the task, as peopledid here, but we are not always explicit about our expectations for quality work.

� Everyone was able to complete the task as directed, so as a teacher I may have feltthat the students had accomplished the goal. Did the feedback the groups received ontheir work (the rank ordering) provide useful feedback on how to improve the work?How could feedback be given so that students could learn from their efforts andimprove their work in the future?

� Specific details may have been provided (the four characteristics), but there may nothave been a shared understanding among the learners about what these details meanor how to demonstrate them.

o What does it mean to be useful?o Does being attractive mean the same thing to me as it does to someone else,

such as the teacher?o Is it more important for the construction to be tall, since that was listed first,

or do all of the characteristics carry the same weight?o What measures do I take to evaluate tallness or sturdiness?o Is the construction a model of something useful and sturdy or should it be

considered the actual product to be evaluated?

� The role of student communication is important. Did the participants ask to have theopportunity to explain their constructions? Would it have helped to know what theconstructing group was thinking? Do we allow students to communicate the thoughtsbehind their work before we evaluate it? What effect might that have on ourassessments of students?

� The evaluation process should be clear. Should we examine and evaluate each of thefour characteristics and then use that information to determine the ranking or should itbe a holistic score?

� Assigning grades based on the spectrum of work produced does not provide usefulinformation for the purpose of helping students learn. How did the participants feelabout the way we shared the rankings? How would this impact future student work?How does a ranking like this relate to experiences the participants have had aslearners?

o Some teachers give A's to the very best work, while the average work in thegroup receives B's or C's. In this case, the grade is dependent upon what isbest or average for this particular group of students.

o What kind of feedback does a letter or number grade give to the learner? Is thepurpose of a grade to be a record of performance or a means of helping thestudent learn?


� Models can provide characteristics of quality work. Looking at examples can providecharacteristics or critical attributes of effective work which can influence the qualityof student work and provide a basis for evaluation.

Reflection and Connection:

Note: The use of a reflection journal throughout this Institute is highly encouraged. Seeinformation about the reflection journal on page xv. Since this will be the first timeparticipants see a reflection prompt, you may want to display Transparency 1A-1 about thebenefit of reflection.

In order to be fair to all students, we must ensure that students know what is expected ofthem. If we want students to become independent learners who do not rely on the teacher totell them when they have done good work, then we must enable students to understand andapply these expectations to their work. By knowing what is expected and what quality worklooks like, students are better able to judge their own work and the work of others in order toimprove the quality of their work in the process of learning.

At this point, what are your thoughts about why teachers need to set learning expectationswith students?

Transparency 1A-1


Benefits of Reflection

“Effective learning is balanced withopportunities for REFLECTION.”

“Kids need time set aside for reflection, andthey need to become consciously aware of itspower and their ability to use it. Addingreflective thinking to school learning is one ofthe simplest of all instructional innovations.Although there are more elegant approaches,many teachers have found that the simpleaddition of a student learning log, with timeset aside each day for responding to well-structured teacher ‘prompts,’ builds reflectioninto the day and moves students to a new levelof thinking.”

Zemelman & Hyde, A Principle of Best Practice Learning from Best Practice: NewStandards for Teaching and Learning in America’s Schools, Heinemann, 1998, p. 11.

Transparency 1A-2


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tthhoouugghh tt ss aabboouu tt wwhhyy tt eeaacchheerr ss nneeeedd tt oo

ss ee tt ll ee aa rr nn ii nn gg ee xx pp ee cc tt aa tt ii oo nn ss ww ii tt hh

ss tt uu dd ee nn tt ss ??

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 1: Section BMaking Evidence of Understanding Clear to Students Page 11

Making Evidence of Understanding Clear to Students

Key Questions:� How do we determine student learning expectations?� How do we establish the Evidence of Understanding with students?� Why should we establish Evidence of Understanding with students?

Frame:Teachers use the Texas Essential Knowledge and Skills (TEKS) for mathematics todetermine “Big Ideas” and learning expectations for students. In order to help studentsunderstand and use these expectations to guide and improve their work, students should playa role in determining what makes quality work. Models of quality work are used to helpdetermine the Evidence of Understanding.

Materials:� Charlie's Checklist by Rory S. Lerman� a variety of everyday containers of various sizes and shapes (examples: boxes or

containers that held pizza, cereal, oatmeal, milk (quart), toothpaste, staples, salt,margarine, animal crackers, yogurt, dry cat food, toothpicks)

� 5” X 7” note cards

Transparencies/Handouts:� Charlie’s Checklist (1B-1)� Mathematics TEKS for Contain It, Grade 3 (1B-2)� Constructing Understandings (1B-3)� Day 1B Reflection Prompt (1B-4)

Procedures Notes

1. Facilitate a discussion with theparticipants about their experiences aslearners regarding how they knew whatwas expected of them.

Think of a time when you were a learner.Think of a grade you received. Was it anumber grade such as 92? Was it a lettergrade such as B+? What did that grade meanto you? How did you know your work wasgood? How could you have improved yourwork? What kinds of feedback did youreceive as a learner?


2. Present the focus questions on thetransparency before you begin reading toprovide a purpose for listening. Thenread and discuss the book Charlie'sChecklist. In the story, Charlie had "musthave's" and "must be's" that he used tosort the letters that he received. He usedthese criteria to sort his letters to knowwhich owner would be good for him.After he recognized that Chester was theowner he wanted, Charlie thought ofmore attributes he wanted that just fitChester, such as having freckles oroveralls.

This book provides a useful definition forthe idea of criteria. Even young studentsare capable of understanding criteria as"must have's" and "must be's." Teacherscan refer to this idea as they ask studentsto identify the criteria or "must have's"and "must be's" for quality work inmathematics.

(From Transparency 1B-1)

What did Charlie call the things he wanted inan owner? (must have's and must be's)

What things did Charlie put on his checklist?(must have room for me; must be kind toanimals; must provide a stable home)

What did Charlie call the things on his list?(hint: a word that begins with c) (criteria)

This book provides examples of severalimportant points:

� With more information (as workprogresses), criteria can be changed oradded.

� We can check on the appropriateness oreffectiveness of something by examiningit in terms of the criteria.

3. "Contain It" is an activity thatdemonstrates how students can play arole in determining the criteria for qualitywork based on models of student work.Participants complete the task aselementary students would in theclassroom. The procedure for the activityis described below.


4. Discuss the restrictions on thedescriptions. (It is helpful to have thesewritten on a poster for people to refer toas they are working.)

Do not tell what it is used for. (I can't sayit's a pizza box.)

Do not tell what it is made of. (I can't sayit's a cardboard box with a lid.)

Do not tell what is printed on it. (I can'tsay it has a red roof or has the letters P IZ Z A on it.)

5. Place four boxes or containers on display.Read a description of one of the objectsand see if the learners can identify whichobject you are describing. Thedescription of a pizza box might besomething like this.

“This container has six faces. The twolargest faces are squares that measureabout two adult hands by two adulthands. These two large faces are on thetop and bottom of the container. Thesetwo large faces are joined by four long,skinny rectangles that measure about twoadult hands by one adult pinky.”

Which container am I describing?

Ask clarifying questions such as: How didyou know which container I was describing?

How did you know it was not this othercontainer?

Have participants think about how providingthis model helps understanding of the task.Encourage them to notice how people will gobeyond the model by adding their owncreative touches to the model when theywrite their descriptions.


6. Instructions: Each group will receive twocontainers. Each group will write onedescription for one container and onedescription for the other container. Thedescription must be complete enough tohelp the audience (other participants) tobe able to distinguish the container beingdescribed from the other containers ondisplay. Each group will read adescription aloud and the otherparticipants will try to identify thatcontainer. Answer any questions aboutthe instructions before groups begin towork.

7. Distribute containers. "In 20 seconds,when I say go, the members of yourgroup will have ten minutes to prepare awritten description of both of yourcontainers. Go." Calm music may be usedfor the writing period.

8. Have all of the containers returned to thedisplay. Direct participants to look at thedisplay of containers. Remindparticipants that you want them to be ableto identify the container being described.They can request more information fromthe writing group or can ask for a piece ofinformation to be read again. Direct theparticipants to "listen purposefully" aseach group reads their description toidentify the features of writing that aremost helpful in determining whichcontainer is being described.


9. Have a group read one description at atime. Follow the lead of the participantsin holding up containers from the displaythey think might fit the description. Askquestions that help them tell about theirthinking. Encourage rereading of certainparts of the description or even addingnew information if necessary. When theentire group agrees on one container,verify that it is the correct container withthe writing group. Then ask theparticipants to identify the features of thedescription that were helpful inidentifying the container. Write thesefeatures on a chart paper.

As you write the effective parts of thedescriptions, you will write the features inthe participants' words. Based on yourknowledge of the TEKS (see transparency),you can add more details to the feature.Example: A participant says, "They usedwords like circumference." You can annotateby writing "use of mathematical language/vocabulary."

10. Continue listening, identifyingcontainers, and writing features withseveral more groups until you have agood list of features. Some of the featuresmay be duplicated from one group toanother (you don't need to write themmore than once), but you can probablywrite 2 or 3 features from eachdescription.

The Evidence of Understanding willdepend upon the actual descriptions andparticipant interaction. Some of these criteriamay be identified.

A good description:

� identifies the three-dimensional solid.

� describes the number of faces.

� identifies the two-dimensional faces.

� uses estimates of non-standardmeasurements.

� connects to something known.

11. Discuss the technique we have just used.By looking at models of student work, wehave determined characteristics that weremost effective in making a qualitydescription. Now armed with thisinformation, students will be better ableto meet the teacher's expectations whenthey independently write a description ontheir own. The same process can be usedagain when examining individual work,and the list of features may be revised byadding new ideas and deleting some ofthe features that students determine arenot as important as they write theirdescriptions.

Point out that the criteria used to identify aquality description has been uncovered bythe participants’ own descriptions.


12. Examine the Math TEKS used in thislesson (see Transparency 1B-2).

13. How does this process fit in with what wedo in the mathematics classroom? Wewant to use this process of learningexpectations for "Big Ideas." Use theTEKS transparency to discuss possible"Big Ideas" in mathematics.

14. Discuss the benefit of involving learnersin constructing their knowledge byidentifying the Evidence ofUnderstanding based on their own work.Use Transparency 1B-3 to initiate thisdiscussion.

Debriefing:

� How did we identify the features of quality work?(We need the teacher's knowledge of TEKS, models of quality work, and students'input into identifying the criteria so students have understanding and ownership of thecriteria.)

� How are these criteria used?(Students use criteria to guide their work. They use the criteria to examine their ownwork and the work of others for the purpose of improving the quality of their work.)

� When do we use this process?(When students are completing significant tasks, routine procedures (such as problemsolving or journaling) or working with "Big Ideas" in mathematics, it is beneficial totake time to identify the criteria or Evidence of Understanding for quality work.)

Reflection and Connection:A key element in facilitating this process is the teacher's knowledge of what students areexpected to learn based on the Texas Essential Knowledge and Skills. With the TEKS inmind, the teacher orchestrates learning situations so that students will have the opportunity toidentify the skills and components necessary for quality work. A teacher can do this bymodeling and by providing models and learning-specific tasks that lead students to recognizethe criteria for quality work.

What impact on student work could result from involving students in knowing theEvidence of Understanding that is expected of good work?

Transparency 1B-1


Charlie’s Checklist

What did Charlie call the things on his list?(hint: a word that begins with "c.")

How did Charlie describe the things he wantedin an owner?

What things did Charlie put on his checklist?

(from Charlie’s Checklist, story by Rory S. Lerman, pictures by Alison Bartlett, OrchardBooks, 1997.)

Transparency 1B-2


Mathematics TEKS for Contain It, Grade 3

Students select and use formal language to describe their reasoning asthey identify, compare, and classify shapes and solids; and they usenumbers, standard units, and measurement tools to describe andcompare objects, make estimates, & solve application problems.Students organize data, choose an appropriate method to display thedata, and interpret the data to make decisions and predictions andsolve problems.

(8) Geometry and spatial reasoning. The student uses formalgeometric vocabulary. The student is expected to name, describe, andcompare shapes and solids using formal geometric vocabulary.

(11) Measurement. The student selects and uses appropriate unitsand procedures to measure length and area. The student is expectedto:

(A) estimate and measure lengths using standard units such asinch, foot, yard, centimeter, and meter;(B) use linear measure to find the perimeter of a shape; and(C) use concrete models of square units to determine the area ofshapes.

(13) Measurement. The student applies measurement concepts. Thestudent is expected to measure to solve problems involving length,area, temperature, and time.

Transparency 1B-2


(15) Underlying processes and mathematical tools. The studentapplies Grade 3 mathematics to solve problems connected toeveryday experiences and activities in and outside of school. Thestudent is expected to:

(A) identify the mathematics in everyday situations;(D) use tools such as real objects, manipulatives, andtechnology to solve problems.

(16) Underlying processes and mathematical tools. The studentcommunicates about Grade 3 mathematics using, informal language.The student is expected to:

(A) explain and record observations using objects, words,pictures, numbers, and technology; and(B) relate informal language to mathematical language andsymbols.

Transparency 1B-3


Constructing Understandings

“Children’s learning always involvesCONSTRUCTING ideas and systems.”

“Kids need encouragement to reflect, to sharetheir emerging ideas and hypotheses withothers, to have their errors and temporaryunderstandings respected – and they needplenty of time.”

Zemelman & Hyde, A Principle of Best Practice Learning from Best Practice: NewStandards for Teaching and Learning in America’s Schools, Heinemann, 1998, p. 15.

Transparency 1B-4


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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 1: Section CThe Process of Identifying Evidence of Understanding Page 22

The Process of Identifying Evidence of Understanding

Key Questions:� How does a teacher connect Big Ideas and learning expectations to the TEKS?� How does a teacher design effective instruction that connects the TEKS to learning

expectations and rigorous learning in mathematics?� What are the components of an effective Lesson Planning Process?

Frame:The process of involving students in identifying Evidence of Understanding can take timeto do well. For this reason, teachers can select certain "Big Ideas" or significant projects inmathematics for which Evidence of Understanding will be identified. In many cases,expectations in the Evidence of Understanding for one learning experience can be usedwith other experiences. By starting with appropriate “Big Ideas” and Evidence ofUnderstanding, teachers are better able to make informed decisions about the design ofmathematics instruction.

Materials:� copies of TEKS for mathematics in Grades 1 through 5� chart paper and markers

Transparencies/Handouts:� Filters for Selecting Big Ideas (1C-1)� Lesson Planning Process Chart (1C-2)� Grade 3 Mathematics TEKS (1C-3)� Examples of “Big Ideas” (1C-4)� Day 1C Reflection Prompt (1C-5)

Procedures Notes

1. What is the difference between anactivity and a "Big Idea"? Measuring thelength of your shoe is an activity, whileusing standard measuring tools (like aninch ruler) could be a "Big Idea."

Brainstorming: What are "Big Ideas" inmathematics? Have participants talk andlist on chart paper at their tables whatthey consider to be "Big Ideas" inmathematics. Allow sufficient time fordiscussion and writing.

You may be able to help participantsunderstand "Big Ideas" by providing anexample such as this one. Number sense is amajor concept that is more broad than a "BigIdea." Finding the number before or after is askill that is one way to demonstrateunderstanding of a "Big Idea" (or anEvidence of Understanding). Comparing andordering whole numbers is a "Big Idea."


2. Have participants take a gallery walk ingroups to examine the different "BigIdeas" generated by groups. Askparticipants to look for patterns in thelists and for "Big Ideas" that may bemissing from their own list.

Use transparency 1C-1 to present a focus forexamining the “Big Ideas.”

3. Following the gallery walk, ask theparticipants to identify common "BigIdeas" they saw. Write these "Big Ideas"on a chart.

Transparency 1C-2, The Lesson PlanningProcess, will be used to summarize the stepsthat follow. You may choose to refer to eachsection of the transparency as you progressthrough the steps or you can simply referback to the sections as you review thetransparency in step 9 below.

4. One of these "Big Ideas" is likely toinvolve problem solving. Use problemsolving as a model for the process ofidentifying the Evidence ofUnderstanding for quality work. Displaythe Transparency 1C-3, the TEKS for theUnderlying Processes strand in Grade 3Mathematics. (Participants should have acopy of the TEKS for mathematics thatthey can refer to at this time.) Look forTEKS that identify learning expectationsfor problem solving.

Participants are likely to say that the strandfor Underlying Processes and MathematicalTools is a useful place to find problemsolving. Is this the only place where learningexpectations for problem solving are located?

(Step 1 of LPP Transparency 1C-2)

5. Based on the mathematics learningexpectations for problem solving, whatare some of the skills and knowledgestudents use to demonstrateunderstanding of these concepts? Asparticipants give ideas, write these onchart paper. To rephrase the question,how do we know students have donegood work in problem solving?

The ideas will vary depending on the group,but here are some examples.

� Students use manipulatives to solveproblems.

� Students can use more than one strategyto find a solution.

� Students can write or draw to show howthey found a solution.

� Students can explain their thinking asthey solved a problem.

� Students find a reasonable solution to aproblem.



6. Now that we have identified what isexpected of students' learning, we canconsider how to design a learningexperience so that students know what isexpected and can use this knowledge todo quality work to further their learning.If we wanted students to do the things onthis chart (refer to expectationsparticipants identified from the TEKS),what kind of learning experience wouldwe need to plan?

The discussion may include thoughts likethese:

� Manipulative materials should beavailable and used by students.

� Students should work in small groups sothey can talk about how they solved aproblem.

� In order to know how students worked,they should write about the process theyused.

� Time needs to be devoted to students'discussion of their solutions andstrategies.


7. Based on what you want to see and hearfrom the students, what directions do youas the teacher give to lead students towhat you want? Lead participantdiscussion about how students willdemonstrate the evidence of theirunderstanding.

Here are some examples:

� Students need to show more than oneway to solve a problem.

� Students need to write words and/orpictures to tell about their thinking.

� Students need to write a number sentenceto tell what they did. Students need toname an estimate.


8. Now we have set expectations withstudents about their learning and theways they will demonstrate theirunderstanding. What will happen toensure that students have the opportunityto revise their work and to learn fromeach other? What will be the process forstudents to talk about their work?

As students share their strategies andsolutions, ask questions that help studentsmake connections with each others' work.

� Who did something like ____ did? Whatdid you do? How is that the same?

� Could someone else tell me what theydid? Why did you do that?

� Who has a different way to solve theproblem? What did you do differently?



9. Review Transparency 1C-2, The LessonPlanning Process, to summarize thesteps that we just completed. As youdiscuss each section of the transparency,relate it to the steps above and to theprocess of designing instructionalopportunities for students in mathclassrooms.

Debriefing:

� What was the process for determining the “Big Ideas” for this lesson.(The teacher's knowledge of mathematics TEKS is a crucial place to begin. Afteridentifying what students are to learn, a teacher determines what students need to doto demonstrate their understanding. By looking at the outcome, we can help studentsunderstand what good work looks like in order to improve the quality of their ownwork.

� How do the “Big Ideas” connect to the Evidence of Understanding for qualitywork?(The “Big Ideas” for a math concept lead a teacher to identify what students do tocomplete a learning task successfully. This Evidence of Understanding is criteria forquality work based on how students show their ability to apply mathematics skills andconcepts. Students use the criteria to guide them. These criteria helps studentsexamine their own work and the work of others for the purpose of improving thequality of their work.)

Reflection and Connection:The TEKS guide teachers in the identification of Big Ideas in mathematics. From these BigIdeas, teachers and students work together to determine appropriate Evidence ofUnderstanding. The teacher designs mathematically powerful learning experiences to enablestudents to demonstrate their knowledge and skills by meeting the Evidence ofUnderstanding.

How does the identification of Big Ideas and Evidence of Understanding impactinstructional design?

Transparency 1C-1


Filters for Selecting Big Ideas

� Has enduring value beyond the classroom

� Gets at the heart of the discipline (i.e. “doingmathematics”)

� Requires “uncoverage” of new informationor ideas

� Offers potential for engaging students

Wiggins & McTighe, Understanding by Design, ASCD, 1998, p. 23.

Transparency 1C-2


Lesson Planning Process Chart

Step Question Context1. Take a “Big Idea.” Why? Why is this concept important to these

students?

2. Identify Evidence ofUnderstanding (what students areexpected to know and do).

What? What will students know and do toproduce good work?

3. Design the lesson so thatstudents are actively engaged inrigorous learning.

How? How does the teacher orchestrate thelearning experience to enable this tooccur?

4. Provide opportunities andsupport for students to talk abouttheir work.

How? How do communication andcollaboration support learning?

Transparency 1C-3


Grade 3 Mathematics TEKS

(3.15) Underlying processes and mathematical tools. The studentapplies Grade 3 mathematics to solve problems connected toeveryday experiences and activities in and outside of school. Thestudent is expected to:

(A) identify the mathematics in everyday situations;(B) use a problem-solving model that incorporatesunderstanding the problem, making a plan, carrying out theplan, and evaluating the solution for reasonableness;(C) select or develop an appropriate problem-solving strategy,including drawing a picture, looking for a pattern, systematicguessing and checking, acting it out, making a table, working asimpler problem, or working backwards to solve a problem; and(D) use tools such as real objects, manipulatives, andtechnology to solve problems.

(3.16) Underlying processes and mathematical tools. The studentcommunicates about Grade 3 mathematics using informal language.The student is expected to:

(A) explain and record observations using objects, words,pictures, numbers, and technology; and(B) relate informal language to mathematical language andsymbols.

(3.17) Underlying processes and mathematical tools. The studentuses logical reasoning to make sense of his or her world. The studentis expected to:

(A) make generalizations from patterns or sets of examplesand nonexamples; and(B) justify why an answer is reasonable and explain thesolution process.

Transparency 1C-4


Examples of “Big Ideas”

� Use appropriate tools to solve problems.(3.15)

� Use a problem solving model to solveproblems everyday. (3.15)

� Communicate strategies used to solve aproblem. (3.16)

� Explain your mathematical thinking. (3.17)

� Analyze information to determine patterns.(3.17)

Transparency 1C-5


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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 1: Section DWorking Through the Lesson Planning Process Page 31

Working Through the Lesson Planning Process

Key Questions:� What are the components of the Lesson Planning Process?� How do these components work together to ensure quality mathematics instruction for

students?

Frame:Participants will engage in a lesson as students. Following the lesson, the facilitator willshare the thinking that went into the design and implementation of the learning experience.By sharing this metacognitive process, the facilitator is helping to clarify the LessonPlanning Process previously outlined.

Materials:� The Great Divide, by Dayle Ann Dodds, Candlewick Press, 1999� manipulative materials, such as linking cubes and counters� calculators� paper and markers for creating books� chart paper and markers

Transparencies/Handouts:� Problem Solving Model (1D-1)� Lesson Planning Process Chart – The Great Divide (1D-2)� Benefits of Discussion (1D-3)� Day 1D Reflection Prompt (1D-4)

Procedures Notes1. Before you begin reading, ask the

participants to be on the lookout for anypatterns they notice in the book. Read thebook The Great Divide aloud to thegroup.

The Great Divide tells the story of a racewith different stages and events. At each ofthe intervals, half of the people in the raceare forced to stop racing. The race beginswith 80 people and continues the pattern (40,20, 10, 5) of dividing in half until 5. Thenone racer must stop and the other 4 continueand divide into 2 groups of 2. Disasterhappens for the 4 racers and the 1 racer whohad been left behind emerges as the winner.


2. Ask participants to identify the patternsthey noticed in the book.

What pattern happens in this book? (Anumber is divided in half each time.)

Does the pattern continue throughout thebook? (No, it stops at 5.)

3. Inform the participants that they willwork in small groups together to create apattern of numbers that are divided inhalf and then write and illustrate a storyusing these numbers. They will use aproblem-solving model to guide them.Review with the participantsTransparency 1D-1, the ProblemSolving Model.

Discuss this Problem Solving Model withyour students.

1. Understanding the Problem (What dowe need to know?)

2. Making a Plan (How can we solve thisproblem? What number do we start with? )

3. Carrying Out the Plan (What will weuse or do to help us find the numbers weneed? How will we show what we know?Who is going to do what when we write ourstory?)

4. Evaluating the Solution (How will weknow if we have numbers that follow thepattern of dividing in half? What can we doto check our work?)

4. Explain the directions to the participants.Each small group will create a book thathas a pattern of numbers that are dividedin half. The pattern must have at least 4steps. The pattern may stop as it did inthe book (5-1=4), but there needs to be atotal of at least 4 steps. Participants mayuse tools (manipulatives, calculators) tohelp them find and check their pattern ofnumbers. After finding the set ofnumbers they want to use (ex: 60 , 30 ,15, 14 , 7), the group will write andillustrate a story based on this numbersequence. The story must provide areason for the numbers to be divided inhalf each time. Groups should be readywhen they share with the class to tell howthey found the numbers in the pattern.

How can I help you understand thedirections? What will members of your groupneed to do? How will you make a record ofthe steps your group members took as youworked through the Problem SolvingModel?


5. Work with the students to determine howthey can do good work on this task. A listof three or four criteria is probablyenough to begin this task. Post theEvidence of Understanding for view bygroup members as they work.

The criteria will be determined by theparticipants, but the Evidence ofUnderstanding they generate might looksomething like this:

I do good work on this project when I…

� make two equal groups to divide anumber in half.

� follow a plan for solving a problem.

� choose the right tools to help me solvethe problem.

� check my answer with another strategy.

� use numbers in a way that makes sense.

6. Before the groups begin, tell participantsthat each group will be expected to sharetheir stories and the way they used theProblem Solving Model with the othermembers of the class. Tell them that youwill allow sufficient time for determiningthe numbers and writing the story. Theremay not be enough time for all groups toillustrate their stories, so some storiesmay be read aloud without pictures toaccompany the story.

Assign a person in each group to recordanything mathematical that happens whilecreating the story.

7. As the groups are working, the facilitatorcan model the types of questioningstrategies that scaffold students'understanding and support rigorousthinking. The facilitator may also wish totake notes while observing to use whenpresenting the metacognitive piece thatfollows.

Listen to and talk with participants as theyare working.

"How is your group going to begin to find apattern?" "With what number will you start?""How do you know when you are dividing anumber in half?" "When is there a time whena number is divided in half like this?" "If oneof your group members is not sure how thepattern of numbers works, what can you sayor do to help that person understand?" "Whatwill you do to check your number pattern?"What happens when you can't divide anumber in half?" "What do you notice aboutthe numbers that you can't divide in half?"


8. After a sufficient period has passed toallow groups to identify their numberpatterns and write their stories, have eachgroup share aloud their stories. Askdifferent members of each group to sharethe process they used as they worked.

How does the number pattern this groupchose fit with their story? How is thisnumber pattern similar to or different fromthe number pattern chosen by your group (oranother group who has already presented)?How does the way this group used theProblem Solving Model compare with theway another group used it? What might havebeen helpful?

9. Review the Evidence of Understandingestablished before the work began. Askeach group to discuss among themselveshow well they were able to meet theseexpectations.

How well did you meet the Evidence ofUnderstanding? Was there anything thatmight have been useful in helping you to dobetter work? What might you choose to dodifferently next time?

10. Review Transparency 1D-2, The LessonPlanning Process, to summarize thelesson planning process used to designthis learning experience. As you discusseach section of the transparency, relate itto the instructional choices made at eachpoint in the lesson. Discuss how this isconnected to the process of designinginstructional opportunities for students inmath classrooms.

11. Discuss the importance of discussion inadvancing learning. Use transparency1D-3 to focus this discussion.

“Learning may occur incidentally as thelearner observes the cognitive processes offellow group members as they develop aninterpretation. Learning may also be moredirect when teachers or peers function asmore knowledgeable participants andscaffold the interaction so that the learnerbecomes capable of achieving more withtheir assistance than they could haveindependently.” (from Lively Discussions,IRA, 1996, p. 14-15)


Debriefing:

� What were the "Big Ideas" in this lesson? Why are these important for students tolearn?(One "Big Idea" involved the students' ability to separate a set into two equivalentsets, which is an important prerequisite for understanding halves and division by two.Another "Big Idea" was the use of a Problem Solving Model, which is a criticalthinking process students should know.

� How are the criteria for good work identified?(In this lesson, we relied on prior experience to help set expectations for good work.The Evidence of Understanding may develop based on previous learningexperiences or by examining models of quality work.

� What things happened in the lesson to facilitate rigorous math learning?(A book was used to provide a context for the math concept. Participants had accessto tools and a Problem Solving Model that could be used to support their work. Thetask was open-ended to some degree to encourage diverse responses. Participantsdetermined expectations for quality work prior to the work time. Participants usedmathematical knowledge and skills in a realistic context.)

� How was communication used to support learning?(Participants worked in small groups to allow for interaction and cooperative work.The teacher monitored the work in small groups and asked questions to supportrigorous thinking. Sufficient time was provided for sharing and discussion about theirwork.)

Reflection and Connection:Many of us are better able to understand a process when we see it demonstrated. The purposeof this learning experience was to help participants see instructional decisions made withinthe context of a mathematics lesson. Too often teachers begin their lesson planning bychoosing activities that appear to be interesting instead of designing experiences based onwhat students need to learn. We will engage in continued exploration of the Lesson PlanningProcess in following activities.

How do the components of the Lesson Planning Process enable a teacher to makeinformed decisions to ensure that students are learning?

Transparency 1D-1


Problem Solving Model

1. Understand the Problem(What do we need to know?)

2. Make a Plan(How can we solve this problem? What number do westart with?)

3. Carry Out the Plan(What will we use or do to help us find the numbers weneed? How will we show what we know? Who is goingto do what when we write our story?)

4. Evaluate the Solution(How will we know if we have numbers that follow thepattern of dividing in half? What can we do to check ourwork?)

Transparency 1D-2


Lesson Planning Process Chart – The Great Divide

Step 1Big Ideas

Step 2Evidence of Understanding

Step 3Orchestrating for RigorousLearning

Step 4Communication to SupportLearning

� identify equivalent sets thatmatch given situations

� explain the process used tosolve a problem

I do good work when I…

� make two equal groups todivide a number in half

� follow a plan for solving aproblem

� choose the right tools tohelp me solve the problem.

� check my answer withanother strategy

� use numbers in a way thatmakes sense

� open-ended problem (morethan one solution and/orstrategy can be used)

� choice of tools available� verify solution (more than

one way to solve)� numbers used in context� person assigned to notice

and report on mathematicalskills used

� person in each group toreport on math skills used

� numbers used in context ofstory.

� opportunity to share andprovide feedback based onEvidence of Understanding

� “What did your group do tomake sure your numberpattern divided the numbersin half?”

� “Why do you think yournumber pattern fits withyour story?”

� “How did you find yournumber pattern?”

Transparency 1D-3


Benefits of Discussion

“Vygotsky (1978) theorized that socialenvironments provide learners with theopportunity to observe higher levels ofcognitive processing. From this perspective,discussions may be viewed as a socialenvironment in which students can witnesshow group members work together tocollaboratively construct meaning while alsoparticipating in the process.”

Lively Discussions!: Fostering Engaged Reading. IRA, 1996, p. 15-16.

Transparency 1D-4


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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 1: Section EApplying the Lesson Planning Process Page 40

Applying the Lesson Planning Process

Key Questions:� How does a teacher use the Lesson Planning Process?� What actions are necessary to connect the TEKS to learning expectations and rigorous

learning in mathematics?

Frame:Classrooms are often organized around activities teachers select from textbooks or otherresources. In planning for instruction, teachers many times begin with the activities studentswill do and hope the learning will occur. By using student outcomes to begin the process ofdesigning instructional experiences, teachers can better focus on what they expect students tolearn and how they expect students will demonstrate their understanding.

Materials:� copies of TEKS for mathematics in Grades 1 through 5� chart paper and markers

Transparencies/Handouts:� What Should I Look For in a Math Classroom? (1E-1)� Why is Rigorous Learning Important? (1E-2)� Lesson Planning Process Chart (1E-3)� Day 1E Reflection Prompt (1E-4)

Procedures Notes1. Present the following scenario to

participants for discussion: Considersomeone you know who teachesmathematics to elementary students.When you walk into this classroom, whatdo you see? What are students doing?What are students saying? What is theteacher doing and saying? How is themathematics learning visible throughwhat is happening in this classroom?

At your table, discuss the evidence ofmathematics learning you might see in anelementary classroom.

Allow an adequate period of time for theparticipants to talk in small groups. Afterseveral minutes, ask participants to sharesome of the evidence of math learning theydiscussed. Write the participants’ ideas on achart paper.


2. Have participants consider the followingquestions in relation to their ideas aboutevidence of mathematics learning: Howdo good teachers know what to do in theclassroom to support an effective learningexperience? What sources of informationguide teachers’ decisions aboutinstruction? How do teachers know whatis expected and how to meet the learningneeds of the particular students in theirclasses?

