Lower Elementary:
Question: Mrs. Carrel is passing out different shapes for a class project. She is passing the shapes out in a pattern: circle, triangle, square, circle. If she continues passing out the shapes using this pattern, what will the 19th shape be?
Answer: square
Solution: The pattern repeats every 4 shapes. The pattern goes circle, triangle, square, circle. One way to find the 19th shape is to count by 4s until we get a number less than 19. 4, 8, 12, 16. That means that the pattern will start again at 17. The 17th shape is a circle, the 18th shape is a triangle, and the 19th shape is a square.
Upper Elementary:
Question: Beverly has 6 identical bottles. One bottle is empty. Which bottles should be used to completely fill the empty bottle if one bottle is filled 1/2 way, one is filled 1/3 of the way, one is filled 1/5 of the way, one is filled 1/6 of the way, and the last one is filled 1/10 of the way?
Answer: The 1/2, the 1/3, and the 1/6 bottles
Solution: One way to solve this problem is to find a common denominator for each of the bottles and see which arrangement will equal one whole. The LCM of 2, 3, 5, 6, and 10 is 30, so the common denominator is 30. 1/2 = 15/30, 1/3 = 10/30, 1/5 = 6/30, 1/6 = 5/30, and 1/10 = 3/30. So, our choices are 15/30, 10/30, 6/30, 5/30, and 2/30. The only combination that makes 30/30 is 15/30 + 10/30 + 5/30 = 30/30. This means that the 1/2, 1/3, and 1/6 bottles combine to fill the empty bottle.
Middle School:
Question: What value of n in the equation nx – 3y = 6 would give the equation of a line with a slope of 12?
Answer: 36
Solution: One way to solve this problem is to first write the equation in slope-intercept form, which is y = mx + b. So, the first thing we should do is subtract nx from both sides.
–3y = –nx + 6
Now, divide both sides by –3.
y = n/3 x – 2
Now, the slope of a line is the value of m, or the number in front of x. So, to find the value of n that makes the slop 12, set n/3 equal to 12.
n/3 = 12
Multiply both sides by 3.
n = 36.
If n is 36, the slope of the line will be 12.
Algebra and Up:
Question: Gary is building a snowman. He needs to make 2 snowballs, one for the body and one for the head. The head has a volume of 2,304π cubic inches. The radius of the body is three times as large as the head. What is the volume of the body?
Answer: 62,208π cubic inches, or 36π cubic feet
Solution: One way to solve this problem is to first find the radius of the head then find the radius of the body so that we can find the volume of the body. The formula for the volume of a sphere is V = 4/3 πr3. To find the radius of the head, set the volume of the head equal to 2,304π and solve for the radius r.
2,304 π = 4/3 πr3
Multiply both sides by 3/4.
1,728π = πr3
Divide both sides by π.
1,728 = r3
Take the cubed root of both sides.
12 = r
The radius of the head is 12 inches, or 1 foot. Since the radius of the body is 3 times as large as the head, the radius of the head is 3 feet, or 36 inches. Substituting the radius into the formula for the volume of a sphere, V = 4/3π (3)3 = 36π cubic feet, or 62,208π cubic inches. So, the volume of the body is 36π cubic feet, or 62,208π cubic inches.
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