Unit 5, Lesson 7
Using Diagrams to Represent Multiplication
Grade 6
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Here are three ways of finding the area of a rectangle that is 24 units by 13 units.
Discuss with your partner:
How are the diagrams the same?
How are the diagrams different?
If you were to find the area of a rectangle that is 37 units by 19 units, which of the three ways of decomposing the rectangle would you use? Why?
Here are two ways to calculate 24 times 13.
Two vertical calculations of 24 times 13. Calculation A, 7 rows. First row: 24. Second row: multiplication symbol, 13. Horizontal line. Third row: 12. Fourth row: 60. Fifth row: 40. Sixth row: plus 200. Horizontal line. Seventh row: 312. Calculation B, 5 rows. First row: 24. Second row: multiplication symbol, 13. Horizontal line. Third row: 72. Fourth row: plus 240. Horizontal line. Fifth row: 312.
Discuss with your partner:
In Calculation A, where does each partial product—the 12, 60, 40, and 200—come from?
In Calculation B, where do 72 and 240 come from?
Which diagram in the first question corresponds to Calculation A? Which one corresponds to Calculation B? How do you know?
Find the product of 18 and 14 in two ways:
Calculate numerically.
Find the area, in square units, of this 18-by-14 rectangle. Show your reasoning.
Here is an area diagram that represents
A rectangle partitioned vertically and horizontally into 4 rectangles. Top left rectangle, vertical side, 1, horizontal side, 2. Top right rectangle, vertical side, 1, horizontal side, 0 point 4. Bottom left rectangle, vertical side, 0 point 3, horizontal side, 2. Bottom right rectangle, vertical side, 0 point 3, horizontal side, 0 point 4.
Find the region that represents
Label the other regions with their areas.
Find the value of
Here are two ways of calculating
Two vertical calculations of 2 point 4 times 1 point 3. Calculation A, 7 rows. First row: 2 point 4. Second row: multiplication symbol, 1 point 3. Horizontal line. Third row: 0 point 1 2. Fourth row: 0 point 6. Fifth row: 0 point 4. Sixth row: plus 2. Horizontal line. Seventh row: 3 point 1 2. Calculation B, 5 rows. First row: 2 point 4. Second row: multiplication symbol, 1 point 3. Horizontal line. Third row: 0 point 7 2. Fourth row: plus 2 point 4. Horizontal line. Fifth row: 3 point 1 2.
Analyze the calculations and discuss these questions with a partner:
In Calculation A, where do the 0.12 and other partial products come from?
In Calculation B, where do the 0.72 and 2.4 come from?
In each calculation, why are the numbers below the horizontal line aligned vertically the way they are?
Find the value of
Label the area diagram to represent
Here are two ways to calculate
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.
In the boxes next to each number, write the letter(s) of the corresponding region(s).
In Calculation B, which two numbers are being multiplied to give 0.5?
Which numbers are being multiplied to give 2.5?
To find the product of two numbers, such as
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. D, vertical side, 1, horizontal side, 3. C, vertical side, 1, horizontal side, 0 point 4. B, vertical side, 0 point 2, horizontal side, 3. A, vertical side, 0 point 2, horizontal side, 0 point 4.
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. First calculation, 7 rows. First row: 3 point 4. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 0 8, A. Fourth row: 0 point 6, B. Fifth row: 0 point 4, C. Sixth row: plus 3, D. Horizontal line. Seventh row: 4 point 0 8. Second calculation, 5 rows. First row: 3 point 4. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 6 8, A plus B. Fourth row: plus 3 point 4, C plus D. Horizontal line. Fifth row: 4 point 0 8.
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:
In both calculations, adding the partial products gives a total of 4.08, which is the product of