Strategies for 6 × 3. In this part of the lesson, students practice doubling and partitioning strategies for the facts that need to be added to the My Multiplication Table students started to complete in Lessons 3 and 4. Display and direct students' attention to the first two Break-Apart Products pages in the Student Activity Book. Tell them that you are going to practice some strategies with rectangles that will help them solve more multiplication facts. Students will need a red and a yellow crayon or colored pencil. Complete Questions 1–4 together as a class. These questions explore different ways to think about solving 6 × 3.
Question 1 asks students to use their own strategies and write a number sentence on a 6 × 3 rectangle.
Question 2 asks students to “break the product” in half, that is, to divide the rectangle in half and think of 6 × 3 as double 3 × 3, or 9 + 9 = 18. Question 3 makes use of known facts. Since many students know the multiplication facts for the fives or can derive them by skip counting, Question 3 asks students to break apart the 6 × 3 rectangle into 5 × 3 and 1 × 3 rectangles. In this way, students have a visual image for thinking of 6 × 3 as 5 rows of 3 and 3 more, or 5 × 3 + 1 × 3 = 15 + 3 = 18.
Figure 1 shows how a 6 × 3 rectangle can be used three different ways to find the product. The following questions provide a visual image of breaking products into smaller or easier facts that students already know.
As you work on these problems together, ask students to match the rectangles to the number sentences.
- How does the rectangle match the problem of 6 × 3? Show me on the rectangle. (There are 6 rows and 3 squares in each row.)
- How can you use the rectangle in Question 1 to find the answer for 6 × 3? (Possible response: I skip counted by 3s.)
- How can you use the rectangle in Question 2 to find the answer for 6 × 3? (The red rectangle and the yellow rectangle are each half of the rectangle. They each have 9 small squares, so you can just double 9 to get 18.)
- How do the number sentences in Question 2C match the rectangle? (In the first sentence, the six was divided into two threes and the rectangle was divided into two rectangles with 3 rows. So, then the two rectangles had 9 small squares each, so you can add the 9 and 9 to get 18.)
- How is the rectangle in Question 3 divided? (into 5 rows and one row)
- Why can it be divided that way? (Breaking the rectangle into 5 rows and 1 row is like breaking six into 5 and 1.)
- How do the number sentences match the rectangle? (The red rectangle is 5 × 3 and the yellow is 5 × 1. Then you add 15 and 3 and you get 18.)
- How are the rectangles in Questions 2 and 3 alike and different (Question 4)? (They all have 6 rows and 3 in each row. They all have 18 squares. The rectangle in Question 2 was divided in half and the rectangle in Question 3 was divided into 5 rows and 1 row.)
Strategies for 4 × 7. Assign Questions 5–9 to student pairs. Help students identify different partitions of a 4 × 7 rectangle by repeating prompts similar to those used for the 6 × 3 rectangle.
Strategies for 8 × 4 and 7 × 5. Assign Questions 10–12 to develop strategies for 8 × 4 and Check-In: Question 13 to develop strategies for 7 × 5.
After students have completed Questions 1–12 compare break-apart strategies using the following prompts.
- Which ways to break apart the products make it easier to find the product or remember the fact? (Possible response: Breaking a rectangle in half means that you can use doubling or multiplying by 2. Breaking the rectangle into 5 rows and whatever is left means that you can use 5s.)
- Which ones can you use halving and doubling? (when one of the factors is even like 6 × 3, 4 × 7, or 8 × 4)
Use Check-In: Question 13 on the Break-Apart Products page in the Student Activity Book to assess students' progress toward the following Expectations:
- Represent multiplication problems with rectangular arrays and number sentences [E1].
- Use strategies to solve multiplication problems [E2].
- Break apart products into the sum of simpler products to solve multiplication problems.
The workshop in Lesson 10 provides targeted practice with representing and using strategies to solve the multiplication facts.
Number sentences with both multiplication and addition follow the Order of Operations. The multiplication operations are completed before the addition operations.
For example: 8 × 4 = 5 × 4 + 3 × 4
= 20 + 12
= 32
