Look at the side length of each square and use the area formula to formulate an expression.
B
b
Subtracting is the mathematical operation that comes from cutting something out. Use the expressions from Part A to develop a subtraction equation.
C
c
The process of moving the bottom rectangle changes the lengths of the sides, but not the area.
D
d
Think about the answer to Part B and how that fits into the special product category of this chapter.
A
a
Large Square: a^2 Small Square: b^2
B
b
a^2-b^2
C
c
Length: a+b Width: a-b Area: a^2-b^2
D
d
We have verified that (a+b)(a-b)=a^2-b^2 which is the Product of a Sum and a Difference.
Practice makes perfect
a
To find the area of a square, we need to apply the area formula, A=s^2, with the lengths of the sides from the image.
Side Length
Square's Area
a
a^2
b
b^2
b
When we cut something out of a shape, the mathematical operation is subtraction. In this case, we are subtracting the smaller square from the larger square. We can use our formulas for the squares' areas from Part A to show the math.
A_(Large) - A_(Small)=a^2-b^2
c
When we rearrange pieces we change the length and width the figure. Let's look at this in more detail to see what's happening to the sides to make the new rectangle.
Now, if we cut along the side indicated and place it to the right next to it's matching color we end up with the following.
From the picture, we can see that the length of this newly constructed rectangle is a+b and its width is a-b. To find the area we can multiply the length and width, then simplify.
To find the area of the new rectangle is a direct application of the Product of Sum and Difference.
Area:(a+b)(a-b) = a^2-b^2
d
The pattern of movement of rectangles demonstrates the rule for Product of Sum and Difference. First, it subtracts one square from another, demonstrating that the area of the L-shape is a^2-b^2. Then it rearranges the area, without changing the area, to show that the area is (a+b)(a-b).
