Lesson 11

This warm-up prompts students to compare four representations of multiplication. It gives students a reason to use language precisely as they talk about characteristics of the items being compared.  During the synthesis, ask students to explain the meaning of any terminology they use, such as strategies, area, and parts.

• 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.
  .

• “What number do the diagrams and the expression in C represent?” (18) “How do you know?” (There are 18 dots in the array. There are 18 squares in the rectangle. If I add up the parts of the expression or the parts of the rectangle, I get 18.) 
• “What might be the length of the unlabeled side of the rectangle in B? How do you know?” (3, because the rectangle is the same one as in A, just not showing a grid. 3, because \(3 \times 2 = 6\) and \(3 \times 4 = 12\).
• Consider asking:
  • “Let’s find at least one reason why each one doesn’t belong.”

The purpose of this activity is for students to find the area of ungridded rectangles using strategies based on the distributive and associative properties. Students represent these strategies on rectangles with no grid. This will be helpful in future lessons as students use area diagrams to represent the multiplication of larger numbers.

• Groups of 2
• “We are going to find the area of more rectangles. How are these rectangles different from the rectangles we worked with in the last lesson?” (They don’t have a grid in them. We can’t see the squares.)
• 30 seconds: quiet think time
• Share responses.

• “Mark or shade each rectangle to help you find its area. Then write one or more expressions that represent your work and show how you found the area.”
• 5–7 minutes: independent work time
• “Share how you found the area of each rectangle with your partner. Be sure to ask and answer any questions you have about your partner’s strategy.”
• 3–5 minutes: partner discussion

• “What numbers are you multiplying to find the area?”
• “How could you decompose one of the factors to help you find the product? How would you show that on the diagram?”

• “How was showing your strategy on a rectangle with no grid different than showing your strategy on a rectangle with a grid?” (I just estimated where I thought I should split the rectangle. I was thinking more about the numbers than counting all the squares.)

In this sorting activity, students identify expressions that could represent the area of the same rectangle and explain their reasoning. To do so, they apply their understanding of properties of multiplication and draw rectangles as needed as they interpret parts of the expressions. Some students may sort expressions based only on the value of the expressions. Encourage them to explain or show how they know, for instance, that \(8 \times 6\) and \(3 \times 6 + 5 \times 6\) can represent the area of the same rectangle (MP2, MP7). Some of the expressions from this activity are used in the synthesis to highlight the commutative, distributive, and associative properties of multiplication.

• Create a set of cards from the blackline master for each group of 2 or 4.

• Groups of 2 or 4
• Give each group a set of pre-cut cards from the blackline master.
• Give students access to grid paper.

• “This set of cards includes expressions that represent areas of rectangles. Group together expressions that can represent the area of the same rectangle.”
• “Work with your partner to explain your sorting decisions. You can draw rectangles if you find them helpful.”
• 8 minutes: partner work time

• Invite students to share their sorting results, drawings (if any), and explanations on how they know those expressions go together.

• Record each group of expressions. Discuss the connections between the expressions, illustrating them on a drawing of a rectangle. For instance, for

  \(8\times3\\(4 \times 3) \times 2)\\4 \times (2 \times 3)\)

  ask questions such as:
  • “Where do we see the 8 in \(4 \times (2 \times 3)\)?”
  • “What’s the area of this rectangle? What could its side lengths be?”
  • (Draw and label a rectangle.)

    

    

  • “Where do we see the \((4 \times 3) \times 2\) in the rectangle?”

    

    

  • “Where do we see the \(4  \times (2 \times 3)\) in the rectangle?”