There are many quality educators who makesound instructional choices to meet students’learning needs. There are also teachers whoare less proficient at designing andfacilitating rigorous learning experiences inmathematics. While teachers have resourcesto help them plan, such as textbooks andactivity guides, these resources may not leadto effective lesson design. Is there a way tohelp teachers plan more effectively for mathinstruction?

3. Distribute Handout 1E-1, “What Should ILook for in a Math Classroom?” Allowapproximately five minutes forparticipants to read the two pagessilently. Then ask small groups to talkabout the information. Following thesmall group discussions, lead participantsin a large class discussion about what amathematically powerful classroom lookslike. What elements will participantswant to consider as they work together towrite their own lesson plan?

Summarize from the discussion the ideasrelated to the question, “What does it mean tohave a mathematically powerful classroom?”

4. Use transparency 1E-2 to discuss somepossible characteristics of rigorouslearning.

5. Present the Lesson Planning Process asa tool to address designing more effectivemathematics instruction. Review thesteps on Transparency 1E-3, the blankLesson Planning Process. Explainthoroughly how each of the steps is usedand why each is important. Provide anexample from the area of number andnumeration that exemplifies what eachcomponent of the Lesson PlanningProcess means.


6. Explain to the participants that they willwork in grade level (or similar) groups.Each group will use the Lesson PlanningProcess to design a lesson based onlearning expectations from the TEKS inthe strand of number and numeration.Individuals can use the handout to recordtheir own ideas, and then the groupshould use chart paper to record theircombined ideas for the lesson. (Youmight choose to have groups record theirideas on a blank Lesson PlanningProcess transparency.) This informationshould include the Big Idea, theEvidence of Understanding, the processto facilitate rigorous learning, andexamples of the student and teacher talkabout mathematics.

The groups should know before they beginthat they will be expected to share theirlessons with the rest of the participants.While the groups are working, you can posequestions and provide support for their work.You should also be observing the groups atwork so you can connect the processes theyuse to the way a teacher can use the LessonPlanning Process in the classroom.

7. After a sufficient working period, haveeach group share their lesson. Encouragereaction (questions, ideas, connections)from participants in other groups.

Ask questions that encourage participants tomake connections between one lesson andanother. Annotate the discussion with someof your own observations of the groups asthey worked.

Debriefing:

� What benefits for instructional design do you see?

� What challenges might teachers have in the use of this process?

� How could you help a new or inexperienced teacher understand how this processworks?


Reflection and Connection:With this Lesson Planning Process, the teacher is not simply the manager of activities, butserves as the facilitator of mathematically powerful learning experiences. Teachers beginwith the Texas Essential Knowledge and Skills to identify appropriate “Big Ideas” formathematics. From these “Big Ideas” student learning expectations are determined and theseexpectations are made clear to students as Evidence of Understanding. The teacherorchestrates the experience to enable students to engage and interact with math concepts andwith each other. Communication among students and the teacher is rich with thinkingopportunities as students’ work is discussed in connections to expectations set forth by theEvidence of Understanding. The reflection inherent in the Lesson Planning Processsupports reflective instructional design.

How does the Lesson Planning Process compare to lesson design models in your district orschool?

Handout 1E-1


What Should I Look for in a Math Classroom?by

The Math ConnectionMathematical Sciences Education Board

A math classroom should provide practical experience in mathematical skills that are abridge to the real world of jobs and adult responsibilities. This means going beyondmemorization into a world of reasoning and problem solving.

Sounds good, but how will I recognize a good math classroom when I see it?

Look for these changes from the traditional classroom, and if you see them, you will belooking at a classroom that is preparing students for the world outside of school.

WHAT ARE STUDENTS DOING?� Interacting with each other, as well as working independently, just as adults do at

work.� Using textbooks as only one of many resources. Manipulatives such as blocks and

scales and technology such as calculators and computers are useful tools, and studentsshould be learning how and when to use them.

� Becoming aware of how math is applied to real life problems, not just learning aseries of isolated skills. And as in real life, complex problems are not solved quickly.

� Realizing that many problems have more than just one "right" answer. Students canexplain the different ways they reach a variety of solutions and why they make onechoice over another.

� Working in groups to test solutions to problems. They are more than only "listeners"and are highly involved.

� Learning how to communicate mathematical ideas with one another.� Working in a physical setting that promotes teamwork and helps them challenge and

defend possible solutions. Even while using computers, they do not always workalone but with other students, helping each other.

WHAT ARE TEACHERS DOING?� Raising questions that encourage students to explore several solutions and challenge

deeper thinking about real problems. They are not just lecturing.� Moving around the room to keep everyone engaged and on track. They are not glued

to the chalkboard.� Allowing students to raise original questions about math for which there is no

"answer in the book," and promoting discussion of these questions, recognizing that itmay be other students who will find reasonable answers.

� Using manipulatives and technology when it is appropriate, not just as "busy work."� Drawing on student discovery and creativity to keep them interested. The teacher

knows that boredom is the enemy of learning.� Encouraging students to go on to the next challenge once a step is learned,

understanding that not all students learn at the same pace.

Handout 1E-1


� Bringing a variety of resources into the classroom from guest speakers to creative useof technology.

� Working with other teachers to make connections between disciplines to show howmath is a part of every other major subject.

� Using assessments that reflect the way math is being taught, stressing understandingand problem-solving skills, not just memory.

� Exploring with students career opportunities that emphasize mathematical conceptsand applications.

LEARNING also takes place outside of school. Thinking mathematically is critical to everylife skill from balancing a checkbook to understanding the newspaper. In every job peopleuse math skills that require the ability to identify a problem, look for information that willhelp solve the problem, consider a variety of solutions and communicate the best possiblesolution to others.

LOOK CLOSELY AT A MATH CLASSROOM IN TODAY'S SCHOOLS.

Is it teaching the same old stuff in the same old way, turning out students grossly unpreparedfor the real adult world?ORIs it teaching skills for life and work in the next century?

Which do we choose for our children?

YOU LOOK AND DECIDE.

Developed by The Math ConnectionMembers of the Math Connection are:American Association of Colleges for Teacher EducationAmerican Association of School AdministratorsMathematical Association of AmericaNational Association of Elementary School PrincipalsNational Association of Secondary School PrincipalsNational Association of State Boards of EducationNational Council of Teachers of MathematicsNational School Boards Association

Coordinated by:Mathematical Sciences Education Board2101 Constitution Avenue NW · HA 476 Washington, DC 20418Phone: (202) 334-1289 · Fax: (202) 334-1453email: [email protected] by: The Annenberg/CPB Math and Science Project

Updated 9/17/97

Transparency 1E-2


Why is Rigorous Learning Important?

� Rigorous learning and content demandattention.

� Rigorous learning increases flexibility inthinking.

� Rigorous learning develops perseveranceand tolerance for new ideas.

� Rigorous learning creates self-confidence.

Teaching What Matters Most, ASCD, 2001, p. 9-10.

Transparency / Handout 1E-3



Step1. What is the Big Idea?

2. What is the Evidence ofUnderstanding formathematics? (What arestudents expected to knowand do?)3. How does the teacherorchestrate the learningexperience so that rigorouslearning will occur?

4. How can the teacherfacilitate opportunities andsupport for students to talkabout their work?

Transparency 1E-4


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Day TwoWhole Number Concepts

Table of Contents

Preparing for Day Two 2

Section A Place Value Puzzles 5

Section B Investigating Whole Number Relationships 12

Section C Whole Number Sampler 31Button, Button, Where’s the Button? 40Expanding Numbers 42In and Out 51It’s the Place that Counts 60Make It Zero (Base-ten Block Version) 65Make It Zero (Calculator Version) 71More 76Number, Number, Where’s the Number? 79See Saw 82

Section D Closure for Day Two: Shaping Up a Summary 89

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2Preparing for Day Two Page 2

Day Two Materials

Reusable Materials Consumable MaterialsSection APlace Value PuzzlesTransparencies:� Place Value Puzzle 1 (2A-1)� Place Value Puzzle 2 (2A-2)� Place Value Puzzle 3 (2A-3)� Day 2A Reflection Prompt (2A-4)

Handouts:� Place Value Puzzle 1 (2A-1)� Place Value Puzzle 2 (2A-2)� Place Value Puzzle 3 (2A-3)

Section BInvestigating Whole Number RelationshipsSamples of the following for each table:� Single counters (beans, straws, teddy

bear counters, unlinked linking cubes,color tiles)

� Two-color counters� Counters that students bundle (straws

and rubber bands, linking cubes, countersin cups)

� Proportional materials that are pre-bundled (base-ten blocks, bean sticks)

� Non-proportional materials (money;colored chips with a key)

� Calculators

Transparencies:� Chart 1 (2B-1)� Chart 2 (2B-2)� Part-Part-Whole Mat (2B-3)� Ten Frame Mat (2B-4)� Place Value Mat (2B-5)� Number Line (2B-6)� Hundred Chart (2B-7)� Charts 3A and 3B (2B-8 and 2B-9)� Day 2B Reflection Prompt (2B-10)

Handouts:� Chart 1 (2B-1)� Chart 2 (2B-2)� Part-Part-Whole Mat (2B-3)� Ten Frame Mat (2B-4)� Place Value Mat (2B-5)� Number Line (2B-6)� Hundred Chart (2B-7)� Charts 3A and 3B (2B-8 and 2B-9)

Section CWhole Number SamplerTransparencies:� Lesson Planning Process Chart / Analysis

Chart (2C-1)� Day 2C Reflection Prompt (2C-21)


Handouts:� Lesson Planning Process Chart / Analysis

Chart (2C-1)


Section CButton, Button, Where’s the Button?For each group of four:� 10 small paper cups labeled with

numerals 1 – 10� 1 buttonSection CExpanding NumbersScissors*Calculators

Transparencies:� Expanding Numbers Activity Page (2C-

3)� Expanding Numbers Recording Sheet

(2C-4)

*Create a reusable set of Expanding NumberCards and scissors won't be needed.

Handouts:� Expanding Number Cards* (2C-2)� Expanding Numbers Activity Page (2C-

3)� Expanding Numbers Recording Sheet

(2C-4)

*Print on cardstock and laminate to create areusable set.

Section CIn and Out� Twelve Ways to Make Eleven by Eve

Merriam� Counters� Small cup for each group of 4� Manila paper for each participant� Scissors

Transparencies:� In and Out Target Board (2C-5)� In and Out Recording Sheet 1 (2C-6)� In and Out Recording Sheet 2 (2C-7)

� Manila paper 11" X 17"

Handouts:� In and Out Target Board per group (2C-

5)� In and Out Recording Sheet 1 (2C-6)� In and Out Recording Sheet 2 (2C-7)

Section CIt’s the Place that Counts� Bags of small beans or centimeter cubes

Transparencies:� Recording Charts (2C-8)� Bean Place Mat (2C-9)

Handouts:� Recording Charts (2C-8)� Bean Place Mat (2C-9)


Section CMake It Zero (Base-ten Block Version)� Base-ten blocks

Transparencies:� Make It Zero Activity Page (2C-10)� Place Value Mat (2C-11)� Make It Zero Recording Sheet (2C-12)

Handouts:� Make It Zero Activity Page (2C-10)� Place Value Mat (2C-11)� Make It Zero Recording Sheet (2C-12)

Section CMake It Zero (Calculator Version)� Calculators

Transparencies:� Make It Zero Activity Page (2C-13)� Make It Zero Recording Sheet (2C-14)

Handouts:� Make It Zero Activity Page (2C-13)� Make It Zero Recording Sheet (2C-14)

Section CMoreFor each pair of participants:� 1 deck of cards with face cards removed� 10 small paper sacks numbered 1 – 10Section CNumber, Number, Where’s the Number?Transparencies:� Hundred Chart (2C-15)

� 5" X 7" note cards

Section CSee Saw� Pinto beans� Paper clip and pencil or clear plastic

spinner per pair of participants

Transparencies:� See Saw Directions (2C-16)� See Saw Game Board (2C-17)� Spinner (2C-18)� See Saw Recording Sheet (2C-19)� See Saw Reflections (2C-20)

Handouts:� See Saw Directions (2C-16)� See Saw Game Board (2C-17)� See Saw Recording Sheet (2C-19)� See Saw Reflections (2C-20)

Section DClosure for Day Two: Shaping Up a Summary

� Square Circle Triangle Charts made fromchart paper

� Sticky notes

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section APlace Value Puzzles Page 5

Place Value Puzzles

Key Question:� What is the role of place value in our number system?

Frame:This activity focuses on ways to use place value concepts to solve a problem. It is included tohelp participants understand the role place value plays in understanding whole numbers.

Transparencies/Handouts:� Place Value Puzzle 1 (2A-1)� Place Value Puzzle 2 (2A-2)� Place Value Puzzle 3 (2A-3)� Day 2A Reflection Prompt (2A-4)

Procedures Notes1. Ask the participants to look at the first

puzzle and build the greatest 4-digitwhole number possible, using any of thedigits from 0 through 9. Repeats of thedigits are allowed.

Since the greatest digit is 9 and repeateddigits are allowed, the greatest 4-digit wholenumber is 9,999.

After participants have had an opportunity tosolve the puzzle, ask for volunteers to readtheir greatest number. Ask the class if itmeets all the criteria. Have volunteersexplain why it is or is not a good answer.

2. Ask the participants to look at the firstpuzzle page and build the least 4-digitwhole number possible, using any of thedigits from 0 through 9. Repeats of thedigits are allowed.

The least 4-digit whole number that meetsthe criteria is 1,000.

After participants have had an opportunity tosolve the puzzle, ask for volunteers to readtheir least number. Ask the class if it meetsall the criteria. Have volunteers explain whyit is or is not a good answer.

3. Ask the participants to look at the secondpuzzle and build the greatest 4-digitwhole number possible, using any of thedigits from 0 through 9. Repeats of thedigits are not allowed.

If no digits are repeated, the greatest 4-digitwhole number possible is 9,876.



4. Ask the participants to look at the secondpuzzle and build the least 4-digit wholenumber possible, using any of the digitsfrom 0 through 9. Repeats of the digitsare not allowed.

If no digits are allowed to be repeated, theleast 4-digit whole number possible is 1,023.

After participants have had an opportunity tosolve the puzzle, ask for volunteers to readtheir least number. Ask the class if it meetsall the criteria. Have volunteers explain whyit is or is not a good answer. Why can't youput 0 in the thousands place?

If 0 is in the thousands place, the numberwould be a 3-digit number.

5. Ask the participants to look at the thirdpuzzle and build the greatest 4-digitwhole number possible, using any of thedigits from 0 to 9 without consecutivenumbers next to each other. Repeats ofthe digits are not allowed.

With these criteria, the greatest 4-digit wholenumber possible is 9,758.


6. Ask the participants to look at the thirdpuzzle and build the least 4-digit wholenumber possible, using any of the digitsfrom 0 to 9 without consecutive numbersnext to each other. Repeats of the digitsare not allowed.

With these criteria, the least 4-digit numberwhole possible is 1,302.

After participants have had an opportunity tosolve the puzzle, ask for volunteers to readtheir least number. Ask the class if it meetsall the criteria? Have volunteers explain whyit is or is not a good answer.

Debriefing:Have participants share answers to the following question:How did you know when you had a good answer?I built a 4-digit number. My number met all of the criteria.

Ask participants to discuss these questions in their groups:� What are the similarities among the puzzles?� What are the differences between the puzzles?� What number concepts do you need to know to solve these puzzles?� What do you have to know about place value to solve these puzzles?� What concepts are needed to understand place value?

Make a chart of the concepts needed to understand place value by asking each group to sharea new idea that is not yet listed until all of the ideas are written. Some of these ideas couldinclude the value of each digit, the value of each place, a digit has more value the further leftit is in the number (a 9 in the ones place is not as “valuable” as a 9 in the tens, hundreds, orthousands place), the concept of 1 more than or 1 less than is important for consecutivenumbers.


Reflection and Connection:These puzzles gave you the opportunity to think about the important components of placevalue. The power of place value is that an infinite set of numbers can be written using a finiteset of digits. It is often difficult for young children to understand that digits have differentvalues depending on their position in numbers. Take a few minutes to reflect in your journalon the following question:

What role does place value play in the understanding of whole numbers?

Transparency / Handout 2A-1


Place Value Puzzle 1

Using the digits, 0 - 9, form the two requested numbers. Digits may be repeated.(Not all digits will be used when forming the two numbers.)

� In the top frame, build the greatest 4-digit whole number.� In the bottom frame, build the least 4-digit whole number.




Using the digits, 0 - 9, form the two requested numbers. No two digits may be the same.(Not all digits will be used when forming the two numbers.)

� In the top frame, build the greatest 4-digit whole number.� In the bottom frame, build the least 4-digit whole number.




Using the digits, 0 - 9, form the two requested numbers. No two digits may be the same.(Not all digits will be used when forming the two numbers.)

� In the top frame, build the greatest 4-digit whole number without consecutive numbers next toeach other.

� In the bottom frame, build the least 4-digit whole number without consecutive numbers next toeach other.

Transparency 2A-4



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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section BInvestigating Whole Number Relationships Page 12

Investigating Whole Number Relationships

Key Question:� How does using different models for whole numbers help our students add to their

knowledge of number?

Frame:Through the exploration of concrete and graphical number models, participants will explorethe value of using these models for teaching basic number comparisons, ways of groupingnumbers, and place value. By comparing and contrasting the use of models in differentsituations, participants build their knowledge for making instructional decisions for teachingnumber.

Materials:Samples of the following materials for each table:� Single counters (such as beans, straws, teddy bear counters, unlinked linking cubes,

color tiles)� Two-color counters� Counters that students bundle (such as straws and rubber bands, linking cubes,

counters in cups)� Proportional materials that are pre-bundled (such as base-ten blocks, bean sticks)� Non-proportional materials (such as money, colored chips with key)� Calculators

Transparencies/Handouts:� Chart 1 (2B-1)� Chart 2 (2B-2)� Part-Part-Whole Mat (2B-3)� Ten Frame Mat (2B-4)� Place Value Mat (2B-5)� Number Line (2B-6)� Hundred Chart (2B-7)� Charts 3A and 3B (2B-8 and 2B-9)� Day 2B Reflection Prompt (2B-10)


Procedures NotesPart 11. Let participants know that we are going

to look at manipulatives and graphicorganizers and decide which works bestto show the numbers 7, 23 and 139.Some of the models work better thanothers. You will have an opportunity todiscover and discuss which models workbest to represent these numbers.

2. Give each group four different concretemodels for number: teddy bear countersor any example of single counters,linking cubes or any example of countersthat bundle, base ten blocks or anyexample of proportional materials thatare pre-bundled, and money or anyexample of non-proportional materials.Assign each member of each group oneof the models.

3. Have each participant use the assignedconcrete model and find as many ways aspossible to model each of these numbers:7, 23, 139. Have each participant useChart 1 to record answers.

4. In their groups, have each person sharethe ways found to model each of thenumbers with the assigned concretemodel.

Depending on your audience, you may haveto introduce the manipulatives used in thisinvestigation.

Ask participants questions to encourage themto notice the important mathematicalcharacteristics of each type of concretemodel. For example: Is color or position animportant part of your model? Does itrequire more than two colors? Doessomeone have to give you information to beable to interpret it? Does your modeldepend on groups or one-to-onecorrespondence? If it depends upon groups,are the groups shown according to their size(a proportional model) or not (a non-proportional model).


Part 21. Give each group four different graphical

models for number: Part-Part-WholeMat, Ten Frame Mat, Place Value Mat,and Number Line. Assign eachparticipant one of the graphical models.

2. Have each participant use the assignedgraphical model to model each of thesenumbers: 7, 23, 139. Have them useChart 2 to record their answers.

3. In their groups, have each participantshare how the numbers were modeledwith the assigned graphical model.

Ask participants questions to encourage themto notice the important mathematicalcharacteristics of each type of graphicalmodel.

Part-Part-Whole Mat: Designed to show aset (the whole) separated into two disjointsubsets (the parts). When used forsubtraction, the set is placed in the “whole”section, then one of the subsets is moved toone of the “parts” sections.

Ten Frame Mat: Designed to show the onesand tens places. There are two frames of tenon the ones side to facilitate regrouping inaddition or subtraction.

Place Value Mat: This mat allows a visualseparation of the place value columns and away to organize both proportional and non-proportional manipulatives.

Number Line: The number line is not veryuseful in showing quantity unless the wholenumber line between the number and 0 canbe viewed all at once. It is an importantconnection to using numbers to describelength.


Part 31. In this activity, participants will focus on

the types of concrete and graphicalmaterials that can be used to examinenumber concepts. The concretemanipulatives just used are examples ofsome of the concrete types. These typesinclude student-created representations,single counters (such as teddy bearcounters), two-color counters, countersthat students bundle (such as linkingcubes), proportional materials that arepre-bundled (such as base-ten blocks),non-proportional materials (such asmoney).

2. Using Chart 3A, have participantsdetermine the strengths and weaknessesof each type of concrete model forcomparing pairs of numbers. Askparticipants to name examples of thevarious types of manipulatives.

Types of concrete models:

Student-created representations(counting out loud, fingers, drawings)

Discuss the advantages and thedisadvantages of each student-createdrepresentation; for example, students“believe” in the relationships betweenrepresentations when they make themthemselves, but the materials may not be neator may take a lot of time to make.

Single counters (beans, straws, teddybear counters, unlinked linking cubes,color tiles, anything that can be countedone at a time)

Think about the fact that these models arenot organized, yet they are great for one-to-one correspondence.

Two-color counters These counters show different ways todecompose numbers into their parts, startingwith the whole and seeing it in terms of itsparts, but you are limited to only two colors.


Counters that students bundle (strawsand rubber bands, linking cubes, countersin cups

Discuss the thinking that’s necessary forstudents to create and use the bundles, whymaking groups is useful (efficiency). Whatmakes groups of 10 so special?

Proportional materials that are pre-bundled (base-ten blocks, bean sticks,paper models)

Think about advantages of students notneeding the time to make the bundles.

Non-proportional materials (money;colored chips w/key.)

Non-proportionality is a step further in termsof mathematical abstraction.

3. Provide time to debrief the strengths andweaknesses of each model. Create agroup chart. Ask each group to share onestrength or weakness. Continue sharinguntil all groups are satisfied with the list.Use the descriptions of the concretemodels as prompts, if necessary.

4. Using Chart 3B, have participantsdetermine the strengths and weaknessesof each graphical model for comparingpairs of numbers. While four modelswere used in Part 2, here participants willexamine six models.

Graphical Models:

� Calculators A calculator shows us the use of place valueso that an infinite set of numbers can berepresented with a finite set of digits.

� Part-Part-Whole Mat Discuss the usefulness of this model forshowing decomposition of numbers.

� Ten Frame Mat Discuss the advantage of this organizedstructure for visual memory; decompositionideas involved; emphasizing groups of ten;benchmarks of 10 and 5. Discuss thedisadvantages for its use with large numbers.


� Place Value Mat This mat allows a visual separation of theplace value columns and a way to organizeboth proportional and non-proportionalmanipulatives.

� Number Line Students often have problems with countingdots instead of the spaces because ofcontinuous vs. discrete model. It is a visualrepresentation of relationships, but it is notvery useful for decomposition.

� Hundred Chart Discuss the advantages and disadvantages ofvarious representations and the need to learnthe relationship positions (0 - 99 and 1 -100).

5. Allow time to debrief the strengths andweaknesses of the graphical models.Using a group chart, have each groupgive one strength or weakness. Continueadding to the chart until the groups haveall participated.


Debriefing:Once the analyses are complete, lead a discussion about the use of these models ininstruction. If the goal is to introduce new concepts concretely, then it is important forteachers to understand when the use of these different models would be advantageous. Haveparticipants consider the following at their tables.

� I am going to teach basic counting strategies: counting in sequence, counting sets, andcounting on. Which concrete manipulatives should I use? Which graphical modelswould be the most helpful?

� I have students who don’t seem to understand place value concepts at all. Whichconcrete manipulatives should I use? Which graphical models would be the mosthelpful?

After groups have had 5 – 10 minutes to discuss, have everyone do a “Walk and Share.” Allparticipants stand, walk 7 steps, find a partner, and share the discussion from their group.This process should be repeated at least once.

Have participants return to their groups. If time permits, ask three or four people to sharesomething new they learned or something they already knew that was confirmed.

Reflection and Connection:This investigation should give participants a working knowledge of the advantages anddisadvantages of using concrete and graphical models to represent whole numbers. Take afew minutes to reflect in your journal on the following question:

How will you use this information to make instructional decisions for the teaching ofwhole numbers to support student learning?


(This page is intentionally left blank.)

Transparency / Handout 2B-1


Chart 1: Investigating Number Relationships with Concrete Models

Make a written record of how you built your three numbers. You may draw your model or writeabout it. Circle the concrete model you used: single counters, counters that bundle, proportionalmaterials that are pre-bundled or non-proportional materials.

7 23 139



Chart 2: Investigating Number Relationships with Graphical Models

Make a written record of how you built your three numbers. You may draw your model or writeabout it. Circle the graphical model you used: Part-Part-Whole Mat, Ten Frame Mat, Place ValueMat, Number Line.

7 23 139



Part-Part-Whole Mat

Part Part

Whole



Ten Frame Mat



Place Value Mat

Thousands Hundreds Tens Ones



Number Line

To make the centimeter Number Line:

� Cut on the dotted lines� Overlap the ends, putting the 20 from the first strip on top of the 20 on the second strip� Continue overlapping in the same way, putting the 40 on the 40, the 60 on the 60, and the 80

on the 80� Make sure the centimeters are accurate at the overlapped edges� Tape the strips together



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140



Hundred Chart

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100



Chart 3A: Strengths and Weaknesses of Concrete Models

Numbers toCompare

Student-createdrepresentations

Single counters Two-colorcounters

Counters thatstudents bundle

Pre-bundledproportional

Non-proportional

6 comparedto 8

15 comparedto 12

162 comparedto 181



Chart 3B: Strengths and Weaknesses of Graphical Models

Numbers toCompare

Calculators Part-Part-Whole Mat

Ten Frame Mat Place ValueMat

Number Line Hundred Chart

6 comparedto 8

15 comparedto 12

162 comparedto 181

Transparency 2B-10



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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CWhole Number Sampler Page 31

Whole Number Sampler

Key Question:� How do we use the Lesson Planning Process to select and orchestrate learning

experiences to develop students’ understanding of whole number meanings andrelationships?

Frame:The participants will explore a sampler of learning experiences designed to support thedevelopment of the understanding of whole number meanings and relationships. Participantswill examine these learning experiences in terms of how the Lesson Planning Process is usedto design effective instruction. By engaging in these learning experiences, participants cangain a sense of how Evidence of Understanding can be built with students over time.

Materials:� Chart paper� Markers

(See Day Two Materials Chart for materials for sample learning experiences)

Transparencies/Handouts:� Lesson Planning Process Chart / Analysis Chart (2C-1)� Day 2C Reflection Prompt (2C-21)

Procedures Notes1. Ask participants to consider their

experiences on Day 1 with Big Ideas andthe Lesson Planning Process. Ask themto share where they can get informationabout what the Big Ideas are (e.g. TEKS).

Based on the participants’ comments, it islikely that someone will refer to the TEKS.

2. Ask leading questions to identify wherein the TEKS to look for Big Ideas.

Participants may identify the KnowledgeStatements.

3. Ask participants to read the Knowledgestatements regarding number for Grades1-5. From these statements, make a list ofBig Ideas regarding numbers based onparticipants’ responses.

4. Then have them turn to the Debriefingsection to compare the lists of Big Ideas.

5. Allow time for discussion about BigIdeas and have this group come toagreement about the Big Ideas to be usedfor the sampler of activities.


6. Hand out the Lesson Planning ProcessChart for the Whole Number Sampler.Have participants review the fourcomponents in the Lesson PlanningProcess.

7. Introduce the Sampler of LearningExperiences for Whole Numbers andexplain to participants that they will beusing the Lesson Planning Process Chartto identify the four components of theLesson Planning Process in relation toeach learning experience.

8. Follow the directions for “See Saw.”After the activity has been completed,review the four steps of the LessonPlanning Process Chart. Allowparticipants time to work in groups offour to complete Steps 1 and 2 on theLesson Planning Process Chart.

As participants review the Lesson PlanningProcess Chart, have them identify the ways“See Saw” promotes Step 3: students activelyengaged in rigorous learning.

Have two or three groups share Evidence ofUnderstanding statements. Write them onchart paper to display for the whole group.

9. Follow the directions for “Make It Zero(Base-Ten Block Version).” Haveparticipants complete Steps 1 and 2 onthe Lesson Planning Process Chart. Havethem compare the two sections of thechart that are completed, looking forsimilarities and differences.

Have participants determine if the Evidenceof Understanding statements from “See Saw”could also apply to ”Make It Zero.” Havethem determine what additional statementscould be added for the new activity.

10. Whole group instruction for the Sampleris possible. The suggested order of theSampler activities for whole groupinstruction is:

� Make It Zero (Calculator Version)� In and Out� Button, Button� Number, Number� More� Expanding Numbers� It’s the Place that Counts

11. After participants have experienced somesubset of the Sampler and filled out theirLesson Planning Process Charts, putthem in small groups to compare anddiscuss their notes.

In these discussions, participants might begrouped by grade level, by learningexperience, by topic, or randomly.


12. Guide participants to create a combinedLesson Planning Process Chart for wholenumber meanings and relationships,identifying overall Big Ideas andEvidence of Understanding for this topic.

As participants look at the activities as awhole group, they may discover that oneactivity aligns with more than one gradelevel. The whole group discussion couldinclude modifications that might be made toan activity to accommodate multiple gradelevels.

Debriefing:The classroom activities presented here are a sampler of learning experiences that could beused to teach whole number concepts. Have participants, with their small groups, look atStep 2 for each activity in the Lesson Planning Process chart and discuss the following: Howmany of these statements are attached to more than one activity? Order the activities fromeasiest to most difficult. Decide how you could add to the Evidence of Understanding as themore complex or difficult activities are completed. Be prepared to share your thoughts withthe whole group.

Discuss with the participants how the Sampler activities address Steps 3 and 4. Use thequestions on the chart from Day 1, Lesson Planning Process Chart, to facilitate thediscussion. You may also want to reference the article in Day 1, “What Should I Look for ina Math Classroom?”

To the facilitator: The following list provides an example of a list of combined Evidences ofUnderstanding that might be created for Whole Numbers.

Step 1“Big Ideas” (connected to TEKS)

Step 2Evidences of Understanding

� Using representations and models toshow whole numbers.

� I can use objects to show a wholenumber.

� I can draw pictures to show a wholenumber.

� I can read and write the number.� Using whole numbers to describe

quantities.� I can count to tell how many are in a set.� I can use a whole number to describe

how many.� I can talk about what a number means.� I can write about what a number means.� I can pick a reasonable number to

describe something.� Using place value to describe a quantity. � I can use 10s and 1s to describe a whole

number.� I can talk about the patterns I see in

numbers.


� Describing relationships between wholenumbers.

� I can compare whole numbers using setsof objects.

� I can draw pictures to compare wholenumbers.

� I can use symbols (and place value) tocompare whole numbers.

� I can talk about the relationships betweenwhole numbers.

� I can write about the relationshipsbetween whole numbers.

Reflection and Connection:The learning experiences in this section allow you to examine important components of theLesson Planning Process. As you completed the tasks involving the meanings of andrelationships among whole numbers, you identified mathematical Big Ideas and Evidences ofUnderstanding that can be established with students over time. In your journal, create arecord of your thinking at the moment: write a paragraph, develop a bulleted list, use agraphic organizer, create a picture…

How will you use these ideas in the design of your own instruction about whole numbers?

Transparency / Handout 2C-1



Step 1Big Ideas

Step 2Evidence ofUnderstanding

Step 3Orchestrating forRigorous Learning

Step 4Communication toSupport Learning




Step 1Big Ideas






Analysis Chart

Step 1Big Ideas




Transparency 2C-21



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aa bb oo uu tt ww hh oo ll ee nn uu mm bb ee rr ss ??

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CButton, Button, Where’s the Button? Page 40

Button, Button, Where's the Button?

Words for the Word Wall:� Before� After� First� Second� Third� Fourth� Fifth� Sixth� Seventh� Eighth� Ninth� Tenth

Materials:� 10 small paper cups labeled with numerals 1-10� 1 button

Procedures Notes1. The teacher should place the 10 labeled

cups upside down in order on a table atthe front of the room and secretly hide abutton under the eighth cup.

