Elementary Investigations 2009

8/11/2019 Elementary Investigations 2009

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Mathematics

Standards

A Parents Handbook Grades K-5

Math & Science Collaborative


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2 What your child should be learning:

The elementary mathematicscurriculum taught in yourchilds school is Investigations inNumber, Data, and Space. Thismeans the classroom activitiesand assignments for kindergar-ten through fth grade shouldre ect those described in thishandbook.

The goal of Investigations is to helpstudents become mathematicalthinkersindividuals with thenecessary skills, strategies, and un-derstanding of mathematical ideasto approach new concepts andproblems with con dence. With

that in mind, the curriculum isorganized around the big ideas(key concepts) of math. Activitiesand assignments are designed toengage students who have a rangeof skills and learning styles.

The content and teaching ap-proaches of Investigationswhat students learn and how they learnitare different from most par-ents experiences with elementaryschool math. While computation(adding, subtracting) is still amajor focus, the curriculum alsoincludes geometry, early algebra,probability, and data. For exam -ple, students begin exploring thefeatures of shapes (geometry)in kindergarten. Throughout thegrades, they work with patternsrecognizing, creating, andextending patterns with objects,shapes, and numbers (algebra).They nd the probability ofevents and learn to carry outa data investigation: posing aquestion, gathering information,showing the data on various

kinds of graphs, describing andinterpreting data, and comparingone set of data to another.

Teaching approaches in Investiga-tions focus on students activeinvolvement in learning. Teachers

ask students to explain their rea-soning and their solutions to theclass, to compare the strategy theyused to solve a problem to theirclassmates strategies, and to thinkabout why a solution makes senseor doesnt make sense. Alongwith building on what studentsalready know, teachers challengethem to think beyond the particular

example. (This triangle has a cornerthat matches the corner of a piece of paper. Do all triangles have to have acorner like that?This strategy works for addition; why doesnt it work formultiplication?) Research showsthat compared to passive learning,such as copying a procedure shown by a teacher, active learning helpsstudents reach deeper levels ofunderstanding.

CLASSROOM ROUTINES ANDTEN-MINUTE MATH

In grades K-3, classroom routinesand daily activities show studentshow math is used in real life andhelp them practice skills. In grades3-5, Ten-Minute Math activitiesare used to reinforce concepts andskills. Examples of routines and Ten-Minute Math activities include:

Attendance (K-1). Studentscount the number who arepresent and the number whoare absent, helping them learnto count accurately (countingeach student only once) and

giving them practice add-ing and subtracting. (If ourclass has 29 students and twostudents are absent, how manyare present today?) They alsolearn number conceptsforexample, beginning the countfrom one side of the room orfrom the other side of the roomresults in the same number.

Calendar (K-1). Students countdays and keep track of ac-tivities on a monthly calendar.(How many days has it beensince our eld trip to the zoo?)

Weather (grade 1). Studentsrecord the weather on a charteach day to practice gatheringand examining data thatchange over time.

Todays Number (grades2-4). Students nd equivalentexpressions for a given number. (For example, equivalentexpressions for 10 include5+5, 100-90, 5x2.) Sometimes

INTRODUCTION

Questions parents can ask their childrenIn the classroom, the teachers goal is to create an environmentwhere students feel comfortable questioning themselves and theirclassmates, and working through the confusion and frustrationwhich can be part of the learning process. At home, parents canprovide a similar kind of support by asking questions that pushtheir childs learning further. You might ask:

What are you being asked to nd out? What does the problem tell you? Have you seen a problem

like this before?

Is there any part of the problem you already know how to do? Is there anything you dont understand? Where can you nd answers to your questions?

Will it help to make a list, a chart, a table, a drawing, a diagram? What will you try rst? Then what? What do you estimate your answer will be? Is your strategy working? Why or why not? Is there another way to check your answer? How do you know if your answer is right or wrong?

End-of-Unit AssessmentFill this shape with as many blocks asyou can.

How many blocks did you use?_______

Now ll this shape with as few blocksas you can.

How many blocks did you use?_______

First grade assessment

Source: Investigations


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3Introduction

students must follow guide-lines, such as starting with

numbers greater than 100, orusing only subtraction.

Quick Images (K-5). The teacherashes an image twice, asksstudents to describe, draw and/or model it from memory, thenfollows up with a discussion.The image may be groupingsof dots (like dots on a domino),shapes, or other things. The goalis to help students form visualimages of quantities ( ve dots,two dots plus eight dots) or torecognize shapes easily. Dis-cussions center around whichfeatures of a shape students re-membered (It had four sides)or the different ways theyvisualized quantities. (I saw

two groups of four dots. Thatseight. There was space for ten but two squares were empty.Ten minus two is eight.)

At each grade level, studentsplay games that give them op-portunities to practice skills andto build strategies. The curricu -lum also includes a computercomponent, geometry software,that schools may or may not use.

STUDENTS WITH SPECIAL NEEDS

All students have the right toparticipate in the learning ac-tivities of the math curriculum.Some students may be able toparticipate fully with the help ofspecial technology or a facilitator;for others, activities may needto be modi ed. Teachers guidesfor Investigations include sugges-tions and support for adapting ormodifying lessons to meet indi-vidual needs. Ask your childsteacher for more information.

WHAT IS IN THIS HANDBOOK?

In this handbook, you will nd anoutline of the math standardsgoals and expectations for stu-dentsfor each grade level. Toillustrate the kinds of activities that build skills and understandingof concepts, information for eachgrade level includes a sample ofstudent work and a game studentsplay (along with directions for

playing at home). Pages 18 and 19provide more information aboutstrategies students use to solveproblems, and examples of graphs.Page 17 contains a glossary ofmathematical terms that may beused by your childs teacher or inhomework assignments.

FOR MORE INFORMATION

This handbook is only an out -line. For more information, tryone of the following sources:

Talk to your childs teacher, theprincipal, or a math coach or su-pervisor for your school district.

The Investigations curriculumprovides two kinds of family

letters for each unit: Aboutthe Mathematics in This Unitand Related Activities to Tryat Home. Ask your childsteacher for copies if youhavent received them.

Each grade level has a Stu -dent Math Handbook with anoverview of the program forthat year, sample problems andstrategies for solving them, andgames. If your child doesnt bring this book home, ask theteacher if you can borrow it.

Visit the developers Web site:http://investigations.terc.edu.

See the glossary on page 17 forfurther explanations of math-ematical terms and concepts.

This publication was developed at the request and with the support of the Math& Science Collaborative located at the Allegheny Intermediate Unit. This materialis based on work supported by the National Science Foundation under Grant No.EHR-0314914. Any opinions, ndings, conclusions, or recommendations expressedin this publication are those of the authors and do not necessarily re ect the viewsof the granting agency.

Writing/editing: Faith Schantz

Design/layout: Julie Ridge

Photographs: Greg Blackman

Contributors and reviewers: Dr. Nancy Bunt, Andrea Miller, Michael Fierle,Corinne Murawski, and Sandra Fowler of the Math & Science Collaborative; LisaBellinotti of the Fox Chapel Area School District; Mindy Harris and Mary Wallaceof the Uniontown Area School District; Dr. Nancy Jacqmin of Carlow University;and parents Kim Killinger and Laurie Wozniak.

Samples of student work were provided by teachers in the Fox Chapel Area,Quaker Valley, Uniontown Area, and Upper St. Clair school districts; and PropelEast charter school.

Examples of math problems, assignments, games, and other text from theInvestigations curriculum were used with permission.

