Chapter 6 Rational Numbers and Proportional Reasoningmath2010eportfolio.weebly.com/uploads/1/6/9/7/16979194/chapter_6.pdf · 86 Chapter 6 • Rational Numbers and Proportional Reasoning

Chapter 6

Rational Numbers andProportional Reasoning

Copyright © 2013 Pearson Education, Inc. All rights reserved.

85

“Students should build their understanding of fractions as parts of awhole and as division. They will need to see and explore a variety ofmodels of fractions, focusing primarily on familiar fractions such ashalves, fourths, fifths, sixths, eighths, and tenths. By using an areamodel in which part of the region is shaded, students can see howfractions are related to a unit whole, compare fractional parts of awhole, and find equivalent fractions. They should develop strategiesfor ordering and comparing fractions, often using benchmarks such as

or 1.”—Principles and Standards for School Mathematics

“Students apply their understanding of fractions and fraction modelsto represent the addition and subtraction of fractions with unlikedenominators as equivalent calculations with like denominators. Theydevelop fluency in calculating sums and differences of fractions, andmake reasonable estimates for them. Students also use the meaning offractions, of multiplication and division, and the relationship betweenmultiplication and division to understand and explain why theprocedures for multiplying and dividing fractions make sense.”

—Common Core Standards for Mathematics

“Facility with proportionality involves much more than setting tworatios equal and solving for a missing term. It involves recognizingquantities that are related proportionally and using numbers, tables,graphs, and equations to think about the quantities and theirrelationship. Proportionality is an important integrative thread thatconnects many of the mathematics topics studied in grades 6–8.”

—Principles and Standards for School Mathematics

12

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Other than whole-number computation, no topic in the elementarymathematics curriculum demands more time than the study offractions. Yet, despite the years of study, most students enter highschool with a poor concept of fractions and an even poorerunderstanding of the operations with fractions. When asked abouttheir memories of fractions, adults will often reply, “Yours is not toreason why, just invert and multiply.”

Rational numbers should be taught as the natural extension of the whole numbers. The fraction can be viewed as the solution to the problem of dividing 3 dollars among 4 people: 3 � 4, or answeringthe question “How many fourths (quarters) will each person receive?”For students to understand this connection between whole numbersand fractions, teaching about fractions and their operations shouldbegin with concrete models, as was the instruction with wholenumbers. A well-developed concept of fractions and a feeling fortheir magnitude must be established before they can be compared andordered meaningfully. This must precede computation.

Number sense involving fractions, and a deeper understanding of theoperations with fractions must be developed prior to formal workwith algorithms for the operations. This chapter provides activitiesthat firmly establish the concepts of fractions. All operations areexplored through a variety of concrete models and reinforced byrepresenting the models pictorially.

Multiple representations will be used to introduce the concepts ofratio and proportion. In later chapters, ratio and proportion will beapplied in real world contexts and used to explore proportionalvariation in geometry.

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86 Chapter 6 • Rational Numbers and Proportional Reasoning

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Chapter 6 • Activity 1 87

Activity 1: What Is a Fraction?

PURPOSE Develop the concept of a fraction.

MATERIALS Pouch: Pattern Blocks and Cuisenaire™ Rods

GROUPING Work individually or in pairs.

Use pattern blocks to solve the following problems.

1. The trapezoid is what fractional part of the hexagon? __________

2. The blue rhombus is what fractional part of the hexagon? __________

3. The triangle is what fractional part of the hexagon? __________

4. The triangle is what fractional part of the blue rhombus? __________

5. The triangle is what fractional part of the trapezoid? __________

What fractional part of each figure is shaded? unshaded?

1. shaded ________ unshaded ________

2. shaded ________ unshaded ________

3. shaded ________ unshaded ________

4. shaded ________ unshaded ________

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Use two red trapezoids and one blue rhombus to construct a shape similar to the one shownbelow.

1. Given that the shape � 1, what pattern block(s) would you use to represent each of the following fractions?

a. ______ b. ______ c. ______

Fill in the same shape using one red trapezoid, two blue rhombuses,and one green triangle.

