## Exploring the Value of the Whole

• Lesson
3-5
1

This lesson focuses on the relationship between parts and the whole. These relationships were developed earlier and require the students to consider the size or value of the same fraction when different "wholes" are compared (i.e., the value of x is relative to the whole; x of a small pie is not equivalent to x of a large pie). This lesson promotes problem solving and reasoning as the students compare similar fractions with different "wholes." Students develop communication skills as they work in pairs and share their understanding about the relationship between the value of a fraction and the whole.

Have students work in pairs or small groups [depending upon the best working arrangement for your students]. The students may have noticed that the same fraction (1/2, for example) is not always equivalent (or the regions are not always congruent). For example, have the students compare the answer to the following questions:

 The green triangle is what fraction of the blue rhombus ? [1/2] The red trapezoid is what fraction of the yellow hexagon ? [1/2] How can and both be equivalent to ______? (Query the students until they are able to articulate that the meaning or value of _____ relative to the whole.)

These and other questions are presented on the Region Relationships 4 activity sheet. Working in pairs, allow students to complete these questions.

 Region Relationships 4 Activity Sheet

On the activity sheet, students will explore other fraction relationships in which the fraction is the same but actually refers to two different wholes. For example,

 The blue rhombus is what fraction of the yellow hexagon ? [1/3] The green triangle is what fraction of the red trapezoid ? [1/3] How can and both be equivalent to 1/3? (Probe students until they are able to articulate that the meaning or value of 1/3 is relative to the whole.)

Have the students make similar comparisons using two yellow hexagons as the whole. Continue investigating relationships with other wholes (e.g., three or more yellow hexagons). You may choose to have them record the relationships in a math journal to which they may refer later. Each group should record relationships on chart paper to share with the whole class. As each group shares, have students add to their journal any relationships that they did not discover on their own.

Assessments

1. At this stage of the unit, it is important to know whether the students can do the following:
• identify the fractions represented by various pattern blocks when "whole" is defined in different ways (e.g., 1 whole = 1 yellow hexagon; 1 whole = 2 or more yellow hexagons)
• demonstrate understanding that the area of equivalent fractions is relative to the whole
2. The students' recordings can be used to make instructional decisions about their understanding of fraction relationships and the notion that the value of the fraction is relative to the whole. Because this entire unit deals with relationships, areas needing additional work can be developed during subsequent lessons. You may choose to use the Class Notes recording sheet to make anecdotal notes about the students' understanding and use those notes to guide your instructional planning.

Extensions

1. Have students repeat the activity using virtual pattern blocks on the computer. They should be directed to The Shape Tool.
2. Another good demonstration of the relationship of the part to the whole is to take two candy bars of the same brand, one bar that is miniature in size and one that is extra large. Ask the students: If you could have one-half of one of the two candy bars, which one-half would you prefer?

Questions for Students

1. The green triangle is what fraction of the blue rhombus?

[1/2.]

2. The red trapezoid is what fraction of the yellow hexagon?

[1/2.]

3. How can both be equivalent to 1/2?

4. The blue rhombus is what fraction of the yellow hexagon?

[1/3.]

5. The green triangle is what fraction of the red trapezoid?

[1/3.]

6. How can both be equivalent to 1/3?

7. The green triangle is what fraction of the yellow hexagon?

[1/6.]

8. What pattern block represents 1/6 when two yellow hexagons are used as the whole?

[The blue rhombus.]

9. Do all parts representing 1/6 have equal size? Explain.

[Student responses may vary.]

Teacher Reflection

• Which students can identify the fractions represented by various pattern blocks when "whole" is defined in different ways? What activities are appropriate for students who have not yet developed this understanding?
• Which students demonstrate understanding that the area of equivalent fractions is relative to the whole? What activities are appropriate for the students who have not yet developed this understanding?
• Which pairs worked well together? Which groupings need to be changed for future lessons?
• If groups were not successful working together, what was the source of the problem? (i.e., were the problems behavior-related or were academic levels not matched appropriately in the groups?)
• What parts of the lesson went smoothly? What parts should be modified for the future?

### Investigating Fractions with Pattern Blocks

3-5
This lesson promotes problem solving and reasoning with fractions as students investigate the relationships between various parts and wholes. It also focuses on representation because students are given multiple opportunities to investigate the relative value of fractions. Students use communication skills as they work in pairs to articulate and clarify their understanding of fraction relationships.

### Virtual Pattern Blocks

3-5
Students use virtual pattern blocks to problem solve and reason with fractions. They investigate relationships between parts and wholes using another representation of a region model, virtual fractions. Students use conversation to explain their understandings in order to extend and clarify their mathematical content knowledge.

### Pattern Block Fractions

3-5
This lesson builds on the previous two lessons by focusing on the identification of fractional parts of a region and by recording them in standard form. Students continue to develop communication skills by working together to express their understanding of fraction relationships and to record fractions in written form.

### Expanding Our Pattern Block Fraction Repertoire

3-5
In this lesson, the students expand the number of fractions they can represent with pattern blocks by increasing the whole. Instead of representing the whole with one yellow hexagon, the students explore fractional relationships when two, three, and four yellow hexagons constitute the whole.

### Learning Objectives

Students will:
• Identify the fractions represented by various pattern blocks when "whole" is defined in different ways (e.g., 1 whole = 1 yellow hexagon; 1 whole = 2 or more yellow hexagons)
• Demonstrate an understanding that the area of equivalent fractions is relative to the whole

### 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.

• 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.