Photographs were taken at Hartwood Elementary School in the Fox Chapel AreaSchool District and at Benjamin Franklin School in the Uniontown Area SchoolDistrict, Pennsylvania.

2009 Math & Science Collaborative of the Allegheny Intermediate Unit

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

Standards by grade level:Kindergarten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4First grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Second grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8Third grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Fourth grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

Fifth grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

Strategies for solving problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

Contents


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NUMBERS

Write numbers from 0 to 10. Re -

late spoken numbers (nine) tocorresponding written numbers(9).

Identify by sight how many objectsare in a small group or set (such as3 dots on a domino).

Count up to 20 objects. Rec -ognize that with each countednumber, the number of objects in

the counted group increases.

Compare quantities up to 10.

Develop visual images of

numbers (as in the illustration).

Make equal groups of objects(such as a set of 3 red chips, aset of 3 white chips).

Compare unequal groups

(3 red chips, 5 white chips;the number of letters in 4students names). Which has

more, fewer, the most, thesmallest number?

Find different combinationsthat make the same number (3and 3, 5 and 1).

ADDING AND SUBTRACTING

Combine 2 amounts and sepa -rate 1 amount from another.

Add and subtract small quanti -ties of objects and show answersin pictures, words, and/ornumbers. For example, lay out5 blue tiles and 1 yellow tile.

KINDERGARTENGRADUATES SHOULD...

Write 5 and 1 are 6.

Make combinations of numbersthat add up to the same number(0+5=5, 1+4=5, 2+3=5, 3+2=5,4+1=5, 5+0=5).

PROBLEM-SOLVING

Solve problems using objectsand pictures. Show solutions by counting and/or writing

numbers.

MEASURING

Measure objects using non-stan -dard units (such as cube blocksor sticks). (My shoe is 7 cubeslong.)

Compare sizes of objects. Whichis longer, wider, taller, heavier?

Order objects from shortest tolongest.

GEOMETRY

Explore shapes, including cir-cles, ovals, squares, rectangles,diamonds, and other polygons(2-dimensional gures made up

of line segments).Describe the features of simpleshapes. (A circle is round. Itlooks like a wheel.)

Make 2-D and 3-D shapes usingpattern blocks, wooden blocks,and clay.

Copy shapes.

Combine or separate shapes tomake new shapes. For example,combine 2 triangles to make arectangle.

PATTERNS AND MATHEMATICALREASONING

Copy simple patterns ( s s s ss s ).


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5Kindergarten

Continue patterns ( l l l l l l ).

Make patterns and identify theunit (the part that repeats).uuu n uuu n

COLLECTING ANDGRAPHING DATA

Sort objects by one of theirfeatures (such as buttons with 3holes and buttons with 4 holes).

With classmates, collect data. For

example, collect answers to thequestion, How many people inour class have a pet?

Find ways to show data (draw -

ing pictures of the buttonswith 3 holes in one box and the buttons with 4 holes in another box; making a chart, table, or listshowing students who have petsand students who dont).

Use data to solve problems oranswer questions. (10 people inour class have a pet.)

Collect 10 TogetherTOOLS: Pennies, a dot cube

HOW TO PLAY:

Play with a partner.

1. The rst player rollsthe dot cube and takes

the number of penniesshown.

2. The second player rolls the dot cube andtakes the number of pennies shown.

3. Partners determine whether they have 10 pennies together. If theydo not, the rst player rolls again. Play continues until players haveat least 10 pennies.

SAMPLE GAME

SAMPLE OF STUDENT WORK

A pattern block picture showing the collection and recording of data

Unit


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words, pictures, numbers, andsymbols.

GEOMETRY

Recognize shapes in the environ-ment (square boxes, rectangular buildings, round cans, diamondsin a wallpaper pattern). Iden -tify some of their features.

Sort 2-dimensional shapes by their

features (4 sides, 3 corners).Identify the features of triangles(all have 3 corners, 3 points, and3 straight lines that may or maynot be equal).

Identify the features of quad-rilaterals (4 corners, 4 points, 4straight lines that may or maynot all be equal).

Make triangles and quadrilater-als of different shapes and sizes.

Make 3-D shapes out of 2-Dshapes (for example, build a boxfrom index cards). Compare 2-Dshapes to related 3-D shapes (seeillustration).

Combine and separate shapes toform new shapes. For example,make a hexagon using triangles.Find differentways of form-ing the sameshape.

MEASURING

Explore measurement.

Measure objects using non-standard units (cut-outshapes, tiles, cubes, sticks).Describe measurements(a little more than 9 cubeslong).

NUMBERS

Count, read,

write, andorder numbersto 105. For example, order cubetowers from least to most, nd themissing numbers in a sequence, orcount on from a known numberon a 100 chart (begin at 37 andcount on 11 more). (See page 17for an example of a 100 chart.)

Skip count by 2s, 5s, and 10sstarting with any familiar number (Six, eight, ten ve, ten,fteentwenty, thirty, forty).

Find strategies for counting groups,such as counting cubes by 5s or10s, or using tally marks.

Count up to 50 objects. For ex -ample, trace ones own foot andnd out how many pennies cant inside the tracing.


Add numbers to make sums upto 20 (5+7=12, 10+9=19).

Break apart numbers up to20 (subtract) and take away onepart (11-8=3, 10-3=7).

Know which combinations ofnumbers add up to 10 (5+5=10,6+4=10).

Show different ways of express -ing a number (5+5+5+5=20, 30-10=20, 2 dimes).

Recognize the relationship be -tween addition and subtraction.

PROBLEM-SOLVING

Determine whether a story

problem requires addition,subtraction, or both.

Solve story problems using strat -egies such as using one problemto solve another.* (There are 9rabbits. 5 hop away. How manyrabbits are left? 9-5. 10-5=5, so9-5=4.)

Solve story problems with more

than one step. (Keeshawn had15 pennies in one pocket and 6pennies in the other pocket. Hespent 5 pennies. How many didhe have left?)

Use objects, a number line,and a 100 chart to solve prob -lems. Show strategies and so -lutions using combinations of

59 60 61 62 64

Rectangular prismRectangle

FIRST GRADEGRADUATES SHOULD...

* See pages 18 and 19 for more examples of strategies.


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7

Compare multiple measure -ments of an object.

Compare measurements of an

object using different units. Compare estimates to actual

measurements.

Understand some real-worldpurposes of measurement.


Explore patterns using shapes,colors, and numbers.

Create patterns.

Continue patterns.

Analyze patterns to identifythe unit (the part that repeats).s s s s s s s s s

Recognize patterns thatinvolve a constant rate ofincrease (such as adding 3

pennies to a jar each day).

Predict how patterns willcontinue (What will be the

color of the 12th square? Howdo you know?)

Compare patterns to discoversimilarities (similar patterns:s s s , uu n , AAB).


Sort objects by one of their fea -tures (shape, color).

Pose survey questions. (Wouldyou rather be invisible or able toy?)

With classmates, collect answersto survey questions.Show data using tally marks,charts, picture graphs, and bargraphs.

Describe data. (18 people wouldchoose to be invisible. 2 wouldchoose to y. 20 people answeredthe question.)

Interpret data. (More peoplechose to be invisible. Not manychose being able to y.)

First Grade

1 2 3 4 65

Double CompareTOOLS: Deck of primary numbercards (to play at home, label theQueens as 0s, the Aces as 1s, andremove the other face cards)

HOW TO PLAY:


1. Deal the cards face down.

2. Both players turn overtheir top 2 cards.

3. The player with the larger total saysMe! and takes the cards. If the totals are the same, both play -ers turn over 2 more cards.