This activity helps students become familiarwith ordinal numbers and to learn to interpretclues in a logical manner. Students guessunder which of the cups labeled withnumbers 1-10 the teacher hid the button. Ifthey make a guess that is incorrect, theteacher will give the class a clue to help themguess correctly. Clues tell whether the buttonis under a cup that comes before or after theincorrect guess.

2. The students are to guess under whichcup the button is hidden. Each time theirguess is incorrect, the teacher will givethe class a clue to help them determinethe correct cup. The clue will tell theclass whether the button is under a cupthat comes before or after the cupguessed.

If the class guessed the 1st cup, the teacherwould give the clue, "The button is under acup that comes after the 1st cup."

For the next guess a student might guess the6th cup. The teacher would give the clue,"The cup the button is under comes after the6th cup.”

If the guess was either the 9th or 10th cup,the teacher would give the clue, "The cup thebutton is under comes before the 9th or 10thcup.”

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CButton, Button, Where’s the Button? Page 41

3. Students should use complete sentencesto make their guesses.

4. Guesses continue until the correct cup ischosen.

5. The class should play the game a secondtime. This time the teacher secretly hidesthe button under the third cup. However,after 2 or 3 clues, the teacher should stopthe game and ask the class what theyknow so far about the clues they havebeen given.

This time the teacher should ask the classwhat information they know from the cluesthey already have.

Examples of clues might include:

"The button is under a cup after the firstcup." "The cup the button is under comesbefore the fifth cup."

Lead the children to decide what cups can beeliminated and what cups the button couldpossibly be under, instead of randomlyguessing. Strategies might include drawing apicture, making a diagram, or making a chart.

6. Continue the game with a new guess.After each guess and clue, talk with theclass about what they learned from theclue.

The teacher should look and listen forevidence of students’ understanding:

� using all the clues before making a newguess

� using the clues to eliminate numbers

� knowing positions of ordinal numbers

7. Play the game several more times untilmost of the children understand how tointerpret clues.

8. After playing the game for several days,let the children take turns hiding thebutton and giving clues.

Discussion:

� Were you able to find the button quickly? How many clues did it take each time?

� Did you find it more quickly after you tried a couple of times?

� Was there a button that was harder to find? Which cup was it under? Why do yousuppose it was harder to find that button?

� What strategies did you use in making your guesses? (For example, always guess the“middle” cup because that will always eliminate half of the cups there.)

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CExpanding Numbers Page 42

Expanding Numbers

Words for the Word Wall:� Ones� Tens� Hundreds� Thousands� Millions� Period

Materials:� Scissors (to make number cards if they are not already made)� Calculators

Transparencies/Handouts:� Expanding Number Cards (2C-2)� Expanding Numbers Activity Page (2C-3)� Expanding Numbers Recording Sheet (2C-4)

Procedures Notes1. Have the students organize the cards into

place-value groups: all of the onestogether, all of the tens together, etc.

2. Have students find the card that shows50. Ask them which other card theywould need to put with 50 to make 54.Discuss with the students how theywould have chosen the two cards. Havethem enter the value of each card into anaddition problem in the calculator: 50 + 4= 54. Have them verify that the stackedcards and the calculator answer are thesame.

As students build numbers, have them put thelargest value on the bottom of the stack andplace other cards on top of the first card indecreasing order. For this problem, thestudents should have the 50 card and the 4card. Have them place the 4 card on top ofthe 50 card, lining up the right edges. 54 willbe showing.

3. Write the number 54 in words on theoverhead (fifty-four). Talk about the twoparts of the written word: fifty and four.

Ask: How many cards did you use to buildthe number 54? (two cards) How does thenumber of cards relate to the words? (Thereare two words with a hyphen.) Do you thinkall numbers work that way?


4. Have students build the number 104 withthe cards. Write the number in words onthe overhead (one hundred four). Discusswith the students that some numberwords work together to create onenumber, such as one hundred, threethousand, etc. Have the students verifywith the calculator by entering the valueof each card: 100 + 4 = 104.

Ask: How many cards did it take? (two) Howmany words? (three, but one hundred is justone number)

5. Write one hundred thousand on theoverhead. Discuss with the students howmany cards they need to build thisnumber. Have them find the card.

6. Write one hundred forty thousand on theboard. Discuss with the students howmany cards they need to build thisnumber. Have them find the cards andbuild the number. Have them verify theanswer by entering the values in thecalculator: 100,000 + 40,000 = 140,000.

7. Write one hundred forty-two thousand onthe board. Discuss with the students howmany cards they need to build thisnumber. Have them find the cards andbuild the number. Have them verify theanswer by entering the values in thecalculator: 100,000 + 40,000 + 2,000 =142,000.

Ask: If fifty needs one card and one hundredthousand needs one card, how can the wordshelp us decide how many cards we need?

Ask: If fifty-four needs two cards and onehundred forty thousand needs two cards, howdo the words help us decide how many cardswe need?

8. Have pairs of students write a rule forusing the number words to helpdetermine the number of cards needed tobuild the number. Have them use the ruleas they complete the activity page.

9. After the activity page has beencompleted, debrief the activity by askingstudents to talk about the rule they usedand whether it always works. If a ruledoesn’t always hold true, have thestudents rewrite the rule and test it bylooking at the results they already have.Have students share the rules they havedeveloped.


Discussion:

� Did your rule work? Why do you suppose that happened?

� Did you get the same answer with the cards and the calculator? Why do you supposethat happened?

� Which number was the most difficult to figure out? Why do you suppose it was moredifficult than the other numbers?

Expanding Number Cards Handout 2C-2


7 0 10 100 1,000

8 1 20 200 2,000

9 2 30 300 3,000

3 40 400 4,000

4 50 500 5,000

5 60 600 6,000

6 70 700 7,000



80 800 8,000

90 900 9,000

10,000 100,000

20,000 200,000

30,000 300,000

40,000 400,000

50,000 500,000



60,000 600,000

70,000 700,000

80,000 800,000

90,000 900,000

1,000,000

2,000,000

3,000,000



4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000



Expanding Numbers Activity Page

Work with a partner. Write a rule about how the number words canhelp you decide how many Expanding Number cards you need tobuild a number.

Our rule is:

� Choose 5 of the numbers written in words.� Write these numbers on the Expanding Numbers Recording

Sheet.� Decide how many cards you will need to build the number. Get

those cards.� Build the number.� Record your answer as a sum on the calculator.

Numbers Written in Words� five hundred sixty-three� seven hundred thirty-four� one thousand, seven hundred� six thousand, twenty-two� ten thousand, five hundred� nineteen thousand, two hundred fifteen� one hundred eighty-two thousand, nine hundred fifty-three� one million, three hundred seventy-eight thousand, four hundred

fifty-two



Expanding Numbers Recording Sheet

Directions:Choose 5 numbers from the Numbers in Words in list. Write the words in column 1. Write how many cards you will need in column 2.Write the cards you used in column 3. Write the addition problem you entered on the calculator in column 4. Write the number indigits in column 5. Look at the example.

Column 1Number in Words

Column 2Numberof Cards

Column 3Cards we used

Column 4Addition problems

Column 5Number in digits

Seventy-eight 2 70, 8 70 + 8 = 78

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CIn and Out Page 51

In and Out

Words for the Word Wall:� Decrease by one� Increase by one� Greater than� Less than� More than� Equal� Total� Number sentences

Materials:� Twelve Ways to Make Eleven by Eve Merriam (ISBN 0-689-80892-5)� Counters� Small cup for each group of 4� Manila paper for each participant

Transparencies/Handouts:� In and Out Target Board per group (2C-5)� In and Out Recording Sheets 1 and 2 for each student (2C-6 and 2C-7)

Procedures Notes1. Read and discuss the book, Twelve Ways

to Make Eleven by Eve Merriam (ISBN0-689-80892-5)


2. Assign the students partners. Model theprocess of this activity with a student. Tobegin, the teacher determines how manycounters to toss and records that numberon the Recording Sheet 1. Next, eachgroup will lay their target paper on a flatsurface and place the determined numberof counters in a cup. Players take turnstossing, all at once, the counters over thetarget. The number of counters that landin and the number of counters that landoutside of the target are recorded on theIn and Out Recording Sheet 1.Participants take turns tossing andrecording.

The object of this activity is to toss countersover a target paper and record how manycounters land in and out of the target. Eachparticipant will make a toss and record his orher findings.

As you model playing the game, ask somequestions to make sure your studentsunderstand the process:

� How many counters do you have in all?

� How many counters landed in the target?

� How many counters landed outside of thetarget?

� How do you record each toss?

After your first toss, ask students somequestions to promote mathematical thinking:

� When you toss again, will you get thesame results?

� What are some possible number pairs youmight toss?

� Why is it helpful to record your results onthe recording sheets? (To remember whathas happened, to communicate to theteacher and others what you have done.)


3. On Recording Sheet 2, participants writethe comparing number sentence thatdescribes the number of counters thatlanded in and the number of counters thatlanded outside the target. The process isrepeated until all recording sheets arecompleted.

Model writing a comparison numbersentence on Recording Sheet 2. As studentscompare the numbers that land in and out ofthe target, suggest that they write two dots(like a colon, :) beside the larger number andonly one dot (like a period) beside thesmaller number. Then they just go from dotto dot to draw the greater than or less thansign. Does the arrow always point to thesmaller number? (Yes.) Point out that the“less than” sign begins with the letter “L”and that the less than sign looks similar to theletter “L”. You may want to make a sign foryour classroom saying “<ess than” (using anexaggerated less than sign instead of theletter “L.” If both numbers are the same,each number gets two dots (like a colon, : )and when you go from dot to dot, you drawthe equal sign.


4. Once all recording sheets are complete,ask the participants to describe what theyhave recorded. They should comparetheir grid with the other participants’recording sheet in their group. Studentsshould be ready to report to the class theirentire group’s different recorded numberpairs.

5. As the number pairs are reported, theteacher will record them on a class recordsheet (like Recording Sheet 1.) Theteacher only records different numberpairs. If a number pair is repeated, theteacher should lightly retrace thesenumbers on the class record sheet.

To assess the student’s understanding of thenumbers they have recorded, choose one ofthe number pairs on the recording sheet andask a student to model it with counters in andout of the target. Ask questions such as:

Do the pairs of numbers on your recordingsheet look the same as the pairs of numberson the recording sheets of the others in yourgroup?

� Do you notice any patterns? What arethey?

� Encourage the students to talk about thepatterns they noticed.

� Based on the student’s comments, theteacher may introduce or reinforce mathvocabulary such as decrease by one,increase by one, greater than, less than,more than, equal, and total.

After all groups have had an opportunity toreport their findings, the teacher should asksome leading questions to guide the studentsto recognize what they did. Were the studentsable to find all of the number pairs for thechosen total?

Remind the class of the book, Twelve Waysto Make Eleven. Were the pictures in thebook only made with two numbers? No.Would there be other ways or pairs ofnumbers that would make our total? Yes, anumber can be composed and decomposed inmany ways.


6. Each student should make a booklet for aselected number (the chosen numbershould be equal to or greater than 6).

To make a book with 8 pages, use 1 largesheet of manila paper per student.

� Fold the paper in half; short end toshort end, and crease.

� Fold back one side halfway andcrease; fold back other side halfwayand crease. (If opened fully, the paperwould have 4 long rectangles.)

� Keeping the paper folded, fold theshort end to short end and crease. (Ifopened fully, the paper would have 8rectangles.)

� Unfold the last fold and allow thesides to flip down (the center foldwill look like a tent peak; the papercan stand by itself on a desk or tableand will make the shape of a “w”).

� Cut down from the peak of the tentwhere the center folds meet. Cut onlydown to the next fold.

� Pick the paper up with one hand oneither side of the cut. Fold down sothe cut is across the top. You willneed to crease one fold again.

� Fold into the shape of a book.

The first page of the book includes the titleand the author. (Example: “Ways to MakeSix” by Author’s Name) On each of thepages, students should record one possiblenumber pair and draw a picture that describesthis number pair.

7. Variation or follow-up activity: Theactivity, In and Out, could be played inthe same manner, recording the fractionalpart of the set of counters that landed inthe target and the fractional part of the setthat landed off the target.


Discussion:Open the discussion by asking participants to describe what they noticed about the differentrecording sheets and why different charts were used. Different charts show the information indifferent ways. What number pairs were repeated? Was there another in/out pair for the totalnumber that was not listed on a recording sheet?

Participants should realize the in/out pairs are different ways to make the total number. Theymay also recognize that as one number in the pair increases, the other number decreases. Forexample, one in/out pair may be 2 and 6 for a total of 8, whereas another pair may be 3 and 5.The 2 increased by one and the 6 decreased by one. With continued exposure to the game,students will learn to think more and more about patterns and numerical relationships.



In and Out Target Board

Out Out

In

Out Out



In and Out Recording Sheet 1

I used _____ counters.

In Out



In and Out Recording Sheet 2


Trial In Out Comparing Parts

A

B

C

D

E

F

G

H

I

J

List all of the different number sentences.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CIt’s the Place that Counts Page 60

It’s the Place that Counts

Words for the Word Wall:� Place value� Equal groups� Square numbers

Materials:� Bags of small beans or centimeter cubes

Transparencies/Handouts:� Recording Charts (2C-8)� Bean Place Mat (2C-9)

Procedures Notes1. Distribute bags of beans to students.

Have each participant reach in the bagand get a handful of beans. Have themdivide the beans into groups of four withones left over using the Bean Place Mat.Discuss with the students how the three-column recording chart might be used. Ifpossible, have them record the “ones leftover” in the right hand column and thegroups in the middle column, reflectingthe positions on the place value mat.

Model for the students by placing all of thebeans in the right hand column, then movinggroups of 4 beans to the middle column.Place 4 beans in a line in each rectangle.

2. Allow students time to repeat the processthree more times, using a differentnumber of beans each time. For each setof beans, have the students divide thebeans into groups of four and ones leftover, recording the numbers on therecording chart.


3. Discuss with the students the numbersthey have recorded. Ask them to recordall of the numbers listed in the “ones leftover” column.

What numbers did you record in the “ones”column?

Why do you suppose only 0, 1, 2, and 3 wereused in the ones column?

If you added four beans to one of the“handfuls” of beans, how would that changethe recording chart? Why do you supposethat happens?

(Show the students a recording chart with 3in the groups column and 3 in the onescolumn.) If these numbers represented piecesof candy, which number would you want?Why?

4. Present them with the challenge, “If thedigits in the ‘ones’ column were the onlydigits you could use, what would youneed to do to show 4 or more groups?How did you show 4 or more beans?”

5. Have students discuss various options.Possibilities would be additional columnswith other size groupings or changing thegroupings for the “groups of four”column. If using the additional column isnot brought up, ask the students how thethird column might be used. Discusswhat the value of the column might be.After a different system is selected, havethe students build each of the numbersusing the different grouping systems.

Did your new system work for every set ofbeans or did you have to make modificationsto accommodate the various numbers?

How could you modify your system to workwith any number of beans?

How would you describe the shape of thegroups of four beans? If you made four ofthose groups, what shape could be made?

6. Have the students investigate other groupnumbers such as 5, 6, 7, or 8. Have themmake a list of the possible numbers in theones column. Discuss with them whenthey would need to move from the two-column display to the three-columndisplay. For each new grouping, havethem answer the questions in 5 above.


Discussion:Based on your explorations, what rules could you write about the number of beans in theones column? In the groups column? In the groups of groups column?

The number 1 2 3 was recorded on my recording chart. Which of the numbers represents themost beans? How do you know? What values could the 1 represent? How do you know?



Recording Charts



Bean Place Mat

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CMake It Zero (Base-ten Block Version) Page 65

Make It ZeroBase-ten Block Version

Words for the Word Wall:� Place value� Base-ten blocks� Ones� Tens� Hundreds

Materials:� Base-ten blocks

Transparencies/Handouts:� Make It Zero Activity Page (2C-10)� Place Value Mat (2C-11)� Make It Zero Recording Sheet (2C-12)

Procedures Notes

1. Provide each student with the Activitypage, a place value mat, a recordingsheet, and base-ten blocks.

2. Have the students make differentnumbers on the place value mat with thebase-ten blocks. Use several different 2-and 3-digit numbers. Make sure studentsunderstand “build the number with thefewest blocks.”

Build 32 with your base-ten blocks. Whichblocks did you use? How many blocks didyou use? What other ways could you build32? How many blocks did that way take?Which way had the fewest blocks?

3. Allow students time to work though the 6puzzles individually. When they havefinished, have them check their answerswith a partner.

Monitor the class and validate “buildingnumbers with the fewest blocks.”

4. With a partner, have them answer thediscussion questions. Have them writetheir rule and test it with the newnumbers.


5. As a whole class, have partners sharetheir rule and tell whether or not itworked for all of the numbers.

Select two or three partnerships thatexhibited a good understanding of theprocess to share their work. Ask questionssuch as: Did any other pair come up with thesame rules? Did any other pair havedifferent rules? Are all of these rulesacceptable? How do we know?

6. Work towards having class consensus onrules that will always work.

Students may come up with a different rulefor each place.

Discussion:

� Did you and your partner have the same answers? How did you decide on the bestanswers?

� Did your rule work? How do you know? Did you find more than one way to showthat your rule worked? What were those ways?



Make It Zero Activity Page, Base-ten Block Version

Materials: base-ten blocks, place value chart, recording sheet

Puzzle A� Build 342 with the fewest number of base-ten blocks.� Remove blocks to change the number to 340. Write or draw

what you did on the recording chart.

Puzzle B� Build 637 with the fewest number of base-ten blocks.� Remove blocks to change the number to 630. Write or draw


Puzzle C� Build 475 with the fewest number of base-ten blocks.� Remove blocks to change the number to 405. Write or draw


Puzzle D� Build 254 with the fewest number of base-ten blocks.� Remove blocks to change the number to 204. Write or draw


Puzzle E� Build 526 with the fewest number of base-ten blocks.� Remove blocks to change the number to 26. Write or draw what

you did on the recording chart.

Puzzle F� Build 189 with the fewest number of base-ten blocks.� Remove blocks to change the number to 89. Write or draw what

you did on the recording chart.



Look at your recording chart and answer the following questions:� How do you know you built each number with the fewest

number of base-ten blocks?� What did you remove to change one of the digits to zero?� How did you decide which blocks to remove?� Did you always remove the same type of block?

Write a rule that tells how to remove blocks to change one digit tozero. Try your rule with these numbers:� Change 234 to 230� Change 324 to 320� Change 432 to 402� Change 243 to 203� Change 342 to 42� Change 423 to 23

Did your rule work? How do you know?



Place Value Mat




Make It Zero Recording Sheet, Base-ten Block Version

I built thisnumber.

I removed these blocks……and changedmy number tothis number

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CMake It Zero (Calculator Version) Page 71

Make It ZeroCalculator Version

Words for the Word Wall:� Ones� Tens� Hundreds� Thousands� Place value

Materials:� Calculators

Transparencies/Handouts:� Make It Zero Activity Page (2C-13)� Make It Zero Recording Sheet (2C-14)

Procedures Notes1. Have students practice entering numbers

and operations into the calculator andrecording the keystrokes. It may behelpful to have them work as partners,one entering the information into thecalculator and the other recording thekeystrokes.

Enter 345 into the calculator. If I wanted tochange this number to 349, I could enter + 4= and get the new number. On my recordingsheet I would write + 4 = in the columnlabeled “Keystrokes.” I would write 3 in thecolumn labeled” Number of keystrokes”because I pressed 3 keys. I would write“349” in the “Changed number” columnbecause that is on my display.

What if I wanted to change 345 into 645?What would I enter? (+ 300 =) What do Iwrite in the “Keystrokes” column? (+ 300 =)How many keystrokes is that? (5 keystrokes)How do I know? (You count the number oftimes you pressed a key on the calculator.)What is the “Changed number?” (645) Isthat the number I wanted? (Yes.) What otherways are there to reach 645? (Allowstudents to come up with other ways. Havethem count the keystrokes.) Which way hasthe fewest keystrokes? (+ 300 =)

2. Have students complete the activitysheet, recording their answers on therecording sheet.

Monitor the work of the students. Askstudents to verify that they have found theway to change a digit to zero with the fewestkeystrokes.


3. Have them share their answers with apartner, comparing recording sheets.Have them individually answer thediscussion questions.

4. With their partner, have them write a rulethat tells how to change a digit to zerowith the fewest number of keystrokes.Have them work with their partner to testthe rule, using the additional problems.

Monitor the class and find two or threestudents that have a good understanding ofthe process. Have them share their work withthe class.

5. Challenge students to find numbers thatwon’t work with their rule. Explain that ifthere are no counterexamples, the rule isvalid.

Discussion:

� Did you and your partner use the same keystrokes? Why do you suppose thathappened?

� What patterns did you find as you solved the puzzles? How did the patterns help youwrite your rule?

� What does place value have to do with this activity?



Make It Zero Activity Page, Calculator Version

Materials: calculators, recording sheet

Puzzle A� Enter 3246 into your calculator.� Using the fewest number of keystrokes possible, change the

number to 3240. Record your steps on the recording sheet.

Puzzle B� Enter 4352 into your calculator.� Using the fewest number of keystrokes possible, change the


Puzzle C� Enter 9784 into your calculator.� Using the fewest number of keystrokes possible, change the


Puzzle D� Enter 7547 into your calculator.� Using the fewest number of keystrokes possible, change the




Discuss the following questions about the puzzles:

Compare your keystroke chart with your partner’s chart and answerthese questions:� How did you decide to solve each puzzle? How do you know

your method worked?� Did you and your partner record the same keystrokes? Why do

you think that happened?� If you and your partner had different keystrokes, which one had

the fewest? Why do you think that happened?

Write a rule that tells you how to change a digit to zero with thefewest number of keystrokes. Try your rule with these numbers andsee if it works.� Change 5396 to 5096� Change 32,756 to 30,756� Change 523 to 503� Change 142,536 to 42,536

Did your rule work? Why do you think it worked? Would it workwith any number? Why do you think so?



Make It Zero Recording Sheet, Calculator Version

NumberEntered

KeystrokesChangednumber

Number ofkeystrokes

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CMore Page 76

More

Words for the Word Wall:� Compare numbers� Equal� Equal amount� Few, fewer, fewest� Greater than� Least� Less than� More than

Materials (for each pair of students):� 1 deck of cards with face cards removed (Aces count as ones)� 10 small paper sacks, numbered 1-10

Procedures Notes1. Participants will work together in pairs.

Model the process of playing the gamethrough several rounds by playing thegame with a student. The teacher willdeal all 40 cards to himself or herself andhis or her partner.

As you demonstrate dealing the cards,emphasize that you are giving your partnerone card and then giving one card toyourself.

When all cards are dealt, ask some questionsto promote mathematical thinking. Examplesmight include:

� Do you think each player got his fairshare of cards? Why or why not?

� Do you each have the same number ofcards?

� How can you be certain?

2. On each round, both players turn over thetop card of his or her stack and determinewhich card is worth more. (The numberon the card determines the value of thecard.) Both cards are placed into the baglabeled with the same number as thehigher number of the two cards. If bothplayers turn over cards with the samenumber, the bag with that numbercaptures the tie.

As you model the process for playing thegame, ask some of these questions to helpilluminate strategies the students might use.

� How do you know which card has thelarger number?

� What might you do if you weren’t surewhich number was more and which wasless?

� How could you check to see who iscorrect if you and your partner don’tagree?


3. Play continues until all the cards areplayed.

4. Players then examine each bag to look forpatterns among the cards in that bag. Askthe class to begin by emptying their"five" bag.

The teacher should model examining a bagby emptying the "five" bag. Encouragestudents to talk about the patterns theynotice. Based on the students’ comments, theteacher may introduce or reinforce mathvocabulary such as equal, more than, lessthan, and other terms as listed in the wordwall list. Some sample questions mightinclude:

� What do you notice about the cards inbag 5?

� Are some of the cards the same?

� Which number do you find most often inthis bag?

� Which numbers are not in this bag?

� Are any of the bags empty?

� How can you explain this?

� Which bags captured the most cards?

� Why do you think this happened?

� Did you notice any patterns when youcompared the numbers from one bag?

5. Ask students to return all cards from the“five” bag back into the bag. Encouragestudents to continue looking for patternsone bag at a time. After an adequateexploring time, have students stop. Askstudents to build a real graph on the floorusing the cards and the sacks. Thenumbered sacks can be placed along thebottom of the graph and the cards fromeach sack can be placed above the sackfrom which they were removed. Havethem reflect on what they noticed. Theteacher may choose to write the students’observations on a chart or other display.

The teacher should ask the class to decide thename and labels for their graph. The titlecould be "More" and the labels could be"Sack numbers" and "Cards in the Sacks".Suggested questions about the graph mightinclude:

� Is there an even or odd number of cardsin each column?

� Can you explain why?

� Which sack has the most/least number ofcards above it?

� Are any of the columns empty?

� Can you explain why?


6. Ask students to describe how they knewthey were doing good work as they wereplaying and discussing the game.

The teacher should write the students’responses on a chart. This is the evidencefrom the students that they were doing goodwork. Possible list of Evidence ofUnderstanding may include:

� I can count to tell how many are in a set.

� I can use pictures to compare numbers.

� I can read the number.

� I can talk about how I compare numbers.

Some leading questions from the teacher maybe useful to guide students to recognize whatthey did.

� How did you know which number wasmore? What did you do to check if youwere unsure about which number wasmore?

� How were you able to compare thenumbers in each of the bags?

Now that we know how to do good workwhen we compare numbers, we can payattention to this list to help us do good workcomparing numbers in other situations.

Discussion:Open the discussion by asking students to tell what they think they learned from the game. Ifone player turns over a 10, can the other player turn over a higher card? What did they thinkwhen they turned over an ace? Did partners need to help each other determine which cardwas "more"? Were there any empty sacks that could have captured cards if there had beenmore cards to play?

Using the bags lessens the sense of competition, but also brings out the idea of hierarchicalinclusion: each bag will capture only numbers that are less than or equal to the amountspecified by the numeral on the bag.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CNumber, Number, Where’s the Number? Page 79

Number, Number, Where's the Number?

Words for the Word Wall:� Before� After� Between� Odd� Even� Prime number� Composite number� Multiple� Factor

Materials:� 5” X 7” note cards

Transparency/Handout:� Hundred Chart (2C-15)

Procedures Notes1. The teacher should place the 100 chart on

the overhead and model the activity. Thisactivity is played with a partner. Tobegin, the teacher secretly selects anumber on the 100 chart. On a note card,the teacher writes a riddle that contains 4clues to help identify the hidden number.The selected number should be written onthe back of the note card.

This activity helps students become familiarwith the 100 chart and to learn to interpretclues given in a riddle format in a logicalmanner. If they make a guess that isincorrect, the teacher will give the classanother clue to help them guess correctly.

2. The students are to guess which numberhas been selected. Each time their guessis incorrect, the teacher will give the classa clue to help them determine the correctnumber. The clue will tell the classwhether the number comes before or afterthe number guessed.

A riddle might look like this:

The number I am thinking of is an oddnumber. It is a number that is less than 30and is greater than 15. My secret number is amultiple of 5. What is my number?

3. Students should use complete sentencesto make their guesses.

4. Guesses continue until the correctnumber is chosen.


5. The class should play the game a secondtime. This time the teacher secretlyselects the number 17. After each clue,the teacher should stop the riddle and askthe class what they know so far about theclues they have been given. Lead thechildren to decide what numbers can beeliminated and what numbers might bepossible instead of randomly guessing.Strategies might include making adiagram or making a chart.

The clues for 17 could be:

� "The number is a number less than 25."

� "The number is an odd number."

� "The number is a prime number betweenthe numbers 11 and 23."

� "The number I am thinking of is a factorof 34."

This time the teacher should ask the classwhat information they know from the cluesthey already have.

6. After each clue, talk with the class aboutwhat they learned from the clue. Discussthe number of possible answers there arewith each clue. Have them answer thequestion, “If you could have only twoclues, which two would give you themost success in choosing the correctnumber?”

The teacher should look and listen forevidences of student understanding:

� using all the clues before making a newguess;

� using the clues to eliminate numbers; and

� knowing mathematical definitions.

7. Play the game several more times untilmost of the children understand how towrite their own riddles. Have eachstudent write a riddle and share it with apartner.

Riddles should be written in completesentences.

Discussion:

� Were you able to guess the number quickly? How many clues did it take each time?

� Were you able to find the hidden number more quickly after you solved severalriddles?

� Was there a number that was harder to find? Why do you suppose it was harder tofind that number?

Transparency 2C-15


Hundred Chart

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section CSee Saw Page 82

See Saw

Materials:� Pinto beans� Paper clip and pencil or clear plastic spinner per pair of participants

Transparencies/Handouts:� See Saw Directions (2C-16)� See Saw Game Board (2C-17)� Spinner (2C-18)� See Saw Recording Sheet (2C-19)� See Saw Reflections (2C-20)

Procedures Notes1. Demonstrate the game to the students.

Using the transparency game board andthe spinner, spin a number and place thatmany beans on the game board. Continueuntil you have covered at least one area(16 beans or more).

2. Have students play the game in partners.Each person needs a game board and arecording sheet. Two people share thespinner and the supply of beans.

3. After students have played one completegame, have them answer the reflectionquestions for the first game.

4. Demonstrate for the students how to usethe recording chart. Play the game again,keeping a record of your moves on therecording sheet. List the number spunand then show the number of people,lines, and areas you have filled.

When completing the recording chart, startwith the largest grouping. For example, ifyour first spin was 3, you would say, “I have0 playgrounds completely filled, 0 areascompletely filled, 0 lines completely filled,and 3 people.” If you spun a 3 again, making6 beans, you would say, “I have 0playgrounds completely filled, 0 areascompletely filled, 1 line completely filled,and 2 people.” Once a bean is part of acomplete line, it is not listed in the peoplecolumn. Once a line is a part of an area, it isnot listed as a line or as people.


5. Once students have played a completegame, recording their results on therecording sheet, they should answer thereflection questions for the second game.

Discussion:You have 2 areas and 3 lines completely filled. How many more lines do you need to fill upthe playground? How do you know?

After you add the number of lines you need to fill up the playground, how would you have towrite it? How do you know?



See Saw Directions

Materials:� 1 See Saw Game Board per player� Spinner� Beans

To play the game:

1. The object of the game is to fill all of the people spots. You maystart anywhere on the game board. As you add beans, fill up oneline of see saws, then an area of see saws, and then the wholeplayground.

2. On your turn, spin the spinner. Get that number of beans andplace them on the game board.

3. To win the game, you have to fill up the see saws with the exactnumber. If you have room for two more beans, then you have tospin 2 to win.



See Saw Game BoardKEY

Playground

Area

Line

People

Transparency 2C-18


Spinner

0 1

3 2



See Saw Recording Sheet

Now I have…On thisspin…

…I gotthis

number.Playgrounds Areas Lines People

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

11th

12th

13th

14th

15th

16th

17th

18th

19th

20th

21st

22nd

23rd

24th

25th

26th

27th

28th

29th

30th



See Saw Reflections

After you play the first game, answer these questions:

� What numbers are on the spinner? Why do you suppose thoseare the only numbers?

� How are the number of people in a line, the number of lines inan area, and the number of areas in a playground related?

After you play the second game, answer these questions:

� Look at the recording chart. What is the highest digit in anycolumn? Why do you suppose this is true?

� If you look at the game board and see six filled lines, how doyou write that on your recording chart? Why do you write it likethat?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 2: Section DClosure for Day Two Page 89

Closure for Day TwoShaping Up a Summary

Materials:� Chart paper for Square, Circle and Triangle charts� Sticky notes

Procedures:

1. Have posted three charts, one with a square, one with a circle, and one with a triangle.

2. Have participants post one sticky note on each chart.

SQUARE: Something I learned today that SQUARES with my belief if…CIRCLE: A question going AROUND in my mind is…TRIANGLE: Three POINTS I want to remember are…

3. At the start of Day Three, select a few examples from each chart to share with thewhole group. Make sure any questions that are raised are addressed in some wayduring the training.


