Learning Math: Measurement
Classroom Case Studies, 68 Part C: Problems That Illustrate Measurement Reasoning (55 minutes)
Session 10: 68, Part C
In this part, you’ll look at several problems that are appropriate for students in grades 68. For each problem, answer the below questions. If time allows, obtain the necessary materials and solve the problems.
Questions to Answer:

Problem C1
Bicycles are equipped with different types of tires. Twentysixinch 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, scaledup cubes.
Examine the surfacearea 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 26inch 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:

Problem C2
Solution:
Scale Factor 
Dimensions 
Surface Area 
Volume 
1:1  1 by 1 by 1  6 square units (un^{2})  1 cubic unit (un^{3}) 
2:1  2 by 2 by 2  2^{2} • 6 = 24 un^{2}  2^{3} = 8 un^{3} 
3:1  3 by 3 by 3  3^{2} • 6 = 54 un^{2}  3^{3} = 27 un^{3} 
4:1  4 by 4 by 4  4^{2} • 6 = 96 un^{2}  4^{3} = 64 un^{3} 
5:1  5 by 5 by 5  5^{2} • 6 = 150 un^{2}  5^{3} = 125 un^{3} 
10:1  10 by 10 by 10  10^{2} • 6 = 600 un^{2}  10^{3} = 1,000 un^{3} 
25:1  25 by 25 by 25  25^{2} • 6 = 3,750 un^{2}  25^{3} = 15,625 un^{3} 
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:

Problem C3
Solution:
Since Dave and Charlene are the same height, the 28degree 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 28degree 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 28degree 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:
