Math Booster Pack - Springfield School District Booster Pack Reading Improvement ... Ideas for...

Student Owned Strategies for Readingas Thinking in the Content Areas

Math Booster Pack

Reading Improvement Seriesoffered through an ODE/OAESD Collaborative

Student

Owned

Strategies

S.O.S.

S. O. S.

What is a Booster Pack,

and How Do I Use It?

This Booster Pack is a collection of “next step” resources you may find useful when providingongoing staff development in the S.O.S. reading series. As a facilitator, select items that youfeel would be most appropriate for the audience and length of time provided for each session.

Think about the teachers you will work with this year:

• Perhaps they need a “refresher” on some of the key reading strategies you shared fromthe first 5 modules – with content-specific examples.

• Are they interested in learning a few new strategies to use with their students to boostreading comprehension?

• Teachers who have worked with the S.O.S. strategies in their classrooms for a yearshould be ready to discuss lesson design and how to incorporate key strategies into theirinstruction.

Resources to help you meet these objectives are included in this packet in four sections:

1. Discussion starters: “What is math literacy?”

2. A refresher of several S.O.S. reading strategies with content specific examples

3. A sampling of new reading strategies and graphic organizers

4. Ideas for lesson planning and sample lessons provided

Remember, the most effective staff development occurs when the strategies aremodeled, practiced and applied. So… have fun with the ideas in this BoosterPack as you help teachers support successful reading in the content areas!

Math Booster Pack

Table of Contents

Page

Discussion Starters and Background Information............................................................. 1

Reading Strategy Refresher and Examples ....................................................................... 5

Pre Reading ...................................................................................................................... 6- Anticipation Guide- Think Aloud- KWL

Vocabulary ...................................................................................................................... 12- KAU- Concept Definition Map- Frayer Model- Word Splash- Multiple Meaning Words/Symbols & Prefix / Suffix

Reading for Information .................................................................................................. 24- Graphic Organizers

Additional Reading Strategies and Graphic Organizers ................................................. 29- Words Sorts- KNWS- Five-Step Problem Solving- Verbal and Visual Word Association- Three-Level Guides- Semantic Mapping- Notetaking Graphic Organizer

Putting it All Together – Creating a Lesson Using Reading Strategies ............................. 45

References ........................................................................................................................... 53

Concept Definition Map

What is it? (category)

What is it like?

IllustrationsWhat are some examples?

Comparisons

Math Literacy

Things to Think About

1. What is math literacy?

2. What components of math instruction help students become science literate?

3. What are the potential obstacles to math literacy in 6-12 education, and whatcan be done to address them?

Reading Mathematics is Challenging!

• Students must read from left to right, but also from right to left (integernumber line), from top to bottom or vice versa (tables), and evendiagonally (some graphs).

• Mathematics texts contain more concepts per word, per sentence, andper paragraph than any other kind of text. The abstract concepts are oftendifficult for readers to visualize.

• Students must be proficient at decoding not only words but also numericand nonnumeric symbols. The math reader must shift from words like“plus” or “minus” to instantly recognizing their symbolic counterparts,+ and – .

Adapted from: Barton, Mary Lee and Clare Heidema.Teaching Reading in Mathematics A Supplement to Teaching Reading in the Content Areas, 2002

Mathematics Literacy

"The development of a student’s power to use mathematics involveslearning the signs, symbols, and terms of mathematics. This is bestaccomplished in problem-solving situations in which students have anopportunity to read, write and discuss ideas in which the use of thelanguage of mathematics becomes natural. As students communicate theirideas, they learn to clarify, refine, and consolidate their thinking"(National Council of Teachers of Mathematics)

"Reading mathematics means the ability to make sense of everything thatis on a page — whether the page is a worksheet, a spreadsheet, anoverhead transparency, a computer screen, or a page in a mathematicstextbook or journal — in other words, any resource that students might useto learn and apply mathematics." (Teaching Reading in the Content Areas)

In addition to general reading skills needed to comprehend narrativetext, readers of math text must also be able to apply the followingknowledge and skills:

⇒ Understand specialized vocabulary and phrases unique to math

⇒ Understand vocabulary terms and phrases that have different meanings whenused in math

⇒ Interpret words, numeric and nonnumeric symbols

⇒ Recognize and understand organizational patterns common to math texts

⇒ Make sense of text using text structure and page lay-out that may not be userfriendly

⇒ Infer implied sequences and recognize cause-and-effect relationships

⇒ Use inductive and deductive reasoning skills

Comprehension Strategies

RespondTo the Ideas in the Text

Think While You ReadTo Keep Track of Whether Things Make

SENSE!

Use Fix-Up StrategiesWhen Things Don't Make Sense

Activate PriorKnowledge andSet A Purpose

for Reading

Figure OutWhat is

Important

OrganizeKnowledge

MakeInference

Find out theMeanings of

Unknown Words

Ask Questions

Visualize

?

from Irvin, J.L. Reading Strategies in the Social Studies Classroom, Holt Rinehart and Winston, 2001

EFFECTIVE READING BEHAVIORS INEFFECTIVE READING BEHAVIORS

Before Reading Preview text Build background information Think about key words or phrases

Before Reading Start reading without thinking about the topic Do not preview text for key vocabulary Do not know purpose for reading Mind often wanders

During Reading Adjust reading for different purposes Monitor understanding of text and use

strategies to understand difficult parts. Integrate new information with existing

knowledge

During Reading Read different texts and for different tasks all the

same Do not monitor comprehension Seldom use any strategies for understanding

difficult parts

After Reading Decide if goal for reading has been met Evaluate comprehension Summarize major ideas in a graphic organizer

or by retelling major points Apply information to a new situation

After Reading Do not know content or purpose of reading Read passage only once and feels finished Express readiness for a test without studying

Adapted from: Irvin,J.L. Reading Strategies in the Social Studies Classroom, Holt, Rinehart and Winston, 2001

S.O.S. Reading Strategies“Refresher”

withContent Specific Examples

Adapted from: Billmeyer, Rachel and Mary Lee Barton. Teaching Reading in the Content Areas: If Not Me, Then Who? Aurora: McREL (Mid-continent Regional Education Laboratory), 1998

Anticipation GuideAnticipation Guides can be used to activate and assess students' prior knowledge, tofocus reading, and to motivate reluctant readers by stimulating their interest in the topic.Because the guide revolves around the text's most important concepts, students areprepared to focus on and pay attention to read closely in order to search for evidence thatsupports answers and predictions. Consequently, these guides promote active readingand critical thinking. Anticipation Guides are especially useful in identifying anymisperceptions students have so that the teacher can correct these prior to reading.