2. What fraction is represented by each of the following?

a. a blue rhombus _____ b. a red trapezoid _____ c. a green triangle _____

1

8

1

2

1

4

Use Cuisenaire rods to solve the following.

1. If the orange rod � 1, each rod is what fractional part of the orange rod?

a. red ________ b. green _______

c. yellow ________ d. purple _______

2. If the purple rod � 1, each rod is what fractional part of the purple rod?

a. brown ________ b. orange _______

c. dark green ________ d. black _______

1. If the which rod � 1? _____________

2. If the which rod � 1? _____________

3. If the which rod � 1? _____________

4. If the which rod ? _____________

5. If the which rod ? _____________

6. If the which rod ? _____________

Explain how you used the rods to arrive at your answers.

= 12

3red rod =

1

3,

= 11

2red rod =

1

2,

= 13

4white rod =

1

4,

white rod =

1

5,

red rod =

1

3,

red rod =

1

2,


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Activity 2: Square Fractions

PURPOSE Reinforce the concept of a fraction and the meaning of equivalentfractions, and illustrate operations with fractions using geometricmodels.

MATERIALS Other: Sheet of paper 20 cm square (colored construction paperworks well) and scissors (optional)

GROUPING Work individually.

GETTING STARTED Work through each section of the activity in order, following thefolding and cutting directions carefully. If scissors are not used, foldand crease the paper sharply so that it will tear cleanly.

Fold the square as shown and cut or tear along the fold to divide thesquare into two congruent parts.

1. Each triangle is what fraction of the original square? ________

Pick one of the two triangles and fold it as shown. Cut or tear alongthe fold and label the two triangles 1 and 2.

2. Triangle 1 is what fractional part of

a. triangle A? ______

b. the original square? ______

Explain your answers.

In the remaining large triangle, fold the vertex of the right angle tothe midpoint of the longest side. Cut along the fold and label thepolygons B and 3 as shown.


a. triangle 1? ______

b. triangle A? ______

c. trapezoid B? ______

A

1

2

B

3

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Fold one of the endpoints of the longest side of trapezoid B to themidpoint of that side as shown. Cut along the fold and label thepolygons C and 4.


a. triangle 3? ______

b. triangle A? ______

c. trapezoid C? ______

Fold trapezoid C as shown. Cut along the fold and label the polygonsD and 5.

5. Square 5 is what fractional part of

a. trapezoid C? ______

b. trapezoid D? ______

c. triangle 3? ______

d. the original square? ______

6. Trapezoid C is what fractional part of

a. triangle A? ______ b. square 5? ______

7. Trapezoid D is what fractional part of

a. triangle 1? ______ b. trapezoid C? ______

D

C

5

C

B

4

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Fold trapezoid D as shown. Cut along the fold and label the two polygons 6 and 7.


a. trapezoid D? ______

b. triangle 3? ______

9. Parallelogram 7 is what fractional part of

a. square 5? ______ b. trapezoid B? ______

c. triangle 1? ______ d. original square? ______

10. Explain how triangle 1, triangle A, and the original square can be

used to illustrate that is equivalent to

11. Explain how these figures can be used to illustrate that of

is which is the multiplication problem

12. If the original square is one unit, explain how trapezoid D andtriangle 3 can be used to model the addition problem

13. If trapezoid B is one unit, explain how triangle 4 and trapezoid D can be used to model the division problem 1

2,

1

6= 3.

3

16+

1

8=

5

16.

1

2*

1

2=

1

4.

1

4,

1

2

1

2

1

2.

2

4

7 6

D

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Activity 3: How Big Is It?

PURPOSE Develop the ability to estimate the magnitude of a fraction.

MATERIALS Online: Fraction Sorting Board and Fraction Cards

GROUPING Work in pairs.

GETTING STARTED Use these rules to complete the following problems.

A fraction is close to 1 if the numerator and denominator are approximately the same size.

if the denominator is about twice as large as the numerator.

0 if the numerator is very small compared to the denominator.

12

THE FRACTION SORTING GAME

This is a game for two players. Cut out the fraction cards and shuffle them. Students taketurns placing a card in the appropriate column on the fraction sorting board and justifyingeach placement to the other player. Reshuffle the deck and play again.