4. Keep turning over 2 cards each time. The player with thelarger total says Me! and takes the cards.

5. The game is over when there are no more cards to turn over.

OTHER WAYS TO PLAY:

The player with the smaller total says Me!

Play with 3 players.

Play with Wild Cards, such as jokers. A Wild Card can beany number.

SAMPLE GAME


Unit

An example of using a picture to solve an addition problem


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Use objects, a number line, a 100chart (see page 17 for an ex -

ample), mental math, and esti-mation to solve problems. Showstrategies and solutions usingcombinations of words, pictures,numbers, and symbols.

GEOMETRY

Draw and build 2-dimensionaland 3-dimensional shapes, usingpattern blocks and wooden blocks.Sort 2-D and 3-D shapes by theirfeatures (such as the number ofsides).

Name the number of sides in anypolygon (a 2-D gure made up ofline segments).

Combine and separate shapes toform other shapes. Find differentways of forming the same shape.

Recognize a right angle (90).

Identify the features of rectangles(all have 4 sides and 4 rightangles).

Identify the congruent faces (sidesthat are the same size and shape)on rectangular prisms (3-D rect-angles).

Build rectangular arrays (forexample, a rectangle made ofsquare tiles or arectangle drawnon grid paper).

Recognize mirror-image sym-metry in objects and designs (forexample, a butter ys wings).

NUMBERS

Find strategies for grouping num -

bers and counting by groups. Count up to 60 objects by 1s, 2s,

5s, or 10s.

Skip count by 2s, 5s, and 10sup to 110 ( fty-two, fty-four,fty-six, ve, ten, f -teen, forty, fty, sixty).

Solve problems with quantitiesthat include 10s and 1s. (Baseballcards come in packs of 10. Kristinhas 2 packs and 3 cards. Howmany altogether?)

Know that even numbers canmake groups of 2, or 2 equalgroups. (We have 12 people. Wecan have 2 teams of 6.) Know

that odd numbers cant makegroups of 2, or 2 equal groups.(We have 13 people. Not every -one can have a partner.)

Understand fractions as equalparts of a whole (half of thepizza is left over, boys make upone-third of our class).

Express fractions in words (one-half, one-fourth) and in writtennumbers (, ). Understand whateach part of a written fraction rep-resents. Express mixed numbers(such as 1) in words (one andone-half) and in written numbers(1). Understand what each part ofa written mixed number represents.

Combine and break apartnumbers with uency.


Use a variety of strategies to add

numbers with sums up to 100,and to subtract 2-digit numbers.*Write an equation to show a solu -tion (2+2=4, 27-3=24).

Know that numbers can beadded in any order with the sameresults (7+4+9=20, 9+7+4=20,4+9+7=20).

Express a number in many ways

(19+1=20, 100-80=20, 2 dimes).Know addition and subtractioncombinations (the numbersthat can be added or subtractedto equal a given number) up to10+10.

Know the values of a penny,nickel, dime, quarter, and dollar.Make exchanges with coins (10

pennies for a dime, 2 dimes and 5pennies for a quarter).

Add even numbers together, oddnumbers together, and even andodd numbers together. Reasonabout the results.

PROBLEM-SOLVING

Determine whether a story problemrequires addition, subtraction, or both. (Maya had 100 pennies. Thenshe lost some. If she has 73 penniesleft, how many did she lose?)

Solve story problems with morethan one step.

Write stories to match problems. Forexample, write a story for 100-73.

Congruent faces

Rectangular array

SECOND GRADEGRADUATES SHOULD...


Use different types of pattern blocks to ll the shape. Record your work by tracing the shapes of the blocks you used onto each shape outline.

Use 1 type of pattern block to ll the shape [above].

Use two types of pattern blocks to ll the shape.



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Create patterns and designs thathave mirror-image symmetry.

MEASURING

Create and use standard measur -ing tools (such as an inch ruler).

Measure in U.S. units (inches, feet,yards) and metric units (centi -meter, meter), with increasingaccuracy.

Recognize the importance ofusing common (standard) units,through experiences with estimat-ing, comparing, and measuringlengths and distances.

Recognize length as a de ningcharacteristic of objects.

Tell time to the quarter hour (3:15p.m.). Solve problems involvingtime measurement. Use a timelineto measure and record the dura-tion of events.

Describe the area (the amount ofspace inside a 2-D shape) of arectangular array in terms of units.(The rectangle has 2 rows with 4tiles in each row.)


Explore constant ratios (ratioswhere the rate of change is thesame, such as one totwo or 1:2, and two tofour or 2:4) using cubesand pattern blocks.(It takes 2 trapezoid

pattern blocks to cover1 hexagon, so for 2hexagons, well need 4trapezoid blocks.)

Use a table to organize numbersexpressing a constant ratio.

Explain the numbers in a table.(Its a count-by-2s pattern.If you double the number of

hexagons you get the number oftrapezoids.)

Extend a table. (How many trap -ezoids will cover 6 hexagons?)

Analyze patterns with repeatingunits (red/green/green, KXX ).Write a number sequence to de -scribe one element in the pattern.

s l n s l n s l n 1 2 3 4 5 6 7 8 9

Number sequence for squares: 3, 6, 9

Extend number sequences.

(What symbol will be over 16?)Recognize the difference betweena repeating pattern (as in the il-lustrations) and a growing pattern(as in the table).


Collect data. (What books havewe read this month?)Organize data by category.(Which books are ction, whichare non- ction?)

Show data using Venn diagrams,line plots, and other representa-tions (see sidebar on page 19).Describe data, including thehighest number, the lowest num - ber, and the mode (the numberthat occurs most frequently).

Interpret data. (Our class readmore non- ction than ctionlast month.)

Compare 2 sets of data (such asthe number of baby teeth lost ineach of 2 second grade classes).

Second Grade

Number of Hexagons

123456

246???

Number of Trapezoids

Close to 20TOOLS: Cubes or other objects, recording paper, deck of primarynumber cards (to play at home, label the Queens as 0s, the Aces as1s, and remove the other face cards)

HOW TO PLAY:


1. Deal 5 cards to each player.

2. Take turns. Each player should:

Choose 3 cards that make a total as close to 20 as possible.

Record the total of the 3 cards and the score. The score is thedifference between the total and 20.

Take that number of cubes.

Put those 3 cards aside and take 3 new cards.

3. After each player has taken 5 turns, total the scores.

4. Each player counts their cubes. They should have the samenumber of cubes as their total scores.

5. The player with the lowest score is the winner.

OTHER WAYS TO PLAY:

Play with Wild Cards, such as jokers. A Wild Card can be any number.

SAMPLE GAME


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What your child should be learning:10

NUMBERS

Read, write, and order numbers

to 1,000.Identify how many groups of 100,groups of 10, and groups of 1 arein a 3-digit number (352 = 3 onehundreds, 5 tens, and 2 ones, or35 tens and 2 ones). Identify theplace value of the digits in any3-digit number.

Use pattern blocks and groups of

objects to name and show frac -tions. Divide a whole into fractionsand name them. Divide a groupinto fractions and mixed numbers.(2 people share 3 brownies. Eachgets 3/ 2 , or 1 brownies).

Identify equivalent fractions (forexample, 3/ 6=2/4=).

Understand decimal fractions inthe context of money ($.25=onequarter, or of a dollar).

ADDING, SUBTRACTING,MULTIPLYING, DIVIDING

Combine numbers that add upto 100 (48+52, 33+67). Makecombinations of coins that equal$1.00 (such as 3 quarters, 2dimes, 5 pennies).

