Learning Math: Measurement
Classroom Case Studies, 6-8 Part C: Problems That Illustrate Measurement Reasoning (55 minutes)
Session 10: 6-8, Part C
In this part, you’ll look at several problems that are appropriate for students in grades 6-8. For each problem, answer the below questions. If time allows, obtain the necessary materials and solve the problems.
Questions to Answer:
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Problem C1
Bicycles are equipped with different types of tires. Twenty-six-inch tires have a diameter of 26 in., whereas 28 in. tires have a diameter of 28 in. You are riding a bicycle with 26 in. tires. If one turn of the pedals moves you forward one tire rotation, how many times must you turn the pedals to ride 1 mile?
Problem C2
Take a unit cube and increase all three dimensions by the scale factor in the table below. For example, to make a new cube that has a scale factor of 2:1, you would double the length, width, and height. The new cube would have dimensions of 2 by 2 by 2, a surface area of 24 square units, and a volume of 8 cubic units. Fill in the chart with the dimensions, surface area, and volume of the new, scaled-up cubes.
Examine the surface-area and the volume columns in your table. What patterns of growth do you notice? Can you determine a general rule?
Problem C3
Charlene is out surfing and catches the eye of her friend, Dave, who is standing at the top of a vertical cliff. The angle formed by Charlene’s line of sight and the horizontal measures 28 degrees. Charlene is 50 m out from the bottom of the cliff. Charlene and Dave are both 1.7 m tall. The surfboard is level with the base of the cliff. How high is the cliff?
Solutions
Problem C1
Solution:
With each rotation, the tire covers the distance of its circumference. So, the circumference of one 26-inch tire rotation = 26 = approx. 81.68 inches = approx. 7 feet. 5,280 feet per mile
7 feet per tire rotation = approx. 754 pedal turns per mile.
Answers to Questions:
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Problem C2
Solution:
Scale Factor |
Dimensions |
Surface Area |
Volume |
1:1 | 1 by 1 by 1 | 6 square units (un2) | 1 cubic unit (un3) |
2:1 | 2 by 2 by 2 | 22 • 6 = 24 un2 | 23 = 8 un3 |
3:1 | 3 by 3 by 3 | 32 • 6 = 54 un2 | 33 = 27 un3 |
4:1 | 4 by 4 by 4 | 42 • 6 = 96 un2 | 43 = 64 un3 |
5:1 | 5 by 5 by 5 | 52 • 6 = 150 un2 | 53 = 125 un3 |
10:1 | 10 by 10 by 10 | 102 • 6 = 600 un2 | 103 = 1,000 un3 |
25:1 | 25 by 25 by 25 | 252 • 6 = 3,750 un2 | 253 = 15,625 un3 |
The surface area of a cube is increased by the scale factor squared. The volume is increased by the scale factor cubed.
Answers to Questions:
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Problem C3
Solution:
Since Dave and Charlene are the same height, the 28-degree angle measures exactly the height of the cliff. Visualize closing off the angle to form a triangle and then sliding that triangle down to water level. The side opposite the 28-degree angle would align exactly with the cliff. Use similar triangles to determine the height of the cliff. Using a protractor, draw a right triangle with a 28-degree angle opposite the vertical leg forming the right angle. Measure the length of the two legs of the right triangle with a ruler. Then set up a proportion between those two sides and the height of the cliff and 50 m length in the surfing triangle. The solution should give you a cliff height of roughly 26.5 m.
Answers to Questions:
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