Lesson 15

Common Denominators to Compare

Warm-up: What Do You Know about 15 and 30? (5 minutes)

Narrative

The purpose of this warm-up is to elicit what students know about the numbers 15 and 30, preparing them to work with fractions whose denominators are factors of 15 and 30 later in the lesson. While students may bring up many things about these numbers, highlight responses that relate the two numbers by their factors and multiples.

Launch

• Display the numbers.
• “What do you know about 15 and 30?”
• 1 minute: quiet think time

Activity

• Record responses.

Student Facing

What do you know about 15 and 30?

Activity Synthesis

• If no students mentioned factors of 15 and 30, ask them about it.
• “What are the factors of 15?” (1, 3, 5, 15)
• “What are the factors of 30?” (1, 2, 3, 5, 6, 10, 15, 30)
• “What factors do they have in common?” (1, 3, 5, 15)
• “Do 15 and 30 have any common multiples? What are some of them?” (30, 60, 90)

Activity 1: Tricky Fractions? (20 minutes)

Narrative

In earlier lessons, students compared fractions by rewriting one fraction as an equivalent fraction with the same denominator as the second fraction. In this activity, students see that—although it’s still possible to compare the fractions—this particular strategy doesn’t work if neither of the denominators of the two fractions is a factor or multiple of each other. Students learn that in such a case, both fractions can be expressed as equivalent fractions with a common denominator, and the denominator is a different number that is a multiple of both of the original denominators.

MLR8 Discussion Supports. Synthesis: Display sentence frames to support partner discussions: “First, I _____ because . . .”, “I noticed _____ so I . . . .”
Representation: Develop Language and Symbols. Activate background knowledge. Provide students with access to a visible display that shows definitions or reminders of the terms “factor” and “multiple.”
Supports accessibility for: Memory, Language

• Groups of 2

Activity

• “Take a few quiet minutes to work on the first two questions.”
• 6–7 minutes: independent work time
• “Share your responses to both questions with your partner. Be sure to explain how you compared the fractions in the first question.”
• 3–4 minutes: partner discussion
• Pause for a brief whole-class discussion. Invite students to share their responses for the first two questions.
• If no students suggested that the second pair of fractions are hard to compare because their denominators have no factors in common (or one does not multiply or divide to make the other), ask them about it.
• “Now work with your partner on the last question.”
• 3–4 minutes: group work time

Student Facing

1. In each pair of fractions, which fraction is greater? Explain or show your reasoning.

1. $$\frac{4}{3}$$ or $$\frac{13}{12}$$
2. $$\frac{4}{3}$$ or $$\frac{7}{5}$$
2. Han says he can compare $$\frac{4}{3}$$ and $$\frac{13}{12}$$ by writing an equivalent fraction for $$\frac{4}{3}$$. He says he can’t use that strategy to compare $$\frac{4}{3}$$ and $$\frac{7}{5}$$. Do you agree? Explain your reasoning.
3. Priya and Lin showed different ways for comparing $$\frac{4}{3}$$ and $$\frac{7}{5}$$. Make sense of what they did. How are their strategies alike? How are they different?

Priya: $$\frac{4 \ \times \ 5}{3 \ \times \ 5}=\frac{20}{15} \hspace{1.2cm} \frac{7 \ \times \ 3}{5 \ \times \ 3}=\frac{21}{15}$$

​​$$\frac{21}{15}$$ is greater than $$\frac{20}{15}$$, so ​​​​​$$\frac{7}{5}$$ is greater than $$\frac{4}{3}$$.

Lin: $$\frac{4 \ \times \ 10}{3 \ \times \ 10}=\frac{40}{30} \hspace{1.2cm} \frac{7 \ \times \ 6}{5 \ \times \ 6}=\frac{42}{30}$$

$$\frac{42}{30}$$ is greater than $$\frac{40}{30}$$, so $$\frac{7}{5}$$ is greater than $$\frac{4}{3}$$.