Use a variety of strategies to addand subtract 2- and 3-digit num - bers, such as breaking numbersapart by place value.* (76+37.70+30=100, 6+7=13, 100+13=113.)

Add multiples of 10 and multiplesof 100 with uency.

Use knowledge of addition combinations (numbers that can beadded to equal a given number) tosolve subtraction problems. (10-6.I know that 6+4=10, so 10-6=4.)

Use pattern blocks and groups of

objects to add fractions.Understand that multiplication in-volves making equal-sized groups.Understand that division involvesequal sharing. Recognize the rela-tionship between multiplicationand division.

Use a variety of strategies to solvemultiplication and division prob -

lems.* (How many groups of 5pennies can I make with 35 pen-nies? I can skip-count the pennies by 5s: 5, 10, 15, 20and count howmany groups I have.) Recognizeand use symbols for multiplicationand division (x, , ).

Know that 2 numbers can bemultiplied in either order with the

THIRD GRADEGRADUATES SHOULD...

same results (3x4=4x3).

Know multiplication facts withproducts up to 50 (7x7=49,11x3=33).

PROBLEM-SOLVING

Describe, analyze, and comparestrategies for solving addition andsubtraction problems.

Analyze a variety of subtractionstory problems to determine whatkinds of problems they are, including:

Taking away. (A pet store had 162gold sh. They sold 25 of them.How many were left to sell?)

Finding the unknown part.(Our class goal is to collect 50cans of food for the food bank bythe end of the week. Its Tuesdayand weve collected 9 cans. Howmany more cans do we need tocollect between now and Fridayto reach our goal?)

Comparing. (Todays tem -perature is 62. A month ago thetemperature was 44. How manydegrees warmer is it today?)

Use objects, a number line, a 100chart (see page 17 for an example),and a 1,000 chart to solve prob -lems. Use arrays (objects or sym - bols laid out in rows or drawn ongrid paper) to visualize multiplica-tion problems (see page 17). Showstrategies and solutions usingcombinations of words, pictures,numbers, and symbols.

GEOMETRY AND MEASUREMENT

Identify 2-dimensional and 3-di-mensional shapes and sort them by their features.

Make 2-D and 3-D shapes.

Describe the features of 2-Dshapes such as polygons (2-D


1. How would you share each of the following? Write about yourthinking or use a drawing to show your solution.

9 brownies shared among 4 people

9 balloons shared among 4 people

2. How much money does each person get? Compare your answer tothe answer you get using a calculator.

9 dollars sharedamong 4 people

9 4 on a calculator

3. Put a circle around each of your four answers above. They are allanswers to 4 people sharing 9 things. How are they different fromeach other? How are they alike?



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11

gures made up of linesegments) and circles. For

example, name the numberof sides, angles, and vertices(the points where 2 sides meet)in a triangle.

Describe the features of 3-Dshapes, such as prisms, cylin-ders, and pyramids. For exam -ple, name the number of faces( at surfaces), edges (line seg -ments where2 faces meet),and vertices,and whether ashape is at orcurved.

Identify shapes in the environ -ment (classroom: rectangularprism, basketball: sphere).

Determine whether or not shapesare congruent (the same sizeand shape) by sliding, ipping, orturning them to seeif they can be madeto coincide.

Explore and create 2-D represen-tations of 3-D shapes (nets),such as anunfolded box.

Identify angles as larger or smallerthan a right angle (90).

Measure length in U.S. units(inches, feet) and metric units

(centimeter, meter). For example,measure how far each studentcan jump.

Identify perimeter (the sum ofthe length of the sides of a shape),and measure perimeter in non-standard units (such as the sideof a block) as well as in U.S. andmetric units. Make and compare

different shapes that have thesame perimeter.

Find the areaof a shape (theamount ofspace insidea 2-D shape) by countingsquare units. Make and compareshapes with the same area.

Understand volume (the amount

of space inside a 3-D shape) in thecontext of lling a box with cubes.Design a box for a given volume.


Read simple graphs and tablesthat show situations involvingchanging data (such as weeklytemperature uctuations fromSeptember to December). De -scribe and interpret graphs. (Theaverage temperature changedalmost every week.)

Read graphs that show a constantrate of change (such as 3 moremarbles added to a collectioneach day).

Create and explain tables that showa constant rate of change. (Mytable shows that on Day 6, therewere 48 marbles in the collection.)Day 1| 2 | 3 | 4 | 5 | 6 Marbles 8| 16| 24| 32| 40| 48Compare tables and graphs thatshow different constant rates ofchange (3 marbles added dailycompared to 2 marbles addeddaily, for example).

Write rules to describe patterns ofchange. (I started with 11 marblesin my collection. I get 3 moremarbles each day. So my rule forhow many marbles I have any dayis Number of days x 3 + 11.)


Pose questions that can be an -swered with data. (What kindsof games do we like to play?)

Organize data by categories.

Show data using line plots, bargraphs, double bar graphs, andother representations (see sidebaron page 19).

Describe the shape of the data,

including the highest and lowestnumbers, where data are concen -trated, the mode, the median, and

outliers (see glossary on page 17for more information).

Interpret data, using appropriatelanguage (less than half, notmany, almost all).

Compare 2 sets of data (such asthird graders and rst gradersanswers to the same question).

Third Grade

Congruent shapes

The area of this figureis 8 square units.

face

vertex

edgeRectangular prism

Capture on the 300 ChartTOOLS: 300 chart (a chart with the numbers from 1 to 300 listed in or -dersee the illustration of the 100 chart in the glossary for a similarexample), deck of plus/minus cards (cards with positive and negativenumbers), 30 chips, game piece for each player, recording sheet*

HOW TO PLAY:

Play in pairs or in 2 teams.

1. Place 30 chips on different numbers on the 300 chart. Deal 5plus/minus cards to each player or team and place the remain -ing cards face down on the table. Players put their game piecesanywhere on the 300 chart to start.

2. On each turn, a player tries to land on a square with a chip byusing any combination of plus/minus cards. For example, Briansgame piece is on 255 and theres a chip on 260. Brian uses a +4and a +1 card to move to 260. Players can use from 1 to 5 cards.

3. A player who lands on a square with a chip captures it by takingit off the board. Players can capture only 1 chip during a turn,and it must be from the square they landed on.

4. Players record their moves as an equation on the recording sheet.For example, if Kara begins on 145 and uses the cards +2, +10,-100, and +3, she records 145 + 2 + 10 - 100 + 3 = 60.

5. Place used plus/minus cards face down in a discard pile andreplace them with cards from the top of the deck. If the deck ofplus/minus cards is used up, shuf e the discard pile and turnit face down on the table to reuse. The rst player or team tocapture 5 chips wins.

* To play at home, ask your childs teacher for photocopies of a 300 chart and plus/minuscards, or make your own chart following the example in the glossary, and your owncards: four each of +1, -1; two each of +2, -2, +3, -3, +10, -10, +20, -20, +30, -30; oneeach of +4, -4, +5, -5, +40, -40, +50, -50, +100, -100, +200, -200.

SAMPLE GAME


12/20


NUMBERS


to 10,000.Identify place value to 5-digits.Recognize the pattern of placevalue (ones, tens, hundreds).

Identify fractional parts of awhole (such as 3/ 5 of a rectangle)and a group (such as 1/ 10 of theclass).

Recognize fractions that are equal

to 1 (such as 4/ 4) and greaterthan 1 (5/ 4), and change fractionsgreater than 1 to equivalent mixednumbers ( 5/ 4=1).Compare fractions with like de -nominators ( 3/ 8 is smaller than 7/ 8)and unlike denominators ( 3/ 4 islarger than 3/ 8). Order fractions ona number line.

