Illuminations: Fun with Fractions

# Fun with Fractions

## Investigating Fraction Relationships with Relationship Rods

 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.

### 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

### Materials

 One set of relationship rods per student (purchased commercially or made by printing the Relationship Rods in color) Investigating Fraction Relationships Activity Sheet Fraction Bars Applet

### Instructional Plan

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.

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 or by using the Integer Bar Applet.

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

Tell students that they will have an opportunity to use the Integer Bar Applet later in the lesson.

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 = 1½, purple = 2, yellow = 2½, dark green = 3, black = 3½, brown = 4, blue = 4½, and orange = 5.]

Continue exploring relationships with each color representing the whole by completing the Investigating Fraction Relationships activity sheet.

Students can repeat part or all of this activity using the Integer Bar Applet.

### Questions for Students

 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.

### Assessment Options

 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 ddentify fraction relationships using different "wholes" as a reference Examining student recordings on the Investigating Fraction Relationships Activity Sheet can be helpful in making instructional decisions about students’ understanding of fraction relationships. Solutions to the activity sheet are available.

### Extensions

 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. 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.

### 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?

### NCTM Standards and Expectations

 Number & Operations 3-5Recognize and generate equivalent forms of commonly used fractions, decimals, and percents. 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.
 This lesson was developed by Tracy Y. Hargrove.

1 period

### NCTM Resources

Principles and Standards for School Mathematics

### Web Sites

 More and Better Mathematics for All Students
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