Day ThreeFraction and Decimal Concepts

Table of Contents

Preparing for Day Three 2

Section A Fraction Meanings with Cuisenaire Rods 7

Section B Understanding Fractions 17

Section C Fraction/Decimal Menu Sampler 37Cookie Sharing 56Fraction Rectangles 57Same Name 58More Same Name 59Shake and Spill 60Fraction Riddles 63Geoboard Fractions 65Tenths 67NOT Tenths 72Hundredths 74Measuring with Decimals 76Real-world Decimals 77Wipe Out ONE! 78Show Me! Tell Me! 79

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3Preparing for Day Three Page 2

Day Three Materials

Reusable Materials Consumable MaterialsSection AFraction Meanings with Cuisenaire Rods� Cuisenaire Rods

Transparencies:� Fraction Meanings with Cuisenaire Rods

Task Card (3A-1)� Fraction Meanings with Cuisenaire Rods

Recording Sheet (3A-2)� Fraction Meanings with Cuisenaire Rods:

Levels of Sophistication (3A-3)� Day 3A Reflection Prompt (3A-4)

� Chart paper� Crayons (pkg of 16)

Task Cards:� Fraction Meanings with Cuisenaire Rods

Task Card (3A-1)

Handouts:� Fraction Meanings with Cuisenaire Rods

Recording Sheet (3A-2)� Fraction Meanings with Cuisenaire Rods:

Levels of Sophistication (3A-3)Section BUnderstanding Fractions� Cuisenaire Rods� Pattern blocks� Linking cubes� Two-color counters

Transparencies:� Task Cards, Levels 1 through 9 (3B-1

through 3B-3, 3B-5 through 3B-7, 3B-9through 3B-11)

� Analysis Charts (3B-4, 3B-8, 3B-12)� Day 3B Reflection Prompt (3B-13)

� Centimeter graph paper

Task Cards:� Task Cards, Levels 1 through 9 (3B-1

through 3B-3, 3B-5 through 3B-7, 3B-9through 3B-11)

Handouts:� Analysis Charts (3B-4, 3B-8, 3B-12)

Section CFraction/Decimal Menu SamplerTransparencies:� Lesson Planning Process Chart / Analysis

Chart (3C-1)� Fraction/Decimal Menu Card (3C-2)� Day 3C Reflection Prompt (3C-34)

Handouts:� Lesson Planning Process Chart / Analysis

Chart (3C-1)� Fraction/Decimal Menu Card (3C-2)

For Menu supply table:� 1 Inch Graph Paper (3C-3)� Centimeter Graph Paper (3C-4)� 10-by-10 Grid Paper (3C-5)� Fraction Circles (3C-6 through 3C-8)� Hundredths Circle (3C-9)� 100-by-100 Grid Paper (3C-10)


Section CAppetizer: Cookie SharingTransparencies:� Cookie Sharing Recording Sheet (3C-11)

� Markers/Crayons

Handouts:� Cookie Sharing Recording Sheet (3C-11)

Section CEntrees: Fraction Rectangles� Color tiles

Transparencies:� Fraction Rectangles Task Card (3C-12)

� 1” graph paper� Markers/Crayons

Task Cards:� Fraction Rectangles Task Card (3C-12)

Section CEntrees: Same Name� Pattern Blocks

Transparencies:� Same Name Task Card (3C-13)

Task Cards:� Same Name Task Card (3C-13)

Section CEntrees: More Same Name� Pattern Blocks

Transparencies:� More Same Name Task Card (3C-14)

Task Cards:� More Same Name Task Card (3C-14)

Section CEntrees: Shake and Spill� Two-color counters� Paper or plastic cup

Transparencies:� Shake and Spill Task Card (3C-15)� Shake and Spill Recording Sheet (3C-16)

� Chart paper for class graphs (see exampleon page 62)

Task Cards:� Shake and Spill Task Card (3C-15)

Handouts:� Shake and Spill Recording Sheet (3C-16)


Section CEntrees: Fraction Riddles� Cuisenaire Rods

Transparencies:� Fraction Riddles Task Card (3C-17)� Fraction Riddles Recording Sheet (3C-

18)

� Construction paper� Markers/Crayons

Task Cards:� Fraction Riddles Task Card (3C-17)

Handouts:� Fraction Riddles Recording Sheet (3C-

18)Section CEntrees: Geoboard Fractions� Geoboards� Geobands

Transparencies:� Geoboard Fractions Task Card (3C-19)� Geoboard Grids (3C-20)

Task Cards:� Geoboard Fractions Task Card (3C-19)

Handouts:� Geoboard Grids (3C-20)

Section CEntrees: Tenths� Scissors

Transparencies:� Tenths Task Card (3C-21)� Tenths Recording Sheet (3C-22)

� Markers/Crayons� Glue sticks

Task Cards:� Tenths Task Card (3C-21)

Handouts:� Tenths Recording Sheet (3C-22)� My Book of Tenths pages (3C-23

through 3C-25)Section CEntrees: NOT Tenths� Scissors

Transparencies:� NOT Tenths Task Card (3C-26)� NOT Tenths Recording Sheet (3C-27)


Task Cards:� NOT Tenths Task Card (3C-26)

Handouts:� NOT Tenths Recording Sheet (3C-27)


Section CEntrees: Hundredths� Scissors

Transparencies:� Hundredths Task Card (3C-28)� Hundredths Recording Sheet (3C-29)


Task Cards:� Hundredths Task Card (3C-28)

Handouts:� Hundredths Recording Sheet (3C-29)

Section CEntrees: Measuring with Decimals� Scissors� Linear measuring tools such as cm

measuring tape or meter stick

Transparencies:� Measuring with Decimals Task Card

(3C-30)

� Markers/Crayons� Glue sticks� Chart Paper

Task Cards:� Measuring with Decimals Task Card

(3C-30)Section CDessert/Assessment: Real-world DecimalsTransparencies:� Real-world Decimals Task Card (3C-31)

� Participants bring in examples ofdecimals from the newspaper, magazines,catalogs, etc.

� Chart paper for posting examples

Task Cards:� Real-World Decimals Task Card (3C-31)

Section CDessert/Assessment: Wipe Out ONE!� Pattern blocks� Fraction die with 1/3, 1/2, 1/6, 1/9, 1/18,

1/18

Transparencies:� Wipe Out ONE! Task Card (3C-32)

Task Cards:� Wipe Out ONE! Task Card (3C-32)


Section CDessert/Assessment: Show Me! Tell Me!� Scissors

Transparencies:� Show Me! Tell Me! Task Card (3C-33)

� Markers/Crayons� Glue sticks� Chart Paper� Cards (with fractions and decimals such

as 5/10, 0.5, 0.50, 8/10, 0.8, 0.80, 35/100,0.35, 2 75/100, 2.75)

Task Cards:� Show Me! Tell Me! Task Card (3C-33)

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section AFraction Meanings with Cuisenaire Rods Page 7

Fraction Meanings with Cuisenaire Rods

Key Questions:� What are rational numbers and how do fractions describe them?� How are fractions used to describe relationships?

Frame:This learning experience is designed to broaden understanding about how to use fractions todescribe relationships. A model that is flexible enough to represent a variety of relationshipsenables teachers to represent unit and non-unit fractions as well as fractions greater than one.Patterns that emerge from a variety of recording strategies further build understanding ofrelationships among rational numbers.

Materials:� Cuisenaire Rods� Chart paper� Crayons (pkg of 16)

Transparencies/Task Cards/Handouts:� Fraction Meanings with Cuisenaire Rods Task Card (3A-1)� Fraction Meanings with Cuisenaire Rods Recording Sheet (3A-2)� Fraction Meanings with Cuisenaire Rods: Levels of Sophistication (3A-3)� Day 3A Reflection Prompt (3A-4)


Procedures Notes1. Ask participants to use fractions to

describe the relationships between theCuisenaire Rods. They will use words,pictures, and/or numbers to record therelationships. Ask participants to describethe patterns they notice.

Suggestions for introducing the task:

� How many red rods does it take to equalthe length of a brown rod? If it takes 4red rods to build the brown rod, then oneof the red rods equals one of 4 equal partsor 1/4 of the brown rod. If the brown rodis one, then a red rod is 1/4. If the brownrod is one, which rod represents 1/2?Which rod equals 3/4? How do youknow?

� If the orange rod equals one, which rod is1/2? How do you know? If it takes 5 redrods to equal one, which rod equals 1/5?How about 3/5? Describe yourreasoning.

� If the blue rod equals one, which rodequals 1/3? Which rod equals 2/3?Explain.

� Your task is to use fractions to describeALL of the relationships between pairs ofCuisenaire Rods.

� You will use words, pictures, diagrams,and/or numbers to record your work.How will you organize your work to becertain you have found ALL of thepossible relationships? How will yourecord and share your discoveries?


2. Orchestrate the sharing of strategies andpatterns by ordering the work by “levelof sophistication”.

� Circulate among the groups as theywork. Listen to them as they describetheir strategies. Observe the waysthey record their solutions.

� Distribute numbers to groups toindicate the order in which they willshare their solutions, strategies andthe patterns they noticed – lesssophisticated to more sophisticated.

Listen for:

� Appropriate mathematical vocabulary(equal parts, fraction names, equivalent,one whole, numerator, denominator)

� Conversation indicating an understandingof the relationship between the linearmodel (Cuisenaire Rods) and thefractions they represent.Example: Orange = one.

� Since it takes 2 yellow rods to equalits length, 1 yellow rod is one of twoequal parts or 1/2.

� It takes five red rods to equal one soone red rod is 1/5; two red rods orone purple rod is 2/5; three red rodsor one dark green rod = 3/5; four redrods or one brown rod = 4/5; and fivered rods = 5/5 or one orange rod(ONE).

� It takes ten white rods to equal oneso 1 white rod is one of ten equalparts or 1/10. Two white rods or onered rod = 2/10; three white rods orone light green rod = 3/10; four whiterods or one purple rod = 4/10; fivewhite rods or one yellow rod = 5/10;six white rods or one dark green rod= 6/10; seven white rods or one blackrod = 7/10; eight white rods or onebrown rod = 8/10; nine white rods orone blue rod = 9/10; and ten whiterods = 10/10 or one orange rod(ONE).

� Discussion of the patterns in equivalentfractions or in the charts/recordingstrategies used to describe solutions.


Look for:

� Participants comparing the rods’ lengthsto one using the rods to find unit andnon-unit fractions.

� Participants changing the value of one tofind fractional relationships.

� Participants comparing the rods’ lengthsto find equivalent fractions.

Debriefing:NOTE: It is important for learners to use a rather unfamiliar model occasionally (such asCuisenaire Rods) to create the context in which discussion can take place about the criticalattributes of effective models for different mathematical concepts.

� What fractions did you use to describe the relationships between the CuisenaireRods?

� Were you able to show halves for every Cuisenaire Rod? Thirds? Fourths? Why orwhy not?

� How did you organize your work?

� How did you record your solutions?

� Do you think you found ALL of the fractional relationships? How can you be sure?

� What patterns did you notice? Did other groups discover different patterns from theones you noticed? Why do you think this happened?

If no group records the relationships as fractions in a grid, probe for this level ofsophistication by asking questions such as: What patterns emerge when you use a grid torecord the fractional relationships? (NOTE: The numerators can be recorded as the rodsacross the top of the grid and the denominators can be recorded as the rods along the side ofthe grid.)


Reflection and Connection:Fractions are a way of describing relationships. It is important to use a model that is flexibleenough to represent a variety of fractional relationships. We use Cuisenaire Rods as onemodel for exploring the importance of identifying the whole in understanding the fraction.They enable us to demonstrate various ways of thinking about and recording relationships.The patterns that emerge from the recording strategies used with this model further buildunderstanding of fractional relationships.

How does this model help you think about fractions?

Transparency / Task Card 3A-1


Fraction Meanings with Cuisenaire RodsTask Card

You need: Cuisenaire Rods, paper/pencil or chart/crayons

How many ways can you use fractions todescribe the relationships between theCuisenaire Rods?

� Use words, pictures and/or numbers torecord the relationships.

� Do you notice any patterns? Write aboutthe patterns you see.



Fraction Meanings with Cuisenaire RodsRecording Sheet



Cuisenaire Rods

white wred rlight green gpurple pyellow ydark green dblack kbrown nblue eorange o



Fraction Meanings with Cuisenaire Rods:Levels of Sophistication

Level 4� Finds all fractional relationships between the rods and when asked,

explains how s/he knows s/he found them all� Uses advanced vocabulary and visual representations (designs, charts,

grid) to record fractional relationships� Uses a variety of patterns to make generalizations or connections to other

math strands or contexts outside the classroom

Level 3� Finds all unit fractions, some non-unit and equivalent fractions� Uses appropriate fractional notation� Uses an organized strategy� Is able to explain strategy, patterns discovered� Appropriately labels work

Level 2� Finds some unit fractions but few equivalent fractions or non-unit

fractions� Attempts to use appropriate fractional notation but makes errors� Attempts to use organized method but efforts remain rather random� Because of random strategy, is unable to describe patterns

Level 1� Attempts task but shows little understanding of it� Focuses on the Cuisenaire Rods, themselves, rather than how they model

fractions� Uses a totally random method to address the task� Shows little understanding of patterns in the relationship between the

rods or in the fractions they represent

Transparency 3A-4



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aa bb oo uu tt ff rr aa cc tt ii oo nn ss ??

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section BUnderstanding Fractions Page 17

Understanding Fractions

Key Questions:� What do students need to do to build an understanding of fractions?� How do different models contribute to building this understanding?

Frame:Students must travel through definite developmental levels to come to a thoroughunderstanding of the meaning and relationships among fractions. In this activity, participantswill use a variety of models to experience this journey. The levels model how learningexperiences can be purposefully sequenced to enable students to build understanding offraction concepts.

Materials:� Cuisenaire Rods� Pattern blocks� Linking cubes� Two-color counters� Centimeter graph paper

Transparencies/Task Cards/Handouts:� Task Cards, Levels 1 through 9 (3B-1 through 3B-3, 3B-5 through 3B-7, 3B-9

through 3B-11)� Analysis Charts (3B-4, 3B-8, 3B-12)� Day 3B Reflection Prompt (3B-13)

Procedures and Notes:Walk participants step-by-step through the developmental sequence for understandingfractions. After each three levels, debrief with the appropriate Analysis Chart. This can be adirected activity in which the presenter works through the questions/tasks on the overhead orone in which participants work through the series of task cards in small groups. It will beimportant for participants to proceed sequentially in order to appreciate how theunderstanding of fraction concepts builds. After each 3 levels, ask participants to record the“Big Idea(s)” and “Evidence of Understanding.”

Note: For Level 9, it will be important for participants to use only ONE model to showBOTH fractions. To help this happen, demonstrate with the first pair of fractions on theoverhead.


Levels Notes1. Given a whole, find a unit

fraction (a fraction with anumerator of 1).

Model: Cuisenaire Rods

� If the brown rod is one, find a rod that wouldrepresent 1/4. Explain how you know the rodequals 1/4. (Example: It takes 4 red rods to equalthe length of the brown rod. One of four equalparts is 1/4. Therefore, one red rod equals 1/4 ofthe brown rod.)

� Orange = 1. Find 1/2. Find 1/5. Find 1/10.

� Green = 1. 1/3 = .

� Orange + red = 1. Find 1/2, 1/6, 1/4, 1/3.

2. Use unit fractions to build anunderstanding of non-unitfractions (a fraction with anumerator other than 1).

Model: Pattern Blocks

� If the yellow hexagon is one, find pattern block(s)that would represent 1/6, 3/6, 2/6. Explain yourreasoning. (Example: It takes 6 green triangles toequal the area of the yellow hexagon. One of sixequal parts is 1/6. It takes 3 green triangles toequal the red trapezoid. Since we named one greentriangle 1/6, 3 green triangles equal 3/6. Therefore,the red trapezoid equals 3/6. It takes 2 greentriangles to equal the blue rhombus. One greentriangle is 1/6 so 2 green triangles equal 2/6.Therefore, the blue rhombus equals 2/6.)

� If yellow + red = 1, find 1/9, 4/9, 2/9.

� 2 yellow hexagons = 1. Find 1/6, 2/6, 3/6, 5/6.


3. Given a unit fraction, identifythe whole.

Model: Linking Cubes

� If 4 cubes = 1/3, what is 1? Explain how youknow. (Example: It takes 3 of 3 equal parts toequal one. Since 4 cubes is 1/3, three sets of 4cubes would be 12 cubes. So 12 cubes equal one.)

� 5 cubes = 1/2. What is 1?



Model: Counters

� If 2 counters = 1/4, what is 1?

� 4 counters = 1/3. What is 1?



Debrief with Analysis Chart:

4. Given a non-unit fraction, findthe corresponding unit fraction.

Model: Centimeter Graph Paper

� On your graph paper, 4 squares is 2/3. What is1/3? Describe how you know. (Example: I knowthat half of 2/3 is 1/3. If 4 squares is 2/3, half of 4squares is 2 squares. Therefore 2 squares equal1/3.)

� 8 squares = 4/5. What is 1/5?




5. Given a non-unit fraction,identify the unit fraction.Identify the whole.


� The dark green rod = 3/4. What is 1/4? What isone? Why? Explain your reasoning. (Example: Iknow it takes 3 red rods to equal one dark greenrod. So one red rod is 1/4. Four fourths equalsone. Four red rods is the length of the brown rod.Therefore, the brown rod is ONE.)

� Brown = 2/3. What is 1/3? What is 1?

� Green = 3/5. What is 1/5? What is 1?

� Blue = 3/4. What is 1/4? What is 1?

Model: Counters

� If 6 counters = 3/4, what is 1/4? What is one?How do you know? (Example: To find 1/4, I haveto divide the 6 counters into three parts. Twocounters would be in each part. So 2 countersequal 1/4. Since it takes four fourths to equal one,4 sets of 2 equals one. ONE is equal to 8counters.)

� 4 counters = 2/3. What is 1/3? What is 1?




6. Find equivalent fractions. Model: Pattern Blocks

� If 2 yellow hexagons = 1, find values for otherpattern blocks. (green, blue, red, 1 yellow, 2 blue).

� Find as many values as you can. Explain how youknow the values are the same. (Example: It takestwo yellow hexagons to make one. One yellowhexagon is one of two equal parts or 1/2 . Since ittakes 6 blue rhombi to equal one, 3 blue rhombiequal 1/2. Since we named each blue rhombus 1/6,three of them equals 3/6. Since it takes 12 greentriangles to equal one and 6 green triangles to equal1/2, 6/12 is the same as 1/2. So I know that 6/12and 3/6 are the same as 1/2.)


� If orange + red = 1, find values for other rods.(orange, brown, dark green, purple, light green,red)

� Find as many values as you can. Explain how youknow the values are equivalent. (Example: I canuse white, red, light green, purple and dark greenrods to build a wall of single-color rows that equalthe length of orange and red. When I compare thebrown rod to the other rods in the wall, I can tellthat the brown rod is equal to 8/12, 4/6, and 2/3.So 8/12, 4/6, and 2/3 are equivalent fractions.)

Debrief with Analysis Chart

7. Given a whole, compare twofractions:

� that are both unit fractions.� with like denominators > 1.� with like numerators > 1.� with unlike numerators and

denominators.


Orange + orange + purple = one

Build a “wall” of Cuisenaire Rods and use it tocompare fractions visually.

� 1/6 [<,>] 1/8

� 3/4 [<,>] 1/4

� 2/6 [<,>] 2/3

� 2/3 [<,>] 3/5


8. Given any fraction symbol, a/b,find models for the whole andthe fraction.

Model: Pattern Blocks

� How can you use the pattern blocks to create amodel for 2/7? Describe your thinking. (You needto make a whole that can be separated into 7 equalparts. You can start with something for 2/7 that canbe separated into two equal parts so that each partis 1/7, and then 7 of those would make the whole.For example: A blue rhombus can be separatedinto 2 green triangles. If the blue rhombus is 2/7,the green triangle is 1/7. Since it takes 7/7 to equalone, a shape with an area of 7 green triangles is awhole that can be made with the pattern blocks toshow sevenths, and the blue rhombus is 2/7.)


� Use the Cuisenaire Rods to create a model for 4/5.What rod = 1/5? What rod = 1?

Model: Counters

� Use counters to create a model for 3/5. How manycounters = 1/5? How many = 1?

Additional Models:

Linking cubes, graph paper, two-color counters, eggcartons and plastic eggs, paper for folding. Choose adifferent model and develop example questions such asthose above.

9. Given any 2 fractions, choosean appropriate unit with whichto model them.

Models: Have all available.

� Use one model of your choice to show the fractions2/3 and 1/6. Explain your reasoning. (Example: atrain of 6 linking cubes could be used to show boththirds and sixths.)

� Use one model of your choice to show the fractions3/5 and 7/10.

� Use one model of your choice to show the fractions1/2 and 2/3. (Example: I can use linking cubes tomake a rectangle that is divisible by both 3 and 2.A 2 x 3 rectangle or 1 x 6 rectangle can be dividedin halves and thirds. I could also build a 3 x 4, a 2x 6 or a 1 x 12 rectangle.)

Debrief with Analysis Chart


Debriefing:In this activity, we used three types of models to represent fractions:

� Linear models (such as Cuisenaire Rods)� Area models (such as pattern blocks and graph paper)� Set models (such as two-color counters)

What advantage/disadvantage does each kind of model have in representing fractions?

Reflection and Connection:To develop a thorough understanding of the meaning and relationships among fractions,learning experiences must be purposefully sequenced to build upon each other. At eachlevel, students must practice using mathematical language to describe the meaning of thefractional relationships demonstrated by each model.

What value do you see in using a variety of models to represent fractions?

Transparency / Task Card 3B-1


Level 1 Task Card

You need: Cuisenaire Rods

� If the brown rod is 1, find 14 .

� Orange = 1. Find 12 . Find

15 . Find

110 .

� Dark green = 1. 13 = _______.

� Orange + red = 1. Find 12 ,

16 ,

14 ,

13.



Level 2 Task Card

You need: Pattern Blocks

� If the yellow hexagon is one, what is 16?

26?

36?

46 ?

� If yellow + red = 1, find 19 ,

49 ,

29 .

� 2 yellow hexagons = 1. Find 16 ,

26 ,

36 ,

56 .



Level 3 Task Card

You need: Linking Cubes

� If 4 cubes = 13, what is 1?

� If 5 cubes = 12 , what is 1?



You need: Counters

� If 2 counters = 14 , what is 1?

� If 4 counters = 13, what is 1?





Analysis Chart

Step 1Big Ideas






Level 4 Task Card

You need: Centimeter Graph Paper, markers

� If 4 squares = 23, what is

13?

� If 8 squares = 45 , what is

15?


14 ?


16?



Level 5 Task Card


� Dark green = 34 . What is

14 ? What is 1?

� Brown = 23. What is

13? What is 1?

� Green = 35 . What is

15? What is 1?

� Blue = 34 . What is

14 ? What is 1?

You need: Counters

� 6 counters = 34 . What is

14 ? What is 1?

� 4 counters = 23. What is

13? What is 1?


15? What is 1?


18? What is 1?



Level 6 Task Card

You need: Pattern Blocks

2 yellow hexagons = 1.

Find as many values as you can for:� 1 green triangle� 1 blue rhombus� 1 red trapezoid� 1 yellow hexagon� 2 blue rhombi


Orange + Red = 1.

Find as many values as you can for:� the orange rod� the brown rod� the dark green rod� the purple rod� the light green rod� the red rod



Analysis Chart

Step 1Big Ideas






Level 7 Task Card


Orange + orange + purple = 1.

a.16 [ < , > ]

18

13 [ < , > ]

12

b.34 [ < , > ]

14

58 [ < , > ]

78

c.26 [ < , > ]

23

412 [ < , > ]

48

d.23 [ < , > ]

36

712 [ < , > ]

56



Level 8 Task Card

You need: a variety of models (such as Cuisenaire Rods, patternblocks, linking cubes, graph paper, two-color counters, egg cartonsand plastic eggs, paper for folding)

� Use pattern blocks to create a model for 27 .

Find 17 . Find 1.

� Use Cuisenaire Rods to create a model for45 . Find

15 . Find 1.

� Use counters to create a model for 35 . Find

15. Find 1.

� Develop models to show other fractions.



Level 9 Task Card

You need: a variety of models (such as Cuisenaire Rods, patternblocks, linking cubes, graph paper, two-color counters, egg cartonsand plastic eggs, paper for folding)

� Use one model of your choice to show the

fractions 23 and

16 .


fractions 35 and

710 .


fractions 12 and

23.



Analysis Chart

Step 1Big Ideas




Transparency 3B-13



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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CFraction/Decimal Menu Sampler Page 37

Fraction/Decimal Menu Sampler

Key Question:� How can we use the Lesson Planning Process to select and orchestrate learning

experiences to develop students’ understanding of the meanings of and relationshipsamong rational numbers?

Frame:Participants will explore a sampler of learning experiences designed to support theunderstanding of the meanings of and relationships among fractions and decimals.Participants will examine these learning experiences in terms of how the Lesson PlanningProcess is used to design effective instruction. The process we use here models how “ChoiceTime” or “Math Menus” can work in the classroom to extend/reinforce/differentiate themathematics being taught as well as to provide an opportunity for teachers to assess—to lookfor/listen for evidence of understanding.

Materials:(See Day Three Materials Chart for materials needed for the activities in theFraction/Decimal Menu.)

Transparencies/Handouts:� Lesson Planning Process Chart / Analysis Chart (3C-1)� Fraction/Decimal Menu Card (3C-2)� For supply table:� 1 Inch Graph Paper (3C-3)� Centimeter Graph Paper (3C-4)� 10-by-10 Grid Paper (3C-5)� Fraction Circles (3C-6 through 3C-8)� Hundredths Circle (3C-9)� 100-by-100 Grid Paper (3C-10)

� Day 3C Reflection Prompt (3C-34)

Procedures Notes1. Assemble a supply table that includes the

materials and copies of the handoutslisted in the Day Three Materials Chartfor the Fraction/Decimal Menu.


2. Set up several stations around the roomthat include copies of the task cardsprinted on cardstock, easels cut from oldfile folders. This will enable yourparticipants to choose tasks to do without“bunching up” in a single location.

When students/participants choose a task,they take the Task Card, easel and anynecessary supplies or materials to anotherpart of the room to do the activity. When theactivity is finished, the task card andequipment are returned and another task isselected. This continues until time expires.

3. Hand out the Lesson Planning ProcessChart for the Fraction/Decimal menu.Have participants review the fourcomponents in the Lesson PlanningProcess.

Participants will be using the LessonPlanning Process Chart to identify the fourcomponents of the Lesson Planning Processin relation to each learning experience.

4. Introduce the Fraction/Decimal Menu.Briefly describe a few of the menu tasks– especially the ones that will becompleted by every participant (such asthe Shake and Spill task in which data isgathered and recorded on a group graph).

� The “Appetizer” is a task that isintroduced to the whole group, butwhich students finish at differenttimes depending on the efficiency ofthe group or the need for discussionor processing of ideas. Whenstudents finish the Appetizer, they arefree to choose any other task exceptthe “Dessert” tasks.

� The “Dessert/Assessment” activitiesmust be done after several/most of the“Entree” activities are completed.Tasks that are in this section are oftenmore challenging than the Entreeactivities, require “homework” orprerequisite activities, or are intendedto be assessment tasks.

Students/participants can keep track of theactivities they have and have not done bychecking them off on their Menu Card.

Menus MUST be followed by a richdiscussion of the activities. Save time forthis and model a summary discussion of oneof the tasks.


5. After participants have experienced somesubset of the Sampler and filled out theirLesson Planning Process Charts, putthem in small groups to compare anddiscuss their notes. It is helpful to remindparticipants that Step 3 of the LessonPlanning Process is achieved by askingrigorous questions that challenge thethinking of students. For example,suggest to participants that they glance atthe assessment questions for Wipe OutOne and Show Me, Tell Me on page 40.

In these discussions, participants might begrouped by grade level, by learningexperience, by topic or randomly.

6. Have participants share their small groupdiscussions in a whole group discussionbased on the debriefing questions.

Guide participants to create a combinedLesson Planning Process Chart for rationalnumber meanings and relationships,identifying overall big ideas and Evidence ofUnderstanding for this topic. (See thesample LPP Chart for Rational NumberMeanings and Relationships.)

7. Ask participants to compare their BigIdeas and Evidence of Understanding forwhole numbers with those they wrote forrational numbers.

Debriefing:

� What Evidences of Understanding did you identify for the activities you reviewed?How did your grade level affect the Evidences of Understanding you identified?

� Which models made the most sense to you as representations for fractions ordecimals? Why?

� How could you use Math Menus to differentiate instruction in your classroom?

When discussing the following Assessment Tasks, have participants brainstorm questionsthat could be asked to assess students’ understanding.


Wipe Out ONE! (Assessment Task)

Example questions to ask students as they play the game:� What fractions are possible for you to roll with your fraction die?� Which is the greatest fraction on your die? How do you know?� What is the likelihood you will roll the greatest fraction? How do you know?� Which is the least fraction on your die?� What is the likelihood you will roll the least fraction? How do you know?� What fraction did you roll? Can you remove that fraction from your “ONE”?

Why/Why not?� What trade did you make? How do you know it was a “fair trade”? How will the

trade you made allow you to remove the fraction you rolled?� What fraction do you need to roll to win? What rolls will NOT allow you to win?

Why?

Show Me! Tell Me! (Assessment Task)

Example questions to ask students as they work:� Why did you choose this model? Why did you NOT choose to use this model?� How do you know your grid picture shows the fraction or decimal on the card you

drew?� How can you make pies that match your grid picture? How do you know they are the

same?� How will you show hundredths? Do you need a different model than you used to

show tenths? Why/why not?� What denominator is used to record tenths? How does your denominator change

when you record hundredths?� How could you prove that twenty hundredths is the same as two tenths?

To the facilitator: The following list provides an example of a list of combined Evidences ofUnderstanding that might be created for Fractions and Decimals.




� Using representations and models toshow rational numbers.

� I can tell you what the whole is.

� I can use different wholes to show afraction or decimal.

� I can use a set of objects to show afraction or decimal.

� I can draw pictures to show a fraction ordecimal.

� I can read the number.

� I can write the number.

� Using rational numbers to describe aquantity in relation to a whole.

� I can use a fraction or decimal to describehow much of something.

� I can talk about what a fraction ordecimal means.

� I can write about what a fraction ordecimal means.

� I can explain (talk about, write about)what the numerator means.

� I can explain (talk about, write about)what the denominator means.

� I can pick a reasonable fraction ordecimal to describe how much ofsomething.

� Identifying equivalent symbolicrepresentations for rational numbers

� I can write different fraction names forthe same rational number.

� I can write different decimal names forthe same rational number.

� I can write the same rational numbereither as a fraction or a decimal.

� Extending place value to representnumbers less than one.

� I can use tenths and hundredths todescribe a decimal.

� I can talk about patterns I see in decimals.


� Describing relationships between rationalnumbers.

� I can compare rational numbers usingobjects.

� I can draw pictures to compare fractionsand decimals.

� I can use symbols (and place value) tocompare fractions (and decimals).

� I can talk about the relationships betweenfractions and decimals.

� I can write about the relationshipsbetween fractions and decimals.

Reflection and Connection:The learning experiences in this section allow you to examine important components of theLesson Planning Process. As you examined tasks involving the meaning of and relationshipsamong fractions and decimals, you experienced how a Math Menu can work in the classroomto extend/reinforce/differentiate the mathematics being taught. Create a record of yourthinking at the moment:� Write a paragraph.� Develop a bulleted list.� Create a picture.� The sky’s the limit.

How will you use these ideas in the design of your own mathematics instruction aboutrational numbers?




Step 1Big Ideas






Analysis Chart

Step 1Big Ideas






Fraction/Decimal Menu Card

AA pp pp ee tt ii zz ee rr� Cookie Sharing

EE nn tt rr ee ee ss

� Fraction Rectangles

� Same Name

� More Same Name

� Shake and Spill

� Fraction Riddles

� Geoboard Fractions

� Tenths

� NOT Tenths

� Hundredths

� Measuring with Decimals

DD ee ss ss ee rr tt // AA ss ss ee ss ss mm ee nn tt

� Real-World Decimals

� Wipe Out ONE!

� Show Me! Tell Me!

1 Inch Graph Paper Handout 3C-3


Centimeter Graph Paper Handout 3C-4


10-by-10 Grid Paper Handout 3C-5


Fraction Circles 1 Handout 3C-6


Hundredths Circle Handout 3C-9


100-by-100 Grid Paper Handout 3C-10



TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CAppetizer: Cookie Sharing Page 56

Cookie Sharing Recording Sheet

Draw 3 cookies above. How could you sharethe 3 cookies fairly with everybody in yourgroup of four?

We think each person will get

because:

Transparency / Task Card 3C-12

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Fraction Rectangles Page 57

Fraction Rectangles Task Card

You need: color tiles, 1” graph paper, markers/crayons

Use tiles to build a rectangle that is:

12 red,

14 yellow,

14 green.

Record the rectangle on graph paper and labelthe parts. Can you do it another way? Buildand record.

Now use the tiles to build each of therectangles below. Build and record each in atleast 2 ways:

� 13 green,

23 blue

� 15 red,

45 yellow

� 16 red,

16 green,

13 blue,

13 yellow

� 12 red,

14 green,

18 yellow,

18 blue


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Same Name Page 58

Same Name Task Card

You need: pattern blocks, paper

Suppose = 1.

