Lower Elementary:
Question: The moon moves through 8 phases in a pattern. The pattern is: new moon, waxing crescent, first quarter, waxing gibbous, full moon, waning gibbous, last quarter, waning crescent, and then it goes back to a new moon. If the moon starts off as a full moon, what phase will the moon be in after 14 phases?
Answer: First quarter
Solution: We know that the pattern repeats every 8 phases, so if we start at the full moon and go 8 phases, we will be back at the full moon. We are asked what phase it will be after 14 phases. After 8 phases, we are back at a full moon. 14 – 8 = 6. So now we have to go 6 more phases after the full moon. 16 phases after the full moon is the first quarter. So 14 phases after the full moon is the first quarter.
Upper Elementary:
Question: It takes Monica 3 minutes to read 2 pages of a book. If a book has 720 pages and she has already read 600 pages, how long will it take Monica to finish the book?
Answer: 180 minutes (3 hours)
Solution: The book is 720 pages and Monica has already read 600 pages. That means she has 120 pages left (720 – 600 = 120). It takes her 3 minutes to read 2 pages. One way to solve this is to find the number of groups of 2 pages there are in 120 pages and then multiply that answer by 3. There are 60 groups of 2 in 120. 60 × 3 = 180. It will take Monica 180 minutes, or 3 hours, to finish the book.
Middle School:
Question: For a particular video game, the formula to calculate the damage dealt to the opponent is Damage = [(Attacker’s Offense – Defender’s Defense)2 + (Attacker’s Offense)] ÷ 2 – (1.5)Defender’s defense, rounded up. If the value is negative, the damage dealt is 1. If the attacker’s offense is 532 and the defender’s defense is 482, how much damage will be dealt?
Answer: 793
Solution: If we plug in the values of the attacker’s offense and the defender’s defense, we have:
[(532 – 482)2 + 532] ÷ 2 – (1.5)482
This is an order of operations problem, so we need to start with the grouping symbols. First we do 532 – 482, which is 50.
[502 + 532] ÷ 2 – (1.5)482
Next we square 50, which is 2,500.
[2,500 + 532] ÷ 2 – (1.5)482
Next we do 2,500 + 532, which is 3,032.
3,032 ÷ 2 – (1.5)482
Next is multiplication and division. We will divide 3,032 by 2 and multiply 482 by 1.5. 3,032 ÷ 2 = 1,516 and 482 × 1.5 = 723.
1,516 – 723
Subtract.
793.
The attack will deal 793 damage.
Algebra and Up:
Question: In the image, the rectangle has a perimeter of 32 inches. The quarter circle has a radius of 6 inches and its center is the lower right corner of the rectangle. What is the perimeter of the shaded region (leave your answer in terms of π)?
Answer: 20 + 3π
Solution: First, let’s label the lengths of the rectangle and the quarter circle. The quarter circle has a radius of 6 inches. Since the quarter circle’s radius is the same as the height on the right side of the rectangle, that side length is 6 inches. In a rectangle, the opposite sides are the same, so the left side of the rectangle is 6 inches. The total perimeter of the rectangle is 32 inches. The formula for the perimeter of a rectangle is Perimeter = 2(length) + 2(width). We know what the width and perimeter are, so we can solve for the length.
32 = 2(length) + 2(6)
Multiply the 2(6).
32 = 2(length) + 12
Subtract 12 from both sides.
20 = 2(length)
Divide both sides be 2.
10 = length
So the length of the rectangle is 10 inches. The quarter circle intersects the bottom side of the rectangle. The whole length has to be 10 inches and the circle has a radius of 6 inches, so the missing part must be 4 inches (10 – 6 = 4).
Now that we have all of the side lengths, let’s calculate the perimeter, or the distance around the shape. The three straight sides are 10 inches, 6 inches, and 4 inches. The curved side is a quarter circle. To find the length, take the circumference of the whole circle and divide that by 4. Circumference = 2 • radius • π = 2 • 6 • π = 12π. Divide by 4, and the length of the curved side is 3π. The perimeter of the shaded region is 10 + 6 + 4 + 3π = 20 + 3π inches.
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