## 11.1: The Shaded Region

A rectangle with dimensions 6 cm and $w$ cm is partitioned into two smaller rectangles.

Explain why each of these expressions represents the area, in cm^{2}, of the shaded portion.

- $6w-24$

- $6(w-4)$

Let's practice writing equivalent expressions by using the distributive property.

A rectangle with dimensions 6 cm and $w$ cm is partitioned into two smaller rectangles.

Explain why each of these expressions represents the area, in cm^{2}, of the shaded portion.

- $6w-24$

- $6(w-4)$

Match each expression in column 1 to an equivalent expression in column 2. If you get stuck, consider drawing a diagram.

Column 1

- $a(1+2+3)$
- $2(12-4)$
- $12a+3b$
- $\frac23(15a-18)$
- $6a+10b$
- $0.4(5-2.5a)$
- $2a+3a$

Column 2

- $3(4a+b)$
- $12 \boldcdot 2 - 4 \boldcdot 2$
- $2(3a+5b)$
- $(2+3)a$
- $a+2a+3a$
- $10a-12$
- $2-a$

The distributive property can be used to write equivalent expressions. In each row, use the distributive property to write an equivalent expression. If you get stuck, draw a diagram.

product | sum or difference | |
---|---|---|

row 1 | $3(3+x)$ | |

row 2 | $4x-20$ | |

row 3 | $(9-5)x$ | |

row 4 | $4x+7x$ | |

row 5 | $3(2x+1)$ | |

row 6 | $10x-5$ | |

row 7 | $x+2x+3x$ | |

row 8 | $\frac12 (x-6)$ | |

row 9 | $y(3x+4z)$ | |

row 10 | $2xyz-3yz+4xz$ |

- Suppose that $x$ is 3. Find the area of each square in the diagram. Then find the area of the large rectangle.
- Find the side lengths of the large rectangle assuming that $x$ is 3. Find the area of the large rectangle by multiplying the length times the width. Check that this is the same area you found before.
- Now suppose that we do not know the value of $x$. Write an expression for the side lengths of the large rectangle that involves $x$.

The distributive property can be used to write a sum as a product, or write a product as a sum. You can always draw a partitioned rectangle to help reason about it, but with enough practice, you should be able to apply the distributive property without making a drawing.

Here are some examples of expressions that are equivalent due to the distributive property.

$$\begin {align} 9+18&=9(1+2)\\[10pt] 2(3x+4)&=6x+8\\[10pt] 2n+3n+n&=n(2+3+1)\\[10pt] 11b-99a&=11(b-9a)\\[10pt] k(c+d-e)&=kc+kd-ke\\ \end {align}$$