Section C Practice Problems

Problem 1

What is the area of this figure in square units? Explain or show your reasoning.

A figure with same size square units.

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Solution

50 square units. Sample response: I can cut the figure into a 2-by-10 rectangle that has an area of 20 square units, and a 5-by-6 rectangle that has an area of 30 square units. The total area is 20+3020 + 30 or 50 square units.

Problem 2

Find the area of this figure. Explain or show your reasoning.

<p><span>6-sided shape. Straight sides. All side lengths meet at right angles. </span><span>Bottom, 5 ft. Right side rises 6 ft, then goes right 5 ft, up 4 ft. Top side length, 10 ft. Left side length, 10 ft. </span><span> </span></p>

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Solution

70 square feet. Sample response: I can cut the figure into a 5 foot-by-6 foot rectangle and a 10 foot-by-4 foot rectangle. The area is 30+4030+40 or 70 square feet.

Problem 3

Find the area of this figure. Explain or show your reasoning.

<span>6-sided shape. Straight sides. All side lengths meet at right angles. Bottom, question mark feet. Right side rises question mark feet then goes left 5 feet, then down 2 feet and left 3 feet. Left side 8 feet.</span><span> </span>

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Solution

74 square feet. Sample response: The side length on the right is 10 feet because it is 8 feet and 2 feet. The shape can be cut into an 8 foot-by-3 foot rectangle that has an area of 24 square feet, and a 10 foot-by-5 foot rectangle that has an area of 50 square feet. The total area is 24+5024+50 or 74 square feet.

Problem 4

Lin says that she knows how to find the area of this figure. Diego says there is not enough information to find the area.

Do you agree with Lin or with Diego? Explain your reasoning.

<p><span>Long description: 6-sided shape. Straight sides. All side lengths meet at right angles. Side lengths: Bottom, question mark ft. Right side rises question mark ft, then goes left question mark ft, down 2 ft, left question mark ft, and down question mark ft. </span></p>

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Solution

Sample responses:

  • Lin is correct. I used the side of length 2 to make a ruler. It takes 4 lengths of 2 or 8 to make the right side, and it takes 5 lengths of 2 or 10 to make the bottom. The two unlabeled sides on top are equal and add up to 10 so they are each 5. I can divide the shape into a part on top that is 2 feet by 5 feet and a part on the bottom that is 6 feet by 10 feet. The total area is 10 square feet and 60 square feet, making 70 square feet total.
  • Diego is correct. There is only one measurement given and no matter how I cut up this shape, I can’t use that to find the area of any rectangles.

Problem 5

  1. Each image shows part of a shape filled with squares.

    A shape partially filled with squares.

    Square with unit squares around perimeter. Center empty. Length, 6 unit squares. Width, 6 unit squares.

    For each image, which do you think is greater, the number of squares in the image or the number of squares still needed to fill the middle?

  2. Check whether or not your answers are correct.
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Solution

Sample response:

  1. I think the number of squares needed in the middle is greater in both cases.
  2. The first image has 7 squares on top and also on bottom, so that’s 14. Then there are 5 more on each side, so that’s 10 more. There are 24 squares around the boundary. The middle is 5 by 5, so there are 25 squares. There is 1 square more in the middle than on the boundary.
    The second image has 6 squares on top and 6 on bottom, so that’s 12, and then 8 more on the two sides giving 20 total. The middle is a 4-by-4 square, so 16 squares are needed. There are more squares showing than are needed in the second image.