Part A Casey is making her own board game. The area of the square board she makes is 196 square inches. What is the length, in inches, of one side of the game board? Explain how you got your answer.

Question

Part B Casey cuts out game pieces from sheets of card stock. Each game piece has a diameter of about 1 inch. What is the approximate area of the game piece? Show all your work.

Part C If the game pieces are cut from a piece of card stock that is 8 inches by 11 inches. How many game pieces can she cut from one piece of card stock? Show or explain all your work.

Answer 1

Part A. Casey's square board is 196 square inches. You need to find the side length of the square. Think about inverse (or "opposite") operations. If the problem were flipped and you knew it had a side length of, say, 3 units, you would square the side length. 3² = 9. Well, what if you knew the area was 9 square units? How would you find the side length? Instead of squaring, you would need to find the square root. To solve Part A, take find the square root of 196 square inches. √196 = ? Check your answer by multiplying it by itself. If you get 196, you know you are correct.

Part B. What is the area of a game piece with a diameter of 1 inch? Since the game piece has a diameter, it's safe to assume that it's a circular piece. This means you need the formula for the area of a circle: π⋅r², where r is the length of the radius. Since the diameter is twice the radius, the radius is 0.5 inches. Plug that into your equation and solve. π(0.5)² square inches.

Part C. How many game pieces could Casey cut out of an 8 inch by 11 inch piece of card stock? Each game piece has a diameter of 1 inch. Assuming no waste of paper, Casey could fit eight game pieces across. She could theoretically fit 11 game pieces down. This would mean she has a array that is 8 wide and 11 long. She can fit 8⋅11 game pieces.

Answer 2

1. Part A:
2. Casey is making a square game board. Squares have equal lengths on all 4 sides. The area of a square can be represented by length x width or (side)².
3. The square root is the length of a square with that area.
4. So we know we need to take the square root of the area (196). √
5. √196 = 14. So the length of one side of the game board is 14 inches.
6. Check: Area = length x width or (side)². (14 x 14) =196.
7. Part B:
8. Here have to assume that the shape of the game piece would be a circle.
9. The area of a circle is  π * r²
10. The radius is one-half of the diameter so d=2r and to find the radius, we just divide the diameter by 2. So the radius is 0.5 inches.
11. The area of the game piece would be π * r² = π * (0.5)² = 0.785 inches² or 0.79 inches²
12. Part C:
13. We want to know how many of the circles can fit into the square paper.
14. First we must find the area of the paper. The area of a rectangle is length x width. So the area of this rectangle is A= 8 x 11 = 88 inches²
15. Next, to find how many circles fit into the paper, we must divide the area of the paper by the area of the circle
16. A_paper ÷ A_circle = 88 ÷ 0.79 = 111.39
17. However, we cannot have a portion of a game piece so we round down.
18. 111 game pieces can be cut from one card stock.