Lesson 16

This Number Talk encourages students to think about the multiplication of one-digit numbers and multiples of 10 and to rely on place value to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students multiply numbers larger than 20.

• Display one expression.
• “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 expressions and work displayed.
• Repeat with each expression.

• “How did the first three problems help you solve the last problem?” (Since I knew \(3 \times 20\) was 60, I just added \(3 \times 5\) to that by adding 15. I broke \(3 \times 25\) into \(3 \times 10\), \(3 \times 10\), and \(3 \times 5\) to make it easier to multiply. Since 25 is half of 50, I took half of  \(3 \times 50\) to find \(3 \times 25\), and half of 150 is 75.)

Previously, students multiplied two factors where one factor was a whole number and the other a teen number. The purpose of this activity is for students to make sense of multiplication of a one-digit number and a two-digit number greater than 20. Students analyze representations used to find \(4 \times 23\) and articulate reasons for using strategies that are based on the distributive property and place value to find products (MP7). Along the way, they see that decomposing the two-digit factors into tens and ones is particularly helpful for multiplying. They also reinforce what they know about the connections between area diagrams and multiplication expressions.

• Groups of 2
• “Take a minute to make sense of how Clare and Andre represented \(4 \times 23\).”
• 1 minute: quiet think time
• Give students access to grid paper and base-ten blocks.

• “Now, work with your partner to complete the first two problems about \(4 \times 23\).”
• 8–10 minutes: partner work time
• Monitor for the diagrams that students choose in Diego’s problem and for their explanations. 
• Pause for a discussion. Invite students to share their responses. 
• If not mentioned by students, clarify that all diagrams can be used to multiply 4 and 23, but not all are equally practical. 
• Select a student to explain why diagram D might be more productive than the other diagrams.
• “Now, work independently to complete the last problem.”
• 2–3 minutes: independent work time

If students choose a diagram in the second problem that doesn't help them find the product of \(4\times23\), consider asking:

• “Tell me about how you chose which diagram you would use to find the value of \(4\times23\).”
• “Is there another diagram that would make finding the value of \(4\times23\) easier? How would it make it easier?”

• “What representations did you use in the last question to show \(3 \times 28\)? How did they help you find the product?” (Base-ten blocks helped me break the 28 into tens and ones. A diagram with a grid helped me break 28 into smaller numbers. A diagram with no grid helped me think about the numbers as I labeled it without worrying about all the little squares.)

The purpose of this activity is for students to continue to multiply single-digit whole numbers and numbers greater than 20. The opening problem encourages students to apply place value reasoning (decomposing two-digit numbers into tens and ones) and properties of operations to reason numerically about products. Because one of the factors is small, however, students may use repeated addition (such as \(43 + 43\)) to find subsequent products. In the synthesis, emphasize strategies that are based on place value, connecting the numerical expressions with diagrams as needed.

• Groups of 2
• “Take a look at how Mai started to multiply \(2 \times 37\). Then, talk to your partner about why you think Mai decided to start multiplying this way.” (Thirty-seven is a large number to multiply, so she broke it into tens and ones. \(2 \times 30\) shows the multiplication of the tens.)
• 1 minute: partner discussion
• Share responses.
• Give students access to grid paper and base-ten blocks.

• “Take a few minutes to work independently on the activity. Afterwards, share your responses with your partner.”
• 5 minutes: independent work time
• 5 minutes: partner work time

If a student says they don’t know how to start the problem, consider asking:

• “What would you do if the factors were 3 and 10? What about 3 and 20?”
• “How could you use base-ten blocks or diagrams to help you find these products?”

• Invite 1–2 students to share their responses and reasoning. Display or record their work for all to see. 
• Discuss responses that highlight use of place value and properties of operations. \(4\times 22\), for instance, can be found using the associative property (\(2 \times (2 \times 22)\) or distributive property (\(4 \times 20 + 4 \times 2\)).
• Consider using diagrams or base-ten blocks to reinforce the meaning of expressions as needed.

The purpose of this activity is for students to play a game in which they are able to apply the strategies they’ve learned for multiplying teen numbers and numbers over 20. Students use digits to create an expression that has a value as close to 100 as possible. The first game includes teen numbers and the second game includes numbers over 20.

This activity is optional because it provides extra practice for multiplying by factors that are teen numbers and factors greater than 20. Depending on the time available, students can play 1 or 2 games.

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

• Groups of 2
• Distribute one set of pre-cut cards to each group of students.
• “We’re going to play a game called Close to 100. Let’s read through the directions together and play 1 round together.”
• Play a round against the class, displaying the numbers from the cards and thinking through decisions aloud.

• “Now, you will play a game of Close to 100 with your partner. The game will have 5 rounds.”
• 5–7 minutes: partner game time
• If time allows, have students play the second game of Close to 100. Inform students that the numbers are different in the second game.

• “What were some strategies that were helpful as you played Close to 100?” (I used rounding to think about how large the product would be. I multiplied the tens and ones, then combined them to find the product.)