Lesson 4

Multiplication and Rectangles

Est. Class Sessions: 3

Developing the Lesson

Part 2. Exploring Turn-Around Facts

Show Turn-Around Facts with Rectangles. For Part 2, students need square-inch tiles and their partially completed tables on the My Multiplication Table page from the Student Activity Book from Lesson 3. You will need a display of the Centimeter Grid Paper Master.

In working with the tiles, students will find that multiplication is commutative—although this term is not used at this point. Students use the commutative property to get new facts from old by “turning around” the facts they already found.

Display a rectangle with 3 rows of 7 tiles in each row on the grid paper.

Ask:

X
  • How many rows are in this rectangle? (3 rows)
  • How many tiles are in each row? (7)
  • How many tiles are in my rectangle? How do you know? (21. Possible response: I can count by 3s.)
  • Write a number sentence for this rectangle. (3 × 7 = 21)
  • What is the answer to this problem: 7 × 3 = ? How do you know? Use tiles to justify your answer. (7 × 3 = 21 because all you have to do is turn the rectangle around and it is the same. The number of tiles doesn't change.)

Draw a 3 × 7 rectangle on the grid paper, cut it out and show that if you “turn it around” and place it on top of the other, the rectangles are the same size and shape. See Figure 4.

Tell students we call 3 × 7 and 7 × 3 turn-around facts. Write 3 × 7 = 7 × 3 and ask:

X
  • Is 3 × 7 = 7 × 3 a true statement? How do you know? (Yes, because both sides of the equal sign are 21.)
  • What does the equal sign tell us in this multiplication sentence? (It tells us that 3 × 7 equals the same number as 7 × 3.)
  • What is different about the number sentences on both sides of the equal sign? (The order of the numbers is different, but that is all.)

Ask a student volunteer to display a rectangle with tiles and write its multiplication sentence. Then, have the student turn its factors around to make its turn-around fact and show the corresponding rectangle. Have students repeat the same process at their desks and then share their sets of turn-around facts.

X

Commutative Operations. Multiplication, like addition, is a commutative operation. This means that the order of terms can be changed without changing the answer. For example, 5 × 4 = 20 and 4 × 5 = 20. Note, however, that division and subtraction are not commutative since, for example, 25 ÷ 5 ≠ 5 ÷ 25.

Find Turn-Around Facts with Rectangles. Direct students' attention to their My Multiplication Table pages. Help students locate 7 × 3 = 21, in the column labeled 3 on a display of the My Multiplication Table page from the Student Activity Book and enter its turn-around fact, 3 × 7 = 21, in the row labeled 3. See Figure 5. The turn-around facts for any column in the table are in the corresponding row—for example, row 3 and column 3 have the same entries.

Assign student pairs the task of filling in all the turnaround facts for the facts that are already recorded on their tables.

Ask:

X
  • Are there any patterns to help us remember turn-around facts? (In any multiplication sentence, you can turn the factors around to get the same product. I can find 3 × 4 on the chart and 4 × 3 on the chart and the product is 12 for both.)
Student Guide:
Student Activity Book:
Master:
Rectangles made with 12 tiles
X
+
Rectangles for turn around facts 3 × 7 = 21 and 7 × 3 = 21
X
+
Adding the turn-around facts for the 1s, 2s, 3s, 5s, and 10s
X
+