How to use:

1. Identify the major concepts that you want students to learn from reading. Determine waysthese concepts might support or challenge the students' beliefs.

2. Create four to six statements that support or challenge the students' beliefs andexperiences about the topic under study. Do not write simple, literal statements that can beeasily answered.

3. Share the guide with students. Ask the students to react to each statement, formulate aresponse to it, and be prepared to defend their opinions.

4. Discuss each statement with the class. Ask how many students agree or disagree witheach statement. Ask one student from each side of the issue to explain his/her response.

5. Have students read the selection with the purpose of finding evidence that supports ordisconfirms their responses on the guide.

6. After students finish reading the selection, have them confirm their original responses,revise them, or decide what additional information is needed. Students may beencouraged to rewrite any statement that is not true in a way that makes it true.

7. Lead a discussion on what students learned from their reading.

Anticipation Guide

Directions: In the column labeled Before, place a check next to anystatement with which you agree. After reading the text, compareyour opinions on those statements with information contained in thetext.

Before After_____ _____ 1. Multiples relate to multiplying and divisors relate

_____ _____ 2. 0 is a multiple of any number.

_____ _____ 3. 0 is a divisor of any number.

_____ _____ 4. Multiples of 2 are called even numbers.

_____ _____ 5. Multiples of 1 are called odd numbers.

(Pre-Reading Module p. 4)

Adapted from: Barton, Mary Lee and Clare Heidema. Teaching Reading in Mathematics:A Supplement to Teaching Reading in the Content Areas, 2002

Examples

Anticipation GuideStatistics

Directions: In the column labeled Before, place a check next to any statementwith which you tend to agree. After reading the text, compare your opinions aboutthose statements with information contained in the text.

Before After

_______ _______ 1. There are several kinds of averages for a set of data.

_______ _______ 2. The mode is the middle number in a set of data.

_______ _______ 3. Range tells how far apart numbers are in a set of data.

_______ _______ 4. Outliers are always ignored.

_______ _______ 5. Averages are always given as percents.

Anticipation GuideIntegers

Directions: In the column labeled Before, place a check next to any statementwith which you tend to agree. After reading the text, compare your opinions aboutthose statements with information contained in the text.

Before After

_______ _______ 1. The sum of two integers is always greater than bothof the numbers being added.

_______ _______ 2. It is possible to add two integers and get a sum lessthan zero.

_______ _______ 3. The sum of zero and any other integer is always theother integer.

_______ _______ 4. The product of two integers is always greater thanboth of the numbers being multiplied.

_______ _______ 5. The product of two positive integers is always positive.

_______ _______ 6. The product of two negative integers is alwaysnegative.

∧ Consider using one or more of these questions as you model your own use of readingstrategies with students.

1. Before you begin a reading assignment for math, do you leaf through the passage andread the headings to see what the passage is about?

2. Why might it be helpful to think about what you already know about a topic before readingabout it?

3. When you have to read something for math, do you make sure you understand thepurpose for reading it? What difference would this make?

4. If you thought a topic in your math text was going to be difficult to understand, what couldyou do before you started reading to help you understand?

5. How is reading in math class different from reading in English class?

6. Should you stop and think about why you are reading? Why? When should you do this?

7. How do you know if you've really understood a reading assignment for math class?

8. What can you do if you are reading and don't understand what a sentence is about? Howwould you decide what to do?

9. What do you do when you come to a big word in your math text that you don't know?

10. Are there times when it becomes difficult to understand what you're reading? What makesyou realize it is becoming more difficult? What strategies do you use to read difficult text?

Think Aloud(Intro Module p. 6)

Adapted from: M.T. Craig and L.D. Yore, "Middle School Students' Awareness of Strategiesfor Resolving Comprehension Difficulties in Science Reading,", 1996

K-W-LThe basic K-W-L uses three columns in which to write down information that we Know(background knowledge), Want to know (establishing purpose and asking questions), andhave Learned (main idea). In addition to teaching students to connect to backgroundknowledge, this activity also can develop habits of summarizing, questioning, predicting,inferring, and figuring out word meanings.

KWhat I know

WWhat I want to

find out

LWhat I learned

Example: Fiboriacci’s Sequence

KWhat I know

WWhat I want to

find out

LWhat I learned

Fibonacci’s Sequence1, 1, 2, 3, 5, 8, 13, 21 …

Fibonacci’s Rabbits

Multiplying populations

How do bees fit in theFibonacci pattern?

What is the connectionbetween the Fibonacci’ssequence and the Goldenratio?

Is there a formula for theFibonacci numbersequence?

What do pineapples andpinecones have to do withFibonacci?

Adapted from: Billmeyer, Rachel and Mary Lee Barton. Teaching Reading in the Content Areas: If Not Me, Then Who?Aurora: McREL (Mid-continent Regional Education Laboratory), 1998.


K-W-L Worksheet: Prime Numbers

KWhat I know

WWhat I want to

find out

LWhat I learned

• A prime number hasexactly two divisors(factors), 1 and itself.

• 2 is the only evenprime number.

• Successive oddnumbers that are bothprimes are twinprimes:

- 3 and 5- 5 and 7- 11 and 13

• Why are primenumbers so important?

• What is the sieve ofEratosthenes, and howdo you use it to getprimes?

• Is there a connectionbetween primenumbers and perfectnumbers?

• What is an emirp?