Find as many values as you can for:

� 1 green triangle� 1 blue rhombus� 1 yellow hexagon� 2 green triangles� 1 blue rhombus and 1 green triangle� 2 blue rhombi

What patterns did you notice in thefractions you wrote?


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: More Same Name Page 59

More Same Name Task Card

You need: pattern blocks, paper

Suppose = 1.

Find as many values as you can for:

� 1 green triangle� 1 blue rhombus� 1 red trapezoid� 2 green triangles� 1 blue rhombus and 1 green triangle� 2 blue rhombi� 1 green triangle and 1 red trapezoid

What equivalent fractions did you write?


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Shake and Spill Page 60

Shake and Spill Task Card

You need: two-color counters, cup, recording sheet, chart paper forclass graph

THE EXPERIMENT:Put 8 counters in a cup. Shake the cup and spill thecounters. How much of the “spill” is red? Write afraction to show the amount. How much of the “spill”is yellow? Record.

PREDICT:Which fractions will come up most often? Record yourprediction.

DO:Shake and spill a total of 15 times. Record yourfractions on the class graph.

ANALYZE:Which fraction came up most often? Why to you thinkit did?

EXTEND:Do the same experiment with 11 counters. Record allpossible fractions that come up each time. Write aboutwhat happens.



Shake and Spill Recording Sheet

Shake and spill your counters 15 times. Record the fraction of thespill that is red and the fraction that is yellow. If a fraction comes upmore than once, use a tally mark to show how many times it did.Organize your data. Write about the patterns you see. Record thedata on the class graph.

Spill = Red Spill = YellowFraction Tally Fraction Tally


Shake and Spill Sample Class Graphs

These sample class graphs are to be copied on chart paper.

Fraction of the Spill that isRed (Continue the tallies)

Fraction of the Spill that isYellow (Continue the tallies)

80

80

18

18

82

82

83

83

48

48

58

58

86

86

78

78

88

88


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Fraction Riddles Page 63

Fraction Riddles Task Card

You need: Cuisenaire Rods, a partner, riddle sheet, constructionpaper, markers/crayons

Use your Cuisenaire Rods to figure out theFraction Riddles. Record your solutions.When you finish the riddles, write someoriginal riddles of your own.

FIRST:decide which rods you will use—no more than4 rods.

NEXT:write clues so others can figure it out. (Writeclues involving fractions!)

COPY:your clues on the front of your folded paper,record the answer inside and paper clip theedges together. Share with a classmate.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Fraction Riddles Page 64

Fraction Riddles Recording Sheet

1. Two rodsSmallest is 2

5 of the largest.

Largest is not bigger than yellow.Smallest is bigger than white.Solution: ______________________________

2. Three rodsTrain equals orange plus red.One of the rods is 1

2 of the train.

One of the rods is 13 of the train.

Solution: ______________________________

3. Two rodsDifference between the two rods is white.Shorter is 3

4 of the longer.

Solution: ______________________________

4. Two rodsOne rod is 1

3 of dark green.

The other is 3 times as long as the shortestSolution: ______________________________

5. Four rodsEach rod is 1

4 of orange plus dark green.

Solution: ______________________________


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Geoboard Fractions Page 65

Geoboard Fractions Task Card

You need: geoboard, geobands, geoboard grids

How many different ways can you divide yourgeoboard in half? Record on dot paper. Writean explanation proving that the pieces arehalves.

� Divide your geoboard in fourths using avariety of ways. Record on dot paper.Write an explanation proving that thepieces are fourths.

� Divide your geoboard into eighths.Record on dot paper and write anexplanation proving that the pieces areeighths.

Combine halves, fourths, and eighths to makea design. Record on geoboard grid. Be sureto label all fractional parts of your design.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Geoboard Fractions Page 66

Geoboard Grids


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Tenths Page 67

Tenths Task Card

You need: 1 inch graph paper, cm graph paper, hundred grids,fraction circles, markers/crayons, scissors, glue sticks, recordingsheet, My Book of Tenths pages

� Use graph paper and fraction circles tooutline and color as many different waysto represent tenths as you can.

� Label your representations in two ways: infraction form and decimal form.

� Post your examples on the recording sheetlabeled “Tenths”.

� On the back, write about the meaning oftenths and how they are labeled.

� Use your ideas to make “My Book ofTenths.”



Tenths Recording Sheet

My Book of Tenths Titles Handout 3C-23


My Book of Tenths

My Book of Tenths

My Book of Tenths Hundred Grids Handout 3C-24


My Book of Tenths Fraction Circles Handout 3C-25



TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: NOT Tenths Page 72

NOT Tenths Task Card

You need: 1 inch graph paper, cm graph paper, hundred grids,fraction circles, markers/crayons, scissors, glue sticks, recording sheet

� Use graph paper and fraction circles tooutline and color NON-examples oftenths.

� Post your non-examples on the recordingsheet labeled “NOT Tenths”.

� Label your representations in two ways:in fraction form and decimal form.

� On the back, write about why yourrepresentations are NOT Tenths.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: NOT Tenths Page 73

NOT Tenths Recording Sheet


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Hundredths Page 74

Hundredths Task Card

You need: 1 inch graph paper, cm graph paper, hundred grids,fraction circles, markers/crayons, scissors, glue sticks, recording sheet

� Use graph paper and fraction circles tooutline and color as many different waysto represent hundredths as you can.

� Label your representations in two ways: infraction form and decimal form.

� Post your examples on the recording sheetlabeled “Hundredths”.

� On the back, write about the meaning ofhundredths and how they are labeled.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Hundredths Page 75

Hundredths Recording Sheet


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CEntrees: Measuring with Decimals Page 76

Measuring with Decimals Task Card

You need: linear measuring tools, chart paper to make a number line

� Use linear measuring tools to measure fivethings in your classroom.

� Record the measures in decimals, andorder the measures from least to greatest.

� Use the chart paper to make a number line.Glue the decimals to the number line inthe correct places and post.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CDessert/Assessment: Real-world Decimals Page 77

Real-world Decimals Task Card

You need: examples of decimals from the newspaper, magazines,catalogs, etc.

� Attach your example to the top of a sheetof paper. Write about how the decimalyou found is used.

� Post your example with those of yourclassmates.

LATER:

� Sort your examples. How are theexamples you grouped together alike?How are the rest of the examplesdifferent? Choose one of your groups.Can you find other examples to fit yourgroup?

� Order your decimals from least to greatest.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CDessert/Assessment: Wipe Out ONE! Page 78

Wipe Out ONE! Task Card

You need: pattern blocks, a fraction die, your group of four

This design = ONE:

Each player starts with ONE.

1. Take turns rolling the die. The fraction thatcomes up tells how much to remove fromyour ONE. (Trades may be necessary.)

2. The player with the die will not pass the dieto the next player until all trades are madeand the turn is finished.

3. A player must roll exactly what is needed toremove the last piece or pieces. TheWINNER is the first to Wipe-Out ONE.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 3: Section CDessert/Assessment: Show Me! Tell Me! Page 79

Show Me! Tell Me! Task Card

You need: 1 inch graph paper, cm graph paper, hundred grids,fraction circles, markers/crayons, scissors, glue sticks, cards withfractions and decimals, chart paper for posting models

� Draw a card and choose models torepresent the fraction or decimal.

� Outline and color two different ways torepresent the number you have drawn.

� On paper, record the fraction or decimalon the card you drew and attach both ofyour models.

� Draw another card and repeat.

Transparency 3C-34



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Day FourAddition and Subtraction

Table of Contents

Preparing for Day Four 2

Section A It’s Simply Addition 5

Section B Investigating Addition and Subtraction 9

Section C Addition and Subtraction Sampler 34Decimal Addition 40Diffyboxes 46Dollar Addition and Subtraction 49Double More 52Doubles, No Trouble! 55Fill in the Blanks 60In and Out Revisited 77Magic Squares 84

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4Preparing for Day Four Page 2

Day Four Materials

Reusable Materials Consumable MaterialsSection AIt’s Simply AdditionTransparencies:� It’s Simply Addition (4A-1)� Day 4A Reflection Prompt (4A-2)

Handouts:� It’s Simply Addition (4A-1)

Section BInvestigating Addition and SubtractionSamples of the following for each table:� Single counters (beans, straws,

teddybear counters, unlinked linkingcubes, color tiles)

� Two-color counters� Counters that students bundle (straws

and rubber bands, linking cubes, countersin cups)

� Proportional materials that are pre-bundled (base ten blocks, bean sticks)

� Non-proportional materials (money;colored chips with a key)

� Calculators

Transparencies:� Viewpoint (4B-1)� Part-Part-Whole Mat (4B-2)� Ten Frame Mat (4B-3)� Place Value Mat (4B-4)� Number Line (4B-5)� Hundred Chart (4B-6)� 6 Problem Set Cards (4B-7)� Instructional Decision-Making Charts 1 –

3 (4B-8 through 4B-10)� Analysis Chart (4B-11)� Day 4B Reflection Prompt (4B-12)

� Pencil� Paper

Handouts:� Part-Part-Whole Mat (4B-2)� Ten Frame Mat (4B -3)� Place Value Mat (4B-4)� Number Line (4B-5)� Hundred Chart (4B-6)� 6 Problem Set Cards (4B-7)� Instructional Decision-Making Charts 1 –

3 (4B-8 through 4B-10)� Analysis Chart (4B-11)

Section CAddition and Subtraction SamplerTransparencies:� Lesson Planning Process Chart (4C-1)� Day 4C Reflection Prompt (4C-20)

Handouts:� Lesson Planning Process Chart (4C-1)


Section CDecimal Addition� Scissors*� Calculators

Transparencies:� Decimal Addition Recording Sheet (4C-

4)

*Create a reusable set of Decimal NumberCards and scissors won’t be needed.

Handouts:� Decimal Number Cards* (4C-2)� Decimal Number Line (4C-3)� Decimal Addition Recording Sheet (4C-

4)

*Print on cardstock and laminate to create areusable set.

Section CDiffyboxesTransparencies:� Sample Student Response (4C-5)

� 5” X 7” note cards� Chart paper

Handouts:� Sample Student Response (4C-5)

Section CDollar Addition and SubtractionFor each pair:� Play money in $10, $1, $0.10, $0.01� 3 decahedral (10-sided) dice� 1 regular die or number cube

Transparencies:� Dollar Addition and Subtraction

Recording Chart (4C-6)

Handouts:� Dollar Addition and Subtraction

Recording Chart (4C-6)

Section CDouble MoreFor each pair:� 1 deck of cards with face cards removed� 20 small paper sacks, numbered 1-20Section CDoubles, No Trouble!Transparencies:� Double Facts (4C-7)� Addition Chart (4C-8)� Addition Chart Showing Doubles and

Near Doubles (4C-9)

� Paper for poster or book� Markers

Handouts:� Double Facts (4C-7)� Addition Chart (4C-8)� Addition Chart Showing Doubles and

Near Doubles (4C-9)


Section CFill in the BlanksTransparencies:� Fill in the Blanks Instruction Card (4C-

10)� Problem Cards (4C-11)� Fill in the Blanks Recording Sheet (4C-

14)

Handouts:� Fill in the Blanks Instruction Card (4C-

10)� Problem Cards (4C-11)� Premade sets of: Number Cards (3 sets -

yellow, blue, green) (4C-12), Noun Cards(4C-13)

� Fill in the Blanks Recording Sheet (4C-14)

Section CIn and Out Revisited� Counters� Small cup for each pair

Transparencies:� In and Out Target Board (4C-15)� In and Out Revisited Recording Sheet A

(4C-16)� In and Out Revisited Directions and

Recording Sheet B (4C-17)� In and Out Revisited Spinner (4C-18)

Handouts:� In and Out Target Board for each group

(4C-15)� In and Out Revisited Recording Sheet A

for each student� In and Out Revisited Directions and

Recording Sheet B for each student (4C-17)

� In and Out Revisited Spinner for eachgroup (4C-18)

Section CMagic Squares� Calculators

Transparencies:� Five Magic Squares Recording Sheets

(4C-19)

Handouts:� Five Magic Squares Recording Sheets

(4C-19)

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section AIt’s Simply Addition Page 5

It’s Simply Addition

Key Question:� How do properties of addition help in solving an addition problem?

Frame:Participants will explore the properties of addition as they work towards solving an additionproblem. This activity is included to help participants realize the importance ofunderstanding the concepts and properties of addition as the basis of procedures for findingsums.

Transparencies/Handouts:� It’s Simply Addition (4A-1)� Day 4A Reflection Prompt (4A-2)

Procedures Notes1. Have participants answer the question,

“What do you know about addition?”Record responses on a transparency.

2. Present the problem to the participants.Ask them to describe the task.

Some of the features that could be identified:each letter represents one and only onenumber; the problem is 5 digits plus 5 digitsequals 5 digits; no digit is represented bymore than one letter

3. As participants work through theproblem, give hints as to the solution. Itis not really important for participants toreach a final solution. However, someparticipants will be very frustrated if ananswer is not reached.

Hints:

� Zero is not one of the digits.

� The digits used are 1, 2, 3, 4, 5, 6, 8.

� An important relationship is T + G = Sand S + T = E.

� There are two places in the problemwhere regrouping takes place.

� When you add a number to itself, you getan even number unless there isregrouping.

� There can be more than one way to get aspecific digit in a specific place.

4. Option: Allow participants to ask yes/noquestions. Don’t answer questions whichmatch a specific letter to a specific digit.


Debriefing:

� What are some properties of addition? Which of these properties might help yousolve this problem? How do you know?

� What do you know about the sizes of the digits? For example, what does T + G = Stell you about the relative sizes of those three digits?

Reflection and Connection:Addition as a process is often confused with knowing the algorithm. While this problem canbe solved using the traditional algorithm, it is more helpful to remember the properties ofaddition and begin the puzzle at a point other than the ones place. Write a sentence or two inresponse to the following question:

How might starting this problem in a place other than the ones place and thinking aboutthe properties of addition be helpful?



It’s Simply Addition

Solve the following problem. Each letterrepresents one digit. The same letter alwaysrepresents the same number. Each differentletter represents a different number from 0through 9.

T E X A S+ G R E A T

S T A T E

Transparency 4A-2



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aa dd dd ii tt ii oo nn bb ee hh ee ll pp ff uu ll ??

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section BInvestigating Addition and Subtraction Page 9

Investigating Addition and Subtraction

Key Question:� How do the characteristics of problem situations impact the development of the students’

conceptual understanding of addition and subtraction?

Frame:Participants will explore the use of counters, grouped manipulatives, and graphic organizersas tools to join and/or separate sets. The activity explores the effectiveness of variousconcrete manipulatives and graphical representations for representing addition andsubtraction problems that have different characteristics. The problems are built on aninstructional sequence that promotes student understanding of addition and subtractionconcepts and leads to effective instructional design.

Materials:Samples of the following materials for each table:� Single counters (beans, straws, teddy bear counters, unlinked linking cubes, color

tiles)� Two-color counters� Counters that students bundle (straws and rubber bands, linking cubes, counters in

cups)� Proportional materials that are pre-bundled (base-ten blocks, bean sticks)� Non-proportional materials (money; colored chips with key)� Calculators

Transparencies/Handouts:� Viewpoint (4B-1)� Part-Part-Whole Mat (4B-2)� Ten Frame Mat (4B-3)� Place Value Mat (4B-4)� Number Line (4B-5)� Hundred Chart (4B-6)� 6 Problem Set Cards (4B-7)� Instructional Decision-Making Charts 1 – 3 (4B-8 through 4B-10)� Analysis Chart (4B-11)� Day 4B Reflection Prompt (4B-12)


Procedures Notes1. Provide participants with the concrete

manipulatives and the graphicalrepresentations used in the numberexploration activity from Section B ofDay 2. Groups should receive one-personsets of each assigned manipulative andone copy of each graphic representation.(Additional counters may be needed forthe graphic representations.)

2. Assign problems to each group in thefollowing way: There are six ProblemSets, A, B, C, D, E, and F. There are sixproblems in each set, numbered 1 – 6.Assign each group a number 1 – 6. Thegroup will work that problem numberfrom each set. For example, Group 1 willwork Problem 1 from each of the six setsof problems.

3. Assign each group of four two concretemanipulatives and two graphicalrepresentations. Have each person solveeach of their group’s assigned problemsusing one of the assigned manipulatives.For example, Participant A from Group 1will solve problem 1 from each set ofproblems using base-ten blocks.Participants should solve problemsindividually. Have each person use Chart1 to record how effective the assignedmanipulative was for solving theproblem.

Suggested assignments:

Groups 1 and 4: single counters, base-tenblocks, part-part-whole mat with counters,ten frame mat with counters

Groups 2 and 5: two-color counters, strawsand rubber bands, place value mat with base-ten blocks, number line

Groups 3 and 6: linking cubes, money,hundred chart, calculators

Repeat assignments to other groups asnecessary.

4. After individuals have completed theanalysis, have each group of fourcomplete Chart 2, summarizing theeffectiveness of each manipulative forsolving each problem. Have each groupdetermine the most effective and the leasteffective manipulative for each problem.Have groups share with each other:Group 1 with Group 2, Group 3 withGroup 4, Group 5 with Group 6.

Ask participants questions to encourage themto notice important mathematicalcharacteristics of each type of representation.For example, Which model(s) seemed towork best for problems involving smallnumbers? Why? Which model(s) seemed towork best for problems involving largernumbers? Why? Why did this model NOTwork well for larger numbers? Whatmathematical ideas to students already needto know to be able to use each model?


5. Have the participants brainstorm a list ofissues that determines the difficulty levelof word problems for students. In groups,have them examine Problem Sets A, B,and C and categorize the sets as todifficulty level.

Problem Sets A, B, and C are subtractionproblems. The problem sets were written sothat Problem Set A is the easiest and ProblemSet C is the most difficult.

6. In groups, have the participants analyzethe similarities and differences betweenthe three sets. Have them use the three-circle Venn diagram on Chart 3 to recordtheir findings. As a whole group, compilethe similarities and differences found.

Similarities: all are subtraction; A and B areall single step problems; each set has twopartition, two comparison, and two take-away problems.

Differences: number sizes are different in thethree sets; sentence complexity is different;difficulty level is different.

7. Using Problems Sets A, B, and C, haveparticipants brainstorm a list of Big Ideasfor subtraction, recording them on theAnalysis Chart. If time permits, a similarlist for addition (Problem Sets D, E, andF) can be made. Participants may refer tothe TEKS, if desired.

This list will be used with the Sampleractivities.

Debriefing:Prepare a piece of chart paper for a placemat debriefing. (Draw a rectangle in the center of alarge piece of chart paper whose length and width are about half of the length and width ofthe chart paper. From the center of each side of the drawn rectangle, draw a line to theoutside edge of the paper. Assign each participant in the group one section on the outside ofthe rectangle. The consensus is written in the center rectangle.)


Pose the following key question to participants and have them write in their section of theplacemat. Then have the groups discuss the various opinions written. Allow groups time toeither come to consensus on one viewpoint or to select the response they believe is the best.Have someone write that viewpoint in the center of the mat.

“The problem sets were designed to show different levels of complexity. Based on theproblems you analyzed, how does the size of numbers used in problem situations aid orinhibit the development of conceptual understanding of addition and subtraction?”

Select several tables (or all tables if time permits) to share the center of their mats. (Forfurther study of this question, participants may read and discuss from the book, Knowing andTeaching Mathematics Elementary Mathematics, by Liping Ma.) Have participants continuewith the Reflection and Connection.

Reflection and Connection:This investigation should give participants a working knowledge of the advantages anddisadvantages of using concrete and graphic models to represent addition and subtraction.Take two minutes to think, without talking, about the following question, then “pair/share”your thoughts with a partner.

How will you use this information to make instructional decisions for the teaching ofcomputation that support student learning? What other issues might impact yourinstructional decision-making in regards to teaching addition and subtraction?

Transparency 4B-1


Viewpoint

“The problem sets were designed to show different levels ofcomplexity. Based on the problems you analyzed, how does thesize of numbers used in problem situations aid or inhibit thedevelopment of conceptual understanding of addition andsubtraction?”



Part-Part-Whole Mat

Part Part

Whole



Ten Frame Mat



Place Value Mat




Number Line


� Cut on the dotted lines� Overlap the ends, putting the 20 from the first strip on top of the 20 on the second strip� Continue overlapping in the same way, putting the 40 on the 40, the 60 on the 60, and the 80

on the 80� Make sure the centimeters are accurate at the overlapped edges� Tape the strips together



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140



Hundred Chart

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100



Problem Set A

1. Ann had 8 pencils. She gave 6 pencils toGeorge. How many pencils does she havenow?

2. George had 7 pencils. Three of thepencils had erasers and the rest did not.How many pencils did not have erasers?

3. Juanita gave 3 of her 9 crayons toBrenda. How many crayons does Juanitahave now?

4. After a trip to the library, Wayne had 8books. He read 3 of them the first day.How many books has he not read yet?

5. Angela found one box with 5paintbrushes and a second box with 9paintbrushes. How many paintbrushesdoes she need so that both boxes have thesame number?

6. Mireya was making paper flowers. Shemade 5 pink flowers and 2 purpleflowers. She needs the same number ofeach color. How many purple flowersdoes she still need to make?



Problem Set B

1. Lucy packed 10 shirts in her backpack.Wanda packed 8 shirts in her backpack.Who packed more? How many more?

2. Gerald counted 17 socks in his drawer.Nine socks were white and the rest wereblack. How many black socks did Geraldcount?

3. Francisco saw a plate of 15 cookies. Hedidn’t want to eat too many, so he tookthree and ate them. Julia came in andcounted the cookies still on the plate.How many cookies did Julia count?

4. Timmy’s dog and Samantha’s dog bothhad puppies. Timmy’s dog had 12puppies and Samantha’s dog had 9puppies. Which dog had more puppies?How many more puppies did she have?

5. Marifrances was looking at the eggs inher refrigerator. She counted 16 eggs.She knew 7 of the eggs were hardboiledbecause she had marked them with an“X.” How many eggs were nothardboiled?

6. Elizabeth brought 12 flowers from hergarden to school. She gave 5 of theflowers to her teacher and saved the restfor the school secretary. How manyflowers does she have for the schoolsecretary?



Problem Set C

1. Mr. Voss is planning a trip for his class.He needs to decide how many are goingto ride in the van and how many in cars.The school van holds 15 people. 24people are going on the field trip. Howmany people will need to ride in cars?

2. Mr. Voss found out that there is a newvan and it carries 21 people. The old vanholds 15 people. 24 people are going onthe field trip. Mr. Voss will put as manypeople as he can in a van and the rest willtravel by car. How many fewer peoplewill ride in cars if he takes the new van?

3. Mr. Voss did such a good job planninghis field trip that he is going to plan a tripfor the whole grade level. 56 people aregoing on the trip. He found a bus thatholds 72 people. How many empty placeswill there be on the bus?

4. The field trip was to the State Aquarium.To get the special rate, Mr. Voss had tobuy 75 tickets. On the first field trip, 24tickets were used. He needs 56 tickets forthe second trip. How many more ticketsdoes Mr. Voss need to buy?

5. The bus company charges an extra fee ifthe round trip is more than 100 miles. Mr.Voss found out that it is 37 miles fromhis school to the State Aquarium. Howmany additional miles can the bus travelbefore Mr. Voss has to pay the extra fee?

6. The teachers decided to take the studentson a picnic as a part of the field trip. Mr.Voss bought canned drinks. He bought18 colas, 18 orange drinks, 18 lemon-lime drinks, and 18 root beers. If each ofthe 56 people has 1 canned drink, howmany drinks will the teachers bring backto school?



Problem Set D

1. Lauren had 4 bows for her hair in herdrawer. Her friend, Megan, gave her twomore. How many bows does Lauren havenow?

2. Kathy found 6 white shells at the beach.She found 3 black shells. How manyshells did Kathy find?

3. Larry's mother had 5 pink roses and 5white roses in a vase on the table. Howmany roses were in the vase?

4. Adam found 3 markers in his closet. Hismother gave him 4 new markers. Howmany markers does he have now?

5. George collects Star Wars toys. He has 7toys in his collection. He asks his motherif he could please buy a new toy. Shesaid, "No, not today." How many StarWar toys does George have now?

6. Jeff took a bite of watermelon. He found4 little black seeds and 3 little whiteseeds. If Jeff spit out all of the seeds inhis mouth, how many did he spit out?



Problem Set E

1. Bonnie has 4 dogs, 6 cats and 3 rabbitsfor pets. How many pets does Bonniehave?

2. Jeff hit 8 homeruns during the pre-seasonbaseball games. He only hit 6 homerunsduring the regular season. What is thetotal number of homeruns Jeff hit for histeam?

3. Maria had 12 rosebushes and 14 gardeniabushes in her garden. She planted 8 morerosebushes. How many rosebushes doesshe now have in her garden?

4. Will has 12 pieces of candy. Some arechocolate and some are caramel. He hasthe same number of each flavor. Howmany of each kind of candy does Willhave?

5. For Charlie's party, his mother ordered 2kinds of pizza. She ordered 6 pepperoniand 8 hamburger pizzas. How manypizzas did Charlie have to serve at hisparty?

6. Bob collects state quarters. In hiscollection he has a quarter from each of12 different states. His Uncle George senthim quarters from 9 other states. Howmany quarters are in his collection now?



Problem Set F

1. A farmer had a fence in his cornfield with5 posts. Two crows came and sat on thefirst post. On the second post, there wasone crow. No crows sat on the third andfourth posts. Somehow, 4 crows sat onthe fifth post. How many crows weresitting on the farmer's fence?

2. The class is collecting cans to donate tothe food bank. Jim brought 6 cans andMaria brought 3 cans. Then Jim brought3 more. How many cans do they havenow?

3. Mrs. Key looked out in her backyard andsaw 3 muddy children and 2 muddy dogs.How many muddy feet did Mrs. Key see?

4. A bunny rabbit hopped down 10 steps.Then it hopped down 10 more steps. Howmany steps did the bunny rabbit hopdown?

5. For the field day picnic, one first gradeclass took a survey of their favoritedrinks. Twelve students prefer coladrinks, 4 prefer diet drinks, and 6 preferlemon-lime drinks. How many drinksdoes this class need for its picnic?

6. Coach Bell needed to order blue ribbonsfor the field day events. He needed 50ribbons for third grade and 60 ribbonseach for fourth and fifth grade. Howmany blue ribbons should Coach Bellorder?



Chart 1:Problem Number ___ Using _____________

Solve each problem using your assigned manipulative. Write thestrengths and weaknesses of your assigned manipulative for solvingeach problem.

Problem Effectiveness of Manipulative

A___

B___

C___

D___

E___

F___



Chart 2: Analysis of Problem Number ___

Manipulatives UsedProblemNumber

Most EffectiveManipulative

Least EffectiveManipulative

A___

B___

C___

D___

E___

F___



Chart 3:Venn Diagram

Set C

Set BSet A



Analysis Chart

Step 1Big Ideas




Transparency 4B-12



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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CAddition and Subtraction Sampler Page 34

Addition and Subtraction Sampler


experiences to develop students’ understanding of addition and subtraction?

Frame:The participants will explore a sampler of learning experiences designed to support thedevelopment of the understanding of addition and subtraction. Participants will examinethese learning experiences in terms of how the Lesson Planning Process is used to designeffective instruction to develop understanding of addition and subtraction. Throughconceptual development and problem-solving experiences, students can gain fluency incomputation.

Transparencies/Handouts:� Lesson Planning Process Chart (4C-1)� Day 4C Reflection Prompt (4C-20)

Procedures Notes1. Present to the participants the terms

Conceptual Development and FactFluency. In addition and subtraction,conceptual development is learning howand when to add or subtract. Fact fluencyis the ability to rapidly recall basic facts.As participants do the Sampler activities,they should keep these two ideas in mind.

2. Hand out the Lesson Planning ProcessChart for the Addition and SubtractionSampler. Have participants review thefour components in the Lesson PlanningProcess.

3. Introduce the Sampler of LearningExperiences for Addition and Subtractionand explain to participants that they willbe using the Lesson Planning ProcessChart to identify the four components ofthe Lesson Planning Process in relation toeach learning experience.


4. Introduce “Diffyboxes” to theparticipants. (Some participants may befamiliar with this activity.) After eachpair of participants has completed 5cards, make posters of the cards,grouping them by “2 box Diffys,” “3 boxDiffys,” etc. Use a gallery walk forparticipants to look at each chart to findpatterns in the numbers. Discuss thevarious patterns that are found.

Some of the patterns that might be found are:

� The next-to-last box often has the samenumber on opposite sides.

� Sometimes the next-to-last box has pairsof numbers on opposite sides.

� The size of the numbers does not make adifference as to the number of boxes.

5. Have participants fill out the LessonPlanning Process Chart for “Diffyboxes.”Spend some time discussing all of theobjectives that could be the focus for thisactivity. Discuss how this activity canprovide needed practice in subtractionthat could lead to fact fluency.

6. Allow participants time to explore therest of the activities in the Sampler. Havethem fill out the Lesson Planning ProcessChart as they complete each activity.




Guide participants to create a combinedLesson Planning Process Chart for additionand subtraction, identifying overall big ideasand Evidence of Understanding for this topic.(See Debriefing section.)

Debriefing:Conceptual development and fact fluency must be developed side-by-side. While conceptualdevelopment alone will allow students to problem-solve, fact fluency makes problem-solvingmore efficient. Fact fluency alone allows students to compute quickly, but doesn’t alwaysprovide students with the problem-solving skills they need to be successful in highermathematics. Review the activities you have completed. Which activities promote bothconcept development and facility with facts? What would need to be added to the activities toprovide a balanced approach? Be prepared to share your ideas with the whole group.

To the facilitator: The following list provides an example of a list of combined Evidences ofUnderstanding that might be created for Addition and Subtraction.




� Applying meanings and properties ofaddition/subtraction

� I can use objects to showaddition/subtraction.

� I can draw pictures to showaddition/subtraction.

� I can talk about why I choseaddition/subtraction.

� I can add in any order (commutativity).� I can add any two numbers first

(associativity).� I can read an addition/subtraction number

sentence.� Using procedures for finding sums/

differences� I can count on to find a sum.� I can count back to find a difference.� I can use landmark numbers (like 5 and

10) to find sums/differences.� I can use patterns to find

sums/differences.� I can use basic facts to find

sums/differences.� I can write a number sentence to describe

the sum/difference.� I can pick a reasonable number for the

sum/difference.� I can estimate the sum/difference.� I can use tens and ones to find a

sum/difference.� I can write about the different ways I

found the sum/difference.� Connecting addition and subtraction � I can check my subtraction with addition.

� I can use addition/subtraction factfamilies to solve problems.

Reflection and Connection:The learning experiences in this section allow you to examine important components of theLearning Planning Process. As you completed tasks involving addition and subtraction, youinvestigated concept development and fact fluency.

How will you use those ideas in the design of your own instruction about addition andsubtraction?




Step 1Big Ideas




Transparency 4C-20



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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CDecimal Addition Page 40

Decimal Addition

Words for the Word Wall:� Decimal� Fraction� Range

Materials:� Scissors (to make number cards if they are not already made)� Calculators

Transparencies/Handouts:� Decimal Number Cards (4C-2)� Decimal Number Line (4C-3)� Decimal Addition Recording Sheet (4C-4)

Procedures Notes1. Give each pair of students a set of cards

(numbers 1-9 and four decimal pointcards)

2. Students will be given a target number.They will use the cards to create anaddition problem whose sum is as closeas possible to the target number. At least8 of the 9 digit cards must be used. Atleast one addend must be a decimalfraction.

Suggested target numbers:

� 6.3

� 7.95

� 5.8

� 8.23

� 9.35

� 13.68

� 9.27

� 10.17

� 15.98

� 25.56

Allow students 3 – 5 minutes to work eachproblem. They should move cards arounduntil they get the closest answer.


3. Students record their answers on therecording sheet and on the number line.Using the number line or a calculator,have students figure how far away fromthe target number their sum is. On thenumber line, have them plot the numberthat is equidistant from the targetnumber, but in the other direction. Forexample, if the target is 6.35 and theirsum is 6.5, they were .15 higher than thetarget. 6.2, .15 less than the target, is theother end of the range. These numbersrepresent + or - .15 from the targetnumber.

Through the process of placing the targetnumber in the middle of a range, studentsshould begin to develop the concept of“acceptable range.”

4. Have students use the cards and createsums for a total of five target numbers.The targets can be given to the wholeclass or each pair can choose which fivetargets they want to use.

5. After students have completed the fivetarget sums, discuss with them acceptableranges. Have students share their closestsum and their sum the farthest away fromthe target. Have students decide what isan “acceptable range” for this activity.