1. Complete the following fractions so that they are close to, but less than,

a. b. c. d.

e. f. g. h.

2. Complete the following fractions so that they are close to, but less than, 1.

a. b. c. d.

e. f. g. h.951139

8751227

81137

14925100

1

2.

EXTENSIONS 1. Given the fraction , what numbers would be acceptable in place

of so that the resulting fraction is close to but less than 1? Justify

your answer.

2. Given the fraction , what numbers would be acceptable in place

of so that the resulting fraction is close to ? Justify your answer.12

23

13

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Activity 4: Equivalent Fractions

PURPOSE Develop the concept of a fraction using concrete models and aproblem-solving approach.

MATERIALS Pouch: Cuisenaire™ Rods and Pattern BlocksOnline: Fraction Strips

GROUPING Work individually or in pairs.

For the following problems, use Cuisenaire rods to construct the trains.

1. Make all of the possible one-color trains the same length as a dark green rod and complete the following. If dark green � 1, then

a.

b.

c.

d.

2. Make all of the possible one-color trains the same length as a brown rod and completeeach of the following. If brown � 1, then

a.

b.

c. red = =

dark green = =

purple = = =

dark green =

6=

purple = =

red =

1=

6

light green =

1

2=

6

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EXTENSIONS 1. Given a set of fractions equivalent to the fraction where

is in lowest terms, what is the relationship among the set of

numerators? the set of denominators?

2. Given a set of equivalent fractions, where is in

lowest terms, what is the relationship between the numerator and

denominator of and the numerator and denominator of any

equivalent fraction?

a

b

a

b

a

b ,

c

d ,

e

f , Á ,

a

b

a

b,

Use pattern blocks to construct a shape similar to the star and complete the following.

If the star shape �1, then

1.

2.

3. hexagon =

6=

6=

1

2 blue rhombuses =

6=

1=

trapezoid =

12=

1

Find the fraction strips that can be folded into parts so that the resulting strip is equal inlength to the fraction given in each problem. Folds may be made only on the lines on thefraction strips.

Write the name of the equivalent fractions in the space provided.

1.______

�______

�______

�______

2.______

�______

�______

3.______

�______

3

4=

2

3=

1

2=

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Activity 5: Fraction War

PURPOSE Reinforce estimation and comparison of fractions in a game format.

MATERIALS Online: Fraction Arrays and Fractions Game BoardOther: Deck of playing cards (remove face cards)

GROUPING Work in pairs.

GETTING STARTED Follow the rules below to play Fraction War.

Example: 1. Shuffle the cards and deal them evenly, face down to each player. Players choose a goal for a game from those listed below.

a. Form a proper fraction by placing the card with the smallernumber in the numerator. Player with the smaller fraction isthe winner.

b. Form a proper fraction by placing the card with the smallernumber in the numerator. Player with the larger fraction is thewinner.

c. Form a proper fraction by placing the card with the smallernumber in the numerator. Player with the fraction whose valueis closest to is the winner.

d. Form a fraction by placing the card with the larger number inthe numerator. Player with the larger fraction is the winner.

e. Place the first card in the numerator, the second in the denominator. Player with the fraction whose value is closestto 2 is the winner.

f. Each player decides where to place each card. Player with thefraction whose value is closest to 1 is the winner.

2. Each player turns up two cards from his or her pile and followsthe directions for the chosen game to form a fraction on theFractions Game Board. The ace represents 1.

3. The winner of each round collects the four cards and places themface up at the bottom of his or her pile of cards. If the fractionsformed are equivalent, each player turns over two additionalcards and forms a new fraction. The winner of the round gets alleight cards.

4. When the players have played all the face-down cards, the playerwith the most face-up cards is the winner of the game. Reshufflethe cards and choose a different game.

12

A

A

3

3

4

4

5

5

Player A Player B

FRACTIONS GAME BOARD

Player A wins a round inGame a: the smallerfraction is the winner.

Note:

When necessary, use the Fraction Arrays to compare the fractions.

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Activity 6: What Comes First?

PURPOSE Compare and order fractions.

MATERIALS Pouch: Cuisenaire™ Rods and Pattern BlocksOnline: Fraction Arrays


Use Cuisenaire rods to build all possible one-color trains that are the same length as a brownrod, and complete the following.

