Lesson 11

Multiplication Strategies on Ungridded Rectangles

Warm-up: Which One Doesn’t Belong: Multiplication in Many Forms (10 minutes)

Narrative

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.

Launch

• 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

Activity

• 2–3 minutes: partner discussion
• Share and record responses.
.

Student Facing

Which one doesn’t belong?

Activity Synthesis

• “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 $$3 \times 4 = 12$$.
• “Let’s find at least one reason why each one doesn’t belong.”

Activity 1: Mark, then Express (15 minutes)

Narrative

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.

MLR2 Collect and Display. Direct attention to words collected and displayed from the previous lesson. Invite students to borrow language from the display as needed, and update it throughout the lesson.

Launch

• 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.

Activity

• “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
• 3–5 minutes: partner discussion

Student Facing

For each rectangle:

• Mark or shade each rectangle to show a strategy for finding its area.
• Write one or more expressions that can represent how you find the area.

Student Response

If students say they aren’t sure where to mark or shade the rectangle because they can’t see the squares, consider asking:
• “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?”

Activity Synthesis

• “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.)

Activity 2: Card Sort: Different Expressions, Same Rectangle (20 minutes)

Narrative

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.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Attention, Focus

Required Materials

Materials to Copy

• Centimeter Grid Paper - Standard
• Card Sort: Different Expressions, Same Rectangle

Required Preparation

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

Launch

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

Activity

• “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

Student Facing

Your teacher will give you a set of cards with expressions that represent areas of rectangles.

Sort the expressions into groups so that the expressions in each group can represent the area of the same rectangle. Be prepared to explain your reasoning.

You can draw rectangles if you find them helpful.

Activity Synthesis

• 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

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

• “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?”

Lesson Synthesis

Lesson Synthesis

“Today we matched expressions that could represent the same rectangle. Let’s think about what some of the matching expressions show us about multiplication.”

“What expressions show us that we can decompose one of the factors, then multiply them separately?” (E and K, C and L, B and I)

Display the expressions on cards F and G.

“What do these expressions show us about multiplication?” (When there are more than 2 factors, we can decide which two factors to multiply first without changing the result).

Student Section Summary

Student Facing

In this section, we learned how multiplication and division are related.

$$6\times 5={?}$$

$$30\div 5={?}$$

$$30\div 6={?}$$

We used strategies to multiply and divide and worked towards fluent multiplication and division within 100.

$$\displaystyle 7\times3$$
$$\displaystyle (5\times3)+(2\times3)$$