Unit 3, Lesson 16
Methods for Multiplying Decimals
Accelerated 6
16.2
Activity
Elena and Noah used different methods to compute . Both calculations were correct.
Elena's method. 2 point 4 times 10 equals 24. 1 point 3 times 10 equals 13. 24 times 13 equals 312. 312 divided by 100 equals 3 point 1 2. Noah's method. 2 point 4 equals the fraction 24 over 10. 1 point 3 equals the fraction 13 over 10. The fraction 24 over 10 times the fraction 13 over 10 equals the fraction 312 over 100. The fraction 312 over 100 equals 3 point 1 2.
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Analyze the two methods, then discuss these questions with your partner.
- Which method makes more sense to you? Why?
- What might Elena do to compute ?
- What might Noah do to compute ?
- Will the two methods result in the same value?
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Compute each product using the equation and what you know about fractions, decimals, and place value. Explain or show your reasoning.
Student Lesson Summary
Here are three other ways to calculate a product of two decimals, such as .
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First, we can multiply each decimal by the same power of 10 to obtain whole-number factors.
Because we multiplied both 0.04 and 0.07 by 100 to get 4 and 7, the product 28 is times the original product, so we need to divide 28 by 10,000.
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Second, we can write each decimal as a fraction and multiply them.
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Third, we can use an area diagram. The product can be thought of as the area of a rectangle with side lengths of 0.04 unit and 0.07 unit.
In this diagram, each small square is 0.01 unit by 0.01 unit. The area of each square, in square units, is therefore , which is .
Because the rectangle is composed of 28 small squares, the area of the rectangle, in square units, must be:
All three calculations show that .
To find the product of two two-digit 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. 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:
- 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.
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