6. Once the acceptable range is established,have students try one or two morenumbers to see if they can stay within theacceptable range.

Discussion:

� How did you decide where to place the digits? If the sum was too large, what did youdo? If the sum was too small, what did you do?

� Once the acceptable range was decided, what did you do differently to reach a sumwithin the acceptable range?

Handout 4C-2


1 2 3 4 5

6 7 8 9

. . . .

Decimal Number Cards

Your goal is to reach thetarget number with oneaddition problem.

1. Listen for the targetnumber.

2. Use your cards to reach thetarget number.

3. You must add numbers toget to the target number.

4. You must use at least 8digit cards

5. You must have at least onedecimal fraction.

6. Keep a record on therecording chart and on thenumber line.

Handout 4C-3


Decimal Number Line


� Cut on the dotted lines� Overlap the ends, putting the 2.0 from the first strip on top of the 2.0 on the second strip� Continue overlapping in the same way, putting the 4.0 on the 4.0, the 6.0 on the 6.0, and the

8.0 on the 8.0� Make sure the centimeters are accurate at the overlapped edges� Tape the strips together

Handout 4C-3


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0

6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0

8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0



Decimal Addition Recording Sheet

Target Number Addition Problem Distance fromTarget Number

Range

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CDiffyboxes Page 46

Diffyboxes

Words for Word Wall:� Subtraction� Diffy� Difference

Materials:� 5” X 7” note cards� Chart paper

Transparency/Handout:� Sample Student Response (4C-5)

Procedures Notes1. Ask four students each to give you a one-

digit number. Write them on theoverhead with one color as if they werefour corners of a square. Connect thenumbers with lines.

2. Tell students you are going to create a“Diffybox.” Ask which operation theythink they are going to use on thenumbers in this box – addition orsubtraction. Discuss reasons for theiranswers.

“Diffy” comes from difference. The boxesuse subtraction. Patterns will develop as theboxes are made.

3. Using a different color marker, write thedifferences inside the box along thecenter of each line. If the differences arethe same, your diffy is complete. If not,draw lines between the differences(making a “diamond”). Then find thedifferences using a new color. (Changecolor each time you are going to write thedifferences.)

Demonstrate several boxes until you are surethe students understand the procedures.


4. Have pairs of students create 5Diffyboxes. Have them record them onindex cards. As students complete theirfive Diffyboxes, have them sort the cardsinto classifications by number of boxes ineach Diffybox:

1-Box Diffys, 2-Box Diffys, 3-BoxDiffys, etc.

Students don’t have to use different colors tocreate the Diffyboxes. However, using twocolors does help keep track of the differentboxes within each Diffybox.

5. After each pair has completed five cards,have them trade cards with another pair.The second pair should check thesubtraction on the completed cards.Students should indicate on the cardwhich pair created it and which pairchecked it.

6. Make a chart of the completedDiffyboxes. Have columns for thenumber of boxes. Have students look atthe problems and see if they can find anypatterns.

Ask, “What do the 2-box Diffys have incommon?” “Do the cards with the mostboxes have anything in common?” “Can youpredict when a set of numbers will make alarge number of boxes?”

Discussion:

� What kind of Diffybox is easiest to make?

� Do you think the order of the first four numbers makes a difference? Why do youthink so?

� What happens if you use odd numbers? Even numbers? Why do you think thathappens?

� What would happen if you use some 3-digit numbers? 2-digit numbers? A mixture ofdifferent kinds of numbers?

� Do you recognize any patterns?



Sample Student Response

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CDollar Addition and Subtraction Page 49

Dollar Addition and Subtraction

Materials:For each pair:� Play money in $10, $1, $0.10, and $0.01� 3 Decahedral (10-sided) dice� 1 regular die or number cube

Transparency/Handout:� Dollar Addition and Subtraction Recording Chart (4C-6)

Procedures Notes1. Provide each pair of students play money

in tens, ones, dimes, and pennies.These denominations are used to emphasizeplace-value concepts. Other values could beused as well.

2. Roll three decahedron dice. Thesenumbers tell the amount of money youare going to spend. The amount is alwaysdollars and cents. The dice may bearranged in any order.

Zeros are possible with these dice. Helpstudents understand that a zero in the dollarsplace will make the task more difficult.

Since only three dice are being thrown, theamount will always be less than $10.00.Later in the activity, four dice are used andthe ten-dollar bills will be needed.

3. Roll one regular die. This tells you howmany items you are going to buy.

Number cubes instead of dice may be used.

4. Using the play money, build the cost ofeach item. Make sure that the sum theitem costs equals the total amount youare to spend. Use all three types ofmoney: dollar bills, dimes, and pennies.Find at least three ways to reach the totalamount with the number of itemspurchased. Each way must use dollarbills, dimes and pennies.

For example:

8, 5, and 3 are rolled. $8.53 could be theamount. A 3 is rolled. There are 3 items. Theamounts could be:

$2.21

$4.56

+$1.76

$8.53

$1.01

$2.21

+$5.31

$8.53

$3.49

$2.65

+$2.39

$8.53

5. Repeat the activity several times. A calculator can be used to record the moneycounting. The numbers can be entered as anaddition problem or the total can be enteredand each number subtracted until 0 isreached. Students may need assistance inentering the decimal forms.


6. Find out how much change you wouldreceive if you paid for all three itemswith a ten-dollar bill. Take turns withyour partner in counting back the changeyou would receive.

This variation gives students practice insubtracting from a number with three zeros.

7. An extension of the activity is to use fourdecahedron dice and build four-digitnumbers: $nn.nn.

Discussion:

� Did you build the largest possible value for the total? How would your answers havebeen different if you had built the largest value or the smallest value?

� How many different ways do you suppose there are to build each total? How couldyou prove your answer?

� How would using quarters and nickels as well as dimes and pennies change youranswers? Would there be more ways to build the totals or fewer? How do you know?



Dollar Addition and Subtraction Recording Chart

DecahedralDice

Regular Die Dollars andCents Total

Number ofItems

First Way Second Way Third Way

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CDouble More Page 52

Double More

Words for the Word Wall:� Compare numbers� Equal� Equal amount� Greater than� Less than� More than� Sum

Materials (for each pair of students):� 1 deck of cards with face cards removed (Aces count as ones.)� 20 small paper sacks, numbered 1-20

Procedures Notes

1. Participants will work together in pairs.Model the process of playing the gamethrough several rounds by playing thegame with a student. The teacher willdeal all 40 cards to himself or herself andhis or her partner.

As you demonstrate dealing the cards,emphasize that you are giving your partnerone card and then giving one card toyourself.

When all cards are dealt, ask some questionsto promote mathematical thinking. Examplesmight include:

� Do you think each player got his fairshare of cards? Why or why not?

� Do you each have the same number ofcards?

� How can you be certain?

2. On each round, both players turn over thetop two cards of his or her stack, find thesum of these two cards, and determinewhich sum is greater. (The number on thecard determines the value of the card.)All four cards are placed into the baglabeled with the same number as thehigher sum. If both players turn overcards with the same sum, the bag withthat number captures the tie.

As you model the process for playing thegame, ask some of these questions to helpilluminate strategies the students might use.

� How do you know which person has thelarger sum?

� What might you do if you weren’t surewhich sum was more and which wasless?

� How could you check to see who iscorrect if you and your partner don’tagree?

� If you don't know the sum, is it morehelpful to count on from the smallernumber or from the larger number?


3. Play continues until all the cards areplayed.

4. Players then examine each bag to look forpatterns among the cards in that bag.Begin with bag 15.

The teacher should model emptying bag 15and encourage students to talk about thepatterns they notice. Based on the students’comments, the teacher may introduce orreinforce math vocabulary such as equal,more than, less than, and other terms as listedin word wall list. Some sample questionsmight include:

� What do you notice about the cards inbag 15?

� Are some of the cards the same?

� Which number do you find most often inthis bag?

� Which numbers are not in this bag?

5. Ask students to return all cards from the15 bag back into the bag. Encouragestudents to continue looking for patternsone bag at a time. After an adequateexploring time, have students stop. Havethem reflect on what they noticed. Theteacher may choose to write the students’observations on a chart or other display.Have the students find the pairs of cardsin each sack that add up to that sum andrecord them on paper.

� Are any of the bags empty?

� How can you explain this?

� Which bags captured the most cards?

� Why do you think this happened?

� Did you notice any patterns when youcompared the sums from one bag?

� Can you find a pair of cards in this bagthat add up to 15? Record the answers ona class chart.

6. Ask students to build a real graph on thefloor using the cards and the sacks. Thenumbered sacks can be placed along thebottom of the graph and the cards fromeach sack can be placed above the sackfrom which they were removed.

The teacher should ask the class to decide thename and labels for their graph. The titlecould be "Double More" and the labels couldbe "Sack Numbers" and "Cards in theSacks". Suggested questions about the graphmight include: Is there an even or oddnumber of cards in each column? Can youexplain why? Which sack has the most/leastnumber of cards above it? Are any of thecolumns empty? Can you explain why?


Discussion:Open the discussion by asking students to tell what they think they learned from the game.Which bags were empty? Why? Were there any empty sacks that could have captured cardsif there had been more cards to play? Using the bags lessens the sense of competition, butalso brings out the idea of hierarchical inclusion: each bag will capture only numbers that areless than the amount specified by the numeral on the bag.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CDoubles, No Trouble! Page 55

Doubles, No Trouble!

Words for the Word Wall:� Doubles� Near doubles� Addition� Facts

Materials:� Chart paper for poster or book� Markers or crayons

Transparencies/Handouts:� Double Facts (4C-7)� Addition Chart (4C-8)� Addition Chart showing Doubles and Near Doubles (4C-9)

Procedures Notes1. Read the poem with the students. Discuss

the meaning of “doubles.”

2. Have the students create a poster or bookto illustrate each fact in the poem. Theyneed to think of things that come indoubles. For example, an egg cartonshows 6 + 6 = 12

3. When the books or posters are complete,have each group share their ideas with theclass. Have them examine all of thedifferent ways each double wasillustrated.

4. Have the students examine an additionchart. Have them locate the “doubles.”Ask, “Where do you find the doubles onyour addition chart? What numbers arenear the doubles? If you wanted toremember 8 + 9, what double facts mighthelp you? What doubles is 8 + 9 near onthe chart? Is it easier for you to remember8 + 8 = 16 and 16 + 1 = 17, or 9 + 9 = 18and 18 – 1 = 17?”


Discussion:

� Did all of the groups show the doubles in the same way? Why do you suppose thathappened?

� What are the “near double” facts? How do the doubles help you remember the neardoubles?

� Is 0 + 0 = 0 a double? Why do you suppose we didn’t include it in the book? Howmany double facts does that make? How do you know?

� How many near double facts are there?



Double Factsby John Starkel

1 and 1 are 2,2 and 2 are 4,3 and 3 are 6,Can you think of any more?

4 and 4 are 8,5 and 5 are 10,6 and 6 are 12,Here we go again!

7 and 7 are 148 and 8 are 16,9 and 9 are 18,Look, I’m an adding machine!

Double facts are easy,As easy as can be.Go back to the beginning,And say this rhyme with me.



Addition Chart

+ 0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

4 4 5 6 7 8 9 10 11 12 13

5 5 6 7 8 9 10 11 12 13 14

6 6 7 8 9 10 11 12 13 14 15

7 7 8 9 10 11 12 13 14 15 16

8 8 9 10 11 12 13 14 15 16 17

9 9 10 11 12 13 14 15 16 17 18



Addition ChartShowing Doubles and Near Doubles

+ 0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

4 4 5 6 7 8 9 10 11 12 13

5 5 6 7 8 9 10 11 12 13 14

6 6 7 8 9 10 11 12 13 14 15

7 7 8 9 10 11 12 13 14 15 16

8 8 9 10 11 12 13 14 15 16 17

9 9 10 11 12 13 14 15 16 17 18

Doubles Near doubles

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CFill in the Blanks Page 60

Fill in the Blanks

Words for the Word Wall:� Add� Subtract� Noun� Number

Materials:� Chart paper for poster or book� Markers

Transparencies/Handouts:� Fill in the Blanks Instruction Card (4C-10)� Problem Cards (4C-11)� Premade sets of the following:� Number cards (3 sets: yellow, blue, green) (4C-12)� Noun Cards (4C-13)

� Fill in the Blanks Recording Sheet (4C-14)

Procedures Notes1. Provide each pair of students with the

materials. Students should follow theinstructions on the instruction card andsolve ten of the problems.

2. After the problems have been solved andanswers recorded, have students work ingroups of four to check their answers.Since the numbers for each problem willbe different, students need to check theaddition or subtraction.

3. Have the groups of four take one set ofproblem cards and sort them into additionproblems and subtraction problems. Havethem take each operation and sort theproblems within the operation, lookingfor similarities and differences.

4. Make a chart, labeling each problem asaddition or subtraction and enumeratingthe attributes of addition problems andsubtraction problems.

Help students recognize the three types ofsubtraction used in the problem cards: takeaway, comparison, and partition.


Discussion:

� Did you answer all of your problems correctly? How do you know?

� Which problems were difficult for you? Why do you suppose that is true?

� When you compared answers, did everyone solve the problems in the same way?Why do you suppose that happened?



Fill in the Blanks Instruction Card

You need:� A partner� Problem Cards� Number Cards (three sets: yellow, blue, and green)� Noun Cards� Recording Sheet

Procedures:1. Sort all of the cards. Put the Problem Cards together face down in a stack. Put the

Noun Cards together face down in a stack. Put each set of Number Cards face downin a stack.

2. Choose the top Problem Card. Read the problem. You will put Number Cards in thesquares and a Noun Card in the rectangle (that is not a square!).

3. If the squares for the numbers are blank, you may use any color Number Card. If thesquare is labeled with a letter, you must use that color Number Card.

means use a yellow Number Card.

means use a blue Number Card.

means use a green Number Card.

means use any color Number Card.

4. Place the Number Cards and the Noun Card in the correct places. Then solve theproblem.

5. Write the Problem Letter on the Recording Sheet. Write your noun and your numbers.Write the number sentence or sentences you used to solve the problem. Write howyou knew to add or subtract.

6. Continue working until you and your partner have solved ten problems.

7. Be prepared to share your work with the class.



Problem Cards

Problem A

Matilda found . Then

she found more. How many does she have now?

Problem B

Julio had . On the way

home from school, he lost . How many does he

have now?



Problem C

On the way home from school, Henry saw

. He turned a corner and saw

more. How many did Henry see?

Problem D

Gretchen has red . She

also has pink ones. Does she have more red or

pink? How many more?



Problem E

The red box has . The

blue box has . Which box has more? How many

more?

Problem F

At school, Miss Jones counted

in the supply closet. She

counted more in the hall. How many did Miss

Jones count?



Problem G

Angela has . Judy has

. Who has more? How many more?

Problem H

Jose owned . He

bought another . How many does he own now?

Problem I

Tommy found . He

gave to Hilary. How many does Tommy have

now?



Problem J

On the way to school, Jan bought

. She gave to Nancy

and to Dora. How many does she have now?

Problem K

Olivia collects . She has

in a drawer. She has on a shelf. She has

in her room. How many does she have?



Problem L

While Lloyd was cleaning his room, he found

. He owes John . How

many did he have after he gave John what he owed

him?

Problem M

There were in the yard.

There were in the garage. And there were

on the porch. How many were there?



Problem N

DeWayne went shopping and bought

. were large and the

rest were small. How many were small?

Problem O

Sarah was on a field trip. She counted

. She noticed that were

yellow and the rest were purple. How many were

purple?



Problem P

Richard opened a box he found in the attic. Inside the

box, he found .

Surprised, he opened another box and found

more. The third box contained . How many did

Richard find?



Problem Q

A box was delivered to school. It contained

. Gerald saw that there were

round ones and square ones. were square. How

many were round?

Handout 4C-12


Number Cards

Copy on yellow paper. Enough for 3 sets.

0 2 3 4 5 6 7 8

9 2 3 4 5 6 7 8

9 2 3 4 5 6 7 8

0 2 3 4 5 6 7 8

9 2 3 4 5 6 7 8

9 2 3 4 5 6 7 8

0 2 3 4 5 6 7 8

9 2 3 4 5 6 7 8

9 2 3 4 5 6 7 8

Handout 4C-12


Number Cards

Copy on blue paper. Enough for 3 sets.

10 11 12 13 14 15 16 17

18 19 20 10 11 12 13 14

15 16 17 18 19 20 17 18

10 11 12 13 14 15 16 17

18 19 20 10 11 12 13 14

15 16 17 18 19 20 17 18

10 11 12 13 14 15 16 17

18 19 20 10 11 12 13 14

15 16 17 18 19 20 17 18

Handout 4C-12


Number Cards

Copy on green paper. Enough for 2 sets.

21 22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44

45 46 47 48 49 50 51 52

53 54 55 56 57 58 59 60

21 22 23 24 25 26 27 28

29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44

45 46 47 48 49 50 51 52

53 54 55 56 57 58 59 60

Handout 4C-13


Noun Cards

apples bananas

toy cars chairs

eggs flowers

books notebooks

tables oranges

crayons paint brushes

pencils pens

paperclips pictures

marbles posters

caps rocks

caps shoes



Fill In The Blanks Recording Sheet

Problem Noun Number Sentence How did you know to add or subtract?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CIn and Out Revisited Page 77

In and Out Revisited

Words for the Word Wall:� Decrease by one� Increase by one� Greater than� Less than� More than� Equal� Total� Family of facts

Materials:� Counters� Small cup for each pair

Transparencies/Handouts:� In and Out Target Board for each group (4C-15)� In and Out Revisited Recording Sheet A for each student (4C-16)� In and Out Revisited Directions and Recording Sheet B for each student (4C-17)� In and Out Revisited Spinner for each group (4C-18)

Procedures Notes1. Assign the students partners. Model the

process of this activity with a student. Tobegin, the teacher determines how manycounters to toss and records that numberon the Recording Sheet A. Next, eachgroup will lay their target paper on a flatsurface and place the determined numberof counters in a cup. Players take turnstossing, all at once, the cup full ofcounters over the target. The number ofcounters that land in and the number ofcounters that land outside of the target arerecorded on the In and Out RevisitedRecording Sheet A. In this activity,students will also record the numbersentence that represents that addition fact.

As you model playing the game, ask somequestions to make sure your studentsunderstand the process.

� How many counters do you have in all totoss?

� How many counters landed in the target?

� How many counters landed outside of thetarget?

� How do you record each toss?

� How do you write a number sentence thatrepresents your toss? In + Out = Total


2. Once Recording Sheet A is complete, askthe participants to describe what theyhave recorded. They should comparetheir recording sheet with the otherparticipants’ recording sheets in theirgroup. Debrief the activity by recordingthe various number pairs, written asaddition facts. Record these facts onanother piece of chart paper.

To assess the student’s understanding of thenumbers they have recorded, choose one ofthe number pairs on the record sheet and askstudents to model it with counters in and outof the target.

Do the pairs of numbers on your recordingsheet look the same as the pairs of numberson the recording sheets of the others in yourgroup?

Do you notice any patterns? What are they?

The teacher should encourage students to talkabout the patterns they noticed. Based on thestudent’s comments, the teacher mayintroduce or reinforce math vocabulary suchas decrease by one, increase by one, greaterthan, less than, more than, equal, total, andfamily of facts.

3. Have students read the directions for Inand Out Revisited Recording Sheet B. Inthis part of the activity, students recordthe addition/subtraction fact family foreach toss of the counters. The number ofcounters is chosen by spinning an 11 – 20spinner.

As students build the fact family, askquestions that emphasize the inverserelationship between addition andsubtraction.

� If you know one of the additionsentences, how do you make the secondaddition sentence?

� If you know one of the additionsentences, how do you know thesubtraction sentences?

Discussion:Discuss the activity with the students. Use these questions to focus their attention on themathematics in the activity.

� How are the two recording sheets different? How are they the same?

� How many different number of counters did you toss? Did each number of countershave a different set of addition and subtraction problems? Why do you suppose thathappened?

� Choose one of the numbers you used. What other addition and subtraction sentencescould you write that works with that number?



In and Out Target Board

Out Out

In

Out Out



In and Out Revisited Recording Sheet A


Trial In Out Number Sentence

1

2

3

4

5

6

7

8

9

10

List all of the different number sentences.

Organize the number sentences. List the number sentencesthat use the same addends.



In and Out RevisitedDirections for Recording Sheet B

Directions:

1. Spin the spinner. Get the number of counters shown on thespinner.

2. Toss the counters on the board.

3. Count the number of counters inside the circle and the numberof counters outside the circle.

4. Write an addition number sentence using those numbers and thetotal number of counters. Write a second addition sentence.

5. Write two subtraction sentences using the total number ofcounters as the minuend.

6. Repeat the steps until you have 5 different sets of numbersentences.



In and Out Revisited Recording Sheet B

Trial Addition Sentences Subtraction Sentences

1

2

3

4

5



In and Out Revisited Spinner

11 12

13

14

15

1617

18

19

20

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 4: Section CMagic Squares Page 84

Magic Squares

Words for the Word Wall:� Magic square� Sum� Row� Column� Vertical� Horizontal� Diagonal


Transparencies/Handouts:� Five Magic Squares Recording Sheets (4C-19)

Procedures Notes1. Have the students examine the 3 by 3

magic square. Have them add thenumbers in each row, each column, andeach diagonal. For this magic square, themagic number is 15.

2. Have the students examine the 4 by 4magic square. Tell them that the magicnumber is 34. Ask them to determinewhy 34 is the magic number. (In thismagic square, 34 is the sum of eachhorizontal row, each vertical column,both diagonals, the four corners, the fourmiddle squares, the four squares in thetop left corner, the four squares in the topright corner, the four squares in thebottom left corner, and the four squaresin the bottom right corner.) Help themfind as many of the combinations aspossible before giving them the entirelist.


3. Show the students the attributes of aMagic Square. Have them determine howmany of the attributes they have alreadydiscovered in the example square. Showthem the Magic Square Puzzle. Havethem determine how they could go aboutsolving the puzzle. After they haveworked on the puzzle for a while, havestudents share the strategies they haveused thus far.

Allow students to use calculators for thisactivity.

4. After students have successfullycompleted the sample puzzle, allow themto try the other puzzles.

The additional puzzles include somevariations not found in the sample puzzle.However, the attributes still apply.

Discussion:

� What patterns did you find in the Magic Squares?

� Did all of the magic squares have the same kinds of patterns? How do you know?

� Try making a Magic Square. How many numbers do you need for a 4 by 4 MagicSquare? How will you place the numbers? How will you check to see if it is a MagicSquare?



Magic Squares Recording Sheet 1

Some Magic squares use only 9 numbers like this one:

When you add the numbers in each column, what sumdo you get?

When you add the numbers in each row, what sum doyou get?

When you add the numbers in each diagonal, what sumdo you get?

7 6 12 9

10 11 5 8

13 16 2 3

4 1 15 14

4 9 2

3 5 7

8 1 6

In other squares, 16 numbers are used. If the magicnumber for this square is 34, what do you suppose itmeans?

Find 34 as a sum of other numbers besides those ineach row, column, and diagonal. What patterns didyou find?

How many different ways did you find 34 as the sumof four numbers? Using colored pencils, connect eachset of four numbers that add up to 34.




Attributes of a 4 by 4 Magic Square

� Every row adds up to the same number.� Every column adds up to that same number.� Both diagonals add up to that number.� The four corners add up to that number.� The four middle squares add up to that number.� The four squares in the top left corner add up to that number.� The four squares in the top right corner add up to that number.� The four squares in the bottom left corner add up to that number.� The four squares in the bottom right corner add up to that number.� The sum will be equal to (least number + greatest number) x 2. In the sample square,

the least number is 1 and the greatest number is 16. So, (1 + 16) x 2 = 34 which is themagic number.

Use the attributes and fill in the blanks on this puzzle.

Use every number from 1 to 16.

All sums are 34.

1 12

10 13

11 16

14 7




Try these Magic Square puzzles. A calculator will help you solve them.

Puzzle 1Use every number from 83 to 98.All sums are 362.

Puzzle 2Use every odd number from 5 to 35.All sums are 80.

Puzzle 3Use every fourth number from 28 to 88.All sums are .

89

92 93

98 84

86

33 15

5

35

7 31

48 40 76

84 56

52




Make your own Magic Squares

� Use the attribute list to help you place a set of numbers in one of the puzzle blanks onthis page.

� Check to make sure all of your sums are correct.� Write 6 of the numbers in the one of the puzzle blanks on the next page. Make sure

they are in the correct squares.� Write the range of your numbers.� Give your puzzle to someone to solve.


Day FiveMultiplication and Division

Table of Contents

Preparing for Day Five 2

Section A Fair Share 6

Section B Investigating Multiplication and Division 18

Section C Multiplication and Division Sampler 39Colored Pencils 48My Flowers 50A Remainder of One 52Leftovers 55Let’s Paint! 58How Long? How Many? 604 In a Row 63Go Figure! 66The Greatest Product Wins 68Going Bananas! 70Multiple Towers 72Fresh Produce 76Fresh Produce Challenge 78What’s In Each Box? 80Tiffany’s Beanie Babies™ 83Marissa’s Garden 85The Greatest Product, Part 2 93Marissa’s Garden Again 97

Section D Closure for Day Five: Conceptual Models of Computational Fluency 102

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5Preparing for Day Five Page 2

Day Five Materials

Reusable Materials Consumable MaterialsSection AFair Share� Base-ten blocks� Color tiles� Pattern blocks

Transparencies:� Multiples of 4 (5A-1)� Interpreting Quotients (5A-2)� 3 Types of Division Problems (5A-3)� Using Multiples to Divide (5A-4)� Day 5A Reflection Prompt (5A-5)

� Chart paper� Markers� Blank transparencies� Overhead pens

Section BInvestigating Multiplication and Division� Counters or color tiles� Base-ten blocks� Bare Bear's New Clothes by Peter S.

Seymour, 1986

Transparencies:� How Many Cherries? (5B-1)� How Many Apples? (5B-2)� How Many Cookies? (5B-3)� How Many Outfits? (5B-4)� My Garden (5B-5)� 6 Problem Set Cards (5B-6)� Analysis Chart (5B-7)� Day 5B Reflection Prompt (5B-8)

� Centimeter grid paper� Markers� Blank transparencies� Overhead pens

Handouts:� How Many Cherries? (5B-1)� How Many Apples? (5B-2)� How Many Cookies? (5B-3)� How Many Outfits? (5B-4)� My Garden (5B-5)� 6 Problem Set Cards (5B-6)� Analysis Chart (5B-7)

Section CMultiplication and Division SamplerTransparencies:� Lesson Planning Process Chart (5C-1)� Day 5C Reflection Prompt (5C-28)

Handouts:� Lesson Planning Process Chart (5C-1)

Section CColored PencilsTransparencies:� Colored Pencils (5C-2)

Handouts:� Colored Pencils (5C-2)


Section CMy FlowersTransparencies:� My Flowers (5C-3)


Handouts:� My Flowers (5C-3)

Section CA Remainder of One� Counters or color tiles� A Remainder of One by Elinor J. Pinczes

Transparencies:� A Remainder of One (5C-4)

Handouts:� A Remainder of One (5C-4)

Section CLeftovers� Dice, 1 per pair of players� Color tiles� 6 small paper plates per pair of players

Transparencies:� Leftovers (5C-5)

Handouts:� Leftovers (5C-5)

Section CLet’s Paint!Transparencies:� Let’s Paint! (5C-6)

Handouts:� Let’s Paint! (5C-6)

Section CHow Long? How Many?� Cuisenaire rods� Dice, 1 per pair of players

Transparencies:� How Long? How Many? (5C-7)� How Long? How Many? Recording

Sheet (5C-8)

Handouts:� How Long? How Many? (5C-7)� How Long? How Many? Recording

Sheet (5C-8)

Section C4 In a Row� Counters, a different color for each player� Paper clips, 2 per pair of players

Transparencies:� 4 In a Row (5C-9)� 4 In a Row Game Board (5C-10)

Handouts:� 4 In a Row (5C-9)� 4 In a Row Game Board (5C-10)


Section CGo Figure!� Color tiles� Base-ten blocks

Transparencies:� Go Figure! (5C-11)

� Graph paper

Handouts:� Go Figure! (5C-11)

Section CThe Greatest Product Wins� Dice, 4 per group of players

Transparencies:� The Greatest Product Wins (5C-12)

Handouts:� The Greatest Product Wins (5C-12)

Section CGoing Bananas!Transparencies:� Going Bananas! (5C-13)

Handout:� Going Bananas! (5C-13)

Section CMultiple Towers� Linking cubes

Transparencies:� Multiple Towers Activity Sheet (5C-14)� Hundred Chart (5C-15), enlarged

Handouts:� Multiple Towers Activity Sheet (5C-14)� Hundred Chart (5C-15), enlarged

Section CFresh ProduceTransparencies:� Fresh Produce (5C-16)

Handouts:� Fresh Produce (5C-16)

Section CFresh Produce ChallengeTransparencies:� Fresh Produce Challenge (5C-17)

Handouts:� Fresh Produce Challenge (5C-17)

Section CWhat’s In Each Box?� Calculators

Transparencies:� What’s In Each Box? (5C-18)

Handouts:� What’s In Each Box? (5C-18)

Section CTiffany’s Beanie Babies� Linking cubes

Transparencies:� Tiffany’s Beanie Babies (5C-19)

� Graph paper

Handouts:� Tiffany’s Beanie Babies (5C-19)


Section CMarissa’s Garden� Color tiles� Scissors

Transparencies:� Marissa’s Garden Activity Sheet (5C-20)� Marissa’s Garden Recording Sheet (5C-

21)� Centimeter graph paper (5C-22)� Number Arrays (5C-23)

� Chart paper or construction paper� Markers� Tape

Handouts:� Marissa’s Garden Activity Sheet (5C-20)� Marissa’s Garden Recording Sheet (5C-

21)� Centimeter graph paper (5C-22)� Number Arrays (5C-23)

Section CThe Greatest Product, Part 2� Calculators (optional)

Transparencies:� Greatest Product, Part 2 Recording Sheet

(5C-24)

Handouts:� Greatest Product, Part 2 Recording Sheet

(5C-24)

Section CMarissa’s Garden AgainTransparencies:� Marissa’s Garden Again (5C-25)� Sixes Chart (5C-26)� Sixes Chart Key (5C-27)

Handouts:� Marissa’s Garden Again (5C-25)� Sixes Chart (5C-26)

Section DClosure for Day Five: Conceptual Models of Computational FluencyTransparencies:� 3 Sample Models of Computational

Fluency (5D-1 through 5D-3)

� Chart paper� Markers� Blank transparencies� Overhead pens

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section AFair Share Page 6

Fair Share

Key Questions:� How are multiplication and division related? What does division mean?� Can you explain the “standard algorithm” for division conceptually?

Frame:Division is a difficult operation for many students. We insist that students know theirmultiplication facts in order to divide but is that really what they need to know or all theyneed to know in order to develop computational fluency with division?

Materials:� Base-ten blocks, color tiles, pattern blocks (any set of materials that have multiple

colors or shapes)� Chart paper� Markers� Blank transparencies and overhead pens

Transparencies/Handouts:� Multiples of 4 (5A-1)� Interpreting Quotients (5A-2)� 3 Types of Division Problems (5A-3)� Using Multiples to Divide (5A-4)� Day 5A Reflection Prompt (5A-5)


Procedures Notes1. Have each group get a set of base ten

blocks, color tiles, or pattern blocks.Instruct them that everyone in the groupneeds the same number of each type (orcolor) of block.

How did you distribute your manipulativesevenly among the members of the group?Responses will vary, but most will say thateach person took a particular number ofblocks and continued in this fashion until noblocks were left, or there were not enoughblocks for everyone to have another one.Probably no group used the standarddivision algorithm!

Did any group have no leftovers? If you hadno leftovers, what does that mean about thenumber of blocks you started with? Thenumber of blocks is divisible by the numberof members of the group; also, the totalnumber of blocks is a multiple of the numberof people in the group.

For those groups that have leftovers, what dothe leftovers represent and what does that tellyou about the number of blocks? Theleftovers represent the remainder when thenumber of blocks is divided evenly among thepeople in the group. The number of blocks inthe remainder is less than the number ofpeople in the group. Having a remaindermeans that the number of blocks is not evenlydivisible by the number of people in thegroup; also, the total number of blocks is thesum of some multiple of the number of peoplein the group and the number of blocks leftover.

2. Say to participants, “We are going to usesome of the ideas related tomultiplication and what you just did insharing your manipulatives to developthe concept of division. Let’s start bythinking of 4 as our divisor.” Draw acircle on chart paper. At the top of thechart paper write the title “Multiples of 4(aka Friends of 4). Ask participants tosuggest numbers to go on the chart, bothin the circle and outside the circle.(Transparency 5A-1)

Knowing the multiples of a number isn’t thesame as knowing the multiplication facts.

