Chapter 6 Rational Numbers and Proportional Reasoning

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1 Chapter 6 Rational Numbers and Proportional Reasoning Students should build their understanding of fractions as parts of a whole and as division. They will need to see and explore a variety of models of fractions, focusing primarily on familiar fractions such as halves, fourths, fifths, sixths, eighths, and tenths. By using an area model in which part of the region is shaded, students can see how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions. They should develop strategies for ordering and comparing fractions, often using benchmarks such as or. Principles and Standards for School Mathematics Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates for them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. Common Core Standards for Mathematics Facility with proportionality involves much more than setting two ratios equal and solving for a missing term. It involves recognizing quantities that are related proportionally and using numbers, tables, graphs, and equations to think about the quantities and their relationship. Proportionality is an important integrative thread that connects many of the mathematics topics studied in grades 6 8. Principles and Standards for School Mathematics 85 Copyright 0 Pearson Education, Inc. All rights reserved.

2 86 Chapter 6 Rational Numbers and Proportional Reasoning Other than whole-number computation, no topic in the elementary mathematics curriculum demands more time than the study of fractions. Yet, despite the years of study, most students enter high school with a poor concept of fractions and an even poorer understanding of the operations with fractions. When asked about their memories of fractions, adults will often reply, Yours is not to reason why, just invert and multiply. Rational numbers should be taught as the natural extension of the whole numbers. The fraction 4 can be viewed as the solution to the problem of dividing dollars among 4 people: 4, or answering the question How many fourths (quarters) will each person receive? For students to understand this connection between whole numbers and fractions, teaching about fractions and their operations should begin with concrete models, as was the instruction with whole numbers. A well-developed concept of fractions and a feeling for their magnitude must be established before they can be compared and ordered meaningfully. This must precede computation. Number sense involving fractions, and a deeper understanding of the operations with fractions must be developed prior to formal work with algorithms for the operations. This chapter provides activities that firmly establish the concepts of fractions. All operations are explored through a variety of concrete models and reinforced by representing the models pictorially. Multiple representations will be used to introduce the concepts of ratio and proportion. In later chapters, ratio and proportion will be applied in real world contexts and used to explore proportional variation in geometry. Copyright 0 Pearson Education, Inc. All rights reserved.

3 Chapter 6 Activity 87 Activity : What Is a Fraction? PURPOSE MATERIALS GROUPING Develop the concept of a fraction. Pouch: Pattern Blocks and Cuisenaire Rods Work individually or in pairs. Use pattern blocks to solve the following problems.. The trapezoid is what fractional part of the hexagon?. The blue rhombus is what fractional part of the hexagon?. 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?. shaded unshaded. shaded unshaded. shaded unshaded 4. shaded unshaded Copyright 0 Pearson Education, Inc. All rights reserved.

4 88 Chapter 6 Rational Numbers and Proportional Reasoning Use Cuisenaire rods to solve the following.. If the orange rod, each rod is what fractional part of the orange rod? a. red b. green _ c. yellow d. purple _. If the purple rod, each rod is what fractional part of the purple rod? a. brown b. orange _ c. dark green d. black _. If the red rod = which rod? _,. If the red rod = which rod? _,. If the white rod = which rod? _ 5, 4. If the white rod = which rod =? _ 4, 4 5. If the red rod = which rod =? _, 6. If the red rod = which rod =? _, Explain how you used the rods to arrive at your answers. Use two red trapezoids and one blue rhombus to construct a shape similar to the one shown below.. Given that the shape, what pattern block(s) would you use to represent each of the following fractions? a. b. c. 4 8 Fill in the same shape using one red trapezoid, two blue rhombuses, and one green triangle.. What fraction is represented by each of the following? a. a blue rhombus b. a red trapezoid c. a green triangle Copyright 0 Pearson Education, Inc. All rights reserved.

