Rectangle Perimeter 3
Task
Sadie computes the perimeter of a rectangle by adding the length, $l$, and width, $w$, and doubling this sum. Eric computes the perimeter of a rectangle by doubling the length, $l$, doubling the width, $w$, and adding the doubled amounts.
- Write an expression for Sadieâs way of calculating the perimeter. Write an expression for Ericâs way as well.
- Use both of the expressions to find the perimeter of a rectangle with length 30 and width 75.
- Explain why Sadie and Eric always get the same answer, no matter what the length and width of the rectangle are.
IM Commentary
The purpose of this task is to ask students to write expressions and to consider what it means for two expressions to be equivalent. This is another variation of the Rectangle Perimeter problem shown in "6EE Rectangle Perimeter 1" and "6EE Rectangle Perimeter 2." It goes one step further than "6EE Rectangle Perimeter 1" by asking students to explore the idea of equivalent expressions in addition to writing and using an expression.
Solution
-
Sadie first adds $l$ and $w$ to obtain $l + w$ and then doubles the sum. So her expression is $2(l + w)$.
Eric first doubles the length and the width, getting $2l$ and $2w$, respectively. He then adds these amounts to obtain the expression $2l + 2w$ for the perimeter of the rectangle.
-
With $l=30$ and $w=75$ using Sadieâs expression we have $2(l+w)=2(30+75)=210$.
Using Ericâs expression we have $2l + 2w = 2(30) + 2(75) = 210$.
- Sadie and Eric will always get the same answer, no matter the length and width of the rectangle, because they are both using correct methods for computing the perimeter of a rectangle. Sadieâs expression is equivalent to Ericâs expression by the distributive property: $2(l + w) = 2l + 2w$.
Rectangle Perimeter 3
Sadie computes the perimeter of a rectangle by adding the length, $l$, and width, $w$, and doubling this sum. Eric computes the perimeter of a rectangle by doubling the length, $l$, doubling the width, $w$, and adding the doubled amounts.
- Write an expression for Sadieâs way of calculating the perimeter. Write an expression for Ericâs way as well.
- Use both of the expressions to find the perimeter of a rectangle with length 30 and width 75.
- Explain why Sadie and Eric always get the same answer, no matter what the length and width of the rectangle are.