• What are somepatterns related toprime numbers?

K-W-L Worksheet: Tessellations

KWhat I know

WWhat I want to

find out

LWhat I learned

• What a tessellation is

• Squares, equilateraltriangles, regularhexagons can be usedfor a tessellation.

• You cannot use aregular pentagon for atessellation.

• What combination ofshapes can be used ina tessellation?

• What is meant by acode for atessellation?

• What are someirregular shapes thattessellate?


K-W-L Worksheet: Estimation

KWhat I know

WWhat I want to

find out

LWhat I learned

I know how to estimateanswers to mathproblems in a rough way.

• What is front-endestimation?

• What is the mentalmath strategy oftrading off?

• Front-end estimationis a way ofapproximating ananswer to a mathproblem.

• Trading off is anaddition strategy thatinvolves rounding thenumbers in a problemto the nearest 10 tomade addition easier.

K-W-L Worksheet: Order of Operations

KWhat I know

WWhat I want to

find out

LWhat I learned

I know how to add,subtract, multiply, anddivide.

I know that these are allperformed from left toright.

• What is anexpression?

• What is a numericalexpression?

• What is a variableexpression?

• What are the rules forordering operations?

• An expression is acollection of numbers,variables, andsymbols.

• A numericalexpression has allnumbers and symbols.

• A variable expressionincludes variables.

• Multiply and dividefrom left to right. Addand subtract from leftto right.

Adapted from: Content Area Guide: Math, Readers Handbook: A Student Guide for Reading and LearningGreat Source, 2002.

Eight Principles of Vocabulary Instruction

1. Be enthusiastic about content area language and the power it can offer tostudents who understand how to use these words effectively.

2. Remember that learning involves making connections between what wealready know and new information. Relate new vocabulary words toexperiences and concepts that students already know.

3. Limit the number of words taught in each unit; concentrate on keyconcepts.

4. Teach concepts in semantically related clusters, so that students canclearly see associations among related concepts.

5. Model how to use graphic organizers.

6. Allow students enough practice in working with strategies and graphicorganizers so that their use becomes habit.

7. Use dictionaries and glossaries appropriately.

8. Repeatedly model how to determine a word's meaning in text materials.Observing the process you use will help students know what to do when theyencounter unfamiliar words outside of the classroom.

Adapted from: Billmeyer, Rachel and Mary Lee Barton. Teaching Reading in the Content Areas: If Not Me, Then Who?Aurora: McREL (Mid-continent Regional Educational Laboratory), 1998

K-A-U Vocabulary Strategy

K = Known ------- A = Acquainted -------U = Unknown

Before students read, the teacher presents a list of key words related to the topic of study. Thestudents analyze what they know about each word individually and the degree to which wordsare known or unknown. It is easy to do this with the symbols of a +, , or — . This activityleads naturally to the preteaching of key vocabulary to be used later in the reading.

Examples:

K A U+ —

mean

median

mode

weighted average

line of best fit

correlation

range

K A U+ —

exponent

intersection

domain

intercept

slope

parabola

origin

K A U+ —

polygon

rectangle

pentagon

trapezoid

prism

polyhedron

cone

(Intro Module p. 1)


K A U+ —

base

power

variable

terms

equivalent

Adapted from: Vacca, R.T. and Vacca, J.L.. Content Area Reading, 1996

Concept Definition Mapping

What is it?

Concept Definition Mapping is a strategy for helping students learn the meaning of keyconcepts, essential attributes, qualities, or characteristics of a word. Students must describewhat the concept is, as well as what it isn’t, and cite examples of it. Looking up the concept’sdefinition in the dictionary is not nearly as effective as this process, which gives students amore thorough understanding of what the concept means, includes, and implies. The mappingprocess also aids recall.

How to use it:

1. Share an example of a Concept Definition Map with students with a key vocabulary wordor concept you are studying.

2. Discuss the questions that a definition should answer:• What is it? What broader category or classification of things does it fit into?• What is it like? What are its essential characteristics? What qualities does it possess

that make it different from other things in the same category?• What are some examples of it?

3. Model how to use the map by selecting a familiar vocabulary term from a previous unitand have students volunteer information for the map. For instance, a science teachermight choose the concept migration. “What is it like?” responses might include“seasonal,” “movement from one area to another,” “animals looking for food andfavorable climate to raise their young.” Examples could include Canadian geese,whales, monarch butterflies, and elk.

4. Have students work in pairs to complete a map for a concept in their current unit ofstudy. They may choose to use a dictionary or glossary, but encourage them to use theirown experience and background knowledge as well.

5. After students complete their maps, instruct them to write a complete definition of theconcept, using the information from their maps.

(Vocabulary Module p. 21)


What is it like?


Comparisons



includes 0but not 1


Properties

ones digit is0, 2, 4, 6, or 8

Examples

474

Comparisons

Even

12

Prime

2 is only evennumber thatis a prime

multiple of 2 skipcount starting at 0

58

Odd

Classification ofnumbers

What is it? (category) Properties

n% of A is the sameas A% of n

Examples

discounts

Comparisons

Percent

interest rate

Fraction

additive when base issame: 70% of 130 = 50% of

130 + 20% of 130

percents can be written infraction or decimal form

Number concept fractionwith denominator 100

(per hundred)

benchmark percents10% 25% 50%

test scores

Ratio

What is it? (category) Properties

has a line ofsymmetry


Comparisons

Isosceles

Triangles

Scalene

pair of equalangles

(congruent)

two sides ofequal length(congruent)

Trapezoids

Equilateral(regular)

Geometric propertyShape classification

2 scale divergentor convergent


PropertiesWhat is it like?

2 side parallel

4 sides


Funnel topChimney on housewith sloped roof

Comparisons

TrapezoidRectangleSquare

Some table tops

Quadrilateral

Square

Diagonals: congruent,bisect each other,& perpendicular


PropertiesWhat is it like?

