Lesson 15

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for using the properties of operations when multiplying whole numbers and mixed numbers. These understandings help students develop fluency and will be helpful later in this lesson when students will flexibly multiply.

• Display one problem.
• “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 problems and work displayed.
• Repeat with each problem.

• Display the last problem.
• “If someone found \(7 \times 4\) first, what might they do next?” (Subtract \(7 \times \frac{1}{10}\) since \(3 \frac{9}{10}\) is \(\frac{1}{10}\) less than 4.)

The purpose of this activity is for students to consider situations and the operations involved in order to select reasonable numbers for each situation. Students are given a set of numbers which must each be used once in the statements. Students may find that their initial thinking does not work with the given constraints and may need to revise their work (MP1). They also need to think carefully about the units involved and consider whether a number is representing a linear unit or a square unit. Students may solve these problems using several different strategies.

• Groups of 2

• 3–5 minutes: independent work time
• 8–10 minutes: partner discussion
• Monitor for students who:
  • draw an area diagram.
  • write multiplication equations.
  • revise their thinking and can explain why their original solution didn’t work, but their revised solution does work.

If students need more of an invitation to enter the task, display the first problem with no numbers. Ask students to name some numbers that would make sense in this situation and explain why.

• Ask previously selected students to share their solutions.
• “Which numbers did you think made sense for the length of the rug in feet?” (I thought any of the numbers 5, \(5\frac{1}{2}\), or \(5\frac{3}{4}\) made sense.)
• “How did you decide which number to use for the length of the rug?” (The product had to be \(16\frac{1}{2}\). So I tried a width of 2 feet and that was too small. Then I tried a width of 3 feet and that worked with \(5\frac{1}{2}\) feet for the length.)

The purpose of this activity is for students to match different diagrams and expressions representing the same product. In the previous several lessons, students have studied different ways to find products of a whole number and a fraction using arithmetic properties such as the distributive property. Earlier in the unit they learned about the connection between fractions and division. They combine these skills as the expressions and diagrams they work with incorporate both the distributive property and the interpretation of a fraction as division (MP7). Students may match diagrams to expressions differently than the ways that are listed in the sample responses. Encourage students to match the expressions and diagrams in a way that makes sense to them, as long as they can accurately explain how the expression is represented in the diagram they chose.

MLR7 Compare and Connect. Invite students to prepare a visual display that shows their thinking about their favorite diagram and expression. Encourage students to include details that will help others understand what they see, such as using different colors, arrows, labels, or notes. If time allows, invite students time to investigate each others’ work.
Advances: Representing

• Groups of 2
• “You are going to match expressions and diagrams that show different ways to find the value of the product \(4 \times 5 \frac{2}{3}\).”

• 5–7 minutes: independent work time
• 2–3 minutes: partner discussion

• Display the diagrams B and D. 
• “How are the diagrams the same? How are they different?” (They both have the same shaded area and the same divisions inside. The top one shows the length of the shaded rectangle. The bottom one does not though it has enough information to find the shaded area.)
• Invite students to share their favorite ways to find the product.