What are multiples?

Why are multiples an important conceptrelated to multiplication?


3. Have groups develop a circle chart for themultiples of each number, 2 through 12,and post the charts around the room.

During this activity, tell participants that ifthey know a “friend” of one of the numbersposted around the room that is notrepresented to please add it to the appropriatecircle.

4. On a blank transparency, write thenumber 28. Lead participants to write amathematical sentence for twenty-eightin terms of multiples of 4.

Since we know that 28 is a multiple of 4, wecould write 28 = 28.

How many different ways can we write 28 asa multiple of 4?

Some suggestions might be:

28 = 24 + 4

28 = 20 + 8

28 = 12 + 12 + 4

28 = 4 + 4 + 4 + 4 + 4 + 4 + 4

28 = 32 – 4

5. Remind the participants that themathematical sentence can involvesubtraction.

Note, in reading, teachers use a term calledchunking. Chunking is when you break aword down. The process of breakingnumbers down has been described aschunking with numbers.

6. Repeat the process of writingmathematical sentences involvingmultiples of 4 for numbers that are notmultiples of 4, such as 35. Have differentgroups present their work.

Some sentences might be

35 = 32 + 3

35 = 8 + 8 + 8 + 8 + 3

35 = 28 + 4 + 4 – 1

35 = 36 – 1


7. Ask participants to design three ways todescribe 41 in terms of multiples of three,four, five, six, seven, eight, or nine.Assign a different multiple to each groupto present.

When teachers start division very early withchildren, the beginning learners don’t have torecord with numerals. Primary teachers canteach Fair Share with kids using patternblocks, linking cubes, or other concreteobjects. Students can record division withpictures or diagrams.

Note: to this point we have usedmultiplication, addition, and subtraction todescribe numbers in terms of a possibledivisor; we have not done the division andactually determined the quotient yet.Participants may want to go back to themathematical sentences done so far and try todetermine how the quotient could bedetermined from these non-division numbersentences.

8. After discussing the problem onTransparency 5A-2, ask participants whatwe should do with remainders? Itdepends on what is being divided(Transparency 5A-3). Have participantsbrainstorm different real world contextsin which the remainder can be shared andshould be written as a fraction as part ofthe quotient and contexts in which theremainder cannot be shared and doesn’tbecome part of the quotient. Challengethem to write problems where theremainder is dropped, where it isnecessary to round up regardless of theremainder and where the remaindershould be expressed as a fraction. Havegroups share their favorites.

Consider the problem on Transparency 5A-2:“Eleven people are going on a field trip. Acar holds four people. How many cars willthey need?”

a) 2

b) 2 34

c) 3

This problem emphasizes that we have tothink about context. The answer is 3, but

many students will answer 2 and 34

because

they are just thinking about the numbers and

not the fact that 34

of a car is not reasonable.

Remember, in math, numbers are adjectives.Students need to know what noun thequotient is describing.

9. Use base-ten blocks to model $49 on theoverhead. Ask four participants to cometo the front and share the “money” fairly.Record what the participants did on atransparency. (Transparency 5A-4 showsone possible approach to using multiplesof 4 to divide by 4.)

Many students will each take a long(representing $10), then each take a smallcube and then a second small cube. Becausewe are dealing with money, the last cube canbe broken up, and each person will receive aquarter of the dollar represented by the smallcube.


10. Ask participants to share $49 fairlyamong 4 people in a different way.Record on the overhead.

11. Have participants describe different waysto share 37 cookies fairly among threepeople and discuss the remainder.

Examples: 3 people sharing (30 + 6 + 1)cookies means 10 + 2 = 12 cookies for eachperson with 1 cookie left over or 3 peoplesharing ( 9 + 9 + 9 + 9 + 1) cookies means 3+ 3 + 3 + 3 = 12 cookies per person with 1cookie left over, etc.

How do you think we should represent theremainder? As a fraction part of the quotientbecause the extra cookie can be broken apartand shared also.

What if we were sharing trading cards, howwould we represent the extra? As aremainder that is not part of the quotient,because no one wants part of a card.

12. Have participants try one more: 79somethings divided among 4 kids.

What could the “somethings” be? Remindparticipants to express the remainder basedon what is being shared.

13. In upper level mathematics, quantities areoften described with rational numbers.Have participants explore the use ofrational numbers in fraction form to

describe the quotient 79 ÷ 4, or 794

.

Give the definition of rational number here.

A number written as a

b where a and b are

integers and 0≠b .

Some possibilities include:

794

404

324

44

34

= + + +

794

10 8 134

= + + +

794

1934

=

or

794

804

14

= −

794

2014

= −

794

1934

=


14. Have participants use the sametechniques to explore dividing a threedigit number by a one digit number, forexample

148 divided by 4.

If students have had lots of practice withtwo-digit numbers divided by one digitnumbers, then dividing a 3-digit number by aone-digit number is a natural extension.Example: 148 (when expressed as multiplesof 4) = 100 + 40 + 8, so

1484

= 100

4+

404

+ 84

= 25 + 10 + 2 = 37.

15. Have participants compare this approachto other ways of teaching the divisionalgorithm.

Use base-ten blocks to illustrate how thedivision algorithm is often taught withoutmeaningful language. “We start by saying 4doesn’t go into one, then we say how manytimes does 4 go into fourteen? We spend somuch time emphasizing place value, yet ourown language does not reinforce place value.One hundred forty becomes fourteen in thismodel! Are we misrepresenting numbers?”

16. Have participants try one final problembefore moving on to division by two-digit numbers:

Divide 397 by 5.

Ask participants to do this problem ontransparencies. Encourage the participants,especially the fifthgrade teachers, to userational numbers in a variety of ways.Example:

3975

=400

5 –

35

= 80 – 35

= 79 25

or

3975

= 300

5+

505

+ 405

+ 55

+ 25

= 60 + 10 + 8 + 1 + 25

= 79 25

.

17. Have participants use the multipleapproach to divide 296 by 13.

In order to divide three-digit numbers bytwo-digit numbers, students should knowhow to double and multiply by 10.

Discuss different strategies such as breaking296 into reasonable chunks related tomultiples of 13 such as

296 = 260 + 26 + 10.


18. Have participants practice with anotherproblem:

779 divided by 17

Use doubling and doubling again withmultiplying by 10 to chunk this number.Doubling 17 gives 34 and doubling 34 gives68.

779 = 680 + 68 + 31 = 40(17) + 4(17) + 17 +14.

So 77917

= 40 + 4 + 1 + 1417

= 45 1417

.

19. Solicit discussion about how theseactivities can lead to a general algorithmfor division.

Participants should explain their thinking.The mathematical definition of the divisionalgorithm: dividend = quotient(divisor) +remainder, with the remainder greater than orequal to zero and less than the divisor.

The activity “How Many?” includesmultiplication and division story problemsfor the participants to solve.

Debriefing:How are multiplication and division related? How did this activity lead to a different way ofdoing division than is usually taught? How might thinking about division in terms of familiesof multiples lead to computational fluency with division? Can you describe alternativealgorithms for division?

Reflection and Connection:Young children naturally use multiplication to solve division problems. This activityacknowledges that approach and encourages it.

What kinds of things in this activity led you to discuss the meaning of division and themathematics behind the algorithms for division?

Acknowledgements:This activity was adapted from an activity by Rachel McAnallen found in the May/June2001, Volume XII, Number 5 of Wonderful Ideas with permission from Julie L. Wilder,Editor.

Wonderful Ideas235 McCullough Hill RoadMontpelier, VT 056021-800-92-IDEAS

Transparency 5A-1


Multiples of 4(aka Friends of 4)

13

101

200

280

800

Transparency 5A-2


Interpreting Quotients

Consider the problem:

Eleven people are going on a field trip. A carholds four people. How many cars will theyneed?

a) 2

b) 2 34

c) 3

Transparency 5A-3


3 Types of Division Problems

Write a problem where

a) the remainder is shared and expressed as afraction,

b) the remainder is rounded up, and

c) the remainder is dropped.

Put your favorite on the transparency provideto share with the group.

Transparency 5A-4


Using Multiples to Divide

4 people $40 + 4 + 4 + 1

$10 + 1 + 1 +14

=12.254 people $40 + 4 + 4 + 1

Transparency 5A-5



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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section BInvestigating Multiplication and Division Page 18

Investigating Multiplication and Division

Key Questions:� What are some of the key concepts involved in understanding multiplication?� What stages do children move through to internalize the concept of multiplication?� How can the concepts of multiplication be used to develop understanding of division?

Frame:Multiplication of whole numbers is a natural extension of repeated addition. However, thereare a variety of models for multiplication that extend the use of multiplication to fractionsand decimals. Understanding what multiplication means and the different situations in whichit applies is essential to operation sense. It is also important to understand the different wayschildren construct ideas about multiplication. Division and multiplication are inverseoperations. Fair Share emphasized this connection between the two operations. This activitywill further support those connections and point out levels of sophistication in usingmultiplication and computing products.

Materials:� Counters or color tiles� Base-ten blocks� Centimeter grid paper� Markers� Blank transparencies� Overhead pens� Bare Bear's New Clothes by Peter S. Seymour, 1986

Transparencies/Handouts:� How Many Cherries? (5B-1)� How Many Apples? (5B-2)� How Many Cookies? (5B-3)� How Many Outfits? (5B-4)� My Garden (5B-5)� 6 Problem Set Cards (5B-6)� Analysis Chart (5B-7)� Day 5B Reflection Prompt (5B-8)

Procedures Notes1. Show the bunches of cherries

(Transparency 5B-1). Ask participants tobrainstorm the different ways studentsmight find the number of cherries.

The discussion should include counting byones, skip counting, repeated addition, andgrouping/unitizing. Each of these is a moresophisticated approach than the previous.Unitizing is thinking of the group as a unit.Example: seven groups of four. The fourcherries have been unitized.


2. List the different approaches in somerandom order. Have participants orderthe techniques by level of difficulty andjustify their choices.

3. Demonstrate different approaches tofinding “how many.”

For each of the approaches, what does a childknow? What does the child not know?

Counting by ones: child “knows” one to onecorrespondence, probably does not knowskip counting or any type of grouping. Skipcounting: child recognizes patterns ofcherries and uses skip counting by twos orfours, probably does not think aboutrepeated addition or regrouping. Repeatedaddition: student sees groups of four andadds, may not think about each group as aunit. Grouping: student may see sevengroups, know that five groups of four istwenty and two groups of four is eight andconclude that there are twenty-eight cherries.This child is unitizing.

Are there other ways children might decide“how many” that we’ve not discussed?

4. Show the array of apples (Transparency5B-2). Ask participants to generate asmany ways as they can to determine howmany apples are displayed.

Ask groups to share ideas. Ideas shouldinclude

� counting by ones

� skip counting by threes

� nine groups of three

� five groups of three and four groupsof three

� three groups of nine

� two groups of nine and one moregroup of nine

� ten groups of three less one group ofthree


5. Use Transparency 5B-3 to present thefollowing problem as an example wheremultiplication can be used to replacerepeated addition: “Tony is camping withtwo friends. He wants to take cookies foreveryone. How many cookies does heneed to pack if he wants everyone to havesix cookies?” Have participants write anexpression for the situation.

What might be a common error in answeringthis question? Students might not count Tonywhen deciding how many groups of six toinclude. What kind of model formultiplication have we developed with thisstory? (A repeated addition model as well asan “equal groups” model.) This is anexample where students might write either6 + 6 + 6 or 3 x 6 and is an illustration ofmultiplication as the sum of equal-sizedgroups, or ___ groups of ____, which isimportant language to develop not only forwhole number multiplication, but also laterfor multiplication with fractions anddecimals.

6. Read Bare Bear’s New Clothes. UseTransparency 5B-4 to present thefollowing problem as an example ofmultiplication as a way to countcombinations: “If Bare Bear had threeshirts and two pair of pants, how manyoutfits could he make?”

Give participants time to solve this problem.Walk around the room and note those groupsthat used different approaches. Ask variousgroups to share their problem solvingapproach. What kind of model formultiplication have we developed with thisstory? A Cartesian Product model: countingthe number of ordered pairs made bymatching each element of one set with eachelement of the other set.)


7. Show Transparency 5B-5 or useoverhead base 10 materials while tellingthe following story as an example ofusing multiplication to find area:

My Vegetable GardenAlmost 20 years ago, we lived in a housewith a huge back yard. We decided toplant a garden. Not thinking about theconsequences, we tilled a huge gardenspot. Of course, it was much more thanwe could keep up with and when thevegetables came in, we had enough tofeed the entire community! Needless tosay, we didn’t have a garden the nextyear. A few years ago, I finally talkedmy husband into planting another garden.The first year it was 10 feet by 10 feet.The second year we added two feet toone side to plant radishes. The third yearwe added three feet to the other side toaccommodate green beans.

How big is our garden?

Have participants model the garden problemwith you using base 10 materials or drawingit on centimeter grid paper.

What kind of model for multiplication havewe developed with this story? An area model

How can this area model lead to an algorithmfor multiplication?

Year one10*10 = 100

Year two12*10 = (10 + 2)*10

= 10*10 + 2*10

= 100 + 20

Year three12*13 = (10 + 2)*(10 + 3)

= (10 + 2)*10 + (10 + 2)*3

= 10*10 +2*10 + 10*3 + 2*3

= 100 + 20 + 30 + 6

Compare this to the partial productsalgorithm and standard algorithm formultiplying two two-digit numbers:

13 13x12 x12

6 (2 x 3) 2620 (2 x 10) 13_30 (10 x 3) 156

100 (10 x 10)156


8. Distribute the six sets of multiplicationand division problems to the participants.Assign each group to work at least oneproblem from each set.

Introduce participants to the AnalysisCharts for Multiplication and Division.After working the problems, each groupof four will fill in the charts for theirparticular problems.

Because there are a variety of difficultylevels in each set, give groups time to readthrough the problems and decide whichproblem from each set each group will work.Make sure each problem is worked by atleast one group.

Set A – multiplication problems involvingrepeated addition or obvious groups of equalsize

Set B – division problems where the size ofeach group is known (where the quotientrepresents how many equal groups)

Set C – multiplication problems involvingcounting combinations

Set D – division problems where the numberof groups is known (where the quotientrepresents the size of each group)

Set E – multiplication problems involvingarea

Set F – division problems with a remainder(where a decision needs to be made aboutwhat to do with the remainder based on thecontext of the problem: (a) just use the wholenumber quotient and ignore the remainder,(b) continue to divide the remainder so that itbecomes part of the quotient by usingfractions or decimals, (c) or increase thequotient by one and drop the remainder.

Debriefing:What are the different meanings and models involved in developing understanding of theconcepts of multiplication and division? Describe the different levels of sophistication, anddiscuss how you as a classroom teacher can determine where your students are in theirdevelopment of understanding the meanings of and models for multiplication and division.


Reflection and Connection:There are many different types of models for multiplication: repeated addition, equal groups,arrays, area, and combinations. What is alike about all of these models that make them allable to be represented mathematically with multiplication?

How do we help students connect the ideas among different models of multiplication tobuild a complete understanding of multiplication and division?



How Many Cherries?

Brainstorm the different ways children mightdetermine how many cherries are pictured.



How Many Apples?

How many apples are shown?

Brainstorm the different ways children mightdetermine the number of apples.



How Many Cookies?

Tony is camping with two friends. He wantsto take cookies for everyone. How manycookies does he need to pack if he wantseveryone to have six cookies?



How Many Outfits?

If Bare Bear has three shirts and two pair ofpants, how many outfits can he make?



My Garden

The first year my garden is 10 feet by 10 feet.

The second year we added 2 feet to one side.



The third year we added 3 feet to the other side.



Problem Set A

1. Mary’s mom is making cookies for theclass. There are 15 children in the classand Mary wants everyone to have 2cookies. How many cookies shouldMary’s mom make?

2. Andrew is planting begonias for hismom. She wants 6 rows of 8 plants.How many plants will Andrew plant?

3. There are 21 children in Mrs. Adamsclass. Each child needs 3 spiralnotebooks. How many spirals are neededfor her class?

4. The Bluebonnet Girls are going to sellcookies to earn money for camp. If thereare 12 girls and each girl has to sell 20boxes of cookies, how many boxes ofcookies does the leader need to order?

5. In Harry Potter’s wizardry world, 1 goldGalleon is worth 17 silver Sickles and 1Sickle is worth 29 bronze Knuts. Howmany Knuts are 5 Galleons worth?

6. A flea can jump 350 times its bodylength. If humans could jump like fleas,how far could you jump?



Problem Set B

1. Sara’s mom made cookies for Sara’sclass. She made 36 cookies and plannedfor each child to have 2 cookies. Howmany children are in the class?

2. Mrs. Garza has 54 plants for her garden.If she puts 6 plants in each row, howmany rows will she need?

3. Ronnie has decided to give away his hotwheels. He has 81 cars and wants eachfriend to get 3 cars. How many friendscan he give cars to?

4. Mrs. O’Hara found 135 doodads in herattic. She will give each of hergrandchildren 15 doodads. How manygrandchildren does she have?

5. Northwest School PTO has purchased628 pencils. If they give 3 pencils to eachstudent, how many students does theschool have?

6. Mr. Surveda has 192 square feet ofpaving stones. He wants to make a patiothat is 12 feet wide. How long will thepatio be?



Problem Set C

1. Jamie is going to camp and can only takeone suitcase. She plans to take 4 shirtsand 3 pair of shorts. How many differentoutfits will she have for camp?

2. Mr. Quicksilver tosses a coin and rolls adie. How many different combinationscan he get?

3. Michael has 5 lizards and 4 frogs. If hewants to take one of each for show andtell, how many choices of pairs to takedoes he have?

4. Marsha has a shirt and skirt in each of 6different colors. If she never pairs a shirtand skirt of the same color, how manyoutfits does she have?

5. The menu has 2 appetizers, 5 entrees and3 desserts. How many different mealsare available if each meal has one itemfrom each category?

6. Bare Bear has 2 hats, 3 pairs of shoes, 4pairs of pants and 6 shirts. How manydifferent outfits can he make?



Problem Set D

1. Jose’s mom sent cookies to school. Thereare 18 children and she sent 36 cookies.How many cookies will each child get ifthey are shared evenly?

2. Mrs. Garza has 54 plants for her garden.If she wants 6 rows, how many plantswill she put in each row?

3. Ronnie has decided to give away his toyhot rods. He has 81 cars and wants togive them to 3 friends. How many carswill each friend receive if he shares themfairly?

4. Rebecca has a collection of panda bears.She has 4 empty shelves in her room. Ifshe has 128 bears and wants the samenumber on each shelf, how many will beon each shelf?

5. The library just received 144 new books.If they have 3 empty shelves and want toput the same number of books on eachshelf, how many books should be on eachshelf?

6. We have 288 Christmas ornaments. Ifwe pack the same amount in each of 6boxes, how many ornaments should gointo each box?



Problem Set E

1. Mica’s sandbox is 4 feet by 6 feet. Whatis the area of the sand box?

2. Mrs. May is planting 5 rose bushes in hergarden. She is putting them side by sidein front of her house. Each rose bush willneed 3 square feet of area. How muchspace does she need for her roses?

3. Mica’s sandbox is 4 feet by 6 feet. Whatis the area of the sand box? If thesandbox is 2 feet deep, what is thevolume of the sand box? Hint, build amodel with cubes.

4. Tiffany is redecorating her room. Herroom measures 14 feet by 15 feet. Howmany square feet of carpet will beneeded?

5. Mr. Branton is going to tile a 9 foot by 15foot kitchen, a 6 foot by 8 foot bathroomand a 16 foot by 23 foot family room.How many square feet of tile should hepurchase?

6. Tiffany is redecorating her room. Herroom measures 14 feet by 15 feet. IfTiffany has eight-foot ceilings (distancefrom floor to ceiling). How many squarefeet of wall (do not include the ceiling)will she need to paint?



Problem Set F

1. George is giving his trading cards to 5 ofhis friends. If he has 128 cards, howmany cards will each friend receive if hegives each one the same amount?

2. Tiffany’s room is 14 feet by 15 feet with8 foot ceilings (distance from floor toceiling). If a gallon of paint covers 250square feet and she wants to put 2 coatsof paint on the walls (not ceiling), howmuch paint should she buy?

3. Mica’s sandbox is 4 feet by 6 feet by 2feet deep. If sand comes in 5 cubic footbags, how many bags will be needed tofill the sandbox? (Hint, build a modelwith cubes.)

4. Ann, Bonnie, Janie, and Dinah workedtogether to clean up an elderly couple’syard. When they finished, the couplegave them $50. How much did each girlreceive if they shared the money fairly?

5. Tiffany is redecorating her room. Herroom measures 14 feet by 15 feet. Howmany square yards of carpet will beneeded?

6. Middletown School is taking the fifthgrade on a field trip. The school requires2 sponsors for each group of 25 students.If the fifth grade has 335 students, howmany sponsors are needed? Each buswill hold 40 people. How many bussesare needed for the field trip?



Analysis Chart

Step 1Big Ideas




Transparency 5B-8



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TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CMultiplication and Division Sampler Page 39

Multiplication and Division Sampler


experiences to develop students’ understanding of multiplication and division?

Frame:The participants will explore a sampler of learning experiences designed to support thedevelopment of the understanding of multiplication and division. Participants will examinethese learning experiences in terms of how the Lesson Planning Process is used to designeffective instruction to develop understanding of multiplication and division. As participantsmove through the sampler activities, the increasing levels of sophistication needed to solvethe problems should become apparent.

Transparencies/Handouts:� Lesson Planning Process Chart (5C-1)� Day 5C Reflection Prompt (5C- 28)

Procedures Notes1. Hand out the Lesson Planning Process

Chart for the Multiplication and DivisionSampler. Have participants review thefour components in the Lesson PlanningProcess.

2. Introduce the Sampler of LearningExperiences for Multiplication andDivision and explain to participants thatthey will be using the Lesson PlanningProcess Chart to identify the fourcomponents of the Lesson PlanningProcess in relation to each learningexperience.

3. If you think the group needs the practice,have the entire group do Marissa’sGarden together. Then, have participantsfill out the Lesson Planning ProcessChart for Marissa’s Garden. Spend sometime discussing all of the objectives thatcould be the focus for this activity.

If participants are comfortable with theLesson Planning Process Chart and its usewith the activities, have them proceed ontheir own with selected learning experiencesin the sampler.


4. Allow participants time to explore therest of the activities in the Sampler.Have them fill out the Lesson PlanningProcess Chart as they complete eachactivity.

There are several ways to deliver theseactivities including the way Day 3 wasorganized.




Guide participants to create a combinedLesson Planning Process Chart formultiplication and division, identifyingoverall Big Ideas and Evidence ofUnderstanding for this topic. (See Debriefingsection.) Be sure the notion of levels ofsophistication come out in the discussions.

Debriefing:Conceptual development and fact fluency must be developed side-by-side. While conceptualdevelopment alone will allow students to solve problems, fact fluency makes problemsolving more efficient. Fact fluency alone allows students to compute quickly, but doesn’talways provide students with the problem-solving skills they need to be successful in highermathematics. Have participants review the activities they have completed, thinking about thefollowing questions: Which activities promote both concept development and facility withfacts? What needs to be added to the activities to provide a balanced approach? Whichactivities help develop conceptual understanding of multiplication?

To the facilitator: The following list provides an example of a list of combined Evidences ofUnderstanding that might be created for Multiplication and Division.




� Applying meanings and properties ofmultiplication/division

� I can use objects to showmultiplication/division.

� I can draw pictures to showmultiplication/division.

� I can talk about why I chosemultiplication/division.

� I can multiply in any order(commutativity).

� I can multiply any two numbers first(associativity).

� I can read a multiplication/divisionnumber sentence.

� Using procedures for finding products/quotients

� I can skip count to find a product orquotient.

� I can use equal groups to find a productor quotient.

� I can use factors and multiples to findproducts/quotients.

� I can use patterns to findproducts/quotients.

� I can use basic facts to findproducts/quotients.

� I can write a number sentence to describethe product/quotient.

� I can pick a reasonable number for theproduct/quotient.

� I can estimate the product/quotient.� I can use tens and ones to find a

product/quotient.� I can write about the different ways I

found the product/quotient.� Connecting multiplication and division � I can check my division with

multiplication.� I can use multiplication/division fact

families to solve problems.


Reflection and Connection:The learning experiences in this section allow you to examine important components of theLesson Planning Process. The sequence of lessons includes a variety of difficulty levels tohelp children move to different levels of sophistication required for a complete understandingof multiplication and division.

How will you use the ideas related to levels of difficulty and sophistication inunderstanding multiplication and division in the design of your own instruction?




Step 1Big Ideas




Transparency 5C-28



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ttoo ll eevvee ll ss oo ff dd ii ff ff ii ccuu ll tt yy aanndd

ss oo pp hh ii ss tt ii cc aa tt ii oo nn ii nn uu nn dd ee rr ss tt aa nn dd ii nn gg

mm uu ll tt ii pp ll ii cc aa tt ii oo nn aa nn dd dd ii vv ii ss ii oo nn ii nn tt hh ee

ddeess ii ggnn oo ff yyoouurr oowwnn iinnss tt rruucc tt ii oonn??

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CColored Pencils Page 48

Colored Pencils

Words for the Word Wall:� total� groups of fives� groups of tens

Transparency/Handout:� Colored Pencils (5C-2)

Procedures Notes1. Present the problem to students and have

them brainstorm ways they woulddetermine the answer to the question.

Possible solution processes include gettingout real pencils and counting them, countingthe individual pencils in the picture, countingthe groups of pencils to see if they are allalike and then counting by fives, puttinggroups of fives together and counting bytens.

2. Have students discuss how their solutionsare alike and different.

Discuss the accuracy and efficiency ofcounting by groups compared to counting byones; counting by tens compared to countingby fives, etc.

Discussion:What are some of the ways that you determined the total number of pencils? (This activity isintended for the early grades. Because the colored pencils are arranged in groups of five,skip counting or grouping by tens is the intended student behavior.)


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CColored Pencils Page 49

Colored Pencils

Mrs. Garza will have 16 children in her first-grade class. Each child will have a set of 5colored pencils. How many colored pencilswill there be?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CMy Flowers Page 50

My Flowers

Words for the Word Wall:� total� groups

Materials:� Chart paper� Markers

Transparency/Handout:� My Flowers (5C-3)



Possible solution processes include countingthe individual flowers in the picture,counting the individual petals, counting theflowers by threes, counting the petals byfives, etc.


Discuss the accuracy and efficiency ofcounting by groups compared to counting byones.

3. Discuss ways to record the varioussolutions.

Students who counted petals by ones couldshow tallies in groups of 5 that could thenlead them to counting by fives. Studentswho counted by threes or fives could recordthe skip counting with word sentences andrepeated addition number sentences: 3 and 3is 6, 6 and 3 is 9; 3 + 3 + 3 = 9. Studentswho count the groups and think 3 fives are15 can write multiplication word and numbersentences: 3 groups of 5 are 15; 3 x 5 = 15.

Discussion:What are some of the ways that you determined the total number of flowers? (Because theflowers are in groups of threes, skip counting is the intended student behavior. Idea fromNCTM, Teaching Children Mathematics, April 2002, p. 464.)


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CMy Flowers Page 51

My Flowers

Let’s draw a flower. On a large sheet of paper draw a longline. This will be the stem of your flower. Now put yourhand at the top of the stem and spread your fingers. Yourhand is the flower and your fingers are the petals. Tracearound your hand.

If you drew three flowers on your stem, how many petalswould there be?

Your friend draws flowers also. If his stem has three flowerson it also, how many petals are on both of your stems?

Draw a flower garden. How many stems did you draw? Howmany flowers are in the whole garden? How many petals arein the whole garden?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CA Remainder of One Page 52

A Remainder of One

Words for the Word Wall:� remainder� column� row

Materials:� Counters or color tiles� A Remainder of One by Elinor J. Pinczes

Transparency/Handout:� A Remainder of One (5C-4)

Procedures Notes1. Read the book A Remainder of One by

Elinor J. Pinczes.

2. Present the problem to students and havethem brainstorm ways they woulddetermine the answer to the questions.

Problem solutions include drawing ants incolumns of two, three, four and five, usingcounters or color tiles to model the ants incolumns.


As students discuss their thinking, model theuse of words and numbers to record theirthinking. For example, the reason there isone ant left over when 25 ants are put in rowsof twos is because you can make 12 groupsof 2 out of 24, which leaves 1, or (2 + 2 + 2 +2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2) + 1 = 25,or 25 = (12 groups of 2) + 1 or 25 – 1 = 12 x2, etc.

Discussion:

� What are some of the ways you determined when the ants would be in even columnswith no leftovers?

� Did you have to build all the other arrangements to get your next answer?

� How can you record your thinking? (This activity is intended for the early grades. Itis a precursor to division, but notice the word division is never used. So recordingsolutions will most likely involve addition and subtraction, and perhaps multiplicationif it has been introduced.)


� What patterns do you see in the numbers that have 1 leftover for rows of 2s, 3s, and4s, but no leftovers for rows of 5? (They are multiples of 5; they are not even; thenumber before them is always a multiple of 2, 3, and 4; etc.)



A Remainder of One

In the story A Remainder of One by Elinor J. Pinczes, 25marching ants lined up in rows of 2, then 3, and then 4, buteach time there was a remainder of 1 ant. When they lined upin rows of 5, the ants were perfectly arranged in columns ofthe same length. What is the next number of ants for whichthese same arrangements are true?

The number after that?

A pattern for all such numbers?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CLeftovers Page 55

Leftovers

Words for the Word Wall:� remainder� division sentence

Materials:� One die� Color tiles� 6 small paper plates

Transparency/Handout:� Leftovers (5C-5)

Procedures Notes1. Use the transparency (5C-5) to explain

the game.It might be necessary to play a sample gameso the students will understand exactly howthey will accumulate tiles.

2. Ask, “What strategy did you use whenselecting the number of tiles to startwith?”

A good strategy is to start with a number thatwill not sort evenly among two, three, four,five or six plates. Lead students tounderstand that this means they are lookingfor a number that is not a multiple of 2, 3, 4,5, or 6.

3. Explain what the division sentence meansin terms of the number of tiles the playergets to keep.

The player keeps the tiles that represent theremainder in each division situation.

Discussion:Ask question such as the following to uncover relationships between numbers and theirfactors:

� How did you select the number of tiles to start with at the beginning of the game?

� How did you select the number of tiles to start with after you had played for awhile?

� If the tiles you pick can be sorted on two plates with no leftovers, can they be sortedon three plates with no leftovers? (It depends, a number that has a factor of 2 doesn’tnecessarily also have a factor of 3, but it might.)


� If the tiles you pick have leftovers when they’re sorted on two plates, can they besorted on four plates with no leftovers? (No, there will always be leftovers with fourif there are leftovers with two because a number that doesn’t have a factor of twocannot have a factor of 4.)

� If the tiles you pick have leftovers when sorted on four plates, can they be sorted ontwo plates with no leftovers? (Maybe, a number that has a factor of 2 might NOThave a factor of 4, e.g. 6.)



Leftovers

You need: one die, color tiles, 6 small paper plates, paper/pencilObject: To be the player with the most tiles at the end of the game.

� Start with a hand full of tiles.

� Take turns. On your turn, roll the die, take that numberof paper plates and divide the tiles among them. Keepany leftover tiles. This becomes your score for the round.

� Both players record the division sentence that describeswhat happened.

For example: 15 ÷ 4 = 3 R 3

In front of each division sentence, write the initial of theperson who rolled the die.

� The next player uses the tiles remaining on the plates forhis/her turn.

� Play until all tiles are gone. Figure your scores. Playagain.

� Which number(s) of tiles do you think is (are) best tobegin with? Why? Write about it.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CLet’s Paint! Page 58

Let’s Paint!

Words for the Word Wall:� distributive property� factor

Transparency/Handout:� Let’s Paint! (5C-6)




This activity highlights multiplication of aone-digit factor by a two-digit factor, 7 x 13and 2 x 13. Grouping is an intended approachleading to the development of the use of thedistributive property of multiplication overaddition. Lead students to think about 13 as10 + 3 and 7 x 13 as 7 x (10 + 3), or (7 x 10)+ (7 x 3).

Discussion:

� What are some of the ways that you determined the total number of paintbrushes?The number of wells of paint?

� Can you find the totals in more than one way? Explain.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CLet’s Paint! Page 59

Let’s Paint!

How many individual wells of paint are in the picture?

How many paintbrushes are in the picture?

Explain your methods of computation. Be prepared to present

your ideas.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CHow Long? How Many? Page 60

How Long? How Many?

