Learning Math: Measurement
Measurement Relationships Homework
Session 9, Homework
Reptiles need the sun to get them going. Many large dinosaurs had to have large fins or plates down their back. How does this fact relate to what you have learned about the proportion of surface area to volume?
Take It Further
A manufacturer is producing half-liter aluminum cans in a cylindrical shape. The volume of the can is 500 cm3.
- Find the radius and height for the can that will use the least aluminum and therefore be the cheapest to manufacture. In other words, minimize the surface area of the can.
- What shape is your can? Do you know of any cans that are made in this shape? Can you think of any practical reasons why more cans are not made in this shape?
Some aspirin-like tablets are said to work “two and a half times faster” than their competitors. What is an obvious way in which this could be accomplished?
Mr. Hobbs had an ugly blob in the middle of his wall. The paint on the rest of the wall looked fresh, so Mr. Hobbs asked the painter to come and paint only the blob. The painter said he would charge according to the area that needed to be painted. To figure out how much the job would cost, Mr. Hobbs ran a string around the edge of the blob so that it covered the border perfectly. Then, to figure out the area, he removed the string, shaped it into a rectangle, and figured out the area of the rectangle.
How close did Mr. Hobbs’s estimate come to the painter’s bill for painting the blob?
In cubes, you found a proportional relationship between volume and surface area. Does the proportional relationship between volume and surface area also exist for spheres?
The volume of a sphere is (4/3)πr3, and the surface area of a sphere is 4πr2.
Dinosaurs’ plates were likely the result of their very low surface area-to-volume ratio: The plates served to increase the dinosaurs’ surface area without increasing their volume very much.
Animals with a great deal of surface area but little volume cool down and heat up faster than animals with larger volumes. This is important in reptiles since they obtain their body heat from the sun.
- The minimum surface area occurs when the height is exactly twice the radius. If the volume is 500 cm3, the radius is approximately 4.30 cm, while the height is approximately 8.61 cm.
- In this shape, the can fits perfectly inside a cube, since the diameter of the can is the same as the height. Some cans are made in this shape. When cans are displayed in stores or placed on shelves, there is often more of a premium on radius (shelf space), so the radius is often smaller than our “ideal” can.
One obvious way is to increase the surface area-to-volume ratio of the tablet. It would dissolve more quickly. A flat caplet dissolves more quickly than a spherical pill.
This is the same type of problem as the “area of the hand” problem in Problem A7. The answer depends on the shape of the rectangle Mr. Hobbs used, since different rectangles with the same perimeter will have different areas. There is no way of guaranteeing that, using this method, the area of the two figures would be the same.
Yes, a similar ratio exists. The ratio for spheres is r:3 rather than s:6 for cubes and can most easily be found by using a table or by dividing the two formulas into one another.
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.