Lesson 11

This Number Talk encourages students to look for structure in multiplication expressions and rely on properties of operations to mentally solve problems. Reasoning about products of whole numbers helps to develop students’ fluency.

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy. 
• Keep expressions and work displayed. 
• Repeat with each expression.

• “How did the earlier expressions help you find the value of the last expression?”
• Consider asking:
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone approach the problem in a different way?”

The purpose of this activity is for students to use diagrams to reason about equivalence and reinforce their awareness of the relationship between fractions that are equivalent.

Students show that a shaded diagram can represent two fractions, such as \(\frac{1}{2}\) and \(\frac{4}{8}\), by further partitioning given parts or composing larger parts from the given parts. Unlike with the fraction strips, where different fractional parts are shown in rows and students could point out where and how they see equivalence, here students need to make additional marks or annotations to show equivalence.

In upcoming lessons, students will extend similar strategies to reason about equivalence on a number line—by partitioning the given intervals on a number lines into smaller intervals or by composing larger intervals from the given intervals.

In the first problem, students construct a viable argument in order to convince Tyler that \(\frac{4}{8}\) of the rectangle is shaded (MP3).

• Groups of 2

• “Work with your partner on the first problem. Discuss whether you agree with Jada and show your reasoning.”
• 3–4 minutes: partner work time
• Pause for a brief discussion. Invite students to share their responses and reasoning.
• “Now, work independently on the rest of the activity.”
• 5 minutes: independent work time
• Monitor for the different strategies students use to show equivalence, such as:
  • drawing circles or brackets to show composing larger parts from the given parts
  • drawing lines to show new partitions
  • labeling parts of the fractions with two names
  • drawing a new diagram with different partitions but the same shaded amount
• Identify students using different strategies to share during synthesis.

If students don’t explain how the pairs of fractions are equivalent, consider asking:

• “What does it mean for fractions to be equivalent?”
• “How could we show both fractions to determine if they are equivalent?”

• Select previously identified students to share their responses and reasoning. Display their work for all to see.
• As students explain, describe the strategies students use to show equivalence. Ask if others in the class showed equivalence the same way.

The purpose of this activity is for students to generate equivalent fractions, including for fractions greater than 1, given partially shaded diagrams. Student may use strategies from an earlier activity—partitioning a diagram into smaller equal parts, or making larger equal parts out of existing parts—or patterns they observed in the numerators and denominators of equivalent fractions (MP7).

• Groups of 2
• Display or draw a diagram with 2 fourths shaded:

  

  

• “Notice there's a 1 below the diagram. This is another way to show which part of the diagram represents 1.”
• “What fractions can the shaded parts of the diagram represent?” (\(\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, \frac{4}{8}\))

• “Now write two fractions that you think represent the shaded parts of each diagram.”
• 3–5 minutes: independent work time
• “Discuss the names you came up with for each fraction with your partner. Be sure to share your reasoning for each fraction.”
• 2–3 minutes: partner discussion
• Monitor for students who make statements like:
  • The first diagram is \(\frac{4}{6}\), because 4 of the 6 equal parts are shaded. It's also \(\frac{2}{3}\) because every 2 sixths is 1 third and there are 3 thirds. Two of the 3 thirds are shaded.
  • The second diagram is \(\frac{2}{8}\) because 2 of the 8 equal parts are shaded. It's also \(\frac{1}{4}\) because every 2 eighths is 1 fourth, and 1 of the 4 fourths is shaded.

If students name a fraction, based only on the given partitions, consider asking:

• “Tell me about how you named the fraction.”
• “How could you use the diagram to find another way you could name the fraction?”

• Select students to share their strategies for writing multiple fractions for each diagram. Display the diagrams they marked or annotated.
• “In what ways was the last diagram different than the first three?” (It shows 2 wholes. The shades parts were greater than 1.)
• “Was your strategy for finding fractions for this diagram different from the first three? Why or why not?” (No, it still involved making smaller equal parts. Yes, I partitioned the first 1 whole and the second 1 whole separately.)
• If no students mention \(\frac{12}{8}\) for the last diagram, ask, “Can you name another fraction other than \(\frac{3}{2}\) and \(\frac{6}{4}\)?”