## Another Look at the Set Model using Attribute Pieces

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.

### Preparation

Note: Attribute pieces should include the following "attributes" or characteristics:

Color: red, blue, and yellow

Shape: square, rectangle, circle, hexagon, and triangle

Thickness: thick and thin

Size: large and small

Directions for making paper attribute pieces: Homemade attribute pieces can be made by printing the Attribute Pieces Activity Sheet using a color printer.

Attribute Pieces Activity Sheet

Printing and cutting both pages of attribute pieces will produce exactly one full set of attribute pieces. These materials can be printed onto thick paper and/or laminated to increase their durability.

### Introducing the Activity

Part I – Introduction to the Materials: If students have not used attribute pieces before, time should be given for them to freely explore. Point out that the pieces have four variable attributes: (1) color, (2) shape, (3) thickness, and (4) size. Tell students that they are going to become familiar with their attribute block set by grouping them in a variety of ways. Have students first group by color, then by shape, and finally by thickness and size.

Choose two attribute pieces and ask students to look for similarities and differences. How are the pieces alike? different? Can you find two pieces that are different in only one way? Select two other pieces that are different in one and only one way.

In order to familiarize students with the various attribute pieces, have students make a different train. Tell the students that you will choose an attribute piece for the engine. Students should pick an attribute piece that is different in one (and only one) way. Have students tell how the block is different in one way. Continue adding to the difference train. Have students make other trains with two, three, or four differences between attribute pieces. Students should explain why they selected a particular attribute piece based on the number of varying attributes.

### Activity

For the next part of the lesson, have students work in pairs to randomly select eight attribute pieces. For example, students might choose the following set. A capital “T” is used to denote a “thick” attribute piece, while a lowercase “t” is used to denote a “thin” attribute piece.

Have students generate a list of fractions to describe this set and record their description on a 3 × 5 index card. For example, students might record the following information:

The set is…

- 2/8 or 1/4 red
- 3/8 yellow
- 3/8 blue
- 5/8 thick
- 3/8 thin
- 2/8 or 1/4 triangular
- 2/8 or 1/4 square
- 2/8 or 1/4 rectangular
- 2/8 or 1/4 hexagonal
- 6/8 or 3/4 large
- 2/8 or 1/4 small

On the reverse side of the card, students should cut out paper attribute pieces representative of their set and glue them to the card.

Have students repeat this process until they have described at least three and attribute sets. Pair students and have them share their descriptions with a partner. The partner should attempt to construct the set based on the characteristics on the index card. When they think they have the correct answer, they should turn over the index card to check. These self-checking cards can be used for reinforcement at a center or for independent work.

- 3 × 5 index cards (5 for each pair of students)
- Attribute Pieces (either commercial or homemade from the Attribute Pieces Activity Sheet)

**Assessment Options**

- At this stage of the unit, it is important to know whether students can:
- Demonstrate understanding that a fraction can be represented as part of a set
- Describe a set of objects based on its fractional components
- Identify fraction relationships associated with the set

- Use the index cards with fractions that describe the set of attribute pieces to make instructional decisions about students’ understandings.

**Extension**

Move on to the next lesson,Class Attributes.

**Questions for Students**

1. Choose two attribute pieces. How are they alike? How are they different?

2. Choose two other attribute pieces. How are they alike? How are they different?

3. Can you find two pieces that are alike in one and only one way?

4. Can you find two pieces that are alike in two and only two ways?

5. Can you find three pieces that are alike in three and only three ways?

6. Can you find two pieces that are different in one and only one way?

7. Can you find two pieces that are different in two and only two ways?

8. Can you find three pieces that are different in three and only three ways?

[Student responses to all of these questions will depend upon the pieces they select.]

9. When using the cards you constructed in this lesson, is it possible to have more than one answer (different sets of attribute pieces) for a given set of characteristics?

[Yes, depending upon the characteristics selected, it is possible to have more than one answer.]

**Teacher Reflection**

- Which students understand that a fraction can be represented as part of a set? What activities are appropriate for students who have not yet developed this understanding?
- Which students can describe a set of objects based on its fractional components? What activities are appropriate for students who have not yet developed this understanding?
- Which students/groups can articulate the relationships between fractions?
- How are 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 where students are in their understanding of reducing fractions.
- 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 a set of items that are not identical (attribute pieces).
- Describe a set of objects based on its fractional components.
- Identify fraction relationships associated with the set.

### 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.

### 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.