4 angles are congruentand right (90 degrees)

All four sides arecongruent


different colorson chess board

Comparisons

computerdisk

Quadrilateral

rectangle

Adapted from: Carol Santa. Project CRISS: Creating Independence Through Student-Owned Strategies

Adapted from: Vacca, R.T. and Vacca, J.L.. Content Area Reading, 1996

Frayer Model

What is it?

The Frayer Model is a word categorization activity. Frayer believes learners develop theirunderstanding of concepts by studying them in a relational manner. Using the Frayer model,students analyze a word’s essential and nonessential attributes and also refine theirunderstanding by choosing examples and non-examples of the concept. In order to understandcompletely what a concept is, one must also know what it isn’t.

How to use it:

1. Assign the concept or word being studied.

2. Explain all of the attributes of the Frayer Model to be completed.

3. Using an easy word, complete the model with the class. (examples follow)

4. Have students work in pairs and complete their model diagram using the assignedconcept or word.

Example:Essential Characteristics

Set of ordered pairs with no twopairs having the same firstelement

Has a domain and range

Nonessential Characteristics

May be one-to-one

May be linear(has a straight line graph)

Inverse may be a function

Examples

f(x)= 2x + 1

y = _ x _

Area of a circle with given radius

Nonexamples

y < x

perimeter of a rectangle withgiven area



Function

Frayer Model

Definition (in own words) Facts/Characteristics

Examples Nonexamples

Example:Definition (in own words)

A simple, closed, plane figuremade up of three or more linesegments

Facts/Characteristics

• Closed

• Simple (curve does notintersect itself)

• Plane figure (2 dimensional)

• Made up of three or more linesegments

• No dangling parts

Examples

• Rectangle

• Triangle

• Pentagon

• Hexagon

• Trapezoid

Nonexamples

• Circle

• Cone

• Arrow (ray)

• Cube

• Letter A

Adapted from: D.A. Frayer, W.C. Frederick, and H.G. Klausmeier, “A Schema for Testing the Level of Concept Mastery”,University of Wisconsin

WORD

Polygon

Frayer Model

Definition (in own words)

A whole number with exactlytwo divisors (factors)


• 2 is the only even primenumber.

• 0 and 1 are not prime.

• Every whole number can bewritten as a product of primes.

Examples

2, 3, 5, 11, 13, . . .

Nonexamples

1, 4, 6, 8, 9, 10, . . .

Definition (in own words)

A whole number that dividesexactly into a given wholenumber

A polynomial by which a givenpolynomial is divisible


Every whole number has atleast two factors.

Every whole number can bewritten as a product of primefactors.

Examples

Factors of 12 are1, 2, 3, 4, 6, and 12

(x + 1) and (x — 1) arefactors of x_ — 1

Nonexamples

• 5 is not a factor of 12.• 0 is not a factor of any whole

number.• (x + 1) is not a factor of x_ + 1

Adapted from: D.A. Frayer, W.C. Frederick, and H.G. Klausmeier, “A Schema for Testing the Level of Concept Mastery”,University of Wisconsin

Factor

Prime

Word Splash(As described in the ASCD video:

Prereading Strategies for the Content Areas)

Word Splash is designed to help students access prior knowledge of words, personally constructmeaning for the words related to the concept, and allows for a repetition of key ideas important to thenew unit of study.

This strategy is used at the beginning of a unit to activate prior knowledge and introduce students tonew words related to the topic.

Step One: Brainstorm, Predict and Write

• Introduce 6-7 words key to developing a conceptual understanding of the unit topic.

• Arrange the words on a board so that they can be rearranged later.

• Students write complete sentences using 3 of the words demonstrating their understanding of thewords.

• Large group share out of a few of the sentences.

Step Two: Explore Word Relationships

• Tell the class that one of the words is the “all-encompassing” word and the rest fit under it.

• Have students arrange the words in a graphic that makes sense to them or use word cards.

• A few students come up and rearrange the words on the board then explain their organization.

(Special education students and ELL students would benefit from having a sheet of words that they cutout and manipulate on the desktop.)

Step Three: Read and Compare

• Students individually read the passage, paying attention to the words on the board.

• Their purpose is to see what new understandings of words develop through reading.

Step Four: Comparative Results

• Students revise three sentences to better portray the words as developed in the passage.

• In small groups, share sentences. Sentences continue to be revised based on group feedback.

• Each student stars strongest sentence then adds to the chart paper for their group.

Step Five: Share Revised Sentences with Class

• As a group, the students share the sentences they developed to represent the new concept.

(The group discussion and sharing help both ELL and Special Education students learn the words inthe context of the new unit.)


WORD SPLASH

Words from the text:

• sum

• quotient

• commutative property

• integers

• product

• order of operations

• zero

• data analysis

* Choose 3 of the words above. On line A below, write acomplete sentence for each word - showing that you knowits meaning. After you read the text, write a sentence online B showing your new understanding of the word.

1. A)

B)

2. A)

B)

3. A)

B)

Key Strategies to Determine Word Meaning

Multiple Meaning Words / Symbols

∪ — x ÷

addadditionpluspositiveincreasedmake largermore thansumolderhigherfaster

subtracttake awayminusnegativedecreasemake smallerless thandiminisheddifferenceyoungerdeeperslowerlower

multiplytimesproductdouble (x2)triple (x3)quadruple (x4)

dividedivisionintodivided byone half (_)one third (_)one forth (_)

Examples of Mathematic Prefixes, Suffixes, RootsMorpheme Math Usage

bi (two) bisect, binomial, bimodal

cent (hundred) centimeter, percent

circu (around) circle, circumference

co, con (with) coefficient, cosine, collinear, congruent

dec (ten) decimal, decagon

dia (through) diagonal, diameter

equi (equal) equilateral, equiangular

hex (six) hexagon

inter (between) intersect, interpolate

kilo (thousand) kilometer, kilogram

milli (thousand) millimeter, milligram

octo (eight) octogan

para (beside) parallelogram


Using Graphic OrganizersImplementation Guide

OverviewGraphic organizers are made up of lines, arrows, boxes, and circles that show therelationships among ideas. These graphic organizers have the potential of helping studentsorganize their thinking and their knowledge. While textbooks contain many types of text, thelargest portion is or informational. Informational text has five major structures: (1) cause andeffect, (2) compare and contrast, (3) description, (4) problem and solution, and (5) sequenceor chronological order.

