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Educational adventures for curious kids

Math 7 · Unit 4, Lesson 3: Expanding · Tile 1 of 3

The Area Picture

Why the number goes to every term

Why the number goes to every term

Think of 3(x + 4) as a rectangle that is 3 tall and x + 4 wide. Cut it into two smaller rectangles: one is 3 by x, and the other is 3 by 4. Their areas are 3x and 12, so the whole area is 3x + 12. Every piece of the width gets multiplied by the height.

The area model for 3 times the quantity x plus 4A rectangle 3 tall. Its width is split into x and 4. The left part has area 3x, the right part has area 12.x3x4123
3(x + 4) = 3x + 12. The height goes to every piece of the width.

Words we use

Elle (math-speak)Finn (plain English)
ExpandTo write it without the parentheses by multiplying it out. Expand means to spread out, and the expression spreads out and gets longer.
Area modelA rectangle picture of multiplying. The sides are the factors, and the area inside is the product.

Elle and Finn

ElleMath-speak

By the distributive property, a(b + c) = ab + ac.

FinnPlain English

The number outside multiplies everything inside. Every term gets a turn.

Watch it work

3(x + 4)3 × x + 3 × 43x + 12
Area model for 3 times x plus 4A rectangle 3 tall, width split into x and 4, areas 3x and 12.x3x4123
5(2x + 3)5 × 2x + 5 × 310x + 15
Area model for 5 times 2x plus 3A rectangle 5 tall, width split into 2x and 3, areas 10x and 15.2x10x3155
2(a + 3b + 5)2 × a + 2 × 3b + 2 × 52a + 6b + 10
Area model for 2 times a plus 3b plus 5A rectangle 2 tall, width split into a, 3b, and 5, areas 2a, 6b, and 10.a2a3b6b5102
Three pieces of width, three small rectangles.

Let me try a few

NC standard: NC.7.EE.1.

Where to go next

Let me try a few6 problems on the area picture. Next: Tile 2, Negatives Get a Turn TooMinus signs inside and outside All tilesAll 3 parts of this lesson. Tell us what you thought about this tileA few quick questions about just this tile. It takes a few minutes, and opens in a new tab so you keep your place.