## Investigating Fractions with Pattern Blocks

• Lesson
3-5
1

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.

For this lesson, the students need a set of pattern blocks. (Only the yellow hexagons, red trapezoids, blue rhombuses, and green triangles are needed. The students do not use the orange triangle or the tan rhombus for this lesson.) If the students are seated at tables, one complete set of pattern blocks should serve an entire group. The most common regions studied at the elementary grade levels are the rectangle and circle. The "region" represents the "whole," and parts of the region are all congruent. The students should be exposed to a variety of shapes and not limited to the rectangle and circle. It is important that the students work with a variety of regions so that they do not think of the region as only "pieces of a pie." For this reason, pattern blocks are an appropriate tool for work with the region model.

Have students work in pairs to explore relationships among the four shapes. The Questions for Students at the end of this lesson facilitate the exploration and help students focus on the mathematical concepts of these lessons.

The students should use pattern blocks to answer the questions. If overhead pattern blocks (for use on overhead projectors) are available, the two pattern blocks being compared can be displayed on the overhead projector.

If you prefer, give printed copies of the Region Relationships 1 activity sheet to all students. An overhead transparency of this worksheet can be made for use with the entire class. You may want to color a transparent overhead of the pattern block shapes with a permanent marker to create overhead pattern blocks to use for demonstration purposes.

 Region Relationships 1 Activity Sheet

Guide the students through each question on the activity sheet. For the first example, you may want to show students how two green triangles can be placed to exactly cover one blue rhombus, thereby showing that they are equivalent. After that, you may wish to have students perform the demonstrations for one another using the pattern blocks on the overhead projector.

 How many green triangles are in one blue rhombus ? [Two.] How many green triangles are in one red trapezoid ? [Three.] How many green triangles are in one yellow hexagon ? [Six.] How many blue rhombuses are in one yellow hexagon ? [Three.] How many red trapezoids are in one yellow hexagon ? [Two.]

The students might notice that there is one blue rhombus and one green triangle in one red trapezoid. This discovery could lead to a rich discussion of equivalency. If the students do not discover this relationship on their own, guide them in seeing this relationship. For example, you could ask, "Is there a way to represent the red trapezoid using blue and green pattern blocks?" The students should state that they could construct the trapezoid with one green triangle and one blue rhombus. You could then ask, "Could we cover the red trapezoid using only one color?" The students should indicate that the red trapezoid could be covered with three green triangles. And you could also ask, "What does this tell us about the relationship between the blue rhombus and the green triangle?" The students should state that there are two green triangles in one blue rhombus. The students may continue discovering other such relationships using two or more pattern blocks and exchanging them for one pattern block.

Have the students record as many fraction relationships as possible. You may choose to have them record the relationships in a math journal to which they may refer later. Each pair should record relationships on chart paper to share with the whole class. As each pair shares, have the students add to their journal any relationships that they may have missed.

As the students work to understand fraction relationships using the region model, it is appropriate to work with concepts on a continuum from concrete to abstract. This lesson first exposes the students to a concrete representation of the region model through work with pattern blocks. As the students move toward more abstract work, it is appropriate to introduce semi‑concrete representations. Having the students record fraction relationships pictorially gives them the opportunity to be exposed to such a model.

Assessments

1. At this stage of the unit, it is important to know whether the students can do the following:
• understand that a fraction is part of a whole
• state the relationship between the pattern block shapes [e.g., that there are three triangles in one red trapezoid]

2. The students' recordings can be used to make instructional decisions about their understanding of fraction relationships. 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.

Questions for Students

1. Is there a way to represent the red trapezoid using blue and green pattern blocks? Can you cover the red trapezoid using only one color? What does this tell us about the relationship between the blue rhombus and the green triangle?

[The trapezoid can be covered with one green triangle and one blue rhombus, or it can be covered with three green triangles. Consequently, there are two green triangles in one blue rhombus.]

2. Are there other ways to represent various pattern blocks (for example, the yellow hexagon) using more than one color pattern block?

[The students should be lead in a discussion of the relationships inherent in these representations.]

Teacher Reflection

• Which students understand that a fraction is part of a whole? What activities are appropriate for those students who have not yet developed this understanding?
• Which students/groups can articulate the relationship between the pattern block shapes? What activities are appropriate for those students who have not yet developed this understanding?
• What parts of the lesson went smoothly? What parts should be modified for the future?

### Learning Objectives

Students will:
• Demonstrate understanding that a fraction is part of a region
• Identify fraction relationships among pattern blocks

### 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.