## Pattern Block Fractions

3-5
1

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.

For this lesson, students need a set of pattern blocks. (Only the yellow hexagons, red trapezoids, blue rhombi, and green triangles are needed. Students do not use the orange squares or the tan rhombus for this lesson.) If students are seated at tables, one set of pattern blocks can be shared by the group.

Have students work in pairs to explore relationships. Guiding questions are provided to facilitate the exploration and concentrate on the mathematical focus of this lesson. Students should use pattern blocks to find relationships and to determine the answer. If overhead pattern blocks are available, the two pattern blocks being compared can be modeled on the overhead projector. Questions may be made available to students in hard copy. Please see Region Relationships 2 Activity Sheet. An overhead transparency of this worksheet can be made for use with the entire class.

Again ask each of the guiding questions from the first lesson, Investigating with Pattern Blocks, but follow each question with another question about the fractional relationship. For example,

 How many green triangles are in one blue rhombus ? [Two.] The green triangle is what fraction of the blue rhombus ? [One out of two, or ½.]

Model the written form of each fraction by recording each fraction on the board or overhead in standard (fraction) form. Have the students record fractions in their math journals. For example,

 2 = 1 Therefore, 1 = ½

The students should have little difficulty expressing this relationship as a fraction. They have used the fraction ½ on numerous occasions even prior to kindergarten. This lesson should focus on the written format and what it really means. Lead the students in identifying and defining the numerator and denominator. Ask the students to explain what the top number in the fraction represents. [Students should indicate that this top number is the numerator and shows the number of parts of the whole.] The students should also identify the purpose of the bottom number, or denominator, as the number that indicates the number of parts into which the whole is divided.

Continue with all other pattern block relationships, recording the fractions. You may choose to have the students 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 the students record in their journal any relationships that they may have missed.

Have the students repeat the activity using virtual pattern blocks on the computer. They should be directed online to The Shape Tool.

Assessments

At this stage of the unit, it is important to know whether the students can do the following:
• identify fractions when the whole (region) and a part of the region are given
• represent the fractional relationship between the pattern block shapes using standard form of the written notation (e.g., the green triangle is x of the blue rhombus.)
• identify the numerator in a fraction and understand that the numerator is the top number in a fraction and indicates the number of parts of the whole
• identify the denominator in a fraction and understand that the denominator is the bottom number in a fraction and indicates the number of parts into which the whole is divided
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 at the end of this unit to make anecdotal notes about the students' understanding and use those notes to guide your instructional planning.
Extension

Questions for Students

1. How many green triangles are in one blue rhombus? [Two.]
The green triangle is what fraction of the blue rhombus? [1/2.]
What part of this fraction is the numerator? [1.]
What does the numerator in this fraction mean or represent? [The one green triangle of the two that it takes to cover a blue rhombus.] What part of this fraction is the denominator?
[2.] What does the denominator in this fraction mean? [The number of green triangles it takes to cover one blue rhombus.]
2. How many green triangles are in one red trapezoid? [Three.]
The green triangle is what fraction of the red trapezoid? [1/3.]
What part of this fraction is the numerator? [1.]
What does the numerator in this fraction mean or represent? [The one green triangle of the three that it takes to cover a red trapezoid.] What part of this fraction is the denominator? [3.]
What does the denominator in this fraction mean? [The number of green triangles it takes to cover one red trapezoid.]
3. How many green triangles are in one yellow hexagon? [Six.]
The green triangle is what fraction of the yellow hexagon? [1/6.]
What part of this fraction is the numerator? [1.]
What does the numerator in this fraction mean or represent? [The one green triangle of the six that it takes to cover a yellow hexagon.] What part of this fraction is the denominator? [6.]
What does the denominator in this fraction mean? [The number of green triangles it takes to cover one yellow hexagon.]
4. How many blue rhombuses are in one yellow hexagon? [Three.]
The blue rhombus is what fraction of the yellow hexagon? [1/3.]
What part of this fraction is the numerator? [1.]
What does the numerator in this fraction mean or represent? [The one blue rhombus of the three that it takes to cover a yellow hexagon.] What part of this fraction is the denominator? [3.]
What does the denominator in this fraction mean? [The number of blue rhombuses it takes to cover one yellow hexagon.]
5. How many red trapezoids are in one yellow hexagon? [Two.]
The red trapezoid is what fraction of the yellow hexagon? [1/2.]
What part of this fraction is the numerator? [1.]
What does the numerator in this fraction mean or represent? [The one red trapezoid of the two that it takes to cover a yellow hexagon.] What part of this fraction is the denominator? [2.]
What does the denominator in this fraction mean? [The number of red trapezoids it takes to cover one yellow hexagon.]

Teacher Reflection

• Which students can identify fractions when the whole (region) and a part of the region are given? What activities are appropriate for the students who have not yet developed this understanding?
• Which students can represent the fractional relationship between the pattern block shapes using a standard form of the written notation (e.g., the green triangle is _ of the blue rhombus). What activities are appropriate for the students who have not yet developed this understanding?
• Which students can identify the numerator in a fraction? Do the students understand that the numerator is the top number in a fraction and indicates the number of parts of the whole? What activities are appropriate for the students who have not yet developed this understanding?
• Which students can identify the denominator in a fraction? Do the students understand that the denominator is the bottom number in a fraction and indicates the number of parts into which the whole is divided? 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?

### 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.

### 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.

### Exploring the Value of the Whole

3-5
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.

### Learning Objectives

Students will:

• Identify fractions when the whole (region) and a part of the region are given.
• Represent the fractional relationship between the pattern block shapes using a standard form of the written notation [for example, the green triangle is x of the blue rhombus].
• Identify the numerator in a fraction and understand that the numerator is the top number in a fraction and indicates the number of parts of the whole.
• Identify the denominator in a fraction and understand that the denominator is the bottom number in a fraction and indicates the number of parts into which the whole is divided.

### 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.

• 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.