Illustrative Mathematics | v.360 Curriculum

7.1

Warm-up

For each multiplication expression, choose the best estimate of its value. Be prepared to explain your reasoning.

- 1.40
- 14
- 140
- 5.6
- 56
- 560
- 1.66
- 16.6
- 166
- 6.5
- 65
- 650

7.2

Activity

7.3

Activity

7.4

Activity

Label the area diagram to represent and to find that product.
1. Decompose each factor by place value and write the ones and tenths in the blank boxes.
2. Label Regions A, B, C, and D with their areas. Show your reasoning.
3. Find the product that the area diagram represents. Show your reasoning.
Here are two ways to calculate . Each number with a box gives the area of one or more regions in the area diagram.

Two vertical calculations of 2 point 5 times 1 point 2. Calculation A, 7 rows. First row: 2 point 5. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 1 0. Fourth row: 0 point 4. Fifth row: 0 point 5. Sixth row: plus 2. Horizontal line. Seventh row: 3 point 0 0. Calculation B, 5 rows. First row: 2 point 5. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 5. Fourth row: plus 2 point 5. Horizontal line. Fifth row: 3 point 0.
1. In the boxes next to each number, write the letter(s) of the corresponding region(s).
2. In Calculation B, which two numbers are being multiplied to give 0.5?
  Which numbers are being multiplied to give 2.5?

Student Lesson Summary

To find the product of two numbers, such as , we can think of finding the area of a rectangle with those numbers, 3.4 units and 1.2 units, as side lengths.

First, we draw a rectangle and partition each side length by place value, into ones and tenths:

Then, we decompose the rectangle into four smaller sub-rectangles and find their areas.

Area diagram. A rectangle partitioned into 4 rectangles, A, B, C, D.

A:

B:

C:

D:

Each multiplication gives a partial product that represents the area of a sub-rectangle. The sum of the four partial products gives the area of the entire rectangle, 4.08 square units.

We can show the same partial-product calculations vertically. Here are two ways:

Two vertical calculations of 3 point 4 times 1 point 2.

The calculation on the left shows four partial products, one for the area of each sub-rectangle.

The calculation on the right shows two partial products:

0.68 is the value of , or the combined area of A and B.
3.4 is the value of , or the combined area of C and D.

In both calculations, adding the partial products gives a total of 4.08, which is the product of and the area (in square units) of the entire rectangle.