## Another Look at Fractions of a Set

- Lesson

This lesson gives students another opportunity to explore fractions using the set model. This lesson is integrated with other areas of the math curriculum including data analysis and statistics.

In
this culminating activity, students examine the set model using colored
candies. Give students an individual bag of colored candies, e.g.,
M&M's^{®} or Skittles^{®}. Have students open their
bag of candies and sort by color. Have students count the number of
each color in their set and record those data on notebook paper. Have
students record the fraction of each color represented in their
individual packet. All fractions should be reduced to lowest form.

Have students log on to the Create a Graph Tool from the National Center for Education Statistics. Students should choose the type of graph they want to create by using the pulldown menu. Once students have created their graph, they should label the data in fractional parts and reduce all fractions to lowest terms.

As a class, create a line plot of the number of candies in each bag. An example is shown below:

Have students determine the fractional representation for each number of candies. For example, for the graph shown above there were:

- 2 students with 22 candies (2/16 or 1/8),
- 4 students had 23 candies (4/16 or 1/4),
- 5 students had 24 candies (5/16),
- 3 students had 25 candies (3/16), and
- 2 students had 26 candies (2/16 or 1/8).

Next, have students log on to the Circle Grapher to create a circle graph for the class data.

Circle Grapher |

Fractional representations should be labeled. Ask students to share their circle graphs with a neighbor. Discuss how a circle graph is useful for showing fractions of a set. Some students may also recognize that percents are also used in circle graphs. This discussion would be a nice tie-in to percents, specifically fractions out of 100.

- One package of colored candies (such as M&Ms® or Skittles®) for each student
- Create a Graph Web Tool
- Circle Grapher

**Assessments**

Since this is the culminating activity for the unit, it is important to use this activity as a summative assessment opportunity. You may wish to examine students' graphs, and have them write a few sentences summarizing each of the graphs.

**Teacher Reflection**

- Do students understand that a fraction can be represented as part of a set?
- Can students identify fractions when the whole (set) and part of the set is given?
- Can students articulate relationships between fractions?
- Do students understand the relationships inherent when comparing equivalent fractions?
- Can students reduce fractions to lowest terms?
- Are there other models of fractions that I could use with these students to extend their repertoire of fraction representations?
- What other experiences can I introduce to students to help them better understand relationships among fractions?
- How can I ensure that students have a solid conceptual understanding of fractions?
- How can I help students relate the concepts in this unit to other areas of mathematics?
- How can I help students relate the concepts in this unit to other areas of the curriculum?

**Questions for Students**

1. Which type of graph did you create when you went to the Create a Graph Tool from the National Center for Education Statistics? Why did you select this type of graph?

[Student responses may vary. They should give a valid justification for their graph selection.]

2. How does a line plot show the number of candies in each bag?

[Each "X" represents one piece of candy. For example, two Xs above the number 22 indicates two students whose bags contained 22 pieces of candy.]

3. Why is a circle graph an appropriate graph to use for fractions of a set?

[A circle graph is a good representation of fractions. The pieces of the circle graph represent a certain fraction (or percent.)]

### Eggsactly with a Dozen Eggs

### Eggsactly with Eighteen Eggs

### Eggsactly Equivalent

### Another Look at the Set Model using Attribute Pieces

### Class Attributes

### Learning Objectives

Students will:

- Demonstrate understanding that a fraction can be represented as part of a set
- Identify fractions when the whole (set) and part of the set is given
- Describe a set of objects based on its fractional components
- Identify fraction relationships associated with the set
- Identify equivalent fractions
- Identify relationships inherent in equivalent fractions
- Reduce fractions to their lowest terms

### NCTM Standards and Expectations

- Recognize equivalent representations for the same number and generate them by decomposing and composing numbers.

- Use models, benchmarks, and equivalent forms to judge the size of fractions.

- Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.

- Recognize and generate equivalent forms of commonly used fractions, decimals, and percents.

- Collect data using observations, surveys, and experiments.

- Represent data using tables and graphs such as line plots, bar graphs, and line graphs.

### Common Core State Standards – Mathematics

Grade 3, Num & Ops Fractions

- CCSS.Math.Content.3.NF.A.1

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Grade 4, Num & Ops Fractions

- CCSS.Math.Content.4.NF.A.1

Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Grade 4, Num & Ops Fractions

- CCSS.Math.Content.4.NF.A.2

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Grade 5, Num & Ops Fractions

- CCSS.Math.Content.5.NF.B.3

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?