Find decimal equivalents forsome commonly used fractions(=.75).

Read and write decimals to thehundredths place (.01). Relatedecimal places to place value inwhole numbers.

ADDING, SUBTRACTING,MULTIPLYING, DIVIDING

Use pictures or drawings on grid

paper to add fractions and mixednumbers (such as +1).

Use a variety of strategies to add3- and 4-digit numbers.* Forexample, change the numbers tomake the problem easier, thenadjust for the change. (1,852+688.Change 688 to 700, add the num - bers, then subtract the extra 12.)

Use a variety of strategies to

subtract 3-digit numbers.* Forexample, subtract in parts. (924-672. 924-600=324, 324-20=304,304-50=254, 254-2=252.)

Know some multiplication factsfrom memory, and mentallycompute other facts, up to 12x12.For example, a student mightthink, 6x12. 6 is half of 12. I know

FOURTH GRADEGRADUATES SHOULD...

that 6x6=36, so 6x12 is the same as36+36, which is 72.

Use a variety of strategies to multi-ply 2-digit numbers.* For example,create an equivalent problem bydoubling one factor and halving theother (factor: a number that dividesevenly into another number). (6x35.Double 35 to get 70, halve 6 to get 3.3x70 is the same as 6x35.)

Find the factors of 2-digit num -

bers and recognize patterns withthe factors (for example, 3x4=12,3x40=120, 30x40=1200). Findmultiples of 2-digit numbers andrecognize patterns with multiples(multiple: the product of a wholenumber and another whole num - berfor example, 8 is a multipleof 4 because 4x2=8).

Add and subtract multiples of 10and 100 (up to 1,000) with uency.

Use a variety of strategies to di-vide 2- and 3-digit numbers (633,33614).*

PROBLEM-SOLVING

Describe a range of strategies foraddition and subtraction and ex -plain why they work.Create and solve multiplicationstory problems.

Create and solve division storyproblems, with and without re -mainders, involving:

Sharing. (There are 55 valen -tine hearts and 22 students in

the class. How many does eachstudent get?)

Grouping. (There are 55 val -entine hearts. I want to put 3 ineach gift bag. How many gift bags can I ll?)

Use objects, models, a numberline, a 100 chart (see page 17 foran example), and a 1,000 chart

0 1 2

4 / 8 5 / 4

Problem: Janie had 24 baseball cards. She gave 1 / 8 of the cards to her sister and 3 / 8 of the cards to a friend. What fraction of

her cards did Janie give away?

1 / 8

3 / 8




13/20

13

to solve problems. Use arrays(objects or symbols laid out in

rows or drawn on grid paper) tovisualize multiplication problems(see page 17). Show strategies andsolutions using combinations ofwords, pictures, numbers, andsymbols.


Sort 2-dimensional shapes into

categories and subcategoriesaccording to their features. Forexample, identifya quadrilateral asa 4-sided closedgure; classifytrapezoids, paral-lelograms, rectangles, and squares(among other gures) as quadri -laterals; classify a square as a kindof rectangle.Know that a right angle measures90. Name other angles as acute(smaller than a right angle) or ob -tuse (larger than a right angle).

Using a right angleas a reference, iden-tify 30, 45, and 60angles.

Find the area (the amount of spaceinside a 2-D shape) of regularand non-regular polygons (2-Dgures made up of line segments)in square units and parts of squareunits.

Investigate the featuresof geometric solids( gures with length,width, and height, suchas a cone). Compare2-D representations ofsolids, such as nets

(see illustration), silhouettes, and/or drawings. Create 2-D repre -sentations from different perspec-

tives. For example, draw the front, back, and top of a building madeof cubes.

Identify lines of symmetry (theline or lines that can divide a g -ure into 2 mirror-image halves).

Find the volume(the amount ofspace inside a

3-D shape) ofcubes and rect -angular prisms.

Build and draw rectangularprisms (make a cube building,use an unfolded box) to visual -ize and calculate volume.

Find the volume by countingcubes and layers.

Mentally calculate volume.(12 cubes will t in the rstlayer of this box. The box willhave 2 layers when its foldedtogether, so the volume is24 cubes.)

Measure distance up to 100feet in U.S. units (inches, feet)and metric units (centimeter,meter). Choose appropriatetools (such as a yardstickrather than a ruler for longerdistances).

Measure perimeter (the sum ofthe length of the sides of a shape)and draw shapes that have agiven perimeter.

Fourth Grade

One kind ofquadrilateral

45

45

Non-regular polygonRegular polygon

area=8

area=8

Pyramid

Silhouette ofa pyramid

One example of a netan unfolded box

Rectangular prism

continued on page 16

Multiple Turn OverTOOLS: Calculators (optional), recording sheet, deck of multiple

cards (Basic game: numbers 2-50; Intermediate game: numbers2-80; Advanced game: numbers 2-113)*

HOW TO PLAY:

Play with a partner or a small group.

1. Deal out 10 multiple cards to each player.

2. Players arrange their multiple cards face up in front of them,visible to each player.

3. The player with the smallest multiple begins. This player callsout any whole number (except 1). Each player records thatnumber on his or her recording sheet.

4. All the players (including the player who called out the number)search for cards in their set that are multiples of that number. Theywrite those multiples on their recording sheet and turn those cardsface down. If a player has no multiples of the number called, thatplayer writes none under Multiple Cards I Turned Over.

5. Players take turns calling out numbers. The game is over whenone player has turned over all 10 multiple cards.

EXAMPLE: Jamal has the lowest number on a card, a 2, so hegoes rst. He calls out nine and all the players write 9 on theirrecording sheets. Sara has a card with 45, so she writes that onher recording sheet and turns the card face down. Jamal reviews hiscards and records a 36. Holly determines that she has no multiplesof 9, so she writes none under Multiple Cards I Turned Over.* To play at home, ask your childs teacher for photocopies of multiple cards and the

recording sheet.

SAMPLE GAME


14/20


NUMBERS


to 100,000.Identify the place value of thedigits in any 6-digit number.

Read, write, compare, and orderdecimals to the thousandthsplace (.001). Shade intenths, hundredths,and thousandths onrectangular grids to

visualize decimalplaces.

Order fractions with like andunlike denominators (, 4/ 8 , ,7/ 8), including halves, thirds,fourths, fths, sixths, eighths,tenths, and twelfths.

Know that fractions, decimals,and percents all represent parts of

a whole.Compare fractions, decimals,and percents and nd equiva -lents (1/8=.125=12.5%). Convert between fractions, decimals, andpercents to solve problems (3 outof 6, 3/ 6==50%).

ADDING, SUBTRACTING,

MULTIPLYING, DIVIDINGUse strategies to add 4-digitnumbers.* For example, createan equivalent problem that iseasier to solve. (1897+6831. Add3 to 1897, subtract 3 from 6831.1900+6828=8728.)

Use strategies to subtract 4-digitnumbers.* For example, change

one number and adjust for thechange. (3726-1584. Change 1584 to1600 by adding 16. 3726-1600=2126.2126+16=2142. 3726-1584=2142.)Add and subtract multiples of 100and 1,000 with uency.

Estimate answers to addition andsubtraction problems with largenumbers (100,000 and beyond).Use familiar strategies to solveproblems with large numbers.

Use strategies to add and subtractfractions and decimals.

Use strategies to multiply 2- and3-digit numbers.* For example, break up the factors (numbers

that divideevenly intoanothernumber) anduse a rectan-gular arrayto visualizethe problem(see illustra-tion).

