Lesson 11

This warm-up prompts students to carefully analyze and compare representations of fractions. To make comparisons, students need to draw on their knowledge about fractional parts, the size of fractions, and equivalent fractions.

• Groups of 2
• Display the image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

Consider asking:

• “Let’s find at least one reason why each one doesn’t belong.”

In this activity, students see that they can find equivalent fractions by dividing the numerator and denominator by a common factor. They connect this strategy to the process of grouping unit fractions on a number line into larger equal-size parts. The result is a fewer number of parts, and smaller numbers for the numerator and denominator of equivalent fractions.

• Groups of 2

• “Work with your partner to answer the first three problems.”
• “Be prepared to explain how you think Andre’s and Kiran’s strategies are related.”
• 7–8 minutes: partner work time
• “Take a few minutes to answer the last problem independently.”
• 2–3 minutes: independent work time for the last problem

• “What did Andre do with the number lines? How would it help him find equivalent fractions?” (Andre grouped the 12 parts into equal groups of different sizes—2s, 4s—to make bigger parts. Then, he counted the number of those new parts.)
• “What did Kiran do? How is his strategy related to Andre’s?” (Kiran divided 12 by 4 and then by 2, similar to how Andre put 12 parts into groups of 4 and then of 2.)
• “Notice that the equivalent fractions \(\frac{2}{3}\) and \(\frac{4}{6}\) both have smaller numbers for the numerator and denominator than the original fraction. Can you use Andre’s number line to show why this might be?” (The size of the parts are bigger, so there are fewer parts in 1 whole.)

If time permits, ask students:

• “How many equivalent fractions can you find for \(\frac{10}{12}\) using Kiran’s way?” (One) “How many can you find for \(\frac{18}{12}\)?” (Three) 
• “What might be a reason that you could find more equivalent fractions for \(\frac{18}{12}\) than for \(\frac{10}{12}\)?” (18 and 12 have more factors in common than 10 and 12.)
• “How would you show \(\frac{9 \ \div \ 3}{12 \ \div \ 3}=\frac{3}{4}\) on the number line?” (Put the original 12 parts into groups of 3 to get 4 parts, each being a fourth. Mark 3 of those 4 parts to show \(\frac{3}{4}\).)

In this activity, students generate equivalent fractions by applying the numerical strategies they learned. (Students might opt to use other strategies, but most of the given fractions have numbers that would make visual representation and reasoning inconvenient.) Depending on the given fractions, students need to decide whether it makes sense to multiply or divide the numerator and denominator by a common number.

• Groups of 2

• “Work on the activity independently. Then, share your responses with your partner and check each other’s work.”
• 8–10 minutes: independent work time
• 3–5 minutes: partner discussion

If students attempt to partition the number line into 30, 60, or 90 parts, consider asking: “How can we use the patterns from the previous activity to help us here?”

• See lesson synthesis.

This activity is optional because it provides an opportunity for students to apply concepts from previous activities that not all classes may need. It allows students to practice using numerical strategies to find equivalent fractions by sorting a set of 36 cards. Students are not expected to find all equivalent fractions in the set. When students look for equivalent fractions they use their understanding of multiples and the meaning of fractions (MP7).

• Create a set of Fraction Galore cards from the blackline for each group of 3.

• Groups of 3–4
• Give each group one set of cards created from the blackline master.

• “Work with your group to sort the cards by equivalence. Find as many sets of equivalent fractions as you can. Some fractions have no equivalent fractions.”
• 8–10 minutes: small group work time

• “What strategy did your group use to find equivalent fractions? How well did the strategy work? How efficient was it?” (We looked at the numerators and denominators to see if they were multiples or factors we recognized.)
• “Did you notice any new patterns in the fractions that are equivalent?”