5 Chapter 6 Activity 89 Activity : Square Fractions PURPOSE MATERIALS GROUPING GETTING STARTED Reinforce the concept of a fraction and the meaning of equivalent fractions, and illustrate operations with fractions using geometric models. Other: Sheet of paper 0 cm square (colored construction paper works well) and scissors (optional) Work individually. Work through each section of the activity in order, following the folding and cutting directions carefully. If scissors are not used, fold and crease the paper sharply so that it will tear cleanly. Fold the square as shown and cut or tear along the fold to divide the square into two congruent parts. A. Each triangle is what fraction of the original square? Pick one of the two triangles and fold it as shown. Cut or tear along the fold and label the two triangles and.. Triangle 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 to the midpoint of the longest side. Cut along the fold and label the polygons B and as shown. B. Triangle is what fractional part of a. triangle? b. triangle A? c. trapezoid B? Copyright 0 Pearson Education, Inc. All rights reserved.

6 90 Chapter 6 Rational Numbers and Proportional Reasoning Fold one of the endpoints of the longest side of trapezoid B to the midpoint of that side as shown. Cut along the fold and label the polygons C and 4. C 4 B 4. Triangle 4 is what fractional part of a. triangle? b. triangle A? c. trapezoid C? Fold trapezoid C as shown. Cut along the fold and label the polygons D and 5. D 5 C 5. Square 5 is what fractional part of a. trapezoid C? b. trapezoid D? c. triangle? 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? b. trapezoid C? Copyright 0 Pearson Education, Inc. All rights reserved.

7 Chapter 6 Activity 9 Fold trapezoid D as shown. Cut along the fold and label the two polygons 6 and D 8. Triangle 6 is what fractional part of a. trapezoid D? b. triangle? 9. Parallelogram 7 is what fractional part of a. square 5? b. trapezoid B? c. triangle? d. original square? 0. Explain how triangle, triangle A, and the original square can be used to illustrate that is equivalent to 4.. Explain how these figures can be used to illustrate that of is which is the multiplication problem * = 4, 4.. If the original square is one unit, explain how trapezoid D and triangle can be used to model the addition problem = If trapezoid B is one unit, explain how triangle 4 and trapezoid D can be used to model the division problem, 6 =. Copyright 0 Pearson Education, Inc. All rights reserved.

8 9 Chapter 6 Rational Numbers and Proportional Reasoning Activity : How Big Is It? PURPOSE MATERIALS GROUPING Develop the ability to estimate the magnitude of a fraction. Online: Fraction Sorting Board and Fraction Cards Work in pairs. GETTING STARTED Use these rules to complete the following problems. A fraction is close to 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. THE FRACTION SORTING GAME This is a game for two players. Cut out the fraction cards and shuffle them. Students take turns placing a card in the appropriate column on the fraction sorting board and justifying each placement to the other player. Reshuffle the deck and play again.. Complete the following fractions so that they are close to, but less than, a. b. c. d e. f. g. h.. Complete the following fractions so that they are close to, but less than, a. 7 b. c. 75 d. e. 9 f. g. h EXTENSIONS. Given the fraction, what numbers would be acceptable in place of so that the resulting fraction is close to but less than? Justify your answer.. Given the fraction, what numbers would be acceptable in place of so that the resulting fraction is close to? Justify your answer. Copyright 0 Pearson Education, Inc. All rights reserved.

9 Chapter 6 Activity 4 9 Activity 4: Equivalent Fractions PURPOSE MATERIALS GROUPING Develop the concept of a fraction using concrete models and a problem-solving approach. Pouch: Cuisenaire Rods and Pattern Blocks Online: Fraction Strips Work individually or in pairs. For the following problems, use Cuisenaire rods to construct the trains.. Make all of the possible one-color trains the same length as a dark green rod and complete the following. If dark green, then a. light green = = 6 b. red = = 6 c. purple = = d. dark green = 6 =. Make all of the possible one-color trains the same length as a brown rod and complete each of the following. If brown, then a. purple = = = b. dark green = = c. red = = Copyright 0 Pearson Education, Inc. All rights reserved.

