Lesson 6

The purpose of this Choral Count is for students to practice counting by 12, 15, and 24, and notice patterns in the count after 10 multiples. This understanding will help students later in this lesson when they represent quantities that are 10 times as many using tape diagrams. 

When they use the words multiple, value, and place students use language precisely (MP6).

• “Count by 12 starting at 12.”
• Record as students count.
• Stop counting and recording after the tenth multiple has been said.
• Repeat these steps with 15 and 24.

• Circle the tenth multiple in from each count.
• “What do you notice about the tenth multiple?” (It has a zero in the ones place.)
• “Why do you think this is?” (The value is now ten times as much as the first value.)
• “Can you predict the twentieth multiple?”(It is the double of the tenth multiple and it also ends in zero.) 
• 1–2 minutes: quiet think time
• 1 minute: partner discussion
• Share and record responses.

• “What would the tenth multiple be if we were counting by 192? 1092?” (1,920, 10,920)

In this activity, students are given a diagram that shows two quantities, one of which is 10 times as much as the other. They identify possible values and possible equations that the diagram could represent. 

Students see that a single unmarked diagram could represent many possible pairs of values that have the same relationship (in this case, one is 10 times the other) and be expressed with many equations.

The activity also reinforces what students previously learned about the product of a number and 10—namely, that it ends in zero and each digit in the original number is shifted one place to the left because its value is ten times as much (MP7).

• Groups of 2
• Read the first problem aloud.
• “Think of two possibilities for the value of A and B.”
• 2 minutes: quiet think time

• 2 minutes: partner discussion
• “Work with a partner on the rest of the activity.”
• Monitor for students who:
  • use the tape diagram to reason about the equations.
  • rely on numerical patterns in multiples of 10 to reason about 10 times as many

Students may assume that it is impossible for the same diagram to represent multiple expressions. Consider asking:

• “How does the value of B change if A represents 6? How do you know?”
• “What expression could we write to represent this?”

Consider repeating with another number to reinforce the idea.

Display possible values for A and corresponding values for B in a table such as this:

• “What do you notice about the values of each set? How do the values compare?” (The values for B are all multiples of 10. Each value for B is ten times the corresponding value for A.)
• Select students to share their responses and reasoning. 
• For students who used the diagram to reason about the equations, consider asking, “How might you label the diagram to show that it represents the equations you selected?” (Sample response: For the equation \(30 \div 3 = 10\) I would label A “3” and B “30.” I know that 30 is ten times as much as 3, so the diagram represents the equation.)
• For students who used their understanding of numerical patterns to support their reasoning, consider asking, “How did you know that the equation could be represented as a comparison involving ten times as many?” (Sample response: I know that when we multiply a number by 10, the product will be ten times the value. I also know that division is the inverse of multiplication, so I looked for equations that were multiplying or dividing by 10 or had ten as a quotient.)

In this activity, students analyze situations in which one quantity is ten times as much as another quantity. Students may use different strategies to determine the missing quantity. For example, they may rely on counting as a strategy, or use place value understanding to explain regularity in the products of numbers with 10 (MP8). The reasoning in this lesson prepares students to consider quantities that are 100 times and 1,000 times as many in the next section.

• Groups of 2

• 5–6 minutes: quiet work time
• 3 minutes: partners discussion
• Monitor for students who:
  • use multiplication expressions to represent the relationships between values of A and B, for example, \(14 \times 10\) to represent the first pair of quantities 
  • use place value relationships to determine the value of each quantity

If students write a value other than 10,000 for B when A is 1,000, consider asking:

• “What do you notice about the pairs of smaller numbers you found that can help you find the value of B?”
• “What about the choral count from today’s warm-up can help you know what number to write?”

• “What were some things you noticed about the values of A and B?” (Sample responses: 
  • A was less and B was always greater.) “How many times as much as A was B?” (Ten times)
  • The digits of A are all in B and are in the same order, but they are not in the same places in B. There's an extra 0 at the end of the value for B. “Why do you think this is?” (Because they are multiplying 10 by A.)
  • “Could we represent both \(4 \times 10\) and \(10 \times 10\) using the same diagram of A and B? Why or why not?” (Yes, because A could represent 4 or 10, and B represents 10 times that value.)