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Understanding Rational Numbers and Proportions

  • Lesson
6-8
3
Number and Operations
Unknown
Location: Unknown

In this lesson, students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving.

The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a bakery. These activities involve several important concepts of rational numbers and proportions, including partitioning a unit into equal parts, the quotient interpretation of fractions, the area model of fractions, determining fractional parts of a unit not cut into equal-sized pieces, equivalence, unit prices, and multiplication of fractions.

Activity 1: Customers Cut the Cake 

Each day the local baker makes several rectangular sheet cakes, which he cuts into eighths. He sells 1/8 of a sheet cake for $1.59. As part of a new promotional campaign for his store, he wants to cut his sheet cakes into eighths a different way each day. Customers who suggest a new way to cut the cakes into eighths win a free piece of cake each day for a week. What are some of the different ways to cut the cake?

Some questions to ask students include:

  • What is the shape of the baker's cakes? [Rectangular cakes.]
  • What are the restrictions on the ways the cakes can be cut? [Pieces must contain the same amount of cake; they do not have to be the same shape.]
  • How can we verify that pieces that are not the same shape contain the same amount of cake? [Cut the pieces into smaller parts and lay one on top of the other.]

This activity gives students opportunities to represent parts of a whole by using an area model of fractions. Many students know that fractions often refer to equal-sized parts of a unit, but they frequently overgeneralize and believe that the pieces have to be congruent rather than merely contain the same area.

Encourage students to solve the problem in pairs or in small groups by using the Cakes Cut Into Eighths activity sheet.

pdficon  Cakes Cut Into Eighths Activity Sheet 

Alternatively, students can use rectangular pieces of paper to model the cakes, sketching the shapes or cutting the paper into eight pieces and verifying the equivalence of the pieces by cutting and overlapping. Some students may have a limited view and think that all the cuts must be parallel to one side of the rectangle. Challenge them to think of other ways to make the cuts. (It is important for students to know that each rectangle on the activity sheet represents a full cake.) 

Have students place their designs on the chalkboard or the overhead projector. Ask the students to decide which designs are the same and which are different. Examples of some diagrams are shown below. Have the students discuss whether the two rectangles in this figure are cut differently.

1110 figure 2.1

What factors should be considered when deciding whether the two designs are different? (The number of pieces, the equivalence of the pieces, and whether the location of the pieces makes a difference should be considered.) The two rectangles in the above figure contain the same eight pieces, but the pieces are arranged differently. The students can decide whether they want to consider these as two different arrangements.

Another way that must be considered is using cuts that are curves or combinations of line segments, such as the examples in the figure below.

1110 figure 2.2

The equivalence of shapes formed by cuts that are curves is difficult to determine but is a good investigation in itself. The equivalence of shapes that are formed by cuts that are combinations of straight line segments is easier to determine. Including these types of shapes, however, again greatly increases the number of possibilities. By interacting with students, decide which designs to include in the final count.

 

Ask each group to choose one design and either to post it on the chalkboard or bulletin board or to draw it on the overhead projector. Students in each group should be prepared to explain how they know that their method shows eighths. One way to verify that a solution does in fact result in eighths is to cut up the individual pieces further and lay them on top of each other to verify the equivalence of their areas.

As a follow-up activity, teachers may choose to discuss with students why or why not each cake on below is cut into eighths.

1110 figure 2.3 

Activity 2: You Can Eat Your Cake and Have It, Too! 

The baker is conducting a second contest, this time for his employees. As part of a new promotional campaign for his store, each day he wants to feature sheet cakes that have been cut into four pieces in a different way. The pieces do not have to be equal for this promotion. The baker has challenged his employees to suggest interesting ways to cut the cakes into four pieces. The employees must also determine the price for each piece. The bakery sells 1/8 of a sheet cake for $1.59. What are some of the different ways the cakes can be cut, and how much should each piece cost?

Some questions for the students to discuss include:

  • What are the restrictions on the ways the cakes are cut? [Each cake must be cut into four pieces, not necessarily equal-sized pieces.]
  • How can we determine the fractional parts of the pieces we cut? [We can use equivalences we know, such as 2/4 is equivalent to 1/2, to find the value of each part, if we partition by finding parts of parts.]
  • What will happen if we just cut four pieces at random? [It will be difficult to determine the size of each piece.]

Instead of focusing on making equal-sized pieces as in the previous activity, this activity explores determining the fractional parts and cost of pieces when a unit is cut into four unequal parts.