10 94 Chapter 6 Rational Numbers and Proportional Reasoning Use pattern blocks to construct a shape similar to the star and complete the following. If the star shape, then. trapezoid = =. blue rhombuses = 6 = =. hexagon = 6 = 6 = Find the fraction strips that can be folded into parts so that the resulting strip is equal in length to the fraction given in each problem. Folds may be made only on the lines on the fraction strips. Write the name of the equivalent fractions in the space provided.. =.. = 4 = EXTENSIONS. a a Given a set of fractions equivalent to the fraction where b, b is in lowest terms, what is the relationship among the set of numerators? the set of denominators?. a a Given a set of equivalent fractions, where is in b, c d, e f, Á, b lowest terms, what is the relationship between the numerator and a denominator of and the numerator and denominator of any b equivalent fraction? Copyright 0 Pearson Education, Inc. All rights reserved.

11 Chapter 6 Activity 5 95 Activity 5: Fraction War PURPOSE MATERIALS GROUPING GETTING STARTED Reinforce estimation and comparison of fractions in a game format. Online: Fraction Arrays and Fractions Game Board Other: Deck of playing cards (remove face cards) Work in pairs. Follow the rules below to play Fraction War. Example: FRACTIONS GAME BOARD. 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 smaller A number in the numerator. Player with the smaller fraction is the winner. b. Form a proper fraction by placing the card with the smaller A number in the numerator. Player with the larger fraction is the winner. 4 5 c. Form a proper fraction by placing the card with the smaller number in the numerator. Player with the fraction whose value is closest to is the winner. 4 5 d. Form a fraction by placing the card with the larger number in the numerator. Player with the larger fraction is the winner. Player A Player B e. Place the first card in the numerator, the second in the denominator. Player with the fraction whose value is closest to is the winner. Player A wins a round in Game a: the smaller fraction is the winner. Note: When necessary, use the Fraction Arrays to compare the fractions. f. Each player decides where to place each card. Player with the fraction whose value is closest to is the winner.. Each player turns up two cards from his or her pile and follows the directions for the chosen game to form a fraction on the Fractions Game Board. The ace represents.. The winner of each round collects the four cards and places them face up at the bottom of his or her pile of cards. If the fractions formed are equivalent, each player turns over two additional cards and forms a new fraction. The winner of the round gets all eight cards. 4. When the players have played all the face-down cards, the player with the most face-up cards is the winner of the game. Reshuffle the cards and choose a different game. Copyright 0 Pearson Education, Inc. All rights reserved.

12 96 Chapter 6 Rational Numbers and Proportional Reasoning Activity 6: What Comes First? PURPOSE MATERIALS GROUPING Compare and order fractions. Pouch: Cuisenaire Rods and Pattern Blocks Online: Fraction Arrays Work individually. Use Cuisenaire rods to build all possible one-color trains that are the same length as a brown rod, and complete the following.. 5 Which is larger, or? 8 Complete the inequality:. Which is smaller, or? 8 4 Complete the inequality:. 7 Which is larger, or? 8 4 Complete the inequality: Use pattern blocks to construct a star shape (see Activity 4) and complete the following.. 5 Which is larger, or? Complete the inequality:. 7 Which is smaller, or? Complete the inequality:. 5 Which is larger, or? 6 4 Complete the inequality: Use the Fraction Arrays to order the following sets of fractions.., 5, 4 7., 4, , 5, ,, 4 5 Copyright 0 Pearson Education, Inc. All rights reserved.

13 Chapter 6 Activity 7 97 Activity 7: Adding and Subtracting Fractions PURPOSE MATERIALS GROUPING Develop algorithms for adding and subtracting fractions. Pouch: Pattern Blocks Online: Fraction Strips Work individually. If the yellow hexagon, then the red trapezoid = the blue rhombus = and the green triangle =,, 6. Use pattern blocks to solve the following:. red + green red + red green + green red + blue green + green + +. blue + green red - blue red - green Copyright 0 Pearson Education, Inc. All rights reserved.

