Discuss Tools and Strategies. Ask students what tools they have been using to represent and solve multiplication and division problems. Students should mention multiplication tables, the patterns chart, square-inch tiles, drawings, rectangles on grid paper, skip counting with or without the 200 Chart, number lines, and counters.
Distribute the small containers or bags of tools you gathered to small groups of students. Ask students to also locate the following tools: their desk number line, the 200 Chart, and a multiplication table. These tools will help students explain their thinking. Have display versions of these tools available to facilitate sharing and discussions. Briefly review the patterns on the multiplication table that the class recorded on the Patterns for Remembering the Facts chart made in Lessons 3, 4, and 5. See Figure 1.
Ask students to name some of the strategies they have been using to solve multiplication and division problems. Make and display a list of student responses. Students should mention skip counting, repeated addition, repeated subtraction, reasoning from known facts, doubling, using break-apart products, turn-around facts, and fact families.
Tools Support Reasoning. All students can benefit from using manipulatives. In this lesson, some students can use square-inch tiles or draw rectangles on grid paper to find solutions. Students who “just do it in their head” can benefit from stepping back and being asked to use manipulatives or drawings to justify their answers. This will deepen their reasoning and communication skills.
- What do you notice about these number sentences? (Answers will vary. Possible response: Some of them have boxes for unknown numbers. In some, there are number sentences on both sides of the equal sign.)
- What does the equal sign in the first problem, 5 × 5 = 25 mean? (It shows that what is on the left side of the equal sign, 5 × 5, is the same value as what is on the right side of the equal sign, 25.)
- What do the equal signs mean in all of the number sentences? (The equations on both sides of the equal sign are the same amount.)
- What do you need to find out in the number sentences with the boxes? (What number needs to go in the box in order to make the equations on both sides of the equal sign the same value.)
- In the first number sentence, do both sides of the equal sign show the same amount? Use your multiplication table or a different strategy to prove it. (Yes. Possible response: I drew 25 tiles into a rectangle that was 5 × 5.)
- In the second number sentence, how does the 15 help you solve the problem? (Possible response: I know that both sides of the equation must equal 15. I think 5 times what number equals 15 to find 3.)
- Are there any other numbers that will make this number sentence true? (no)
- How can the equation on one side of the equal sign help you solve the equation on the other side? (Both sides must be equal. After you solve one equation, you have to find the unknown number that will make the equation on the other side result in the same amount.)
Complete Open Number Sentences. Ask students to work with a partner to complete the number sentences displayed. Let students decide which strategies and tools they are going to use. When students are finished select a few students to share their strategy and solution to illustrate the use of a variety of tools and strategies.
Explain to students that when both sides of an equation are equal, we call this a true number sentence.
Display the following number sentences and ask:
- What has to happen for a number sentence to be true? (Both sides have to equal the same number.)
- Which of these are true number sentences? Show or tell how you know. (All of them are true except for the second one. 5 × 6 does not equal 45. I know because in the first sentence, 9 × 5 = 45, so its turn-around fact 5 × 9 equals 45.)
- Show or tell how you know that 4 × 10 = 8 × 5 is true. (Possible response: I know 4 × 10 = 40 by skip counting by 10s. I know eight +5 hops on a number line is 40. Both sides of the equation equal 40, so it is a true number sentence.)
Assign Questions 1–11 on the Multiplication with 5s and 10s pages in the Student Activity Book to student pairs.
Reasoning with tools. In Question 1, students can make a rectangle with 5 rows of 4 tiles and count the total number of tiles and write a number sentence for the total: 5 × 4 = 20. Then they can turn the rectangle so that it shows 4 rows of 5 tiles and write the corresponding number sentence: 4 × 5 = 20. A student who solves this problem quickly without manipulatives can be asked to prove that his or her answer is correct using the tiles or a drawing.
Similarly, to support students who cannot solve Question 9A, suggest that they draw a 7 × 5 rectangle, color in the first six rows, and write a number sentence for the colored rectangle. See Figure 2. Then, they can solve the problem by seeing that there is just one row of five left. Check students' understanding by having them match the numbers on both sides of the number sentence with the drawing of the rectangle. Other students may be able to solve the problem using just the numbers on both sides of the sentence. Challenge these students to explain their thinking as if they were a teacher explaining how to solve the problem using tiles or pictures.
When students have finished, have them justify their answers and explain their thinking by asking:
- How did you solve Question 4B? (Possible response: I remembered from the chart that any number times 0 is 0, so 5 times 0 is 0. I put a 5 in the box.)
- Are there any other numbers that will make the sentence true? What about 10? What about 100? (Any number times zero is zero.)
- How did you solve Question 8A? (Possible response: I knew that 8 × 5 = 40 so I thought of what you need to multiply by 10 to make 40. It's 4. 4 × 10 = 40.)
- Can you use the class chart or the multiplication table to help you? If so, how? (Possible response: I was looking for a number on the multiplication chart that is the same as 5 × 8. It's 40 and the pattern for the 10s helped me know that 40 would be 4 × 10.)
- Show how to use square-inch tiles or a drawing on grid paper to solve Question 9A, 7 × 5 = 6 × 5 + . (See Figure 2.)
- Show or tell a different way to solve this problem. (Possible response: I skip counted by 5s to solve 7 × 5 = 35. I knew the other side of the equation had to equal 35, too. 6 × 5 is one less five so I put a 5 in the box to show that I added one more. Both sides equal 35.)
- How did you rewrite the number sentence in Question 11 to make it true? Show or tell how you know the new number sentence is true. (One possible response: For 10 × 2 = 5 × 4, if I skip count by 10s two times I get 20 on one side of the equal sign. If I skip count by 5s four times, I get 20 on the other side of the equal sign. Both sides of the equation equal 20 so it is a true number sentence.)
- Did anyone make an error and then correct it? (Possible response: For 6B, I first wrote that 6 × 5 = 36 but then I looked on the class chart and I saw that it should end in 0 because it is 5 times an even number. So, I went back and skip counted by 5s until I got it right.)
Assign Check-In: Questions 12–15.
Use Check-In: Questions 12–15 and the Feedback Box on the Multiplication with 5s and 10s pages in the Student Activity Book and to assess students' abilities to solve multiplication problems by:
- finding and using strategies [E2, MPE2].
- using the multiplication properties of 0 and 1 [E3].
- using turn-around facts [E4].
- using patterns for the multiplication facts for the 5s and 10s [E5].
- breaking products into the sum of simpler products [E6].
These questions can also be used to assess students' abilities to show their work [MPE5].
The Floor Tiler game from Lesson 4 and the Workshop in Lesson 10 provide targeted practice.
Assign the Multiplication with 5s and 10s Homework section pages in the Student Activity Book for homework. Have copies of the Centimeter Grid Paper Master available for students who choose to use them.
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