Distributive Property with area models

Recall the distributive property of multiplication over addition . . . symbolically: a × (b + c) = a × b + a × c and pictorially (rectangular array area model): b c a a × b a × c

using your mental math skills . . . An example: 6 x 13 using your mental math skills . . . symbolically: 6 × (10 + 3) = 6 × 10 + 6 × 3 and pictorially (rectangular array area model): 10 3 6 6 × 10 6 × 3

Factoring out the gcf with distributive property

3x + 12 Find the gcf of 3x and 12 Divide each term by the gcf Re-write using distributive property Answer: 3 (x + 4)

Examples 12y – 36 9z + 81 20j – 32 6h + 15k

Example answers 12 (x – 3) 9 (z + 9) 4 (5j - 8) 3 (2h + 5k)

What about 12 x 23? Mental math skills? (10+2)(20+3) = 10×20 + 10×3 + 2×20 + 2×3 20 3 200 30 40 + 6 276 10 10 × 20 10 × 3 2 2 × 20 2 × 3

And now for multiplying binomials (a+b)×(c+d) = a×(c+d) + b×(c+d) = a×c + a×d + b×c + b×d c d a a × c a × d b b × c b × d

Because this product is composed of the We note that the product of the two binomials has four terms – each of these is a partial product. We multiply each term of the first binomial by each term of the second binomial to get the four partial products. F + O + I + L ( a + b )( c + d ) = ac + ad + bc + bd Product of the FIRST terms of the binomials Product of the OUTSIDE terms of the binomials Product of the INSIDE terms of the binomials Product of the LAST terms of the binomials Because this product is composed of the First, Outside, Inside, and Last terms, this pattern is often referred to as FOIL method of multiplying two binomials. Note that each of these four partial products represents the area of one of the four rectangles making up the large rectangle.