Lesson 15

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.

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

• Record responses.

• 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)

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.

• Groups of 2

• “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

• 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.)

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

• “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).

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.

• See lesson synthesis.