Lesson 9
The Distributive Property, Part 1
Let's use the distributive property to make calculating easier.
9.1: Number Talk: Ways to Multiply
Find each product mentally.
\(5 \boldcdot 102\)
\(5 \boldcdot 98\)
\(5 \boldcdot 999\)
9.2: Ways to Represent Area of a Rectangle
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Select all the expressions that represent the area of the large, outer rectangle in figure A. Explain your reasoning.
- \(6 + 3 + 2\)
- \(6 \boldcdot 3 + 6 \boldcdot 2\)
- \(6 \boldcdot 3 + 2\)
- \(6 \boldcdot 5\)
- \(6 (3+2)\)
- \(6 \boldcdot 3 \boldcdot 2\)
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Select all the expressions that represent the area of the shaded rectangle on the left side of figure B. Explain your reasoning.
- \(4 \boldcdot 7 + 4 \boldcdot 2\)
- \(4 \boldcdot 7 \boldcdot 2\)
- \(4 \boldcdot 5\)
- \(4 \boldcdot 7 - 4 \boldcdot 2\)
- \(4(7-2)\)
- \(4(7+2)\)
- \(4 \boldcdot 2 - 4 \boldcdot 7\)
9.3: Distributive Practice
Complete the table. If you get stuck, skip an entry and come back to it, or consider drawing a diagram of two rectangles that share a side.
| column 1 | column 2 | column 3 | column 4 | value |
|---|---|---|---|---|
| \(5 \boldcdot 98\) | \(5 (100-2)\) | \(5 \boldcdot 100 - 5 \boldcdot 2\) | \(500 - 10\) | 490 |
| \(33 \boldcdot 12\) | \(33 (10 + 2)\) | |||
| \(3 \boldcdot 10 - 3 \boldcdot 4\) | \(30-12\) | |||
| \(100 (0.04 + 0.06)\) | ||||
| \(8 \boldcdot \frac 1 2 + 8 \boldcdot \frac 1 4\) | ||||
| \(9 + 12\) | ||||
| \(24 - 16\) |
- Use the distributive property to write two expressions that equal 360. (There are many correct ways to do this.)
- Is it possible to write an expression like \(a(b+c)\) that equals 360 where \(a\) is a fraction? Either write such an expression, or explain why it is impossible.
- Is it possible to write an expression like \(a(b-c)\) that equals 360? Either write such an expression, or explain why it is impossible.
- How many ways do you think there are to make 360 using the distributive property?
Summary
A term is a single number or variable, or variables and numbers multiplied together. Some examples of terms are 10, \(8x\), \(ab\), and \(7yz\).
When we need to do mental calculations, we often come up with ways to make the calculation easier to do mentally.
Suppose we are grocery shopping and need to know how much it will cost to buy 5 cans of beans at 79 cents a can. We may calculate mentally in this way:
\(5\boldcdot {79}\)
\(5\boldcdot {70}+5\boldcdot {9}\)
\(350+45\)
\(395\)
In general, when we multiply two terms (or factors), we can break up one of the factors into parts, multiply each part by the other factor, and then add the products. The result will be the same as the product of the two original factors. When we break up one of the factors and multiply the parts we are using the distributive property.
The distributive property also works with subtraction. Here is another way to find \(5\boldcdot 79\):
\(5\boldcdot 79\)
\(5\boldcdot {(80-1)}\)
\(400-5\)
\(395\)
Glossary Entries
- equivalent expressions
Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.
For example, \(3x+4x\) is equivalent to \(5x+2x\). No matter what value we use for \(x\), these expressions are always equal. When \(x\) is 3, both expressions equal 21. When \(x\) is 10, both expressions equal 70.
- term
A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression \(5x + 18\) has two terms. The first term is \(5x\) and the second term is 18.