14 98 Chapter 6 Rational Numbers and Proportional Reasoning 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 one whole = =. 4 - = = 5. + = = 7. + = = Copyright 0 Pearson Education, Inc. All rights reserved.

15 Chapter 6 Activity 7 99 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 strip must then be found that has fold lines in common with the two fractions. Example for Addition: + 4 = Example for Subtraction: + = - 4 = = = 5 Use your fraction strips to solve the following problems = 4 = = 4 = = 4-6 = EXTENSION From your observation in this activity, write rules for adding and subtracting fractions with like and unlike denominators. Copyright 0 Pearson Education, Inc. All rights reserved.

16 00 Chapter 6 Rational Numbers and Proportional Reasoning Activity 8: Multiplying Fractions PURPOSE MATERIALS GROUPING Develop an algorithm for multiplying fractions. Pouch: Pattern Blocks Other: Paper for folding Work individually. Example: 4 of means two of three equal parts of 4. one whole 4 of 4 * 4 = = 6 Place pattern blocks on Figure A to solve the following multiplication problems. Record your solutions pictorially and numerically. Figure A. * =. 4 * =. 4 * = 4. 4 * = * = Copyright 0 Pearson Education, Inc. All rights reserved.

17 Chapter 6 Activity 8 0 Example: of means one of the two equal parts of two thirds. of one whole 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.. 4 * 6 =. 8 * =. 7 * = * = Copyright 0 Pearson Education, Inc. All rights reserved.

18 0 Chapter 6 Rational Numbers and Proportional Reasoning of means one of the three equal parts of. Divide a piece of paper into thirds with vertical folds. of means one of the two equal parts of thirds into halves with a horizontal fold. of of means two of the three equal parts of of = 6 = 6 =. Now, divide the. Fold sheets of paper to solve the following multiplication problems. Record each step of your solutions pictorially and numerically.. * 4 =. 4 * =. * = 4. 8 * = EXTENSION From what you have observed in this activity, write a rule for multiplication of fractions. Copyright 0 Pearson Education, Inc. All rights reserved.

19 Chapter 6 Activity 9 0 Activity 9: Dividing Fractions PURPOSE Develop understanding of division of fractions. MATERIALS Pouch: Pattern Blocks GROUPING Work individually. GETTING STARTED Recall the use of the multiplication and division frame for the division of whole numbers. Example: 6 can mean how many groups of are there in 6? In the following example represents. Example:, means: How many groups of are there in??? one In each of the following, complete the sentence and then use pattern blocks to solve the related division problem.. means, 6, 6 =. means, 4, 4 = 5. means 6, 5 5 6, 5 = 4. means 4, 4 4, 4 = 5. means, 4, 4 = Copyright 0 Pearson Education, Inc. All rights reserved.

20 04 Chapter 6 Rational Numbers and Proportional Reasoning 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 = In each of the following, use pattern blocks to solve the division problems , 6, = =. 4.,, = = EXTENSIONS. Describe how you would use fraction strips to solve the previous problems. Draw at least one illustration of your method.. From what you have observed in this activity, write a rule for division of fractions. Copyright 0 Pearson Education, Inc. All rights reserved.

21 Chapter 6 Activity 0 05 Activity 0: Professors Short and Tall PURPOSE MATERIALS GROUPING Introduce the concepts of ratio and equivalent ratios. Pouch: Colored Squares Other: 5 pennies Work individually or with a partner.. The man at the left is Professor Short. a. How many pennies tall is he? b. How many squares tall is he?. a. Professor Tall looks exactly like Professor Short, but he is 0 pennies tall! Without measuring, predict Professor Tall s height in squares. b. Explain your thinking.. a. Even though you do not have a picture of Professor Tall, you can still measure his height in squares. Explain how you could do this. b. How many squares tall is he? c. Is this the same as your prediction? If not, can you explain why? A ratio is a comparison of two numbers or measures by division. The ratio of Professor Short s height to Professor Tall s height can be written in four ways. 8 to , 0 4. Write the ratio of Professor Short s height in squares to Professor Tall s height in squares in four different ways. 5. The following table contains the heights for some other people who look exactly like Professor Short. a. Fill in the missing numbers in the table. Height in Pennies Height in Squares 6 b. Explain how you found the missing numbers. 8 : 0 Copyright 0 Pearson Education, Inc. All rights reserved.

