## Investigating Fraction Relationships with Relationship Rods

• Lesson
3-5
1

Students use relationship rods to explore fraction relationships. This work with relationships lays the foundation for work with more challenging fraction concepts.

Relationship rods are wooden or plastic rods in ten different colors. They range in length from one to ten centimeters. Each length is a different color.

To assess students' prior knowledge, give students a chance to explore the relationship rods. Engage them in a conversation about the likenesses and differences between the fraction strips and the rods.

To begin the lesson, give students one set of relationship rods (either homemade or commercial) and a copy of the Investigating Fraction Relationships Activity Sheet.

 Fraction Relationship Activity Sheet

Give students a few minutes to explore the materials. Ask them to try to determine how the pieces are related to one another.

Most students will quickly figure out that the various colors can be stacked one on top of the other to create a staircase. This configuration makes comparing the various fractions much easier and is illustrative of the linear model of fractions.

This staircase will appear as follows:

Have students consider the first question on the Investigating Fraction Relationships Activity Sheet: If white = 1, what value would you assign to all the other rods?

[Students should determine that red = 2, light green = 3, purple = 4, yellow = 5, dark green = 6, black = 7, brown = 8, blue = 9, and orange = 10. Discuss student responses as a class.]

Demonstrate the correct answer by lining up the relationship rods being compared on an overhead projector.

Have students complete the following table from the Investigating Fraction Relationships Activity Sheet.

 Integer Bars Size Color 1 White 2 Red 3 Light Green 4 Purple 5 Yellow 6 Dark Green 7 Black 8 Brown 9 Blue 10 Orange

When comparing the two lengths for demonstration purposes, the smaller length should be duplicated to simulate the length of the longer rod. For example, when comparing white with yellow, where white is one and yellow is five, five white relationship rods should be used to directly compare to one yellow. Students can easily see that it takes five whites to make one yellow; therefore, if white = 1, then yellow = 5.

Have students consider the second question on the Investigating Fraction Relationships Activity Sheet: If red = 1, what value would you assign to all the other rods?

[Students should determine that white = ½, light green = , purple = 2, yellow = , dark green = 3, black = , brown = 4, blue = , and orange = 5.]

Continue exploring relationships with each color representing the whole by completing the Investigating Fraction Relationships Activity Sheet.

Assessments

1. At this stage of the unit, students should be able to do the following:
• demonstrate understanding that a fraction can be represented as part of a linear region
• describe part of a linear region using fractions
• identify fraction relationships using different "wholes" as a reference

2. Examining student recordings on the Investigating Fraction Relationships Activity Sheet can be helpful in making instructional decisions about students’ understanding of fraction relationships.

Extensions

1. Have students combine two relationship rods to represent the whole. For example, students might use orange followed by red to create a new piece (orange/red).

Have students consider the value of all the other relationship rods if orange/red = 1. Students should determine that white = 1/12, red = 2/12 or 1/6, light green = 3/12 or 1/4, purple = 4/12 or 1/3, yellow = 5/12, dark green = 6/12 or 1/2, black = 7/12, brown = 8/12 or 2/3, blue = 9/12 or 3/4, and orange = 10/12 or 5/6.

2. Have students continue to work with other color combinations to create new "wholes." Find the value of all the other relationship rods given the new whole.

Questions for Students

1. Suppose you create a new relationship rod using the orange/red rod. This orange/red rod is now the whole. In this case, what value would you assign to all the other rods?

[White = 1/12, red = 2/12 or 1/6, light green = 3/12 or 1/4, purple = 4/12 or 1/3, yellow = 5/12, dark green = 6/12 or 1/2, black = 7/12, brown = 8/12 or 2/3, blue = 9/12 or 3/4, and orange = 10/12 or 5/6.]

Teacher Reflection

• Which students understand that a fraction can be represented as part of a linear region? What activities are appropriate for students who have not yet developed this understanding?
• Which students can describe part of a linear region using fractions? What activities are appropriate for students who have not yet developed this understanding?
• Which students can identify fraction relationships using different “wholes” as a reference? What activities are appropriate for students who have not yet developed this understanding?
• What parts of the lesson went smoothly? What parts should be modified for the future?

### Fun With Fractions: Making and Investigating Fraction Strips

3-5
Students make and use a set of fraction strips to represent the length model, discover fraction relationships, and work with equivalent fractions.

### More Fun with Fraction Strips

3-5
Students continue to work with fraction strips to compare and order fractions. This lesson builds on the work done with fraction relationships in the previous lesson. Students develop skills in problem solving and reasoning as they make connections between various fractions.

### Investigating Equivalent Fractions with Relationship Rods

3-5
Students investigate the length model by working with relationship rods to find equivalent fractions. Students develop skills in reasoning and problem solving as they explain how two fractions are equivalent (the same length).

### Inch by Inch

3-5
In this lesson, students use a ruler to represent various fractions as lengths. This lesson builds on the work done in the previous lessons with as students use a standard instrument to measure a variety of items, including items which can be measured to the nearest half and quarter of an inch.

### Learning Objectives

Students will:

• Demonstrate understanding that a fraction can be represented as part of a linear region
• Describe part of a linear region using fractions
• Identify fraction relationships using different "wholes" as a reference

### 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.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.