1. Which is larger, or ? ______ Complete the inequality: ______ � ______

2. Which is smaller, or ? ______ Complete the inequality: ______ � ______

3. Which is larger, or ? ______ Complete the inequality: ______ � ______3

4

7

8

1

4

3

8

1

2

5

8

Use pattern blocks to construct a star shape (see Activity 4) and complete the following.

1. Which is larger, or ? ______ Complete the inequality: ______ � ______

2. Which is smaller, or ? ______ Complete the inequality: ______ � ______

3. Which is larger, or ? ______ Complete the inequality: ______ � ______3

4

5

6

7

12

2

3

1

2

5

12

Use the Fraction Arrays to order the following sets of fractions.

1. _________ � _________ � _________

2. _________ � _________ � _________

3. _________ � _________ � _________

4. _________ � _________ � ________5

6,

11

12,

4

5

5

12,

2

5,

3

7

2

3,

3

4,

7

8

1

2,

3

5,

4

7

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Activity 7: Adding and Subtracting Fractions

PURPOSE Develop algorithms for adding and subtracting fractions.

MATERIALS Pouch: Pattern BlocksOnline: Fraction Strips


If the yellow hexagon � 1, then the the and the

Use pattern blocks to solve the following:green triangle =16.

blue rhombus =13,red trapezoid =

12,

1. � �

______

2.

3.

4.

5. 1 red - 1 green

1

2 -

1

6

1 red - 1 blue

1

2 -

1

3

1 blue + 1 green

1

3 +

1

6

1 red + 1 blue

1

2 +

1

3

3 green + 3 green

3

6 +

3

6

1 red + 1 red

1

2 +

1

2

1 red + 3 green

1

2 +

3

6

�

� � �

�

�

�

�

�

�

�

�

______�

______�

______�

______�

______�

______�

______�

______�

______�

______�

green + green___ ___

______ + ______

______ + ______

______ + ______

______ - ______

______ - ______

______ - ______

______ - ______

______

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Use pattern blocks to solve the following problems. Write your answers in simplest form,that is, the number represented by the least number of blocks of the same color.

Let

1.

2.

3.

4.

5.

6.

7.

8. 15

12-

5

6=

11

2+

2

3=

3

4+

2

3+

1

6=

2

3+

1

2=

5

6-

3

4=

3

4-

2

3=

3

4+

1

2+

1

6=

1

2+

3

12=

one whole12

14

16

112

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EXTENSION From your observation in this activity, write rules for adding andsubtracting fractions with like and unlike denominators.

When adding fractions using fraction strips, you must fold the strips to show only the fractions that are needed. Strips are placed as shown in the following figures. A longer stripmust then be found that has fold lines in common with the two fractions.

Example for Addition:

Example for Subtraction:

Use your fraction strips to solve the following problems.

1. 2. 3.

4. 5. 6.3

4-

1

6=

2

3+

3

4=

5

12+

1

3=

7

10-

2

5=

7

8-

1

4=

1

2+

3

5=

– =

– = 512

312

812

14

23

14

13

112

112

112

112

112

112

112

112

112

112

112

112

13

2

3-

1

4=

+

+ = 712

312

412

14

13

13

14

112

112

112

112

112

112

112

112

112

112

112

112

1

3+

1

4=

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Activity 8: Multiplying Fractions

PURPOSE Develop an algorithm for multiplying fractions.

MATERIALS Pouch: Pattern BlocksOther: Paper for folding


Example: of means two of three equal parts of

Place pattern blocks on Figure A to solve the following multiplication problems. Recordyour solutions pictorially and numerically.

1.

2.

3.

4.

5.5

6*

1

2=

3

4*

2

3=

1

4*

1

3=

3

4*

1

3=

1

2*

1

3=

2

3*

1

4=

2

12=

1

6

of 23

14

14

one whole

14.1

423

Figure A

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Example: of means one of the two equal parts of two thirds.

Use four hexagons to construct a figure similar to the one shown above, and solve the following problems. Record each step of your solutions both pictorially and numerically.

1.

2.

3.

