## Class Attributes

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.

This activity may require two sessions to complete each of the tasks. First, the students collect data on the class and use that data to generate fractions that describe the class. Because this unit emphasizes data analysis, the students use knowledge of graphing and statistics to complete this lesson.

Before the students can represent class characteristics using fractions, classroom data must be collected. To begin the data collection process, have the students brainstorm a list of the many ways the class or group might be described, for example, by gender, hair color, height, those who own pets, and so forth. This list can be used to create a classroom survey for data collection or you may choose to survey the class with questions from the Class Survey Activity Sheet.

Organize the students into five or six groups and have each group select a question from the survey. Give each group an envelope that contains as many scrap pieces of paper as there are students in the class. Have each group record its question and answer choices, if appropriate, on the envelope. For example, if the students ask about gender, they should include two choices, male or female. If it is not possible to identify all the possible choices, the students should leave their question in an open-ended format. For example, a question about types of pets should be left open-ended, as one might not be able to anticipate the variety of pets represented in the class.

Conduct the survey by passing the envelopes around the room and giving each student a chance to respond. Before starting the survey, have the students remove all the paper from their group’s envelope and leave it at their table. They will use these slips to record and submit their answer to each survey question. Begin the survey by having group members respond to the question on their envelope first, writing their answer on a slip of paper and placing it in the envelope. When the groups are finished with that question, they should pass their envelope to the next group, and so forth, until all the students have had a chance to respond to all the questions. (If the students in your class would benefit from getting up and moving around the room, instruct the students to leave the envelopes at each table and move from table to table to answer the questions.)

Once data are collected, the groups should tally the responses in their envelope, record the number and represent the quantity as a fraction, for example, 12 out of 24 students (12/24 or 1/2) have blue eyes. Have each group reduce their fractions to lowest terms by finding the greatest common factor. For example, suppose 18/24 (or 3/4) of the class owns a pet. The greatest common factor for 18 and 24 is 6. The students might find it helpful to list all the factors for the numerator and the denominator, 18 and 24 in this example, and locate the greatest common factor. This can be done strategically by checking in order each pair of factors that when multiplied yield a particular product. For example, to exhaust all the factors of 18, one would begin with 1 × 18, then 2 × 9, then 3 × 6. Since 4 is not a factor, the student would move on to 5 and then to 6. Six has already been generated with 3 × 6. When the student begins to duplicate factors, they know they have exhausted the list.

For organizational purposes, it is helpful to write the sets of factors in the following manner. The students should list factors on opposite sides (following the format below) with 1 × 18, then 2 × 9 on the inside of the other factors, then 3 × 6 in the middle. When factors begin to repeat, e.g., 6 × 3, the students know that the list of factors has been exhausted. This list of factors for 18 would be recorded as follows:

Step 1: | 1 | 18 | ||||

Step 2: | 1 | 2 | 9 | 18 | ||

Step 3: | 1 | 2 | 3 | 6 | 9 | 18 |

Similarly for 24, the list of factors would evolve as follows:

Step 1: | 1 | 24 | ||||||

Step 2: | 1 | 2 | 12 | 24 | ||||

Step 3: | 1 | 2 | 3 | 8 | 12 | 24 | ||

Step 4: | 1 | 2 | 3 | 4 | 6 | 8 | 12 | 24 |

Next, ask the students to list the factors for both numbers one on top of the other so they can easily recognize the common factors. The greatest common factor should be circled. For example:

The students should divide the numerator and denominator by the greatest common factor to reduce the fraction. For example, for 18/24, the students should divide both the numerator and denominator by 6 to reduce the fraction to 3/4.

If it becomes necessary to divide the lesson into two segments, this might be a logical beginning point for the second part of the lesson. Have group members organize their data in a chart and share it with the class. The students should record all fractional representations and may choose to record appropriate statistics on their chart, for example, mean, median, range, and mode for numerical data.

Groups may choose to create their bar graph using the Bar Grapher by selecting the option to input their own data.

An example of a bar graph of previously collected student data is shown below:

Once the students have created their graph, they should label the data in fractional parts and reduce all fractions to lowest terms. For example, this chart should be labeled with dog being 15/26, cats being 8/26 or 4/13, birds being 2/26 or 1/13, and 1/26 iguana. Ask students to share their graphs with the class and discuss how they used fractions in collecting the data depicted on each graph. If necessary, remind students to consider what the fractions represent, how the data was collected, how categories were established, and how finding the lowest common factor simplified the process of reducing the fraction.

- Five to six envelopes (one envelope for each small group)
- Scrap paper the size of a standard adhesive note (enough for each envelope to contain one slip for each student in the class)
- Class Survey Activity Sheet
- Bar Grapher

**Assessment Option**

At this stage of the unit, it is important to know whether the students can do the following:Use the students' graphs with fractional representations to make instructional decisions about the students' understanding.

- Demonstrate understanding that a fraction can be represented as part of a set
- Describe a set of objects on the basis of its fractional components
- Identify fraction relationships associated with the set
- Reduce a simple fraction to lowest terms

**Extensions**

- The students might want to compare some of their fractional representations [for example, gender] to statistics representative of their state or the United States as a whole. To find out the percent of males and females in each state, the students could visit the U.S. Census site. Engage students in a discussion about how to represent the percent of males and females in each state as a fraction.
- Move on to the last lesson,
*Another Look at Fractions of a Set*.

**Questions for Students**

1. How can you take classroom data and record it as a fraction?

[Record the number of people who fit a particular characteristic as the numerator, and the total number of students in the class as the denominator.]

2. What can you tell about your class based on the data from your survey?

[Student responses will depend upon the data collected.]

3. What fractions need to be reduced?

[Student responses will depend upon the data collected.]

4. Do you notice any patterns when reducing fractions?

[The numerator and denominator were divided by the same number.]

5. How do you know whether a fraction is in lowest terms?

[The GCF (greatest common factor) of the numerator and denominator is 1.]

**Teacher Reflection**

- How are the students recording fractions of the set? In reduced form? Or do all fractions use the number in the set as the denominator? (This information is helpful for documenting students' understanding of reducing fractions.)
- Which students understand how to reduce a fraction to lowest terms? What activities are appropriate for the students who have not yet developed this understanding?
- What parts of the lesson went smoothly? What parts should be modified for the future?

### Eggsactly with a Dozen Eggs

### Eggsactly with Eighteen Eggs

### Eggsactly Equivalent

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

### Class Attributes

### Another Look at Fractions of a Set

### Learning Objectives

Students will:

- Demonstrate understanding that a fraction can be represented as part of the set, given some number of items.
- Develop and conduct a survey.
- Use fractional components to describe their class attributes, for example, gender, hair color, and so forth.
- Identify fraction relationships associated with the set.
- Reduce fractions to their lowest terms.

### NCTM Standards and Expectations

- 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 3, Geometry

- CCSS.Math.Content.3.G.A.2

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

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?

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.