Words for the Word Wall:� rectangle� multiplication sentence

Materials:� Cuisenaire rods� One die per pair of students

Transparencies/Handouts:� How Long? How Many? (5C-7)� How Long? How Many? Recording Sheet (5C-8)

Procedures Notes1. Use Transparency 5C-7 to explain the

game.It might be necessary to play a sample gameso the students will understand exactly howto record their moves.

2. After students have played the gameseveral times ask, “What strategy did youuse to cover the most squares?”

Strategies might include making all longskinny rectangles so that they can fit next toeach other on the graph paper or placing yourrectangles around the edges of the paper first,etc.

Discussion:Ask questions such as the following to help students connect multiplication to area:

� What was the greatest area you could make in the game? How many differentrectangles could you make with that area?

� What was the least area you could make? How many different rectangles could youmake with that area?

� Were there any areas you couldn’t make in the game?

� How could knowing which areas you can make and which you can’t help you playthe game?

� How can you use your multiplication equations in the rectangles to help you knowhow much area you’ve used and how much you have left?



How Long? How Many?

You need: Cuisenaire Rods, One die, How Long? How Many?Recording Sheet

1. Each player uses a different 10-by-10 grid on therecording sheet.

2. On your turn, roll the die twice to find out whichCuisenaire rods to take. The first roll tells “how long” arod to use. The second roll tells “how many” rods totake.

3. Arrange the rods into a rectangle. Trace it on your grid.Write the multiplication sentence that describes the areaof the rectangle inside the rectangle.

4. The game is over when one of you cannot place yourrectangle on the grid because there’s not enough room.

5. Figure out how many of your squares are covered andhow many are uncovered. Check each other’s answers.The person with the most squares covered wins.

6. Play again.



How Long? How Many? Recording Sheet

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section C4 In a Row Page 63

4 In a Row

Words for the Word Wall:� factors� multiples

Materials:� 2 paper clips per pair of players� Counters in two different colors

Transparencies/Handouts:� 4 In a Row (5C-9)� 4 In a Row Game Board (5C-10)


game.It might be necessary to play a sample gameso the students will understand the rules forselecting factors.

2. After students have played the gameseveral times ask, “What strategy did youuse to get 4 in a row?”

3. Have students discuss how theirstrategies are alike and different.

Discussion:Ask questions such as the following to help students explore ideas of factors and multiples:

� If one of the paper clips is on 1, where can you put the other paper clip to create aproduct on the board?

� How many products are on the board that you can get with a paper clip on 5? On 9?

� How does knowing which multiples are on the board for each factor help you planyour game?



4 In a Row

You need: game board, 2 paper clips, different counters for eachplayer, group of 2Objective: To get 4 markers in a row horizontally, vertically, ordiagonally.

� First player places 2 paper clips on factors at the bottomof the game board, multiplies the numbers, and places acounter on the product on the game board. (Two paperclips may be on the same factor.)

� The second player moves only one of the paper clips to anew factor, multiplies the numbers, and places a counteron that product on the game board.

� Continue playing until one player has four counters in arow.

� Play at least 3 games.

� Write about the strategies that helped you win.



4 In a Row Game Board

1 2 3 4 5 6

7 8 9 10 12 14

15 16 18 20 21 24

25 27 28 30 32 35

36 40 42 45 48 49

54 56 63 64 72 81

1 2 3 4 5 6 7 8 9

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CGo Figure! Page 66

Go Figure!

Words for the Word Wall:� distributive property

Materials:� Graph paper� Base-ten blocks� Color tiles

Transparency/Handout:� Go Figure! (5C-11)



This activity is intended to lead to an areamodel that can be generalized to analgorithm.

2. If the groups do not develop it be sure todebrief this one.

Use the distributive property multiple times.

17 X 17 = (10 + 7) X (10 + 7)= 10(10 + 7) + 7(10 + 7)=10 X 10 + 10 X 7 + 7 X 10 + 7 X 7= 100 + 70 + 70 + 49= 289

Discussion:

� How would you help clarify Monica’s thinking?

� How could this analysis lead to an algorithm (procedure) for multiplying two- andthree-digit numbers?


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CGo Figure! Page 67

Go Figure!

You need: Graph paper, color tiles, base-ten blocks

Students have been working on multiplication by breakingapart the larger number. For example, 8 x 12 = 8 x (10 + 2) =8 x 10 + 8 x 2 = 80 + 16 = 96. Monica wants to multiply 17 x17 and asks her partner if both numbers can be broken apart.They decide to experiment. Monica knows that 17 = 10 + 7,so 17 x 17 = (10 + 7) x (10 + 7). She reasons that this is10 x 10 + 7 x 7 = 100 + 49 = 149.

Analyze Monica’s reasoning. Is she correct? If so, help herjustify her results; if not, help her clarify her thinking.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CThe Greatest Product Wins Page 68

The Greatest Product Wins

Words for the Word Wall:� factor� product

Materials:� 4 dice

Transparency/Handout:� The Greatest Product Wins (5C-12)


game.It might be necessary to play a sample gameso the students will understand exactly howthey will place the digits.

2. Ask, “What strategy did you use whendeciding where to put the differentdigits?”

This activity requires participants to analyzeplacement of digits in a multiplicationproblem to obtain the greatest product.Interestingly, in multiplying a 3-digit factorby a one-digit factor, the largest product iscreated by having the greatest of the fourdigits be the one-digit factor and the secondlargest digit be in the hundreds place of thethree-digit factor.

Discussion:

� What strategy did you use to place the digits? Did it work?

� How do you know you’ve created the greatest product?

� What pattern do you think will always work? (If the digits are a, b, c, and d such that� a < b< c< d, then d x cba gives the greatest product.)

� Why do you think the greatest digit doesn’t work in the hundreds place? (Anexample can illustrate the reason for this. Suppose the digits are 2, 3, 4, and 5. Then4 x 532 = 4 x 500 + 4 x 30 + 4 x 2. But, 5 x 432 = 5 x 400 + 5 x 30 + 5 x 2. Theproduct in the hundreds place ends up being the same in both problems (4 x 500 = 5 x400), but the rest of the partial products end up being greater when the one-digitfactor is greater: 4 x 30 < 5 x 30 and 4 x 2 < 5 x 2.)


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CThe Greatest Product Wins Page 69

The Greatest Product Wins

You need: 4 dice and a partner or group of 4Object: The object of the game is to get the greatest possibleproduct.

� Take turns rolling 4 dice.

� Use the four numbers to make a multiplication problemwith a 3-digit factor and a 1-digit factor.

� The winner of the round is the person who gets thegreatest product.

� Play 5 rounds.

Given any 4 numbers, develop a strategy that will always giveyou the greatest product. Write about your strategy.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CGoing Bananas! Page 70

Going Bananas!

Words for the Word Wall:� groups� multiples

Transparency/Handout:� Going Bananas! (5C-13)




There are five groups of six bunches. Note,each bunch has three bananas. The importantthing here is that you cannot tell how manyindividual bananas there are in the picture bycounting the bananas that are showing. If youcount only the bananas seen, each group has13 bananas!

Discussion:

� Since you can’t actually see all of the bananas, how can you prove the total you got isthe correct one?

� Would you have to be able to see ANY of the bananas to determine how many thereare in all? What information would you need to know?


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CGoing Bananas! Page 71

Going Bananas!

The bananas are in bunches of three. Thereare six bunches of bananas on each of fiveshelves.

How many bananas are there in all?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CMultiple Towers Page 72

Multiple Towers

Words for the Word Wall:� multiple� factor� prime

Materials:� Linking cubes

Transparencies/Handouts:� Multiple Towers Activity Sheet (5C-14)� Hundred Chart (5C-15), enlarged

Procedures Notes1. Have the students build towers of cubes

on a hundred chart. Explain that thesetowers will be built using the directionson the Multiple Towers Activity Sheet(5C-14).

A variation (if you do not have coloredlinking cubes) is to give students HundredCharts on transparencies and have them coloreach multiple on a different chart. After allmultiples are colored, then overlaytransparencies to see which numbers havemore and which have fewer colored insquares. (For those numbers with three ormore colors, the total effect will be brown.Children will need to unstack thetransparencies and use another HundredChart to record the multiples.)

2. Discuss the first step: Put a blue cube oneach multiple of 2.

Ask, “How will you decide on whichnumbers to put the cubes? Who can explainwhat we mean by a multiple? Can you giveother examples?” Then place the cubes onthe appropriate numbers.

3. Let partners work together following thedirections on the Activity Sheet andanswering the questions.

Circulate around the room, discussing withpairs of students these questions:

� What is your strategy for findingmultiples?

� Can you identify a pattern to help you?� Do you think every number will have a

tower?

4. As pairs finish their towers andquestions, ask them to design a recordingsystem to show which cubes ended up oneach number, then record their findings.

When students build the towers, the numbers2, 3, 5, and 7 (primes on top row) will haveone cube on them. All other primes will nothave a cube.


5. When all groups have completed theactivity and have recorded their results,pose these questions to the class:

Some students will want to include thenumber one in the set of primes. Marissa’sGarden activity should help clarify that error.

� How did you decide where to put eachcolor of cube? Skip counting, etc

� Which numbers had the tallest towers?Why? These numbers have the mostfactors. Which colors were in them?

� Which numbers had no cubes? Why?They are prime numbers.

� Which numbers had only one cube? 2, 3,5, 7 Why?

� Did you notice that when you put a blackcube on a number, it already had a greencube on it? Why do you suppose thathappened? If a number is a multiple of10, it is also a multiple of 5.

� What happened when you removed thetowers and tried to put them back on thecorrect numbers? What did you do to tryto figure out where they belonged? Didanyone try anything different? How did itwork?

� Did you discover anything else from theactivity that we have not alreadydiscussed?

6. Ask, “How many of each color linkingcube will be needed? If you use divisionto decide, what do you do with theremainder?”

For this activity you will need the followingfor each group:

50 blue 14 white

33 green 12 orange

25 yellow 11 black

20 red 10 maroon (pink)

16 brown

The remainder is dropped.



Multiple Towers Activity Sheet

You need: Linking cubes, Hundred Chart, a partner

Put a blue cube on each multiple of 2.Put a green cube on each multiple of 3.Put a yellow cube on each multiple of 4.Put a red cube on each multiple of 5.Put a brown cube on each multiple of 6.Put a white cube on each multiple of 7.Put an orange cube on each multiple of 8.Put a black cube on each multiple of 9.Put a maroon (pink) cube on each multiple of 10.

� Which numbers have the tallest towers?

� Which numbers have no cubes?

� Which have only one cube?

� Now remove the towers. (Be careful not to break any.)Try to put them back on the correct numbers. Writeabout how you did this.



Hundred Chart

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CFresh Produce Page 76

Fresh Produce

Transparency/Handout:� Fresh Produce (5C-16)



Guess and check is the most frequentapproach used on this problem.


This problem could require all four of thebasic operations. Solution:

1 – lb. lettuce

1 – lb. spinach

2 – lb. eggplant

Discussion:

� How many pounds of produce did the customer first select? Six pounds How do youknow?

� How many pounds of produce did the customer finally purchase? Four pounds Howdo you know?


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CFresh Produce Page 77

Fresh Produce

A farmer at a local market sells much of his produce for $2.00per pound. A customer first selects eggplant, spinach, and redleaf lettuce, for a total of $12.00. This indecisive customerthen asks to leave off the lettuce, for a total of $10.00. Thenthe customer finally puts the lettuce back in and removes halfof the eggplant and pays a total of $8.00. How many poundsof each vegetable did the customer take home?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CFresh Produce Challenge Page 78

Fresh Produce Challenge

Transparency/Handout:� Fresh Produce Challenge (5C-17)



Guess and check is the most frequentapproach used on this problem.


This problem involves decimals, but can beworked without using decimal operations byusing concrete materials. Solution:

1 lb. of lettuce

2 lb. of spinach

2.5 lb. of eggplant

Discussion:If the final cost is $8.25, what does that tell you about the number of pounds of producepurchased? There is a partial part of a pound; one-half pound would cost 75¢.


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CFresh Produce Challenge Page 79

Fresh Produce Challenge

A farmer at a local market sells much of his produce for $1.50per pound. A customer first selects eggplant, spinach, and redleaf lettuce, for a total of $12.00. This indecisive customerthen asks to leave off the lettuce, for a total of $10.50. Thenthe customer finally puts the lettuce back in and removes halfof the eggplant and pays a total of $8.25. How many poundsof each vegetable did the customer take home?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CWhat’s In Each Box? Page 80

What’s In Each Box?

Words for the Word Wall:� factor� product� quotient


Transparency/Handout:� What’s In Each Box? (5C-18)

Procedure Notes1. Present the problem to students and have



Solution: Since the number 27 is in one ofthe boxes below 324, it is a factor. The otherfactor is 12. 21 and 14 share a factor of 3 sothat must go in the common box below them.That forces 4 into the first box on the lastline, and 9 into the remaining box on the lastline, giving

393660

324, 1215

12, 27, 45

4, 3, 9, 5

It’s easy to generate these puzzles by doingwhat? Kids might be able to create thesepuzzles using a calculator. Here’s another

92610

294, 315

14, 21, 15

2, 7, 3, 5

Put the numbers in bold in the puzzle.


Discussion:

� Where did you start with this puzzle? Divided 324 by 27.

� What did this tell you? 27 and 12 are factors of 324.

� What do 27 and 12 have in common? A factor of 3.

� Is this helpful information? Yes, since 12 and 27 have a common factor of 3, the 3must go in the “shared” box below those two numbers.

� What strategy would you use to build a puzzle for others in your group? Put singledigit numbers in the bottom row of boxes, then multiply pairs to complete the contentsof each box. Then decide which numbers will be placed in the template for the puzzle.

� Is there more than one solution to the puzzle? Why or why not?

� Can you place numbers in a template so that there is more than one answer? Explainhow the numbers would be placed. Would this make the puzzle easier or harder?Why?



What’s In Each Box?

Write a number in each box below so that each numberis the product of the two numbers beneath it. Onlywhole numbers greater than 1 may be used.

324

27

5

Build a puzzle for someone in your group to solve.

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CTiffany’s Beanie Babies Page 83

Tiffany’s Beanie Babies™

Words for the Word Wall:� Associative Property� Commutative Property� Area� Volume

Materials:� Graph paper� Linking cubes

Transparency/Handout:� Tiffany’s Beanie Babies (5C-19)

Procedures Notes1. Instruct students to build boxes for

Tiffany’s Beanie Babies out of linkingcubes.

To keep up with how many different boxeshave been constructed, have students tracethe base of each box on graph paper and thenwrite the dimensions of the entire box undereach copy of the base with the dimensions ofthe base in parentheses.

2. Be sure to discuss what the notationmeans: (3 x 6) x 2

This box has a bottom with three rows of sixand two layers.

3. Have students determine how muchcardboard is needed to construct eachbox.

Discuss with students that the amount ofcardboard needed to make each box is ameasure of surface area, while the number ofbeanies each could hold is a measure ofvolume.

Discussion:

� Is a (4 x 3) x 3 box the same box as a (3 x 3) x 4? (Students should notice that a boxwith dimensions of 4 x 3 x 3 could hold 3 layers of 12 beanies or 4 layers of 9beanies, depending on how the box is turned. Mathematically , the (4 x 3) x 3 isactually the same box as the (3 x 3) x 4 box. Notice that both the commutativeproperty and the associative property come into play here to ensure that the boxeshave the same volume.)

� What are the dimensions of the boxes that require the least cardboard to make? Themost cardboard?


TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CTiffany’s Beanie Babies Page 84

Tiffany’s Beanie Babies™

Tiffany has quite a collection of Beanie Babies™. She hasdecided that she should pack them away for awhile. Toprotect them, each will be in an individual box, and each ofthese boxes will be packed into a larger box. Investigate thedifferent shapes of large boxes that she can use to store 36Beanie Babies™ per large box.

Materials: Linking cubes, Graph paper

To keep up with the different options for boxes, draw the baseof each box on graph paper and record the dimensions of thebox, putting the dimensions of the base in parentheses.

How many different boxes can be made?

How do you know when you have them all?

What observations can you make about the boxes?

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CMarissa’s Garden Page 85

Marissa’s Garden

Words for the Word Wall:� prime� composite� factor� array

Materials:� Color tiles� Markers� Scissors� Tape� Chart paper or construction paper

Transparencies/Handouts:� Marissa’s Garden Activity Sheet (5C-20)� Marissa’s Garden Recording Sheet (5C-21)� Centimeter graph paper (5C-22)� Number Arrays (5C-23)

Procedures Notes

1. Introduce the activity by explaining to theparticipants that they are going to helpMarissa with her garden by makingarrays with color tiles.

2. Using 6 color tiles, make differentrectangular arrays.

One way to do this is to line up all 6 squaresin a straight row across the paper (6 squareswide and one square long). Is this horizontalarray different from one that is vertical? (Onesquare wide and 6 squares long) Yes, witharrays, orientation matters.


3. Guide participants to find all arrays forthe number 6.

There are two additional arrays for 6, 3squares wide and 2 squares long and 2squares wide by 3 squares long.

Can we make an array for 6 that is 4 squareswide? No, arrays must be rectangular, andthere are not enough squares to completelyfill an array 4 wide (or there are squares leftover).

Some students want to make an “oblique”array. Explain that an array is defined as arectangle with horizontal rows and verticalcolumns. Show how coloring an obliquearray on the grid paper would not be arectangle.

4. Assign each group a different number (ornumbers) from 1 to 40 and askparticipants to make as many arraysrepresenting that number as possible.When participants have arranged theirsquares into the appropriate arrays, havethem color the arrays on the centimetergraph paper, cut out the arrays, and attachthem to construction paper. (Participantscan also present their arrays on the graphpaper.)

Some numbers will have only two arrayswhile others will have several.

Wait for groups to complete their work.When arrays are completed, place them on awall in some random order.

5. Have students describe relationshipsfound in the arrays.

What relationships exist between the numberof rows and columns of each array and theoriginal number? The number of rows is oneof the factors of the original number. Thenumber of columns is also a factor of theoriginal number. If you multiply the numberof rows by the number of columns, you getthe original number. The largest number ofrows or columns less than the number itselfis never larger than half the number.


6. Sort the arrays. Let’s sort these arrays. Does anyone have aproposal for sorting? There are several waysto sort the arrays: even and odd numbers ofarrays; those numbers with only two arrays,those numbers with more than two arrays,and the number one, the only number withless than two arrays.

Guide participants to sort with the rule “onlytwo arrays” and “more than two arrays”.This sorting rule leaves the number one out.Emphasize that the number one does not fitthese categories.

The numbers with exactly two arrays have aname. What is that name? Prime numbers

The numbers with more than two arrays alsohave a name. What are they called?composite numbers

The number one with only one array plays aspecial role in multiplication—it is themultiplicative identity element (n x 1 = 1 x n= n).

7. Sort the arrays using the rule “odd oreven number of arrays”.

Let’s look at the arrays that have an oddnumber of arrays. Why does a number endup with an odd number of arrays? One of thearrays has the same number of rows ascolumns so the number of rows and columnscannot be reversed to create another array.

What are these numbers called? Squarenumbers


8. Have students complete the NumberArrays (5C-23).

Guide students to organize findings in achart. Groups should record the number, listthe arrays, then list the factors and finallystate the number of factors. For the number6 the following would be recorded.

Arrays: 1 by 6, 6 by 1, 2 by 3, and 3 by 2

Factors: 1, 2, 3, 6

Number of factors: 4

Here you can introduce additionalterminology such as perfect numbers (thesum of the factors less than the number itselfis equal to the number), abundant numbers(the sum of the factors less than the numberis greater than the number itself, e.g. 12 hasfactors 1, 2, 3, 4 , and 6 less than 12 whichsum to 16), and deficient numbers (the sumof the factors less than the number itself isless than the number, e.g. 9 has factors 1and 3 less than 9 which sum to 4).

Discussion:

� How did this activity help develop the definition of prime and composite numbers?

� If you teach prime and composite numbers in class, do you have your students do theSieve of Eratosthenes? How might this activity help with the concept of prime andcomposite while completing the Sieve?

� Explain what parts of this activity you might use with your students.

� Describe how the remainder when dividing by 6 helps determine whether a numbermight be prime.

� Where does the number 1 fit into the scheme of things?

� What does having an odd number of factors tell you about a number?

Additional resources:A Hundred Angry Ants; A Remainder of One; I Can Count the Petal from NCTM; SeaSquares



Marissa’s Garden Activity Sheet

Marissa is an avid gardener. She wants to redesign her gardenfor a new look. She has enough money to buy 36 plants, butis not sure she needs that many. She wants to explore thedifferent ways she can plant different numbers of flowers. Inher design efforts she realizes she will get a different look if 6flowers are planted one in front of another or side by side.She can also plant 3 rows of 2 or 2 rows of 3.

Help Marissa explore all the possibilities for her garden.



Marissa’s Garden Recording Sheet

Which number of plants will give Marissa the greatestnumber of choices?

Do any numbers give her an odd number of choices?

Why do you think there are an odd number of choices?

Do any numbers give her only two choices?

Why do you think this happens?

Centimeter Graph Paper Transparency / Handout 5C-22




Number Arrays

Number Arrays FactorsNumber of

Factors

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CThe Greatest Product, Part 2 Page 93

The Greatest Product, Part 2

Words for the Word Wall:� Factor� Product� Greatest

Materials:� Calculators (optional)

Transparencies/Handouts:� Greatest Product, Part 2 Recording Sheet (5C-24)

Procedures Notes1. Make sure the students understand the

following directions: Given the fivedigits—1 , 2, 3, 4, and 5—place one digitin each box in the template to create thefactors using these five digits that givethe greatest product. Use each digitexactly one time. Record your first,second, and third attempts in theappropriate places below, using yourcalculator to compute the products, andanswer the accompanying questions.

Most people, on their first attempt to solvethis problem, place the greatest digits in thepositions of greatest place value: 532 x 41 =21,812. However, when encouraged to trysomething else, or in comparing it to anotherperson’s attempt, they soon find that this isnot the greatest product. For example, 531 x42 = 22,302. An unexpected finding is that432 x 51 = 22032, which is greater than theother two products with the 5 in the hundredsplace. The greatest product is obtained withthe factors 431 x 52 = 22,412.

2. Have students try the problem again withthe digits 2, 4, 6, 7, 9.

If you follow the pattern learned in theprevious problem, your first attempt is 762 x94 = 71,628. Attempts of other factors showsthat this is the greatest product. From thesetwo problems, one can conjecture that thegreatest product is formed by placing thegreatest digit in the tens place of the two-digit factor, the next greatest digit in thehundreds place of the three-digit factor, thenext greatest digit in the tens place of thethree-digit factor, the next greatest digit inthe ones place of the two-digit factor, and thesmallest digit in the ones place of the three-digit factor. If the digits are labeled andordered such that a < b < c < d < e, then thefactors conjectured to create the greatestproduct can be represented as dca x eb.


3. Have students try the problem again withthe digits 0, 3, 5, 6, 8 and compare theirresults to the conjecture they made inQuestion 2c. Have them consider thefollowing questions: How is this set ofdigits different from the others? Does itfit the conjecture or not?

If the conjecture is tested with this set ofdigits, 650 x 83 = 53,950 is the greatestproduct. But with some experimentation, itcan be shown that this same product is alsoobtained with 830 x 65. Does this discoverydisprove the conjecture? No, because theconjecture that dca x eb produces the greatestproduct is still true. It just isn’t the ONLYset of factors that does in this situation. Whydoes eba x dc work in this situation and notin the others? Could it be because of the 0digit? Students who test other sets of digitsthat contain 0 will find two pairs of factorsthat give the greatest product.

4. Encourage students to use mathematicalproperties to explain why the pattern inthe conjecture works.

The problem in Question 3 can help lead tothe understanding of the mathematicalprinciples involved here. For example, 650 x83 = (65 x 10) x 83 = 65 x (10 x 83) = 65 x830, based on the associative property ofmultiplication. However, if there are morethan 0 ones in the ones place, therepresentation changes. For example, 651 x83 = (650 + 1) x 83 = (65 x 10 x 83) + (1 x83), based on the distributive property ofmultiplication over addition. However, 831x 65 = (830 + 1) x 65 = (83 x 10 x 65) + (1 x65). Note that the first part of the sum is thesame in both products; however, the secondpart of the sum is different, and its size isdetermined by the size of the two-digitfactor. Therefore, although 650 x 83 = 830 x65, 651 x 83 > 831 x 65.



Greatest Product, Part 2 Recording Sheet

1. Given the five digits—1, 2, 3, 4, and 5—place one digit in eachbox in the template to create the factors using these five digits thatgive the greatest product. Use each digit exactly one time. Recordyour first, second, and third attempts in the appropriate placesbelow, using your calculator to compute the products, and answerthe accompanying questions.

First Attempt Second Attempt Third Attempt

a. For what reasons did you place the digits where you did in yourfirst attempt?

b. Based on what reasons did you place the digits in your secondattempt?

c. Did putting the greatest digits in the greatest place valuepositions give you the greatest product?

d. How do you know that you have the greatest product by thethird attempt?

XX X



2. Try the problem again with the digits 2, 4, 6, 7, 9.

First Attempt Second Attempt Third Attempt

a. For what reasons did you place the digits where you did in yourfirst attempt this time?

b. Did you need to make a second attempt? Why or why not?

c. What conjecture can you make about placing the digits to createthe greatest product?

d. How can you test your conjecture?

3. Try the problem again with the digits 0, 3, 5, 6, 8. Compare yourresults to the conjecture you made in Question 2c. How is this setof digits different from the others? Does it fit the conjecture ornot?

4. Use mathematical properties to explain why the pattern in theconjecture works.

XX X

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section CMarissa’s Garden Again Page 97

Marissa’s Garden Again

Words for the Word Wall:� prime� composite� factor

Materials:� Markers

Transparencies/Handouts:� Marissa’s Garden Again (5C-25)� Sixes Chart (5C-26)� Sixes Chart Key (5C-27)

Procedures Notes1. Have students use the Sixes Chart and

circle the prime numbers.Do you see anything interesting in thelocation of prime numbers? Except for theprime numbers 2 and 3 all the primenumbers are in the first or fifth column.

Looking at the chart, can you determine thatthis will always be the case? Yes, thenumbers in the second column are all even,in the third column they are multiples of 3,etc. Prime numbers can only be in columnsone and five.

If prime numbers can only be in columns oneand five of a six-column chart, what couldthis mean in checking to see if large numbersare prime? Divide the number by 6 and if theremainder is 1 or 5 it might be a primenumber. Any remainder other than 1 or 5means the number is a composite for sure.


2. Go over the numbers on the Sixes ChartKey and determine whether each isprime.

149 has a remainder of 5 – prime

207 has a remainder of 3 – composite

319 has a remainder of 1 – but not primebecause it is divisible by 7

437 has a remainder of 5 – but not primebecause it is divisible by19.

523 has a remainder of 1 – prime

543 has a remainder of 3 – composite

To determine which numbers are prime thathave remainders of 1 or 5 when dividing by 6each prime number less than the square rootof the number in question must be dividedinto the candidate to see if it is a factor.

Discussion:

� How did this activity help develop a method for eliminating composite numbers bydivision?

� Describe how the remainder when dividing by 6 helps determine whether a numbermight be prime.

� If you get a remainder of 1 or 5 when dividing by 6, does that guarantee you havefound a prime number? Explain.



Marissa’s Garden Again

Use the chart with numbers in six columns. Circle all theprime numbers.

What do you notice about the location of prime numbersgreater than three?

Can you describe your finding mathematically in terms ofdivision?

Use your discovery to determine if the following numbers arepossibly prime.

149 209

317 437

523 543

How do you determine whether the numbers you’ve identifiedare really prime?



Sixes Chart

Circle the prime numbers. Do you see anything interesting?

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

55 56 57 58 59 60

61 62 63 64 65 66

67 68 69 70 71 72

73 74 75 76 77 78

79 80 81 82 83 84

85 86 87 88 89 90

91 92 93 94 95 96

97 98 99 100 101 102

Transparency 5C-27


Sixes Chart Key

Circle the prime numbers. Do you see anything interesting?

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

55 56 57 58 59 60

61 62 63 64 65 66

67 68 69 70 71 72

73 74 75 76 77 78

79 80 81 82 83 84

85 86 87 88 89 90

91 92 93 94 95 96

97 98 99 100 101 102

TEXTEAMS Rethinking Elementary School Mathematics Part 1 Day 5: Section DClosure for Day Five Page 102

Closure for Day FiveConceptual Models of Computational Fluency

Materials:� Chart paper� Markers� Blank transparencies� Overhead pens

Transparencies/Handouts:� 3 Sample Models of Computational Fluency (5D-1 through 5D-3)

Procedure:Have participants work in small groups to design a visual model to illustrate the relationshipsamong the ideas of number, operations, and computational fluency that they have beenexploring during the past five days.

(Note to the facilitator: Transparencies 5D-1, 5D-2, and 5D-3 are examples of simpleconceptual models that might help participants get started in their thinking and discussion.)

Have each group present their conceptual model to the whole group.

Transparency 5D-1


Problem Solving

Number OperationsComputationalFluency

Transparency 5D-2


Problem Solving

Number Operations

ComputationalFluency

Transparency 5D-3


Computational Fluency

Operations

Number

TEXTEAMS Rethinking Elementary School Mathematics Part 1 AppendixPage A-1

Related Books

Charles, Randall and Joanne Lobato, Future Basics: Developing Numerical Power,Colorado: NCSM, 1998.

Dodds, Dayle Ann, The Great Divide, Cambridge, MA: Candlewick Press, 1999.ISBN 0-7636-0442-9

Fosnot, Catherine Twomey and Maarten Dolk, Young Mathematicians at Work ConstructingNumber Sense, Addition, and Subtraction, Portsmouth, NH: Heinemann, 2001.ISBN 0-325-00353-X

Gambrell, Linda B and Janice F. Almasi, Lively Discussions! Fostering Engaged Reading,Newark DE: International Reading Association, 1996.ISBN 0-87207-147-2

Kilpatrick, Jeremy, Jane Swafford, and Bradford Findell, Adding it Up, Washington, DC:National Academy Press, 2001.ISBN 0-309-06995-5

Lerman, Rory S., Charlie’s Checklist, New York: Orchard Books, 1997.ISBN 0-531-07173-1 (paperback), ISBN 0-531-30001-3 (hardcover)

Ma, Liping, Knowing and Teaching Elementary Mathematics, Mahwah, NJ: LawrenceErlbaum Associates, Publishers, 1999.ISBN 0-8058-2909-1

Merrim, Eve, 12 Ways to Get 11, New York: Aladdin Paperbacks, 1993.ISBN 0-689-80892-5

Mackain, Bonnie, A Remainder of One, Boston: Houghton Mifflin, 1995.ISBN 0-395-69455-8

Mokros, Jan, Susan Jo Russell, and Karen Economopoulos, Beyond Arithmetic, White Plains,NY: Dale Seymour Publications, 1995.ISBN 0-86651-846-0

Seymour, Peter, Bare Bear’s New Clothes, Los Angeles: Price Stern Sloan, Inc, 1986.ISBN 0-8431-1824-5

Strong, Richard W., Harvey F. Silver, and Matthew J. Perini, Teaching What Matters Most,Alexandria, VA: ASCD, 2001.ISBN 0-87120-518-1


Wiggins, Grant and Jay McTighe, Understanding by Design, Alexandria, VA: ASCD, 1998.ISBN 0-87120-313-8

Zemelman, Steven, Harvey Daniels, and Arthur Hyde, Best Practice New Standards forTeaching and Learning in American’s Schools, Portsmouth, NH, Heinemann, 1998.ISBN 0-325-00091-3


Quotes

Worthwhile tasks should be intriguing, with a level of challenge that invites speculation andhard work.Principles and Standards for School Mathematics (NCTM) 2000, p.19

Learning is socially constructed and interactional. Teachers need to create classroominteractions that “scaffold” learning.Best Practices, 1998.

By engaging in (frequent opportunities to talk and write as learners and thinkers) and bydiscussing their reflections with others, students develop a sense of their ownresourcefulness…and are better able to set and work toward their own goals.Standards for the English Language Arts (NCTE, IRA) 1996, p. 17

Smart isn’t something you are, smart is something you get.Jeff Howard

When students are expected to challenge their peers and defend their own explanations, theyacquire more flexible conceptual understanding than they do in isolated learning contexts.Lively Discussions, 1996, p.99

Equity requires high expectations and worthwhile opportunities for all.Principles and Standards for School Mathematics (NCTM) 2000, p. 12

When children become reflective learners, they acquire an important skill that enables themto judge their performance in terms of external standards.Apprenticeship to Literacy, 1998, p. 23


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