Strategy in ActionStudents should complete the following steps to practice the strategy:

Step 1: Preview the Text. What did you notice while previewing this section (such as anysignal words, text structure, or graphic signals)?

Step 2: Read the Text. Now have the students read the passage.

Step 3: Determine Which Graphic Organizer Would Best Display the Information. Havestudents decide which of the graphic organizers might organize the ideas in the text best,depending on their purpose for reading. Be sure to remind students that the organizers canbe modified to suit their purposes. They can complete this part either individually or in smallgroups.

Step 4: Create a Graphic Organizer. Working in small groups, have students create agraphic organizer that displays the ideas in the text.

Step 5: Present the Graphic Organizer. Small groups then present their graphic organizersto the class using an overhead transparency or chart. Remember there is no one bestanswer. Students may display their work differently depending on their purpose for readingand what they chose to emphasize.

DiscussionOnce students have finished the activity, you may want to have a brief discussion with themabout the assignment. Encourage students to probe why they chose the graphic organizerthey did and how graphic organizers can help them organize ideas.

Adapted from: Modified from Judith Irvin, Reading and The Middle School Student.Needham Heights: Allyn & Bacon, 1998

(Reading for Info Module p. 15)

Sample Graphic Organizers

Concept Definition Webs

Polygons

Pentagons

Hexagons

Obtuse

Acute

Right

Scalene Isosceles

Equilateral

ParallelogramTrapezoid

Rectangle Rhombus

Square

Triangles

Quadrilaterals

Range QuartilesModeMean Median

StatisticalMeasures

Measures ofLocation

Measures ofDispersion


Sample Graphic OrganizersGeneralization/Principle Diagrams

Generalization or PrincipleEvery composite number can be written asa product of prime numbers.

Example20 = 2 x 2 x 5

Example39 = 3 x 13

Example154 = 2 x 7 x 11

Example126 = 2 x 3 x 3 x 7

Generalization or PrincipleEvery square number can be written asthe sum of two triangular numbers.

Example16 = 6 + 10

Example36 = 15 + 21

Example144 = 66 + 78

Example100 = 45 + 55


Sample Graphic Organizers

Pyramids Prisms

3-dimensionalsolid

X X

One base X

Pair of parallelbases X

All triangular facesexcept base X

Polyhedron X X

cube X

Compare/Contrast Venn Organizers

213

286

10

4 9

25

16

1

36

1225

TriangularNumbers

SquareNumbers

Different(Triangular but not square)

Different(Square but not square)

Alike(Both triangular and square)


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iffe

ren

ces

Compare/Contrast Organizer

Adapted from: Reading Strategies for the Content Areas: During-Reading Strategies, ASCD, 2003

AdditionalReading Strategies

andGraphic Organizers

Word SortsWhat is it?

Word sorts help students recognize the semantic relationships among key concepts. Studentsare asked to sort vocabulary terms into different categories. The strategy can be used in twodifferent ways. In a “closed sort,” the teacher provides the categories into which students are toassign the words. In an “open sort,” students group words into categories and identify theirown labels for each category. Word sorts help students develop a deeper understanding of keyconcepts, and also are an excellent method of teaching the complex reasoning skills ofclassification and deduction.

How to use it:

1. Students copy vocabulary terms onto 3” x 5” cards, one word per card - or the teacherhas words printed on a handout that students can tear into cards.

2. Individually or in groups, students then sort the words into categories. With youngerstudents or complex concepts, the teacher should provide students with the categoriesand have students complete a “closed sort.”

3. As students become more proficient at classifying, teachers should ask them tocomplete “open sorts”; that is, students sort words into labeled categories of their ownmaking. At this stage, students should be encouraged to find more than one way toclassify the vocabulary terms. Classifying and then reclassifying helps students extendand refine their understanding of the concepts studied.


polygon

perimeter

pentagon

rectangle

parallelogram

trapezoid

Words beginningwith a “P”

DimensionalFigures

polyhedron polygon

Word Sort

Word Sorts


perimeter

volume

radius

cubic

linear

quadratic

prime

scalene

equilateral

length acute

Geometry Word Sort

variable

width right

reflection

rotation

translation

similar

Geometry Word Sort

angles

verticles

edges

triangle

square

sphere

cube

prismcircle

lines cone

length

perimeter

volume

parallel

perpendicular

adjacent

symmetry

intersecting

congruent

opposite

bisector

circumference

radiuscylinder

hexagon

parallelogrampoints

rays

diagonals

rhombus pyramid area

similar

Plane figures

Parts ofShapes Solid figures

Measures Relations

Shapes

Number Sorts(a variation of Word Sorts)

Provide students with a set of number cards. Ask them to place them in thecorrect spot on this graphic organizer.


0

50

105

36

35

72

5

3

41

14

53

Multiples of 5 Prime

Less than 5051

K-N-W-S(K-W-L for Word Problems)

What Is It?

In this pre-reading strategy students use a process similar to K-W-L to analyze and plan howto approach solving a word problem. Students answer what facts they KNOW, whatinformation is NOT relevant, WHAT the problem asks them to find, and what STRATEGY theycan use to solve the problem.

How to Use It?

1. Introduce students to the four-column K-N-W-S worksheet, or have them draw thegraphic organizer on their own paper.

2. Present students with a word problem, and model how to fill in information in each of thecolumns. Explain how you knew what information should be included in each column.

3. Ask students to work in groups to complete a K-N-W-S for other word problems.Students should discuss with their groups how they knew what to put in the columns.

4. Give students ongoing independent practice using this strategy to solve word problems.

KWhat facts do IKNOW from the

information in theproblem?

NWhich informationdo I NOT need?

WWHAT does the

problem ask me tofind?

