# Lesson 14

Ways to Represent Multiplication of Teen Numbers

## Warm-up: Notice and Wonder: Seeing Groups (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea that while there are multiple ways to represent 2 groups of 12, some ways are more useful than others. While students may notice and wonder many things about the images, how 2 images show the groups of 12 have been organized using place value and how this type of decomposition can be helpful in finding the total are the important discussion points.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

• “The image on the left is a drawing of equal groups. The other images are base-ten diagrams. What is the same and different about these representations?” (They all show 12. They all show 2 groups of the same size. In the base-ten diagrams you can see the tens easier. It’s harder to see the tens in the first drawing.)

## Activity 1: A Factor Greater than Ten (20 minutes)

### Narrative

The purpose of this activity is for students to see how, when multiplying a number larger than ten, the distributive property can be used to decompose the factor into tens and ones, creating two smaller products. Base-ten blocks are used to help students visualize what is happening when a factor is decomposed to make two more easily known products. Factors slightly larger than ten can be naturally decomposed into a ten and some ones using place value. This will be useful in subsequent lessons as students progress towards fluent multiplication and division within 100.

When students see that you can decompose a teen number into tens and ones and use this to multiply teen numbers, they look for and make use of structure (MP7).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students base-ten blocks.

### Activity

• “Take a few minutes to look at Tyler’s strategy and decide if you agree or disagree with it.”
• 2–3 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for students who connect the expressions $$7 \times 10$$ and $$7 \times 3$$ to the tens and ones portion of the place value diagram and $$7 \times 13$$ to the entire diagram.
• Have students share why they agreed or disagreed with Tyler’s strategy with a focus on using the place value diagram as a justification.
• “Where do we see the $$7 \times 13$$ in the place value diagram?”
• “Where do the 10 and the 3 come from?”
• “How could you use the place value diagram to figure out how to find the value of $$7 \times 13$$?”
• “Now use Tyler’s method on your own to find the value of $$3 \times 14$$.”
• 2–3 minutes: independent work time
• 2–3 minutes: partner discussion

### Student Facing

1. Tyler says he can use base-ten blocks to find the value of $$7 \times 13$$ because he knows $$7 \times 10$$ and $$7 \times 3$$. He says this diagram proves his thinking.

Do you agree or disagree? Explain your reasoning.

2. Use Tyler’s method to find the value of  $$3 \times 14$$. Explain or show your reasoning.

### Student Response

If students say they don’t see $$7\times10$$ and $$7\times3$$ in Tyler's diagram, consider asking:
• “Where do you see $$7\times13$$ in the diagram?”
• “If we separate the tens and ones, what expression could we use to describe the tens? The ones?”

### Activity Synthesis

• Display base-ten blocks or place value diagrams that students used to solve. As students explain their work, write multiplication expressions to represent them.
• “How does this diagram (or the base-ten blocks) show how Tyler’s method could be used to multiply $$3 \times 14$$?” (We can see there are 3 tens which is 30. We can see that there are 3 groups of 4 ones which is 12. $$30 + 12$$ is 42. The whole diagram represents $$3 \times 14$$.)
• If there is time, ask students to find the value of $$4 \times 12$$ and $$5 \times 16$$ using the base-ten blocks and Tyler’s strategy.

## Activity 2: Ways to Represent (15 minutes)

### Narrative

The purpose of this activity is for students to make sense of different ways of representing multiplication of a teen number. Students analyze a gridded area diagram, base-ten blocks, and an area diagram labeled with side lengths. When they discuss how the different diagrams represent the same product, students reason abstractly and quantitatively (MP2).

MLR8 Discussion Supports. Synthesis: Show a visual display of the diagrams. As students share their observations, annotate the display to illustrate connections. For example, on each diagram, annotate the decomposition of 15 into 10 and 5 by circling the groups of 10 and the groups of 5.
Representation: Access for Perception. Begin by showing a demonstration explaining how you see the product in each of the 3 different models using a different problem to support understanding of the context.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Launch

• Groups of 2
• “We’re going to look at three different ways students showed the same expression. What do you notice? What do you wonder?” (Students may notice: You can see all the squares in the first 2 diagrams, but not in the last one. The middle diagram looks like base-ten blocks. Students may wonder: Why would you choose to use one of these diagrams? What numbers were they multiplying?)
• 1 minute: quiet think time
• Share responses.

### Activity

• “Work with your partner to tell how you see the factors in each diagram and how you see the product in each diagram.”
• 5–7 minutes: partner work time

### Student Facing

Andre, Clare, and Diego represented the same expression. Their representations are shown below.

1. Where do you see the factors in each diagram?

2. Where do you see the product in each diagram?

### Activity Synthesis

• “How are these ways of representing $$3 \times 15$$ the same?” (They all represent 3 times 15. They all show the 15 being decomposed into 10 and 5. They are all shaped like a rectangle.)
• “How are these ways of representing $$3 \times 15$$ different?” (Clare used base-ten blocks, but Andre and Diego used rectangles. Diego didn’t show the squares in his rectangle, but Clare and Andre did.)
• “How could we represent the strategy shown in all the diagrams with expressions?” ($$3 \times 10$$ and $$3 \times 5$$ or $$10 \times 3$$ and $$5 \times 3$$.)

## Lesson Synthesis

### Lesson Synthesis

Display:

$$7 \times 6$$

$$(5 \times 6) + (2 \times 6)$$

$$3 \times 15$$

$$(3 \times 10) + (3 \times 5)$$

“Today we saw some different ways to represent strategies we can use to multiply teen numbers. How are the strategies we use to multiply teen numbers like the strategies we used to multiply smaller numbers in past lessons?” (We can use facts that we know to find facts that we don’t know. We can break down one of the factors into smaller parts to make it easier to multiply.)