4.5

8*

1

3=

7

12*

1

2=

3

8*

2

3=

3

4*

1

6=

of

one whole

�

23

23

12

13

23

12

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EXTENSION From what you have observed in this activity, write a rule formultiplication of fractions.

of 1 means one of the three equal parts of 1. Divide a piece ofpaper into thirds with vertical folds.

of means one of the two equal parts of Now, divide the thirds into halves with a horizontal fold.

of

of means two of the three equal parts of

of

Fold sheets of paper to solve the following multiplication problems. Record each step ofyour solutions pictorially and numerically.

1.

2.

3.

4.3

8*

1

3=

2

3*

2

3=

1

4*

2

3=

1

3*

3

4=

1

2=

2

6=

1

3

2

3

12.1

223

1

3=

1

6

1

2

13.

13

12

13

13

12

12

13

13

13

13

13

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Activity 9: Dividing Fractions

PURPOSE Develop understanding of division of fractions.

MATERIALS Pouch: Pattern Blocks


GETTING STARTED Recall the use of the multiplication and division frame for thedivision of whole numbers.

Example: can mean how many groups of 3 are there in 6? 3

2

3�6

In the following example represents 1.

Example: means: How many groups of are there in 1?

In each of the following, complete the sentence and then use pattern blocks to solve therelated division problem.

1. means _____________________________________________________

2. means _____________________________________________________

3. means _____________________________________________________

4. means _____________________________________________________

5. means _____________________________________________________

3

2,

3

4=

3

2,

3

4

3

4,

1

4=

3

4,

1

4

5

6,

5

12=

5

6,

5

12

1

2,

1

4=

1

2,

1

4

1

3,

1

6=

1

3,

1

6

one 1

? ?

� � 212

12

12

12 1

2

121 ,

12

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To model the problem let

How many sets of (two blue rhombuses) are there in (one hexagon)?

There is one group of (two blue rhombuses), plus a remainder.

The remainder is equal to one blue rhombus, which is of

Therefore one set of two blue rhombuses � one half set of two blue rhombuses

� 1 �

In each of the following, use pattern blocks to solve the division problems.

1. 2.

3. 4. 11

3,

1

2=

2

3,

1

2=

3

4,

1

2=

5

6,

1

3=

= 11

2

1

2

12 ,

13 =

13.

12

13

?

13

12

13

112

13

12 ,

13,

EXTENSIONS 1. Describe how you would use fraction strips to solve the previousproblems. Draw at least one illustration of your method.

2. From what you have observed in this activity, write a rule fordivision of fractions.


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Activity 10: Professors Short and Tall

PURPOSE Introduce the concepts of ratio and equivalent ratios.

MATERIALS Pouch: Colored Squares Other: 15 pennies

GROUPING Work individually or with a partner.

1. The man at the left is Professor Short.

a. How many pennies tall is he? __________

b. How many squares tall is he? __________

2. a. Professor Tall looks exactly like Professor Short, but he is 10 pennies tall! Without measuring, predict Professor Tall’sheight in squares. __________

b. Explain your thinking.

3. a. Even though you do not have a picture of Professor Tall, youcan still measure his height in squares. Explain how you coulddo this.

b. How many squares tall is he? __________

c. Is this the same as your prediction? If not, can you explainwhy?

A ratio is a comparison of two numbers or measures by division. Theratio of Professor Short’s height to Professor Tall’s height can bewritten in four ways.

8 to 10

4. Write the ratio of Professor Short’s height in squares to ProfessorTall’s height in squares in four different ways.

5. The following table contains the heights for some other peoplewho look exactly like Professor Short.

a. Fill in the missing numbers in the table.

b. Explain how you found the missing numbers.

Height in Pennies 4 12 16

Height in Squares 21

8 4020

6

8 : 108 , 108

10

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6. Use the information in the table in Exercise 5(a) to answer thefollowing questions:

a. If a person is 27 squares tall, what is his/her height inpennies? __________

b. If a person is 10 pennies tall, what is his/her height insquares? __________

c. How does each pair of numbers in the table appear to berelated?

In the table in Exercise 5(a), the ratios of the heights in pennies to theheights in squares are equivalent. Ratios that can be expressed asequivalent fractions or quotients are equivalent.