SWhat STRATEGY/operation/tools willI use to solve the

problem?


Example:

Problem:The ends of a rope are tied to two trees, 500 feet apart. Every 10 feet an8-foot post is set 2 feet into the ground to support the rope. How manysupport posts are needed?

K-N-W-S Worksheet: (K-W-L for word problems)

KWhat facts do IKNOW from the

information in theproblem?

NWhich informationdo I NOT need?

WWHAT does the

problem ask me tofind?

SWhat STRATEGY/operation/tools willI use to solve the

problem?

Trees are 500 feetapart.

Posts are placedat 10-foot intervalsbetween the trees.

The posts are 8feet tall.

The posts are set2 feet into theground.

How manysupport posts areneeded?

Draw a model tounderstand how toplace posts.

Solve the problemwith the treescloser and find apattern.

There are 50(500 ÷ 10) 10-footintervals betweenthe trees.


Five-Step Problem Solving

What is it?

Students’ comprehension of word problems can be enhanced by teaching them to read wordproblems as meaningful passages — as short stories from which students can constructmeaning based on their prior knowledge and experience. Teachers use this strategy bypresenting students with a graphic organizer that leads them through a five-step problem-solving process.

How could it be used in mathematics instruction?

This strategy gives students a graphic organizer to use in the problem-solving process. It canhelp students understand the steps and explain their reasoning throughout the process.

How to use it:

1. Introduce students to the layout and design of the graphic organizer. Point out that thediamond shape of the graphic reinforces the fact that all students begin with the sameinformation about a problem and should arrive at the same conclusion, if they aresuccessful at solving the problem. Explain each of the steps outlined in the graphic.

2. Present students with a word problem, reading it aloud and asking students about theirprior knowledge of the situation and elements included in the “story.”

3. Model for students how to complete the first step of the organizer, restating the questionin a number of ways. Ask students to identify which version is the clearest and to explainthe reasoning behind their choice. Once students know how to approach the problem,they will feel more confident about solving it.

4. Model how to complete the remaining steps in the graphic organizer.

5. When students understand the steps in the graphic organizer, offer them opportunitiesfor guided practice. Select another word problem, and lead them through each step ofthe process. Ask students to discuss their thinking as they read the problem and toarticulate the reasons for the responses they give.

6. Let students work in small groups to discuss and complete several more problems usingthe five-step graphic organizer.

See Next Page ∨


Graphic Organizer for Five-Step Problem Solving

1. Restate theproblem/question:

2. Find the needed data:

3. Plan what to do:

4. Find the answer:

STEP 1 STEP 2 STEP 3

Answer:

5. Check. Is your answer reasonable?


Verbal and Visual Word Association (VVWA)

What is it?

The VVWA strategy puts together a vocabulary word and its definition with both a visual and apersonal association or characteristic of the term. This strategy helps students learnvocabulary on their own and helps them retain the new vocabulary through visualcharacteristic associations. This strategy has been shown to be especially effective for low-achieving students and for second language learners in content-area classes.

How could it be used in mathematics instruction?

Much of the vocabulary of mathematics can be represented visually. This strategy may beused by students as they are introduced to new vocabulary to make immediate visualassociations. As students discover the critical characteristics of a concept or make personalassociations, they put these together with the definitions and visuals to deepen theirunderstanding of the concept.

How to use it:

1. Select vocabulary terms that would be appropriate for using VVWA.

2. Direct students to draw a rectangle divided into four sections for each term.

3. Instruct students to write the vocabulary word in the upper-left box of the rectangle.Instruct them to write the text definition of the term or give them a definition to write inthe lower-left box.

4. Direct students to draw a visual representation of the vocabulary word (perhaps found ina graphic in the text) in the upper-right box of the rectangle. Then suggest that theymake their own personal association, an example of characteristic, to put in the fourthbox at the lower right.

VocabularyTerm

VisualRepresentation

Definition

PersonalAssociation

orCharacteristic


Verbal and Visual Word Association (VVWA)

Examples:

RightTriangle

A triangle with oneright angle (90_ )(square corner)

ramp or slide

NormalDistribution

Distribution ofstatistical measures

(data) that has asymmetrical graph

Bell shapedThink of Liberty Bell

Measures are closeto middle like

people’s heights


Three-Level Guides

What is it?

The Three-Level Guide helps students analyze and solve word problems. Using a teacher-constructed graphic organizer, students must evaluate facts, concepts, rules, mathematics,ideas, and approaches to solving particular word problems.

How to use it:

1. Construct a guide for a given problem similar to the one shown on the next page. Thefirst level (Part I) should include a set of facts suggested by the data given in theproblem. The students’ goal will be to analyze each fact to determine if it is true and if itwill help them to solve the problem.

2. The second level (Part II) of your guide should contain mathematics ideas, rules, orconcepts that students can examine to discover which might apply to the problem-solving task.

3. The third level (Part III) should include a list of possible ways to get the answer.Students will analyze these to see which ones might help them solve the problem.

4. Introduce students to the strategy by showing them the problem and the completedthree-level guide. Explain what kind of information is included at each level.

5. Model for students how you would use the guide in solving the problem.

6. Present students with another problem and guide. Have them analyze the informationyou have included to determine its validity and usefulness in solving the problem.

7. With advanced students, ask them to select a word problem from the text and completea three-level guide to be shared with the class.


Example: Three-Level Guides

A Three-level guide to a math problem

Read the problem and then answer each set of questions, following the directions givenfor the set questions.

Problem: Sam’s Sporting Goods has a markup rate of 40% on Pro tennisrackets. Sam, the store owner, bought 12 Pro tennis rackets for $75 each.Calculate the selling price of a Pro tennis racket at Sam’s Sporting Goods.

Part I

Directions: Read the statements. Check Column A if the statement is true according to theproblem. Check Column B if the information will help you solve the problem.

A (true?) B (help?)Sam’s markup rate is 40%.

Sam bought 12 Pro Tennis rackets.

Pro tennis rackets are a good buy.

Sam paid $75 for a Pro tennis racket.