Activity Synthesis

• Display Priya and Lin’s reasoning for all to see. Select students to share their observations on how the two are alike and how they are different.
• Highlight the fact that both students rewrote the two fractions so that they have a common denominator.
• “Why might it be helpful to write equivalent fractions with the same denominator?” (It is easier to compare the fractions when the fractional part is the same size.)
• “Can we choose any number to be the common denominator?” (No, it must be a multiple of both of the original denominators.)
• “Does it matter if we choose a smaller or a larger common multiple?” (No, but it could work better to multiply by a smaller number.)

Activity 2: Use a Common Denominator, or Not (20 minutes)

Narrative

This activity serves two main goals: to prompt students to rewrite pairs of fractions into equivalent fractions with a common denominator, and to consider this newly developed skill as a possible way to compare fractions.

To write equivalent fractions, many students are likely to reason numerically (by multiplying or dividing the numerator and denominator by a common number). Some may, however, find equivalent fractions effectively by continuing to reason about how many of this fractional part is in that fractional part.

To compare the fractions in the second question, students may choose to write equivalent fractions with a common denominator because they were just learning to do so. The fractions, however, were chosen so that students have opportunities to choose an approach strategically, rather than writing equivalent fractions each time. For instance, students may notice that:

• In part a, one fraction is $$\frac{1}{12}$$ away from $$\frac{1}{2}$$ and the other is $$\frac{1}{8}$$ from $$\frac{1}{2}$$.
• In part b, one fraction is greater than 2 and the other is greater than 1.
• In part c, writing an equivalent fraction for only one given fraction (rather than for both) is sufficient for comparing.
• In part d, one fraction is less than $$\frac{1}{2}$$, and the other is greater than $$\frac{1}{2}$$.

• Groups of 2

Activity

• “Work with your partner to write equivalent fractions for the first set of questions.”
• 7–8 minutes: group work time on the first set of questions
• Pause for a brief whole-class discussion.
• Poll the class on the common denominator they chose for each pair of fractions. Record their responses. (Likely denominators for each part:
1. 24 or 12
2. 24
3. 60 or 30
4. 40 or 20)
• Some students are likely to suggest multiplying one denominator by the other. Discuss whether there are other ways to find a common denominator.
• “Work independently to compare the fractions in the second set of questions. Be prepared to explain how you know which fraction is greater. ”
• 7–8 minutes: independent work time on the second set of questions
• Monitor for students who are strategic in how they compare the pairs of fractions in the second question (not exclusively writing equivalent fractions).

Student Facing

1. For each pair of fractions, write a pair of equivalent fractions with a common denominator.

1. $$\frac{5}{6}$$ and $$\frac{3}{4}$$
2. $$\frac{2}{3}$$ and $$\frac{5}{8}$$
3. $$\frac{2}{6}$$ and $$\frac{4}{10}$$
4. $$\frac{7}{4}$$ and $$\frac{17}{10}$$
2. For each pair of fractions, decide which fraction is greater. Be prepared to explain your reasoning.

1. $$\frac{5}{12}$$ or $$\frac{3}{8}$$
2. $$\frac{13}{5}$$ or $$\frac{11}{6}$$
3. $$\frac{71}{10}$$ or $$\frac{34}{5}$$
4. $$\frac{7}{12}$$ or $$\frac{49}{100}$$

Student Response

When working on the second set of problems, some students might be inclined to immediately find a common denominator for each pair of fractions. They might get stuck if they don’t recognize a common factor or multiple of the denominators (for instance, 12 and 100), or if they aren’t sure how to multiply large numbers. Encourage students to consider other strategies they know for gauging the size of fractions and for comparing fractions.

Activity Synthesis

• See lesson synthesis.

Lesson Synthesis

Lesson Synthesis

Invite students to share their responses to the last set of questions of Activity 2 and how they went about making comparisons. Record their responses.

Select students who made strategic choices when making comparisons to share their thinking.

Emphasize that, while it is possible to compare every pair of fractions by rewriting them so that they have a common denominator, all the fractions could be compared by reasoning in other ways.