7. Is the ratio of Professor Short’s height in squares to ProfessorTall’s height in squares (the ratio you found in Exercise 4)equivalent to the ratio of their heights in pennies? Explain whyor why not.

8. Graph the pairs of numbers in the table on the following coordinate grid.

a. What do you notice about the points?

b. Use the graph to answer the following questions:

If a person is 30 pennies tall, what is his/her height insquares? __________

If a person is 18 squares tall, what is his/her height inpennies? __________

Height in Pennies

Hei

gh

t in

Sq

uar

es

0 4 8 12 16 20 24 28 32 36 400

3

6

9

12

15

18

21

24

27

30



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Activity 11: A Call to the Border

PURPOSE Apply ratio and proportion, measurement, scale interpretation,estimation, and communication in a problem-solving context.

MATERIALS Other: Almanac or atlas, state map, ruler, string, scissors, and calculator

GROUPING Work in groups of four.

GETTING STARTED

1. Do you think the governor’s claim is true?

2. If the residents of your state lined up along the state’s border, do you think peoplestanding next to each other would be able to:

• see each other? • hold hands?

• carry on a conversation? • stand shoulder-to-shoulder?

• touch fingertip-to-fingertip? • stand belly-to-back?

3. List the information you will need to verify the governor’s claim and your predictionof how close people will be able to stand.

4. Explain how the perimeter of your state is affected by irregularities such as coastline,islands, river boundaries, mountains, etc.

5. Record the following:

State __________ Population __________ Perimeter of state __________

6. Is the governor’s claim correct? Why or why not? Explain your reasoning.

EXTENSION In which states might this solution to state spending cuts NOT work?Which states pose special problems in implementing your method forsolving the problem? Name some states where residents might bestanding closer together or farther apart than in your state.

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Chapter Summary

The activities in this chapter emphasized development of the conceptual understanding of rational numbers (fractions), and theiroperations. Concrete materials and structured lessons illustrated how the operations with fractions are an extension of the operationswith whole numbers.

A curriculum developmentally appropriate for students is one of the central themes of the Principles and Standards for SchoolMathematics and the Curriculum Focal Points for Prekindergartenthrough Grade 8 Mathematics. Lessons should progress from theconcrete to the representational (pictorial) to the abstract. The activities in this chapter illustrated such a curriculum through careful development of the operations with fractions.

Activity 1 developed the concept of a fraction using a variety of models. In different problems, you (a) determined the fractional part of a whole, (b) compared two areas or the length of two strips to determine a fraction, and (c) determined the whole unit given thefractional part.

In Activity 2, you used geometric representations of fractions that you constructed by folding and cutting a square. The pieces of thesquare helped you to explore the relationship among various piecesthat made up the whole, and reinforced the understanding of addition,subtraction, multiplication, and division of fractions.

Activity 4 introduced equivalent fractions, a concept that is critical toordering fractions, and adding and subtracting fractions. Activities 3, 5,and 6 may be the most important ones in the chapter. They addressedanother central theme of the Principles and Standards for SchoolMathematics: developing number sense. Only when the concept of afraction is understood can one develop a sense of its size. Knowingterms like about the same size, half as much, and very small as com-pared to is critical when estimating the magnitude of a fraction.

Activity 7 used two models to explore addition and subtraction offractions. Each one used the concept of equivalence as developed inprevious activities. The importance of common denominators wasconnected to dividing the whole into equivalent parts that could then be added or subtracted. Activity 8 illustrated the language relationship between the word of and multiplication of rational numbers through a geometric model for fractions, and Activity 9modeled division of fractions in the same way that division of whole

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numbers was shown. That is, how many groups of one factor (thedivisor) are there in the product (the dividend)? By modeling divisionthis way, you can come to understand the familiar rule, “invert andmultiply.”

Activity 10 used multiple representations to introduce the concepts of ratio and equivalent ratios. Activity 11 involved a rich problem-solving situation in which proportional reasoning was applied to scale interpretation on maps. Using reference materials to determinethe state population, estimating the space a person can occupy, andfinding the best estimate for the perimeter of a state were all requiredto solve the Border problem.

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