# Lesson 5

Products Beyond 100

## Warm-up: Number Talk: A Number Times Some Multiple of 10 (10 minutes)

### Narrative

This Number Talk encourages students to decompose factors and to rely on the distributive property to mentally solve. The strategies elicited here will be helpful later in the lesson when students multiply up to four-digit numbers by one-digit numbers, and later in the section when they multiply 2 two-digit numbers by decomposing factors.

### Launch

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

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$8 \times 30$$
• $$5\times30$$
• $$10\times30$$
• $$15 \times 30$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How did the first three expressions help you find the value of $$15 \times 30$$?” (I know that $$(5 \times 30) + (10 \times 30)$$ is $$15 \times 30$$. I doubled the value of $$8 \times 30$$ to get the value of $$16 \times 30$$ and then I subtracted 30 from it to get the value of $$15 \times 30$$.)

## Activity 1: Elena’s Sticky Gift (15 minutes)

### Narrative

In this activity, students build on grade 3 work with arrays to consider how to find the total number in an array without counting by 1. Students are not asked to find the answer, but instead share their strategies for doing so. This allows teachers to observe how students make sense of multiplying larger numbers.

Students may decompose the larger array of stickers into two smaller arrays using the distributive property to determine the product (MP7). They may also use the idea of doubling and tripling to find the product. (For instance, they may start with $$13 \times 2$$ and triple the result to get $$13 \times 6$$.)

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

MLR2 Collect and Display. Circulate, listen for, and collect the language students use as they create a display of their strategies. On a visible display, record words and phrases such as: decompose, partition, associative property, distributive property, and array. Invite students to borrow language from the display as needed, and update it throughout the lesson.

### Launch

• Groups of 2
• Give each group tools for creating a visual display.

### Activity

• “Take a few quiet minutes to answer the question. Then, compare your strategy with your partner’s.”
• 3 minutes: independent work time
• 2 minutes: partner discussion
MLR7 Compare and Connect
• “Create a display that shows both of your ideas. Record your thinking so that it can be followed by others.”
• 5 minutes: partner work time
• Monitor for students who:
• decompose the two-digit factor by place value and use a drawing or an expression to show the decomposition (for example: partition the 13 columns in the array into 10 and 3 columns)
• write expressions that involve the distributive or associative properties (as noted in Student Responses)
• 3 minutes: gallery walk

### Student Facing

How many stickers are there? Explain or show how you would find out without counting every sticker.

### Student Response

For access, consult one of our IM Certified Partners.

If students choose to count the number in the first column or the first row and then skip-count by that number, consider asking them how they might record that strategy (other than writing “skip-count by 6” or “skip-count by 13”).

### Activity Synthesis

• “Which ideas did you see the most as you walked around today?”
• 30 seconds: quiet think time
• Invite students to share their strategies and reasoning.
• To highlight similarities and differences, consider comparing and contrasting the strategies using the term “decompose.”

## Activity 2: More and More Stickers (20 minutes)

### Narrative

In this activity, students use strategies and representations that make sense to them to find products beyond 100. As before, the context of stickers lends itself to be represented with an array. The factors are large enough, however, that doing so would be inconvenient, motivating other representations or strategies (MP2). Look for the ways that students extend or generalize previously learned ideas or representations to find multiples of larger two-digit numbers. While many of the student responses are written with expressions, students are not expected to represent their reasoning using equations and expressions as this time. Teachers may choose to represent student reasoning using equations and expression so students can start connecting representations.

After students work on the first problem, pause to discuss some possible representations for finding the number of stickers. Each of the representations show different ways to represent the decomposition of a factor and students may decompose the factors in a variety of ways.

A. I created an array and decomposed it into smaller arrays.

B. I drew a diagram and decomposed it into smaller sections.

C. I decomposed the 21 and wrote one or more expressions or equations.

$$(9 \times 20) + (9 \times 1)$$
$$(9 \times 10) + (9 \times 10) + (9 \times 1)$$

D. I decomposed the 9 and wrote one or more expressions or equations.

$$(3 \times 21) + (3 \times 21) + (3 \times 21)$$
$$(4 \times 21) + (4 \times 21) + (1 \times 21)$$

Consider using the “four corners” structure to allow for movement and for interactions among students who might not typically interact. Post each of the four strategies in a different corner of the classroom. For the representations that use arrays or rectangular diagrams, it may help to give examples of decomposing based on 12 samples of student work that you observe during the activity. Then, ask students to move to a corner based on their reasoning strategy and representation.

Representation: Develop Language and Symbols. Before displaying the four strategies shown in the Activity Narrative, activate background knowledge. Ask, “What does it mean to decompose a number?” While reviewing the strategies, ask students to engage with their classmate’s explanations by asking, “How did your classmate decompose 21 (or 9)?” For students who need extra support approaching question 2, begin by asking, “How might you decompose 48?”
Supports accessibility for: Memory, Language

### Required Preparation

• Create 4 posters showing the 4 representations shown in the activity narrative.

### Launch

• “Make an estimate: Do you think Elena has fewer than 100 stickers, between 100 and 200, or more than 200?”
• 30 seconds: quiet think time
• Poll the class on their estimates (fewer than 100, between 100 and 200, or more than 200).
• 1 minute: partner discussion

### Activity

• “Take a few quiet minutes to find the exact number of stickers Elena has and explain or show your reasoning.”
• 2–3 minutes: independent work time
• Display the four representations shown in the activity narrative.
• “Which representation best describes your approach? If none of them does, create a display that shows your thinking.”
• Poll the class on their representations. Select a student who uses each strategy to explain more fully how they solved the problem.
• “Now answer the last question using any of these representations or another one that makes sense to you.”
• 5–7 minutes: group work time
• Monitor for the strategies students use to find $$3 \times 48$$ and $$7 \times 23$$.

### Student Facing

1. Elena has another sheet of stickers that has 9 rows and 21 stickers in each row. How many stickers does Elena have? Explain or show your reasoning.

2. Noah’s sticker sheet has 3 rows with 48 stickers in each row. Andre’s sticker sheet has 7 rows with 23 stickers in each row.

Who has more stickers? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

If students start to draw arrays to represent $$3 \times 48$$ and $$7 \times 23$$, ask them if they could represent the factors in another way that doesn’t require drawing individual dots for the stickers.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we multiplied a two-digit number by one-digit number.”

Display $$3 \times 48$$ and $$7 \times 23$$ for all to see. Invite students to share their strategies for finding the value of each product.

“To find the value of $$3 \times 48$$, some of you started by finding $$3 \times 40$$—with or without drawing diagrams—and others started by finding $$3 \times 50$$. If you started with $$3 \times 40$$, what did you do next?” (Add $$3 \times 8$$.) “If you started with $$3 \times 50$$, what did you do next?” (Subtract $$3 \times 2$$.)

“To find the value of $$7 \times 23$$, some of you found $$7 \times 20$$ first and then $$7 \times 3$$. Why did you decide to decompose the 23 into 20 and 3?” (It makes it possible to multiply the 7 by a multiple of 10, which is easier than multiplying 7 by a number that is not a multiple of 10.)

## Cool-down: Rows of Seats (5 minutes)

### Cool-Down

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