22 06 Chapter 6 Rational Numbers and Proportional Reasoning 6. Use the information in the table in Exercise 5(a) to answer the following questions: a. If a person is 7 squares tall, what is his/her height in pennies? b. If a person is 0 pennies tall, what is his/her height in squares? c. How does each pair of numbers in the table appear to be related? In the table in Exercise 5(a), the ratios of the heights in pennies to the heights in squares are equivalent. Ratios that can be expressed as equivalent fractions or quotients are equivalent. 7. Is the ratio of Professor Short s height in squares to Professor Tall s height in squares (the ratio you found in Exercise 4) equivalent to the ratio of their heights in pennies? Explain why or why not. 8. Graph the pairs of numbers in the table on the following coordinate grid. Height in Squares Height in Pennies a. What do you notice about the points? b. Use the graph to answer the following questions: If a person is 0 pennies tall, what is his/her height in squares? If a person is 8 squares tall, what is his/her height in pennies? Copyright 0 Pearson Education, Inc. All rights reserved.

23 Chapter 6 Activity 07 Activity : A Call to the Border PURPOSE MATERIALS GROUPING Apply ratio and proportion, measurement, scale interpretation, estimation, and communication in a problem-solving context. Other: Almanac or atlas, state map, ruler, string, scissors, and calculator Work in groups of four. GETTING STARTED. Do you think the governor s claim is true?. If the residents of your state lined up along the state s border, do you think people standing 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?. List the information you will need to verify the governor s claim and your prediction of 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 for solving the problem? Name some states where residents might be standing closer together or farther apart than in your state. Copyright 0 Pearson Education, Inc. All rights reserved.

24 08 Chapter 6 Rational Numbers and Proportional Reasoning Chapter Summary The activities in this chapter emphasized development of the conceptual understanding of rational numbers (fractions), and their operations. Concrete materials and structured lessons illustrated how the operations with fractions are an extension of the operations with whole numbers. A curriculum developmentally appropriate for students is one of the central themes of the Principles and Standards for School Mathematics and the Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. Lessons should progress from the concrete 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 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 the fractional part. In Activity, you used geometric representations of fractions that you constructed by folding and cutting a square. The pieces of the square helped you to explore the relationship among various pieces that 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 to ordering fractions, and adding and subtracting fractions. Activities, 5, and 6 may be the most important ones in the chapter. They addressed another central theme of the Principles and Standards for School Mathematics: developing number sense. Only when the concept of a fraction is understood can one develop a sense of its size. Knowing terms like about the same size, half as much, and very small as compared to is critical when estimating the magnitude of a fraction. Activity 7 used two models to explore addition and subtraction of fractions. Each one used the concept of equivalence as developed in previous activities. The importance of common denominators was connected 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 9 modeled division of fractions in the same way that division of whole Copyright 0 Pearson Education, Inc. All rights reserved.

25 Chapter 6 Summary 09 numbers was shown. That is, how many groups of one factor (the divisor) are there in the product (the dividend)? By modeling division this way, you can come to understand the familiar rule, invert and multiply. Activity 0 used multiple representations to introduce the concepts of ratio and equivalent ratios. Activity involved a rich problemsolving situation in which proportional reasoning was applied to scale interpretation on maps. Using reference materials to determine the state population, estimating the space a person can occupy, and finding the best estimate for the perimeter of a state were all required to solve the Border problem. Copyright 0 Pearson Education, Inc. All rights reserved.

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