SOLVING MATH PROBLEMS
Math problems! These are two words that strike terror in the hearts of some parents and children alike. Far better to term them “math magic” or “math tricks.” It’s not just a matter of semantics; but a whole thought process. Break down math problems into small, easily understood steps and you’ll magically transform those “problems” into fun solutions. Using objects, looking for patterns, acting out problems, making charts and visualizing are just a few of the many keys.
By Jody Wright
1. Your mom has decided to buy a stained glass window. The artist says that they charge $150.00 per square foot. Your mom wants a window that is 24 inches across and 24 inches down. How much will it cost?
Hint: Break this down into easy steps:
How many feet are in 24 inches?
Multiply the number of feet across by the number of feet down.
Take that sum and multiply it by $150.00
Step One: 24 ÷ 12 = 2 ft.
Step Two: 2 ft. x 2 ft. = 4 ft.
Step Three: 4 ft. x $150.00 = $600.00
Answer: The square stained glass window will cost: $600.00.
2. Your dad has said he likes the idea of having a stained glass window but he wants to have a round window. How much will a 24 inches-diameter round window cost?
Hint: You first need to know the circle’s area. Area = π (3.14) x r².
Remember that the radius is 1/2 of the diameter. Divide the final sum by 144 to get the square feet.
Step One: 24 ÷ 2= 12 inches radius
Step Two: 12 inches x 12 inches = 144 inches
Step Three: 3.14 x 144 inches = 452.16
Step Four: 452.16 ÷ 144 = 3.14 (This is the square foot.)
Step Five: 3.14 x $150.00 = $471.00
Answer: The circular stained glass window will cost: $471.00
3. Look at the two problems above. Why is the cost so different when both windows are 24 inches across?
Hint: draw a picture of the two different windows on graph paper
Step One: On graph paper draw out a 24 inches ´ 24 inches window (you’ll have to scale it down - each graph square may equal 4 inches)
Step Two: Go to the center of the of the 24 ´ 24 window you’ve graphed and make a dot. Use a compass and draw a circle that would be scaled to a 24 inches diameter window. See how it fits exactly in your box?
Step Three: Darken the area outside the circle. This is how much more the square window has in area than the circle. It’s also why your mom is going to pay less for the circle than the square.
Answer: The circle has less square feet. of area when you compare it to the square.
Junior High Worksheet:
1. Juan can run 4km in 30min. At this speed, how far can Juan run in 45min?
Hint: Use the equation speed = distance ÷ time (s = d ÷ t)
Step One: We can calculate Juan’s speed because we are given ‘distance = 4km’ and
‘time = 30min.’
Juan’s speed = 0.133km/min
Step Two: Now that we know Juan’s speed, we can find out how far he runs in any given time. All we have to do is multiply his speed by the time he spends running. If the time he spends running at 0.13km/min is 45min, the distance he travels is:
distance = 5.99km
Answer: Juan can run 5.99km in 45min.
2.) Add 1/2 + 2/3
Step One: Multiply the numerator and the denominator of the first fraction by the number in the denominator of the second fraction: 1/2 = (1 x 3) ÷ (2 x 3) = 3/6
Step Two: Multiply the numerator and the denominator of the second fraction by the number in the denominator of the first fraction: 2/3 = (2 x 2) ÷ (3 x 2) = 4/6
Step Three: Add the numbers in the numerators of these new equivalent fractions:
Answer: 1/2 + 2/3 = 7/6 = 1 1/6
3.) Find the lowest common multiple of 8 and 12.
Step One: Find the prime factors of number 8 (A prime number is a positive integer with exactly two positive factors, 1 and itself).
Factors of 8 = 1, 2, 4, 8. Prime factors are 1 and 2.
Step Two: Find all the factors of 12 and then determine its prime factors.
1, 2, 3, 4, 6 and 12. Therefore the prime factors are 2 and 3
Step Three: Write number 12 as a product of its prime factors.
12 = 2 x 2 x 3 x 1
Step Four: Underline each occurrence of the prime factor of 8 that also is present as the prime factor of 12.
8 = 2 x 2 x 2
12 = 2 x 2 x 3
Step Five: Make a list (List-A) of the underlined numbers: 2, 2
Step Six: Make a list (List-B) of the numbers from Step Four that are not underlined: 2, 3
Step Seven: Multiply the numbers from List-A and List-B to get the LCM:
2 x 2 x 2 x 3 = 24
Step Eight: Verify if 24 is divisible by 8 and 12.
Answer: The LCM is 24.
High School Worksheet
1. The massive Saturn V rocket was used in the historic Apollo 11 mission that landed a man on the moon. Suppose that an observer on the ground at a distance of one kilometer (1000 meters) from the launch pad measures that the angle, α, between the horizon and the top of the rocket is 6.3 degrees. What is the height of the Saturn V rocket?
Hint: This is a simple trigonometry problem that makes use of the tangent function. Remember that the tangent is the ratio of the opposite side to the adjacent side of a right triangle.
Step One: Draw a right triangle with a base d and a height h, where d is the distance of the observer from the launch pad and h is the height of the rocket. The angle between the base and the hypotenuse of the triangle is α = 6.3 degrees.
Step Two: The tangent of the angle a is equal to the height of the rocket, h, divided by the distance, d, and so we can write the equation:
Step Three: We know that α = 6.3 degrees and d = 1000m, and so we solve for h by multiplying both sides by 1000:
Tan (6.3) = h/1000
1000 x tan(6.3) = h
h = 1000 x tan(6.3)
Step Four: Calculate tan(α) using a scientific calculator (the Microsoft Windows calculator can be converted to a scientific calculator by clicking on view and selecting scientific). Make sure your calculator is set to degrees and not radians.
Step Five: Calculate the height h:
h = 110m
Answer: The Saturn V rocket is 110m (more than 300 feet!) tall.
2. The distance from the earth to the sun is 1.5 x 1011m, and the distance from the earth to the moon is 3.8 x 108m. If the sun and moon form a 90 degree angle at the earth, what is the distance from the sun to the moon?
Hint: Use the Pythagorean theorem, A2 + B2 = C2
Step One: Draw a right triangle with the earth, moon and sun at the corners. The earth should be at the right angle.
Step Two: Label the side between the earth and the sun A, the side between the earth and the moon B, and the side between the sun and the moon C. Side C should be the hypotenuse of the right triangle.
Step Three: We know that A = 1.5 x 1011m, and that B = 3.8 x 108m. Substitute these numbers into the Pythagorean theorem:
2.25 x 1022m2 = C2
Step Four: Take the square root of both sides to get the answer:
Answer: The distance from the sun to the moon is 1.5 x 1011m.
3. The radius of the earth is 6.3 ´ 106m. How fast is a point on the equator moving, given that the earth makes one full revolution is 24 hours?
Hint: Use the equation speed = distance/time, s = d/t
Step One: The distance that a point on the equator travels in one full revolution is equal to the circumference of the earth, C:
We know that r = 6.3 x 106m, and so we substitute to find C:
C = 2 x 3.14 x 6.3 x 106m
C = 39564000m
Step Two: The time is given as 24 hours, and so we divide the distance traveled by the time to get the speed:
speed = 1648500 m/hour
Answer: A point on the earth’s equator is moving at 1648500 meters per hour.