Unit 6, Lesson 10
The Distributive Property, Part 2
Grade 6
10.2
Activity
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Here are two rectangles. The length and width of one rectangle are 8 and 5 units. The width of the other rectangle is 5 units, but its length is unknown so we labeled it .
Write an expression for the sum of the areas of the two rectangles.
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The two rectangles can be composed into one larger rectangle, as shown.
What are the length and width of the new, larger rectangle?
- Write an expression for the total area of the new, larger rectangle as the product of its width and its length.
10.3
Activity
For each rectangle, write an expression for the width, an expression for the length, and two expressions for the total area. Record them in the table. Check your expressions in each row with your group and discuss any disagreements.
Rectangle A is partioned by a vertical line segment into two smaller rectangles. The vertical side is labeled 3 and the top horizontal side lengths are labeled "a" and 5.
Rectangle B is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled one third and the top horizontal side lengths are labeled 6 and x.
Rectangle C is partitioned by 2 vertical line segments into three equally sized rectangles. The vertical side is labeled r and the top horizontal side lengths are each labeled 1.
Rectangle D is partitioned by 3 vertical line segments into 4 equally sized rectangles. The vertical side is labeled 6, and the top horizontal side lengths are each labeled 4.
Rectangle E is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled m and the top horizontal side lengths are labeled 6 and 8.
Rectangle F is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled 5 and the top horizontal side lengths are labeled 3 x and 8.
| rectangle | width | length | area as a product of width times length |
area as a sum of the areas of the smaller rectangles |
|---|---|---|---|---|
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A |
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B |
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C |
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D |
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E |
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F |
Student Lesson Summary
The distributive property can also help us write equivalent expressions with variables. We can use a diagram to help us understand this idea.
Here is a rectangle composed of two smaller Rectangles A and B.
A rectangle is partitioned by a vertical line segment creating two smaller rectangles, A and B. Rectangle A has a vertical side length of 3 and horizontal side length of 2. Rectangle B has a horizontal side length of x.
Based on the drawing, we can make several observations about the area of the large rectangle:
- One side length of the large rectangle is 3 and the other is , so its area is .
- Since the large rectangle can be decomposed into two smaller rectangles, A and B, with no overlap, the area of the large rectangle is also the sum of the areas of rectangles A and B: or .
- Since both expressions represent the area of the large rectangle, they are equivalent to each other. is equivalent to .
We can see that multiplying 3 by the sum is equivalent to multiplying 3 by 2 and then 3 by and adding the two products. This relationship is an example of the distributive property.
When working with expressions of all kinds, it helps to be able to talk about the parts. In an expression like , we call the 6 and “terms.”
A term is a part of an expression. A term can be a single number, a single variable, or a product of numbers and variables. Some examples of terms are 10, , , and .
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.
- 7, , and are examples of terms.
- The expression has 2 terms: and .