Explore factors and multiples (mul-tiple: the product of a whole numberand another whole number).

Find all the factors of a num - ber. For example, the factors of

FIFTH GRADEGRADUATES SHOULD...

36 are 1, 2, 3, 4, 6, 9, 12, 18, and36all the numbers that divide

evenly into 36. Recognize a number as prime

(a number greater than 1 withexactly 2 factors: 1 and itself) orcomposite (a number with morethan 2 factors). Find the primefactorization of a number(expressing a number as theproduct of its prime factors). Forexample, the prime factorizationof 36 is 2x2x3x3.

Recognize square numbers(numbers multiplied by them -selves, such as 5x5 or 52 , and 6x6or 62).

Find multiples of a number andidentify patterns with multiples.For example, 20 is a multiple of4, so 200 is a multiple of 40.

Use strategies to divide numberswith 2-digit divisors (23756,33812).* For example, use knowl-edge of multiplication facts (up to12x12) to solve division problems.

0.1

1 / 2=5 / 103 / 5=6 / 10 6 / 10

1 / 2 + 3 / 5 = 11 / 10, or 11 / 10 Adding 1 / 2 and 3 / 5 with paper strips

Problem: 38x26

30

20 6

38

26

30x20600

8

600+180+160+48=988

30x6

180

8x648

20x8

160



Different ways to solve 18 x 14


15/20

15

PROBLEM-SOLVING

Describe and compare strategiesfor solving addition, subtraction,multiplication, and divisionproblems, and explain why thestrategies work.*

Use objects, models, a numberline, a 100 chart (see page 17 foran example), and a 1,000 chart tosolve problems.

Use arrays (objects or symbols laidout in rows or drawn on grid pa-per) to visualize multiplication anddivision problems (see page 17).

Use a clock face, a rectangle, or anumber line to visualize equiva -lent fractions and to add andsubtract fractions.

Show strategies and solutionsusing combinations of words, pic -tures, numbers, and symbols.


Explore the de ning character -istics of 2-dimensional shapesand classify some quadrilaterals(4-sided closed gures) in morethan one way. For example, con -sider whether or not all rectanglesare parallelograms (quadrilateralswith 2 pairs of parallel sides).

Use known angle measures (suchas 90) to identify angles that are30, 45, 60, 120, and 150.

Investigate area (the amount ofspace inside a 2-D shape) andperimeter (the sum of the lengthsof the sides).

Measure the perimeter of rect -angles and determine the area

by counting square units andparts of units.

Identify gures that have thesame area but different perim -eters, and the same perimeter but different areas.

Compare area and perimeter insimilar gures ( gures with thesame shape but not necessarilythe same sizesee page 17 foran example).

Explore volume (the amount ofspace inside a 3-D shape), usingobjects, folded paper, and draw -ings on grid paper.

Find the volume of pyramids,cylinders, and cones in cubicunits. (The volume of this cyl-inder is 12 cubic centimeters.)

Relate the volume of a rectan -gular prism to the volume of apyramid with the same heightand base size.

Recognize that changing vol -ume requires changes in thedimensions of length, width,and/or height.


Create graphs showing rates ofchange that are not constant (such

as the growth of a plant overtime). Compare them to graphsthat show a constant rate ofchange (such as adding 3 penniesto a jar each day). Reason aboutthe different shapes of the graphs.

Write a rule that describes asituation with a constant rate ofchange. For example, write a rule

to describe how the perimeter inthe illustration changes as eachsquare tile is added, such as P[perimeter] = (1 + n) x 2. (Theletter n stands for a changingnumber, the number of squares.)

Represent the same situation in atable, in a graph, and in an equa -tiona statement showing that2 mathematical expressions areequal, such as 2+2=4, or P =(1+n) x 2.

DATA AND PROBABILITY

Pose a question and conductan investigation that involvesexperiments with 2 groups or 2kinds of objects. (How long canadults and fth graders standon 1 foot? How much weightin pennies can 2 different kindsof paper bridges hold?) Designa procedure that can be appliedconsistently, and carry outrepeated trials.

Organize data (for example, usea table) and show data using

Fifth Grade

12

6 57

39 210

111

48

8 out of 12 hours

40 out of 60 minutes2 / 3=8 / 12=40 / 60

base

Close to 1TOOLS: Decimal cards sets A and B, recording sheet*

HOW TO PLAY:

Play with 1 or 2 other players.

The object of the game is to choose cards whose sum is as close to 1as possible.

1. Deal 5 cards in the middle. Each player uses any or all of thesame 5 cards to make a total that is as close to 1 as possible.

2. Taking turns, each player chooses cards and writes the numbersand the sum on the recording sheet.

3. Players total their scores. The score for the round is the differ-ence between a players sum and 1. (The sum can be greater orless than 1.)

4. When all players have a sum and a score, they compare results.

5. Put all 5 cards in the discard pile and deal 5 new cards.6. After 5 rounds, total the scores. The player with the lowest

score wins.

OTHER WAYS TO PLAY:

Follow the rules above with one or both of these variations:

Make and use 4 Wild Cards. Wild Cards can be any number. Give each player his or her own 5 cards.* To play at home, ask your childs teacher for photocopies of both sets of decimal cards and

the recording sheet.

SAMPLE GAME 0. 0 2 5 t w en t y

- f i v e

t ho usa nd t hs 0 .8 7 5 e i g h t h u n d r e d s e v e n t y - f i v e t h o u s a n d t h s

continued on page 16


16/20



Read and analyze graphs on acoordinate grid (see sidebar onpage 19).

Describe information on agraph, including where thegraph shows increases, decreas-es, or no difference in the rateof change. (The graph showsthat the bean plant grew slowlyfor the rst 3 days, then its rateof growth increased quickly.)

Understand what the steepnessof the line represents.

Make a graph on a coordinategrid using information from atable. Show different startingpoints on a graph.

Plot, analyze, and compare dataon a graph showing different ratesof constant change, and predictfuture data points. For example,Penny Jar A started with 6 pen -nies and 4 were added each day.Penny Jar B started with 0 pen -nies and 6 were added each day.

Will the jars ever have the sameamount of pennies?

Write a rule that describes thesituation shown on a graph or in atable, using words and/or num - bers and symbols. For example,a rule to describe the number ofpennies in Penny Jar A on anygiven day would be: 6 + (4 x n).(The letter n stands for a changingnumberthe number of the day.)

DATA AND PROBABILITY

Pose a survey question andgather data about 2 groups.(How many books did our classand the other fourth grade classread last month?)

Organize data (for example, usea table) and show data using lineplots, bar graphs, double bargraphs, and other representations.

Describe the shape of the datafor each group, including:

The highest and lowest numbers

The range (difference between

the highest and lowest numbers) Where data are concentrated

(Most of our class read be -tween 8 and 10 books.)

The mode (the number thatoccurs most frequently)

The median (the number thatwould fall in the middle if allthe numbers are listed in order)

Outliers (unusual data: Some -one in the other class read 20 books.)

Compare the 2 groups and make astatement supported by data. (Thedata show that our class readmore books overall. Even thoughsomeone in the other class read 20 books, that wasnt typical.)

Predict the likelihood of an event(tomorrow will be sunny, rollingan even number on a number cube,a spinner pointing to a 3).

Show the probability of events ona Likelihood Line (ImpossibleUnlikelyMaybeLikelyCertain)and on a 0-1 number line, with0 representing impossible and 1representing certain.

