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Another Look at Fractions of a Set

  • Lesson
3-5
1
Number and OperationsData Analysis and Probability
Tracy Y. Hargrove
Location: unknown

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.

1305 wrapped candy

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:

1305 bags

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.

appicon 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.

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.)]

1287icon
Number and Operations

Eggsactly with a Dozen Eggs

3-5
Students begin to examine fractions as part of a set. This lesson helps students develop skill in problem solving and reasoning as they examine relationships among the fractions used to describe part of a set of 12.
1291icon
Number and Operations

Eggsactly with Eighteen Eggs

3-5
Students continue to examine fractions as part of a set. This lesson helps students develop skill in problem solving and reasoning as they examine relationships among the fractions used to describe part of a set of eighteen.
1296icon2
Number and Operations

Eggsactly Equivalent

3-5
Students use twelve eggs to identify equivalent fractions. Construction paper cutouts are used as a physical model to represent various fractions of the set of eggs, for example, 1/12, 1/6, and 1/3. Students investigate relationships among fractions that are equivalent.
1298icon
Number and Operations

Another Look at the Set Model using Attribute Pieces

3-5
The previous lessons focused on the set model where all objects in the set are the same size and shape. Students also need work with sets in which the objects “look” different. In the real world, we are often faced with fraction situations where the objects in the set are not identical. For this lesson, students use fractions to describe a set of attribute pieces. Students develop skill in problem solving and reasoning as they think about their set and how to create new sets given specific fractional characteristics.
1301icon
Data Analysis and Probability

Class Attributes

3-5
During this lesson, students create their own classroom survey or use previously generated questions to study the class and describe the set [class] in fractional parts. This lesson requires that students identify fractions in real-world contexts from a set of items that are not identical. This lesson is integrated with other areas of the math curriculum, including data analysis and statistics.

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?