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:
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?
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?
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?
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:
|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:
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:
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.