Lower Elementary
Question: Emily made 1 dozen chocolate chip cookies. If her recipe calls for ½ a cup of chocolate chips, how many chocolate chips does she need for 3 dozen cookies?
Answer: 1½ cups
Solution: Because 1 dozen calls for a ½ cup of chocolate chips, in order to bake 3 dozen we would need to triple the amount of chocolate chips.
½ + ½ + ½ = 1½ cups
Upper Elementary
Question: Recycling centers pay $1.90 per pound for aluminum and 90 cents for every pound of plastic. Anthony has collected 10½ pounds of plastic bottles and 13 pounds of aluminum cans. If he turns it in at the recycling center, how much money will Anthony get in return?
Answer: $34.15
Solution: Anthony collected 13 pounds of aluminum, if turns it in to the recycling center for $1.90 a pound, Anthony will be paid $24.70 (13 × 1.90). If Anthony turns in 10½ pounds of plastic bottles for $0.90 per pound he will be paid $9.45 (10.5 × .90).
Combining the two amounts we get a grand total of $34.15.
Middle School
Question: Two trains started form the same spot traveling in opposite directions. Train 1 had a constant speed of 40 mph while train 2 traveled at a constant speed 25 mph less than twice the speed of train 1. How long would it take for train 1 and 2 to be a total distance of 665 miles apart?
Answer: 7 hours
Solution: Train 2 travels 25 mph less than twice train 1. That is 25 mph less than twice 40 mph. Doubling 40 mph and subtracting 25 mph, we find that train 2 travels at a rate of 55 mph.
A number sense solution for this problem would be to notice that because the trains are traveling in opposite directions, every hour the trains are 95 miles farther apart.
Dividing 665 miles by 95 mph, we find how long it takes for the two trains to be 665 miles apart.
665 ÷ 95 = 7 hours
An algebraic solution would be to use the distance formula d = rt, where d = distance, r = rate of train, and t = time in hours.
For train 1: d1 = rate of train 1 • time it takes trains to be 665 miles apart
d1 = 40t
For train 2: d2 = rate of train 2 • time it takes trains to be 665 miles apart
d2 = 55t
The combined distance the trains travel apart is 665 miles, so d1 + d2 = 665
40t + 55t = 665
95t = 665
t = 7 hours
Algebra and Up
Question: Holding a laser 5ft off the ground, Jeremy pointed it at a 32Ëš angle. If the laser beam hits a building 65 ft away, at what height off the ground does the laser beam hit the building?
Answer: About 45.6 feet
Solution: For the moment ignore how high the laser is being held off the floor. With the given information we can draw a triangle with a base spanning the horizontal distance from the tip of the laser to the building, a hypotenuse modeling the path of the laser, and a height spanning the rise of the laser from its initial height.
Using trigonometric function tangent we can solve for x which is the distance from the initial height of 5 feet above the ground to the spot where the laser touches the building.
tan(θ)= opposite/adjacent
tan32 = x/65
x = 65(tan32) ≈ 40.6 feet
Since x only covers the vertical rise of the laser from its initial height, to find the final answer we need to add 5 feet (the initial height above the ground) to x.
x + hi = 40.6 + 5 = 45.6 feet
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