Bell Work: The dimensions of a wallet-size photo are about half the dimensions of a 5 inch by 7 inch photo. The area of a wallet-size photo is about what fraction of the area of a 5 by 7 photo?

Answer: ¼

Lesson 92: Areas of Rectangles with Variable Dimensions, Products of Binomials

Suppose the floor of a square closet with sides measuring 3 ft. 6 in. will be covered with one-foot square tiles. We can sketch a plan to show a way the tiles can be arranged and use the sketch to help us fin the area of the floor.

We see that one part of the area is covered with full tiles, two parts are covered with half tiles, and one part is covered with a quarter tile. We count the tiles in each part. 9 full + 3 half + 3 half + 1 quarter

We can combine the two groups of half tiles. 9 full + 6 half + 1 quarter Six half tiles equal three full tiles. We add to find that our sketch shows 12 full tiles and one quarter tile, meaning the area of the floor is 12 ¼ feet.

Sketching can also help us visualize the areas of rectangles with dimensions expressed as variables.

Example: Suppose a rectangular rug measures 2 ½ feet by 4 ¼ feet. We can represent its area by writing: (2 + ½)(4 + ¼)  Sketch the rectangle on a grid to depict the product  What is the total area of the rug?

Answer: 10 5/8 feet squared

Example: A rectangle is x + 3 units long and x + 2 units wide. What is the area of the rectangle?

Answer: x + 5x + 6 2

Products of binomials: We can find the area of the rectangle in the last example by multiplying the binomials. In this part of the lesson we will practice techniques for multiplying binomials.

The product of two binomials has four terms, some of which may be like terms. Each term of one binomial is multiplied by each term of the other binomial.

We will use an arithmetic model to show the multiplication of two binomials. Breaking out the partial products of the problem on the left, we show that multiplying 43 x 42 is equivalent to multiplying (40 + 3) x (40 + 2).

43 x 42 = 1806 (40 + 3) x (40 + 2) = 1806

Example: Expand: (x + 3)(x + 2)

Answer: x + 5x + 6 2

We must pay attention to the signs when multiplying binomials. Multiplying binomials involves multiplying terms. Every term has a sign. Like signs result in a positive product. Unlike signs result in a negative product.

The acronym FOIL can help us remember the four multiplications that occur when multiplying two binomials. Product of binomials = products of First + Outside + Inside + Last

Example: Expand: (x – 3)(x – 4)

Answer: x – 7x

Practice: Expand using any method: (x + 8)(x – 3)

Answer: x + 5x – 24 2

Practice: Expand using any method: (x – 5)(x + 2)

Answer: x – 3x – 10 2

HW: Lesson 92 #1-25