Lesson 19

The purpose of this True or False is to reinforce the relationship between tens and ones (that 1 ten is equal to 10 ones, or 1 group of 10 is 10 groups of 1). This will be helpful when students use base-ten blocks to represent division and decompose tens into ones to facilitate the process of dividing. It also allows students to practice finding the product of a one-digit whole number and a multiple of 10.

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• “How can you justify your answer without finding the value of both sides?”
• Consider asking:
  • “Who can restate _____’s reasoning in a different way?”
  • “Does anyone want to add on to _____’s reasoning?”

The purpose of this activity is for students to use strategies based on place value to find quotients greater than 10. Students use base-ten blocks to represent quotients with single-digit divisors, for which it is intuitive to think of the divisor as the number of groups. In a later activity, students will be reminded that the divisor can also be interpreted as the size of each group. 

Working with base-ten blocks encourages students to divide out the tens and then the ones, and to see that sometimes it is necessary to decompose one or more tens to finish putting the dividend into equal groups. When students represent a quotient using base-ten blocks they reason abstractly and quantitatively (MP2).

• Groups of 2
• Give each group base-ten blocks.
• “Use base-ten blocks to represent \(39 \div 3\).”
• 1–2 minutes: independent work time
• Select a student who divided the blocks into 3 groups of 13 to share their final representation, such as:

• “Why are there 3 groups?” (We are dividing by 3.)
• “How could the blocks have been divided to end up like this?” (The tens were put into 3 groups and then ones placed one by one into 3 groups until none were left.)
• Highlight that the blocks could have been divided up by the tens and then the ones.

• “Work with your partner on the first problem.”
• 5 minutes: partner work time
• Pause for a discussion.
• “What was different about using the blocks to find \(45 \div 3\) and using them to find \(55 \div 5\)?” (For \(45 \div 3\), it was necessary to decompose 1 ten to finish putting 45 into 3 equal groups. That’s not necessary for \(55 \div 5\) because there was already the right number of tens and ones to make the 5 groups.)
• “Now, work independently to find the value of quotients in the second problem.”
• 6–8 minutes: independent work time

• Invite students to share their responses and reasoning for the last set of quotients.
• Ask students who used base-ten blocks or drew diagrams: “Was it necessary to decompose any of the tens into ones to divide?” (It wasn’t necessary for \(63 \div 3\) because there was already the right number of tens and ones to put into 3 groups. It wasn’t necessary for \(100 \div 5\) because I started with 10 tens and there was already the right number of tens to put into 5 groups.)
• “Why was it necessary or helpful to decompose the tens in 84?” (After putting 7 tens in 7 groups, there’s still 1 ten and 4 ones. The 1 ten couldn’t be split into 7 groups.)

The purpose of this activity is to show that the two meanings of division still apply when dividing larger numbers and that, in some cases, one interpretation may be more helpful than the other.

Students first analyze two ways of using base-ten blocks to represent \(65 \div 5\) and see that the divisor, 5, can be interpreted to mean the number of groups or the size of one group. They then consider how they might interpret and represent the divisor in other quotients. The reasoning here prepares students to reason more strategically as they divide larger numbers. 

• Groups or 2–4
• Give base-ten blocks to each group.
• Ask students to keep their materials closed.
• “Use base-ten blocks to find the value of \(60 \div 5\).”
• 1–2 minutes: independent work time

• “Now take a look at Jada and Han's work in the activity. Which of them represented the division the same way you did?”
• “Work with your partner to make sense of Jada’s and Han’s work and complete the first problem.”
• Pause for a brief discussion.
• “How was Jada’s and Han’s representation different? How did each of them interpret \(60 \div 5\)?” (Jada saw the 5 as the number of groups. Han saw the 5 as the number in each group.)
• Poll the class on how they interpreted \(60 \div 5\) when they represented it during the launch.
• “Now, work independently on the second set of problems.”
• 5 minutes: independent work time

• Invite students to share their responses and reasoning for the last set of problems.
• “How did you decide whether the divisor, the number we’re dividing by, is the number of groups or the amount in each group?” (It depends on the number. In the first two problems, the divisor was 4 and 6, so it was easier to think about 4 groups and 6 groups. In the last problem, the divisor was 15. It was easier to think about how many groups of 15 are in 75 than to think about making 15 groups from 75.)