Encourage students to solve the problem in pairs or small groups by using the Cakes Cut Into Fourths activity sheet or rectangular paper to model the cakes. They sketch the pieces or cut the rectangles into four pieces and determine the value of each piece.

pdficon  Cakes Cut Into Fourths Activity Sheet 

One way to find the value of each piece is to add partitioning lines so that the whole is partitioned into equal-sized pieces. Students may remember some of the ways they cut the cakes into eighths in the first activity, which may help them. Once again, weighing could solve this problem.

The following overhead can be projected after students have had time to create their designs.

1110 overhead  Making Four Pieces Overhead 

Have students share and discuss their designs. The rectangles shown below show a few possible ways to cut the rectangular cakes into four parts. The pieces have been labeled to show the fraction of the cake they represent.

1110 figure 2.4

Students must then find the cost for each piece, if 1/8 of a cake costs $1.59. What should the total cost of one whole cake be? (If 1/8 of a cake costs $1.59, then a whole cake should cost eight times as much, or $12.72.) 

Challenge students to explain and verify their solutions.

1110 chart

Activity 3: That's the Way the Cookie Crumbles! 

You bought a baker's dozen (13) of cookies that you want to share equally with your family. How many cookies will each person get?

Ask students to compare this problem with the one posed in the first activity. Give them time to think about the similarities and differences between this problem and the problem posed in Activity 1.

[This problem is similar to the first problem, that of cutting rectangular sheet cakes into eighths in different ways, because they both involve partitioning a unit into parts. This problem is different in several ways: The whole or unit in this group of 13 cookies; the problem does not specify exactly into how many pieces to cut each cookie or how many people are sharing the cookies; the problem has different solutions for students who have different-sized families; and this problem involves a different interpretation of rational numbers — the quotient interpretation. The quotient interpretation refers to the fact that in this problem, in which thirteen cookies are being shared by n people (n is the number of people in the family), the number of cookies each person receives is the quotient, 13 ÷ n.]

Consider grouping students according to the number of people in their families. Students can draw 13 circles on a piece of paper to practice dividing the cookies. After each group has solved the problem, share the solution processes with the whole class. Have each student complete a table similar to the sample below.

1110 urnap chart 

The table should include all the different-sized families of students in the class and also contain a few other family sizes, including one or two families that are larger than the largest family in the class. Students should begin completing the table by including solutions from the groups in the class and then working on solving the problems for other family sizes.

Discuss the solutions, focusing on the patterns students see in the number of cookies for each person as the size of the family changes. The goal for students is to generalize that the number of cookies for each person is equal to the number of cookies divided by the number of people sharing them.

Ask students to write a rule that represents the number of cookies each person receives if thirteen cookies are shared by n people (where n is the number of people in the family).

[The number of cookies each person receives is the quotient, 13 ÷ n.]

Ask students to describe the way they solved the problem in their groups (e.g., by drawing a picture, using long division, and so on) so that a student in another class would understand what problem was solved and how it was solved.

Assessments 

Assess student understanding of naming fractions and fractional equivalents by focusing on how students solve this problem. For example, are students able to recognize that 4/8 is equivalent to 1/2? Do students use their knowledge that 6/8 is equal to 6 one‑eighths to help them find the cost of 6/8 of a cake (i.e., since 1/8 of a cake costs $1.59, then 6/8 of a cake costs 6 ×:1.59 = $9.54)?

Extensions 

  1. Students find the cost of various-sized pieces, given that 1/8 of a cake costs $1.59 and a whole cake costs $12.72. The following table is a sample. (Tables may include other fractional parts and need not be limited to eighths, fourths, and halves.) Students may wish to use calculators with fraction capability to help them find the various prices. Using calculators may help students focus on the reasonableness of their solutions rather than on the calculations.
  2. How many cookies would each person get if...
    • three people shared twenty cookies? [20/3, or 6 and 2/3 cookies for each person]
    • eight people shared twenty cookies? [20/8, or 2 and 1/2 cookies for each person]
    • x people shared twenty cookies? [20/x cookies for each person]

    What is a rule for finding the number of cookies each person will get if a people share b cookies? [b ÷ a cookies for each person]

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Learning Objectives

Students will:

  • Represent parts of a whole using an area interpretation of fractions
  • Determine the fractional part of a whole when parts are not cut into equal-sized pieces
  • Develop an understanding of the quotient interpretation of fractions
  • Find the unit cost of items that are part of a set
  • Determine the relationship among parts of a whole that are unequal-sized pieces
  • Express fractional parts of a whole as decimal equivalents

Common Core State Standards – Mathematics

Grade 6, Ratio & Proportion

  • CCSS.Math.Content.6.RP.A.2
    Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, ''This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.'' ''We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.''

Grade 7, Ratio & Proportion

  • CCSS.Math.Content.7.RP.A.1
    Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.