Compare experimental prob -ability (probability determinedthrough data collection, such astossing a coin 100 times to ndthe likelihood of getting tails) totheoretical probability (calculat -ing probability: the likelihood ofgetting tails is 50%).

line plots, bar graphs, double bargraphs, and other representa-tions (see sidebar on page 19).Group data into intervals whenappropriate (0-20 seconds, 21-40seconds).

Describe the shape of the datafor each group, including:

The highest and lowest num - bers

The range (difference betweenthe highest and lowest numbers)

Where data are concentrated(10 out of 15 times, Bridge Aheld between 20 and 39 pen -nies. Thats 2/ 3 of the time.)

The mode (the number thatoccurs most frequently)

The median (the number thatwould fall in the middle if allthe numbers are listed in order)

Outliers (unusual data: 1 timeBridge B held 70 pennies.)

Summarize the data for the 2groups, compare them, and makestatements supported by data.

Compare theoretical prob -ability (probability determinedthrough calculations) to theactual outcomes of many trials(experimental probability). Forexample, determine the likelihoodof getting an even number whenrolling a number cube, then rollthe number cube and record howmany times the result is an evennumber.

Show probability as a fraction, adecimal, or a percent. (There are

3 even numbers and 3 odd num - bers on the number cube. So thereis a 3 out of 6 chance of getting aneven number, or 1 out of 2, or .)

Fourth Grade continued from page 13 Fifth Grade continued from page 15


17/20

17Glossary

100 chart, 1,000 chart: A table in whichconsecutive whole numbers are arrangedin rows.

area: The measure, in square units, of thesurface of a 2-dimensional shape.

array: An arrangement of objects orsymbols laid out in rowsor drawn on grid paper tovisualize multiplication.

combination: Two numbers that arecombined through an operation to equalanother number. For example, 6 and 4 areone addition combination for 10; 7 and 5

are one multiplication combination for 35.composite number: A whole number withmore than 2 different whole-number fac -tors, such as the number 14. (The number1 is neither prime nor composite.)

congruent, congruence: Having the samesize and shape, althoughpossibly in differentpositions. Such shapeshave congruence.

coordinate graph: A 2-dimensional graphmade up of a horizontal x axis and avertical y axis and ordered pairssuch as(3,2) in the illustration on page 19rep -resenting values. Coordinate graphs areused to show the relationship between 2variables, such as length and width, ordistance traveled and time.

data: Information, in the form of facts orgures, that can help answer questions.

edge: The line seg-ment where 2 facesof a 3-dimensionalgure meet.

equation: A state-ment showing that 2 mathematicalexpressions are equal, such as 2+2=4, or P= (1+n) x 2.

equivalent: Of equal value. For example,, .75 and 75% are equivalent.

face: A at surface (side) on a 3-dimen -sional shape (see illustration above).

factor: A number that divides into anothernumber without leaving a remainder. Forexample, 1, 3, 5, and 15 are all factors of15 because all divide evenly into 15.

geometric solid: A gure with length, width,and height, such as a cone or a cube.

grid: Any pattern of crisscrossing linesthat creates squares, such as the patternon graph paper.

line plot: A graph that shows data on anumber line with an x or another mark toshow frequency (see page 19).

median: The number that would fall in themiddle if all the numbers in a data set arelisted in order.

mixed number: A number that includes awhole number and a fraction, such as 1.

mode: The number that occurs most fre -quently in a data set.

multiple of a whole number: The product

of the whole number and another wholenumber. For example, the multiples of 4include 4 (4x1), 8 (4x2), 12 (4x3), 16 (4x4).

n: A letter commonly used to representan unknown or changing number. Forexample, if 4 pennies are added to apenny jar each day, a rule for the number of pennies in the jar on any givenday could be 4 x n , where n stands forthe number of the day.

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

GLOSSARYThis glossary is intended to provide information for parent readers. It is notintended to provide complete, technical de nitions of mathematical terms.

net: A 2-dimensional repre-sentation of a 3-dimensional

shape, such as the unfolded box in the illustration.

number line: A model in which numbers areshown as marked points.

operation: A process or action performed ona number or numbers. Addition, subtrac -tion, multiplication, and division are allexamples of operations.

outlier: An unusual data point, such as avalue much higher or lower than the othervalues in the data set.

parallelogram: A 4-sided gure(a quadrilateral) with oppo-site sides that are parallel.

perimeter: The sum of the length of the sidesof a shape.

polygon: A 2-dimensional, closed shapemade up of 3 or more connected line seg-ments that do not cross over each other.

prime factor: A factor that cant be dividedevenly into smaller whole numbers. Forexample, 2 and 3 are prime factors of 12.

prime factorization: Showing a whole numbergreater than 1 as the product of its primefactors. For example, the prime factoriza -tion of 12 is 2x2x3. Each whole number

can be written as the product of its primefactors in only one way.

prime number: A whole number greater than1 with exactly 2 different factors, 1 and thenumber itself, such as the number 7. (Thenumber 1 is neither prime nor composite.)

probability: A number that indicates the rel -ative likelihood that an event will happen.Experimental probability is determinedthrough data collection. For example, if a basketball player succeeded in making 20 baskets out of 80 free throws, her experi -mental probability of making a basketwould be 20 out of 80, or 25%. Theoreti-cal probability is found by analyzing thepossibilities. Because we know there are 2possible outcomes for a coin toss and 1 ofthose is tails, we calculate the probabilityof getting tails to be 1 out of 2, or 50%.

product: The result of multiplying 2 numbers.For example, 12 is the product of 6 and 2.

proportional: Having the same proportions, but not necessarily the same size.

quadrilateral: A 4-sidedclosed gure.

range: The difference between the highest and lowest numbers in

a data set.ratio: A comparison between 2 quantitiesthat shows the size of 1 relative to the sizeof the other. For example, if a cake recipecalls for 2 eggs, 2 cakes would require 4eggs. The ratio is 2:1 (2 eggs to 1 cake).

representation: The format for displayinga problem and/or its solution, such as agraph, a table, a model, a picture, an equa -tion, or a written description.

similar, similarity: Thesame shape butnot necessarily thesame size. Shapesare similar, or havesimilarity, whentheir corresponding sides are proportionaland their corresponding angles are equal.

square number: A number that is the productof a whole number multiplied by itself. Forexample, 9 is a square number because 3x3equals 9.symmetry: A characteristic of shapes thathave balanced or corresponding elements,such as a butter y or a windmill. Line ormirror-image symmetry refers to shapeswith mirror images (a butter ys wings).Rotational symmetry refers to shapes thatwould match exactly if they were rotated(the blades of a windmill).

unit: When referring to a pattern, the part

that repeats. For example, in the patterns s s s s s , the unit is ss s .

vertex: The point where 2 or more lines,rays, or line segments meet (intersect) toform an angle. Plural form: vertices.

volume: The measurement of the spaceinside a 3-dimensional shape.

face

vertex

edge

Array for 2 x 4

Proportional rectangles

2 c m

5 c m4 cm

10 cm

r a yvertex

0 21 43 65 87 109


18/20


A student is given this problem:

There are 38 rows in an audito-rium with 26 chairs in each row.How many people can sit in theauditorium?