The selling price of a Pro tennis racket ismore than 75%

Part II

Directions: Read the statements. Check the ones that contain math ideas useful for thisproblem. Look at Part I, Column B to check your answer.

Markup equals cost times rate.

Selling price is greater than cost.

Selling price equals cost plus markup rate.

Markup divided by cost equals markup rate.

A percent of a number is less than thenumber when the percent is less than100%.

Part II

Directions: Check the calculations that will help or work in this problem. Look at Parts Iand II to check your answers.

0.4 x $75 12 x $75

$75 x 40 40% x $751.4 x $75 $75 + ( _ x $75)


Semantic Mapping

What is it?

A semantic map is a visual tool that helps readers activate and draw on prior knowledge,recognize important components of different concepts, and see the relationships amongthese components.

How Could It Be Used In Mathematics Instruction?

This strategy can be incorporated into the introduction of a topic to activate students’ priorknowledge and then used throughout a unit or series of lessons on the topic. Students will beable to visualize how terms are connected and/or related. This strategy can be used to buildconnections between hands-on activities and reading activities.

How to use it:

1. Write the major topic of the lesson or unit on chart paper. Let students brainstorm a list ofterms that relate in some way to this major topic.

2. Write the major topic in the center of another sheet of chart paper and circle it.

3. Ask students to review the brainstormed list and begin to categorize the terms. Thecategories and terms should be discussed and then displayed in the form of a map orweb.

4. Leave the chart up throughout the series of lessons or unit so that new chapters andterms can be added as needed.


TermTermTermTerm

Category

TermTermTermTerm

Category

TermTerm

TermTerm

Category

Category

MajorConcept

Examples:

linearquadraticcubic

2 x 3 = 3 x 22 x (5 + 3) = 10 + 672 – 58 = 74 – 60

Number Relations

A = lwC = 2πrP = 2(l + w)

simultaneous equationsconsistent/inconsistentdependent/independent

Equations

Formulas

SystemsDegree

(Graphing)

numericaldegreedependent

Unknowns/Variables

counting pricescomparing scoresordering measures

labelssizesdates

Uses

additionsubtractionmultiplicationdivisionsquare rootabsolute value

positive prime triangularnegative composite squarefraction odd perfectdecimal even abundantpercent

Numbers

Operations

Types

ones placetens place place valuetenths place

etc.numeratordenominator

Parts

Adapted from: Barton, Mary Lee and Clare Heidema, Teaching Reading in Mathematics:A Supplement to Teaching Reading in the Content Areas, 2002

Example:

Types

rectangle: A = IwP = 2(I + w)

circle: A = πr_C = 2πr

sphere: V = 4/3πr_cylinder: V =πr_h

Measurement

Formulas

Units

Metric Customary Nonstandardmeter foot pencilcm inch paper clipkm mile glassliter quartgram ouncekg poundCelsius Fahrenheit

ruler, tape measurescalecupclockthermometerprotractor

Tools

Length Cover Volume Other(1-dim) (2-dim) (3-dim) capacitywidth area volume weightheight surface massperimeter area timecircumference temperature

angle measure

Adapted from: Barton, Mary Lee and Clare Heidema, Teaching Reading in Mathematics:A Supplement to Teaching Reading in the Content Areas, 2002

NOTESInformation Worth Noting Questions?

(I wonder….)

Summary of Key Ideas Graphic Representation ofKey Ideas

↵

↵

↵

↵

↵

(Reading for Info Module p. 11)

Created by: MaryBeth Munroe, Southern Oregon S.D.

Putting it All Together

~ Lesson Planning ~

The Strategic Teacher Shares Reading Strategies

Adapted from: McEwan, E.K. Raising Reading Achievement in Middle and High School: 5 Simple-to-FollowStrategies for Principals. 2001 by Corwin Press, Inc. International Reading Association

Belief inAbility to

AffectLearning

A StrategicLearner

Repertoire ofEffectiveTeachingMethods

Knowledge andUnderstanding of

Students

ContentKnowledge

ExplicitInstruction

TheStrategicTeacher

PersonalCharacteristics

Benefits of aStrategy

How andWhen toUse a

Strategy

Modeling

ThinkingAloud

Practice andFeedback

StrategicKnowledge and

Expertise of ReadingStrategy

S.O.S. Reading Strategies

Session Pre-Reading During Reading After Reading

Introduction Session

K.A.U. X X XThink Aloud X X XThink-Pair-Share X X X

Pre-Reading Session

Give One, Get One X XAnticipation Guide X X XK.W.L. X X XD.R.T.A. X X XWord Splash X X XPredicting Nonfiction XTHIEVES XP.A.C.A. X

Vocabulary Session

Modified K.A.U. X X XVocab Alert! X X XContext Clues X XPrefix – Suffix X XConcept Definition Map X X XFrayer Model X X X3 + 3 X X X

Reading for Info Session

Determine Text Features XDetermine Text Structures X XGraphic Organizers X X XRead, Cover, Remember, Retell X X XI.N.S.E.R.T. XS.C.A.N. & R.U.N. X X XP.R.I.M.E. X

Questioning Session

Visualizing Information X X XQuestion Answer Relationship X XQuestion Around X XThick and Thin Questions X XReciprocal Teaching X X XRe Quest XCubing X X

The LessonResearch Suggests a New Format

Readingassignment

given

Independentreading

Discussion to see ifstudents learned mainconcepts, what they

"should have" learned

Prereading activitiesDiscussionPredictionsQuestioning

BrainstormingSetting purpose

Guided ACTIVEreading• silent• pairs• group

Activitiesto clarify,reinforce,

extendknow-ledge

Traditional Format New Format

Adapted from: Billmeyer, Rachel and Mary Lee Barton, Teaching in the Content Areas: If Not Me, Then Who?Aurora: McREL (Mid-continent Regional Education Laboratory), 1998.

Putting it all TogetherLesson Planning

1. Determine your objectives for the lesson. What do you want students to know or to beable to do at the end of the lesson?

