Valerie and Seth are measuring the outside walls of a house using two different instruments. They want to find the house’s perimeter. Valerie measured the north and east sides while Seth measured the west and south sides. Their results are listed in the table.

Question

Valerie's Measurements Seth's Measurements
4 walls at
feet each 1 wall at
feet
3 walls at
feet each 2 walls at
feet each
1 wall at
feet 1 wall at
feet

Complete the steps below to help Valerie and Seth find the perimeter of the house.

Part A
Use decimal numbers to write an expression that represents the length of the walls that Valerie measured.

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Part B
Now use decimal numbers to write an expression that represents the length of the walls that Seth measured.

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Part C
Write an expression for the perimeter of the house. Use the work you completed in parts A and B to guide you.

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The total length of Valerie’s walls is (2.5 × 4) + (21.25 × 3) + 32.

The total length of Seth’s walls is 3.5 + (22.75 × 2) + 58.

To find the expression for the perimeter of the house, add Valerie’s expression to Seth’s:
(2.5 × 4) + (21.25 × 3) + 32 + 3.5 + (22.75 × 2) + 58.

Part D
Simplify the expression from part C to find the perimeter of the house in feet.

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(2.5 × 4) + (21.25 × 3) + 32 + 3.5 + (22.75 × 2) + 58

= 10 + 63.75 + 32 + 3.5 + 45.5 + 58

= 212.75

The perimeter of the house is 212.75 feet.

Part E
Suppose that you evaluate the expression for Valerie’s walls and the expression for Seth’s walls separately. Then you add the two expressions together. Will you get the same answer that you arrived at in part d? Why or why not? Show your work to prove your case.

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Self-Evaluation
How did you do? Rate your work on a scale of 1 to 5, with 5 as the highest score. Then write a brief evaluation of your work below. Note what you learned and what challenged you.

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Answer 1

Let’s walk through the steps to complete each part of the problem regarding Valerie's and Seth's measurements of the house's perimeter.

Part A

For Valerie's measurements:

• 4 walls at 2.5 feet each: \( 4 \times 2.5 \)
• 3 walls at 21.25 feet each: \( 3 \times 21.25 \)
• 1 wall at 32 feet: \( 32 \)

So, the expression for the total length of the walls that Valerie measured is: \[ (2.5 \times 4) + (21.25 \times 3) + 32 \]

Part B

For Seth's measurements:

• 1 wall at 3.5 feet: \( 3.5 \)
• 2 walls at 22.75 feet each: \( 2 \times 22.75 \)
• 1 wall at 58 feet: \( 58 \)

So, the expression for the total length of the walls that Seth measured is: \[ 3.5 + (22.75 \times 2) + 58 \]

Part C

To find the expression for the perimeter of the house, we add Valerie's and Seth's expressions together: \[ (2.5 \times 4) + (21.25 \times 3) + 32 + 3.5 + (22.75 \times 2) + 58 \]

Part D

Now, we simplify the expression from part C step-by-step:

• Calculate each component:
  • \( 2.5 \times 4 = 10 \)
  • \( 21.25 \times 3 = 63.75 \)
  • \( (22.75 \times 2) = 45.5 \)

Now, substitute these values into the expression: \[ 10 + 63.75 + 32 + 3.5 + 45.5 + 58 \]

Add them:

• \( 10 + 63.75 = 73.75 \)
• \( 73.75 + 32 = 105.75 \)
• \( 105.75 + 3.5 = 109.25 \)
• \( 109.25 + 45.5 = 154.75 \)
• \( 154.75 + 58 = 212.75 \)

So, the total perimeter of the house is: \[ \text{Perimeter} = 212.75 \text{ feet} \]

Part E

If you evaluate Valerie’s walls and Seth’s walls separately and then add the two expressions together, you will indeed get the same answer as in part D. Here is the breakdown of the calculations:

1. First, evaluate Valerie’s expression:

   • \( 10 + 63.75 + 32 = 105.75 \)

2. Next, evaluate Seth’s expression:

   • \( 3.5 + 45.5 + 58 = 107 \)

3. Now, add these two results together:

   • \( 105.75 + 107 = 212.75 \)

Thus, adding the individual results gives the same total perimeter: \[ \text{Perimeter} = 212.75 \text{ feet} \]

The answer is the same because addition is associative and commutative, meaning that it does not matter in which order you add the lengths; you will end up with the same total perimeter.

Self-Evaluation

On a scale of 1 to 5, I would rate my work a 5. I clearly understood how to break down the measurements from both Valerie and Seth systematically and combined them accurately. The challenge was in ensuring that I stayed organized while maintaining accuracy in the calculations. I learned the importance of stepwise evaluation to avoid confusion in multi-part problems like this.