If we could look into her mind, wemight see something like this:

Hmmmwhat is this problem askingme to do? Multiply. I can make iteasier by changing the numbers anddrawing an array[see illustration

below] to keep track of what Imdoing. 30 x 20 is an easier problem.3 x 2 = 6, so 30 x 20 = 600. Thats30 rows with 20 people in each row.There are 30 more rows with 6 chairs,30 x 6. 3 x 6 = 18, so 30 x 6 = 180.There are still 8 rows. 8 rows with 20 people would be 8 x 20, which is 160.Then 8 more rows with 6 peoplethats 8 x 6 = 48. My array helps me

check that I multiplied all the parts ofthe problem. Now I have to add all thenumbers together. 600 + 180 + 160+ 48 = 988. 988 people can sit in theauditorium. I knew that the answerwould be a big number because I wasmultiplying double-digit numbers.Also I could picture all those chairs inan auditorium.

While there are many ways tosolve this problem, the exampleabove shows the kind of mathemat -

ical thinking that students developin Investigations classrooms.Throughout their elementaryyears, students try out strategies toadd, subtract, multiply, and dividenumbers (operations), and de -velop problem-solving skills (suchas identifying the type of problem,comparing strategies, or usingmodels). During class, they areasked to share their reasoning andtheir solutions so everyone sees avariety of ways. By the end of fthgrade, the goal is for each studentto be comfortable using a range ofstrategies, including mental strate-gies, to solve problems accuratelyand ef ciently.

Parents may wonder why teach-

ers dont simply show studentssteps to follow that produceright answers, as in theexample on the right. InInvestigations , traditionalmethods for operationsare taught after studentshave an understanding of hownumbers are represented in manyways. Rather than memorizing

a procedure, the students focusis on the underlying mathemat-ics. When students understandwhy a strategy works based onmathematical principles, they canuse the strategy for more complexproblems, or adapt the strategyfor a different operation. By con -trast, a student who is taught tofollow the steps of a traditional

procedure has only learned torepeat those steps with a problemthat looks exactly the same.

In the example above, considerwhat the students thinkingshows about her knowledgeof operations, place value, andproblem-solving, all of which can be applied to solving other prob -lems. The student:

Figured out what kind of prob -lem it was (multiplication)

Chose a strategy (changing thenumbers by breaking apart thefactors38 and 26)

Chose a conceptual tool to visu -alize the problem (an array)

Used an understanding of placevalue to further simplify theproblem (3 x 2 = 6, so 30 x 20

= 600) Used the tool to check work

midway (My array helps mecheck that I multiplied all theparts of the problem.)

Understood that the strategy of breaking apart the factors re -quires multiplying all the parts

Compared the answer to anestimate, and used a real-worldreference, to see if the answerseemed reasonable (estimatingthat multiplying multi-digitnumbers would result in a largenumber, visualizing chairs inan auditorium)

Some of the strategies studentslearn are outlined here, with

STRATEGIESFOR SOLVING PROBLEMS

examples. For more information,ask your child to tell you what

he or she is doing while solving aproblem, ask your childs teacherto show you and explain, or seethe Student Math Handbook foryour childs grade. Note:Childrenare not necessarily required towrite out all the steps shown here.

ADDING

Count on from one number. Example: 3 + 3

4 5 6 3 + 1, 2, 3 3 + 3 = 6

Add by place value.Example: 258 + 392 258

+392

500 (hundreds added)140 (tens added)

+ 10 (ones added)650

ORAdd the hundreds: 200 + 300 = 500 Add the tens: 50 + 90 = 140 Add the ones: 8 + 2 = 10 Add them together: 500 + 140 + 10 = 650

Add one number in parts.Example: 321 + 258 258 = 200 + 50 + 8 321 + 200 = 521 521 + 50 = 571 571 + 8 = 579

Change one number to makethe problem easier and adjustfor the change. Example: 1852 + 688 Add 12 to 688 to get 700. 1852 + 700 = 2552 Subtract 12: 2552 - 12 = 2540

Change both numbers to createan equivalent problem. Example: 1897 + 6831 Add 3 to 1897 to get 1900. Subtract 3 from 6831 to get 6828. 1900 + 6828 = 8728, so 1897 + 6831 = 8728.

Problem: 38x26

30

20 6

38

26

30x20600

8

600+180+160+48=988

30x6

180

8x648

20x8

160

38x26

228

988

76

41


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SUBTRACTING

Count back.Example: 9 - 5

1 2 3 4 5 9 8 7 6 5 4 9 - 5 = 4

Subtract in parts. Example: 57 - 23 57 - 20 = 37

37 - 3 = 34Add up from the lower number.Example: 72 - 37 Add 3 to 37 to get to the

closest tens: 40 Add 30 to get to the tens closest

to the target number: 40 + 30 = 70 Add 2 to get to the target

number: 70 + 2 = 72 Add the added numbers

together: 3 + 30 + 2 = 35 72 - 37 = 35

Subtract by place value.Example: 4355 - 2216 4000 - 2000 = 2000 300 - 200 = 100 55 - 16 = 39 2000 + 100 + 39 = 2139 4355 - 2216 = 2139

MULTIPLYING

Multiply smaller numbers andadd the results. Example: 6 x 8 5 x 8 = 40 Add one more group of 8 (1 x 8). 40 + 8 = 48, so 6 x 8 = 48.

Change one number to makethe problem easier and adjustfor the change. Example: 27 x 30 30 x 30 = 900 30 - 27 = 3 (amount of the change) 3 x 30 = 90 (extra that must be adjusted) 900 - 90 = 810 (subtracting the extra) 27 x 30 = 810

Change both numbers to createan equivalent problem. Example: 6 x 35 Double one factor and halve the other:double 35 to get 70, halve 6 to get 3.3 x 70 = 210, so 6 x 35 = 210.

DIVIDING

Multiply to reach the target

number. Example: 156 13 13 x 10 = 130 156 - 130 = 26 (26 remains after

multiplying by 10) 13 x 2 = 26 10 + 2 = 12 (adding the numbers that

were multiplied by 13: 10 and 2) 12 x 13 = 156, so 156 13 = 12.

Divide smaller numbers and addthe results. Example: 156 13 156 = 130 + 26 130 13 = 10 26 13 = 2 10 + 2 = 12 (adding the numbers found

by dividing: 10 and 2) 156 13 = 12

19Strategies for solving problems

1 2 3 4 50

1

2

3(3,2)

FICTION

fairy talesscience fiction

NONFICTION

books aboutmath

stories aboutreal people

0 1 2 3 4 5 6 7 8

X X X X X X X X X

XX

Number of pockets worn by students

GRAPHING Students learn to read, interpret, compare, and show data onvarious kinds of graphs, including the examples below.

Outside Inside

Location

Where do we like to play?

Grade 3

Grade 1

N u m

b e r o

f s t u d e n t s

0

5

10

15

20

Venn diagram Ordered pair on a coordinate graph

Double bar graph

Line plot

Coordinate graph showingconstant rate of change

Table and coordinate graph showing rate of change that is not constant

0T F S S M T W T F S S M

5

10

15

20

25

DayThursdayFriday

MondayTuesdayWednesdayThursdayFridayMonday

Growth of abean plant

Height3 cm4 cm

6 cm9 cm11 cm14 cm16 cm24 cm

12

3

2

2

8

3

3

0

6

9

12

Day 1 N u m

b e r o f p e n n

i e s

Day 2 Day 3 Day 4


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Published by the Math & Science CollaborativeThis material is based on work supported by the National Science Foundation under

Grant No. EHR-0314914. Any opinions, ndings, conclusions, or recommendationsexpressed in this publication are those of the authors and do not necessarily reect

the views of the granting agency.

Math & Science CollaborativeAllegheny Intermediate Unit

475 East Waterfront DriveHomestead, PA 15120

Phone: 412.394.4600Fax: 412.394.4599

www.aiu3.net/msc

Elementary Investigations 2009

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