2. Select a strategy for accessing students’ prior knowledge of the general topic.Examples: KWL, Anticipation Guide, etc.

3. Preview the text material for vocabulary.

a. Identify critical vocabulary students will need to know prior to reading.

b. Select vocabulary strategies appropriate for the text and lesson.Examples: Frayer Model, Word Splash, Concept Definition Map, etc.

4. Preview the text material for organization.

a. Determine the organizational pattern(s) used in the text.1. Note text features to point out to students prior to their reading.2. Note signal words to which students should pay attention.3. Select a graphic organizer that aligns with the pattern.

b. To provide students with strong guidance in organizing the text information, deviseprereading questions that1. Align with or emphasize the organizational pattern2. Will aid comprehension by focusing students’ attention to their purpose for

reading (Quiz? Performance task? Discussion?)3. Will help students meet your original objectives for the lesson.

5. Select “during reading” questions (process questions that focus on metacomprehensionstrategies such as making predictions, confirming or revising those predictions, andnoting graphic aids that signal important ideas). If students are prompted to focus on theirreading process, their metacomprehension will improve.

6. Select post-reading questions and activities that help students meet your objectives,reflect on and apply what they have learned, and revise existing schema (e.g., writing-to-learn; performance activity; discussion).

Adapted from: Billmeyer, Rachel and Mary Lee Barton, Teaching in the Content Areas: If Not Me, Then Who?Aurora: McREL (Mid-continent Regional Education Laboratory), 2000.

Teacher's Checklist

YES NO

• Have I identified my objectives for this lesson – what I wantstudents to know and be able to do?

• Have I previewed the text and determined keyconcepts/vocabulary students need to know?

• Have I included activities and strategies that will help studentsdevelop a clear understanding of these key concepts?

• Have I selected activities to assess, activate, and buildstudents' background knowledge?

• Have I identified the text's organizational pattern(s) andwhether it highlights information I consider most important?

• If the organizational pattern does not highlight keyinformation, have I determined the frame of mind or pattern Iwill tell students to use while reading?

• Have I selected a suitable graphic organizer students can useto organize key concepts?

• Have I decided the purposes students should keep in mindwhile reading (e.g., whether they will be using the informationin a discussion, performance activity, on a quiz)?

• Have I developed "during reading" questions that will promptstudents to employ metacognitive skills?

• Have I selected post-reading questions and activities thatrequire students to make meaningful connections, and todeepen their understanding by applying what they havelearned?

Adapted from: Strategic Teaching, McREL (Mid-Continent Regional Education Laboratory)

Example Lesson #1

Using M.C. Escher to Teach Geometry Concepts

Pre-Reading

Reading Purpose: To give the student the opportunity to become familiar with theartist/mathematician M.C. Escher and to gain an awareness of how he uses concepts ingeometry, such as tesselations, polygons and tilings, to create imaginative works of art whichinclude surprising puzzles and paradoxes.

KWL: To provide students with background information and to prepare them to participate in adiscussion about what they already know, they first view a film on the life of M.C. Escherand look at examples of works of art that he created.

After seeing the video students discuss what they know about Escher and record thisinformation either visually, in a list, or by using sentences in the column K-Know.

Students continue to reflect on what they want to know and record their ideas andquestions in the W-Want to Know column.

Know Want to know What I Learned

During Reading: INSERT Notes

Students read the article independently, and use the INSERT strategy to take notes in themargin, highlighting important or interesting information.

⇑ = I already knew this ! = Wow

+ = New information ?? = I don’t understand

After-Reading

Students reconvene and share what they have learned. They record new information on theL-Learned section of the KWL organizer.

Video and Reading Selections• The Fantastic World of M.C. Escher (video)• Agnesi to Zeno: Over 100 Vignettes from the History of Math, by Sanderson Smith, (1996)• M.C. Escher, Artist and Geometer, Key Curriculum Press, (1996)

Adapted from: Pam Mathews, Corvallis School District

Sample Lesson #2

Full Circle: A Geometry Lesson

Purpose: To teach students how to read mathematical text, interact with examples, learn newvocabulary, concepts and techniques involved with attributes of circles.

Pre-Reading

1. Each student will record 5 terms that they associate with circles.

2. Each student will compare their list with a partner and add new ideas to their list.

3. The whole class will help generate a class list of terms.

During Reading

In small groups, students read the assigned article. Each group has a different article oncircles.

After Reading

1. Students who read the same article will meet to create a presentation for the class.

2. One student will be selected to present the material to the class and other students willbe available to field questions from the audience.

Reading Selections

• Circles: Definition of a circle, chords, tangent and secant lineshttp://www.math.psu.edu/geom/koltsova/section7.html

• Circles and Angleshttp://www.math.psu.edu/geom/koltsova/section8.html

• Circle Formulashttp:forum.swarthmore.edu/dr.math/faq/formulas/faq.circle.html

Adapted from: Pam Mathews, Corvallis School District

Special thanks to the following educatorsfor contributing samples of classroom reading strategies:

• Reynolds High School teachers• Centennial High School teachers• Pam Mathews, Corvallis School District

References

Classroom Strategies for Interactive Learning (1995), by Doug Buehl

Guiding Reading and Writing in the Content Areas (1998), by M. Carrol Tama andAnita Bell McClain

Invitations: Changing as Teachers and Learners K-12 (1994), by Regie Routman

Raising Reading Achievement in Middle and High School (2001), by Elaine McEwan

Real Reading, Real Writing: Content-Area Strategies (2002), by Donna Topping andRoberta McManus

Teaching Reading in Mathematics: A Supplement to Teaching Reading in the Content Areas(2002), by Mary Lee Barton and Clare Heidema

Teaching Reading in the Content Areas: If Not Me, Then Who? (1998), by Rachel Billmeyerand Mary Lee Barton

Tools for Thought: Graphic Organizers for Your Classroom (2002), by Jim Burke

Yellow Brick Roads: Shared and Guided Paths to Independent Reading 4-12 (2000),by Janet Allen

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