In Twos, Threes, and More

Est. Class Sessions: 3

Developing the Lesson

Part 1. Show Solutions to Multiplication Problems

Find Things That Come in Groups.

Begin by involving the class in a discussion:

How many shoes are there on six children? (12)
How many wheels are there on five cars? (20)
How many arms are there on three octopuses? (24)

Use larger numbers as students seem ready for them.

As students answer the questions, ask how they arrived at their answers. Encourage others to share additional solution methods. As students describe their methods, write number sentences that display their results.

For instance, one way to solve the question “How many shoes are there on six children?” is to use repeated addition: 2 + 2 + 2 + 2 + 2 + 2 = 12 shoes. This problem can also be solved by skip counting by twos. In one class, students suggested acting out the skip counting: each stood up one at a time while the class counted their shoes—2, 4, 6, 8, 10, 12. When discussing the solutions, focus on the fact that the shoes can be counted in six groups of two. Use the related multiplication sentence: 6 × 2 = 12 shoes.

Represent Solution Strategies. After discussing these problems, ask the class to turn to In Twos, Threes, and More in the Student Guide. Read and discuss the vignette with the class. Direct the students to solve Question 1. After students have worked on the question, ask a volunteer to give his or her solution while you display the number sentence stated.

Elicit additional solutions by asking:

Did anyone solve this problem another way?

Display all responses. Possible responses might include:

3 + 3 + 3 + 3 = 12
I skip counted: 3, 6, 9, 12
I did times: 4 × 3 = 12
I did times, too, but mine's different: 3 hands × 4 clocks = 12 hands
I counted each hand.

Ask:

Does anyone find any similarities in any of the responses?

Be sure to point out the first and third solution, calling attention to how the addition of 4 threes is related to 4 × 3. Next highlight 4 × 3 and 3 × 4.

How are these solutions the same? How are they different? (They both equal 12, but one makes me think of 4 groups of 3 things and the other makes me think of 3 groups of 4 things.)
Is there a way to make it clear what the answer 12 represents?

Lead students to understand that adding labels such as 3 hands and 4 clocks would make the thinking in the solution more easily understood by others.

Then ask:

How should I label the answer? (hands)
How did you know the answer should be labeled “hands”? (The question asks, “How many hands”?)

Practice Representing Solution Strategies. Direct students' attention to a display of the Math Practices page in the Student Guide Reference section.

Review and ask students to focus on the following expectations:

Know the problem. I read the problem carefully. I know the questions to answer and what information is important [MPE1].
Check my calculations. If I make mistakes, I correct them [MPE4].
Show my work. I show or tell how I arrived at my answer so someone else can understand my thinking [MPE5].
Use labels. I use labels to show what numbers mean [MPE6].

Ask students to solve Questions 2A–E using the math practices above. Discuss solutions using similar questions as suggested for Question 1. Be sure to point out solutions that highlight the connection between addition and multiplication. Check for use of the math practices stated for each response.

Direct student pairs to solve Questions 3–5. Ask them to review and revise their own solutions checking their work for the expected math practices. Discuss solutions aloud with the class. Then display the following solution to Question 5:

4 + 4 + 4 + 4 + 4 = 20 legs

Discuss this solution by posing the following questions to review math practices appropriate to this problem:

Did this student show that he or she knew the problem [MPE1]? Why or why not? (No. He might have read the problem wrong because he showed 5 horses. He left out the one that was bought, so he should show 4 + 4 + 4 + 4 − 4 or just 4 + 4 + 4 + 4.)
Do you think he or she checked his or her calculations [MPE4]? (Not sure. The calculation for the number sentence he wrote is correct, but he did not show how he checked his calculations.)
Did he or she show his or her work [MPE5]? Could you understand what he or she was thinking? Could you tell if his or her strategy makes sense? (Yes, I can understand his thinking, but his strategy didn't make sense because one horse isn't in the store now.)
Did he or she use labels to show what numbers mean [MPE6]? (Yes, he labeled the answer, “legs.”)

Use Symbols to Show Solution Strategies. Write a few open number sentences on the board and discuss what should go in the boxes to make the sentences true.

5 + 5 + 5 + 5 = × 5
2 + 2 + 2 + 2 = 4 ×
+ 7 = 2 × 7
3 + 3 + 3 + 3 + 3 + 3 = × 3

Do not place too much emphasis on the order of the factors in a multiplication sentence. It is standard practice to associate the number sentence 3 × 4 = 12 with the sentence “3 groups of 4 equal 12.” However, some children might write 4 × 3 = 12. Since multiplication is commutative, this is acceptable.

Then ask students to solve the open number sentences in Question 6 and analyze the true/false number sentences in Question 7 in the Student Guide. Discuss their answers.

For example, ask:

How did you decide what to put in the box in the number sentence in Question 6B? (I saw that there were three 5s on one side, so I filled in the box with 3, for 3 times 5.)
Why did you say that 4 × 3 = 3 + 3 + 3 is false? (I did times on one side and got 12, and when I added on the other side I got 9.)
How did you correct it? (4 × 3 means that there should be four 3s, so I added a three to the other side. When I added, that side equaled 12, too.)

Assign Check-In: Questions 8–9 in the Student Guide. Remind students to use the Expectations on the Math Practices page when giving solutions. Question 8 provides an opportunity for students to represent a solution to a multi-step multiplication problem. In Question 9 students analyze another student's solutions.

Use In Twos, Threes, and More Check-In: Questions 8–9 in the Student Guide and the corresponding Feedback Box in the Teacher Guide to document students' abilities to solve [E3] and represent solution strategies [E5] for problems involving the multiplication facts. Also monitor students' performance with the following Math Practices Expectation: