Equivalent Fractions on Number Lines

Est. Class Sessions: 2–3

Developing the Lesson

Part 1. Equivalent Fractions on Number Lines.

Use Number Lines and Patterns to Find Equivalent Fractions. In this lesson, students work with number lines to find equivalent fractions. Number lines offer a visual tool to determine equivalencies. When fractions are represented on number lines, two fractions are equivalent if they are the same distance from zero. Equivalent fractions can label the same point on the number line.

Direct students to the Equivalent Fractions on Number Lines pages in the Student Guide. Ask them to look at the number lines in the opening section. These lines show that 1/2 and 2/4 are equivalent fractions because they are the same distance from zero.

Ask:

Use the number line to find other equivalent fractions. (0/2 = 0/4 and 2/2 = 4/4)

Record students' findings as number sentences as they report them. Ask students to consider the number lines in Questions 1–2 and to list all the equivalent fractions they find.

After solving these problems as a class, display the following number sentences and ask:

Look at the number lines again. Which of these number sentences are true and which are false?
3/6 = 1/2 (true)
0/3 = 0/2 (true)
2/3 = 3/6 (false)
2/3 = 4/6 (true)
1/3 = 2/6 (true)
Describe any patterns you see within the true number sentences. (Possible response: For 1/3 = 2/6, look at the numerators. If you double the 1, you get 2. For the denominators, if you double the 3, you get 6.)

Next have students work individually or with partners to answer Questions 3–8. To solve the problems, students can use a ruler or the edge of a piece of paper to line up the tick marks on the Fractions on Number Lines Chart. Fractions that lie on the same vertical line are the same distance from zero and are therefore equivalent.

The Fractions on Number Lines Chart is also in the Student Guide Reference section. Encourage students to use this chart as a tool throughout the unit as they solve problems and complete Daily Practice and Problem items. Using the chart will help them develop visual images of the number lines that will help them reason about fractions.

Question 3 asks students to find fractions equivalent to 1/2. If they line up a ruler to make a vertical line through the one-half mark on the first number line, they will find fractions on other number lines that are the same distance from zero: 2/4, 3/6, 4/8, 5/10, and 6/12. The question asks for a pattern. Some students might notice that each fraction equivalent to 1/2 can be obtained by multiplying the top and bottom of 1/2 by the same number. See Content Note. For example,

Multiplying by 1 to Find Equivalent Fractions. Students will likely observe that multiplying the numerator and denominator of a fraction by the same number results in an equivalent fraction. For example, as in our discussion of Question 3, we can multiply the numerator and denominator of 1/2 by 4 to find a fraction equivalent to 1/2:

This is the same as multiplying 1/2 by 1. Since 4/4 is equivalent to 1 and multiplying by 1 (the identity for multiplication) does not change the value, the resulting fraction, 4/8, must have the same value as 1/2. Since students have not yet worked with multiplication of fractions, this lesson does not discuss this reasoning at this time. However, students who are aware of it should be encouraged in their thinking.

Another pattern students explored in Lesson 2 is that for all fractions equivalent to 1/2, the numerator is always one-half of the denominator.

In Question 4, students find patterns for fractions equivalent to 1. Probably the most common response will be that for fractions equivalent to one, the numerator and denominator are equal.

Students continue to use the Fractions on Number Lines Chart to find equivalent fractions in Questions 5–7. Question 7 asks them to find equivalent fractions using number sentences in the form: 1/3 = n/6.

In Question 8, students are asked to use multiplication or division to find a fraction equivalent to 2/3.

Share Strategies for Finding Equivalent Fractions. Upon completion, display the Fractions on Number Lines Chart page from the Student Guide Reference section. Discuss different ways to find equivalent fractions.

To facilitate discussion, ask:

In Question 7, how did you use the number lines on the chart to make the number sentences true? (Possible response: For 1/3 = n/12, I found 1/3 on the number line and used my ruler to follow a straight line down to 4/12. 1/3 and 4/12 are the same distance from 0 on the number line so I know they are equivalent.)
Did you use patterns to solve any? Give examples. (Possible response: For 1/3 = n/6, I looked at the denominators. 3 times 2 is 6 so I multiplied the numerator 1 times 2. 1/3 = 2/6.)
How can you find equivalent fractions without using number lines? (Possible response: If I see that the fraction is equivalent to 1/2, I know the numerator should be half the denominator.)
How can you use multiplication or division strategies to find equivalent fractions? (You can find equivalent fractions by multiplying the numerator and denominator by the same number or by dividing the numerator and denominator by the same number.)

Record students' different strategies for finding equivalent fractions on chart paper with examples such as 1/3 × 2/2 = 2/6 and 1/2 = 2/4 = 3/6 = 4/8. Encourage students to share other strategies as well. Then assign Question 9. Students will apply these patterns and strategies to find equivalent fractions that are not on the number lines.

Use Check-In: Questions 7–9 on the Equivalent Fractions on Number Lines pages in the Student Guide to assess students' abilities to find equivalent fractions using a number line and multiplication and division strategies [E4].