Learning Math: Measurement
What Does It Mean To Measure? Part C: Nonstandard Units (30 minutes)
Session 1, Part C
Print this PDF and cut out the small and medium triangles to fit into the shapes.
Using two small triangles and one medium triangle, build each of these polygons.
Without using measuring tools, can you determine which of the polygons has the greatest area? Explain.
Without using measuring tools, place the polygons in order from least to greatest by the length of their perimeters. How did you determine which shape has the smallest perimeter?
Notice that the length of the hypotenuse of the small triangle and the length of the legs of the medium triangle are equal. This length can be used as one of your “nonstandard” units. What other nonstandard unit can be established by examining the lengths of the sides? Use these nonstandard units to establish the perimeters of the shapes.
If we can measure without using standard units, why do we need standard units?
In this video segment, Lori demonstrates the method her group used to figure out the perimeter of the five different figures, using a nonstandard unit of measure. As you watch the segment, think about the possible difficulties one may encounter when using nonstandard units of measure.
Would Lori’s answer be different if she had chosen a different side or a different triangle to measure with?
You can find this segment on the session video approximately 22 minutes and 29 seconds after the Annenberg Media logo.
Take It Further
Use the Pythagorean theorem to determine the lengths of the sides of the small and medium triangles. Recall that the Pythagorean theorem states that a2 + b2= c2 for the legs (a and b) and hypotenuse (c) of a right triangle. If the length of a leg of one of the small triangles is assigned a value of 1 unit, determine the length of the hypotenuse of the small triangle and the lengths of the legs and hypotenuse of the medium triangle. Use this information to determine the perimeters of the polygons in Problem C1.
If you are working in a group, first try working individually to make each of the five polygons (using all three triangles each time). Provide hints to colleagues who are having difficulty making a particular shape by starting the puzzle for them; put two of the three triangles in place and let them figure out where the third triangle belongs. Resist simply showing someone a finished puzzle. Once the five polygons have been completed, discuss Problems C2 and C3.
It may appear to some people that the areas of the five shapes are different. Others will argue that the areas must be the same since each polygon is constructed from the same identical pieces. Reflect on this question, or discuss it if you are working in a group.
There are different ways to order the polygons by perimeter, all of which require us to first establish a nonstandard unit of measure (e.g., the side of one of the triangles; the length of an eraser) and then to take some measurements in terms of the nonstandard unit. If you are working in a group, share your approaches. Be sure that everyone agrees on the order in which to place the polygons (in terms of perimeter), and that individuals justify their conclusions.
All five polygons have the same area, since they are made up of the same three smaller polygons. Since there is no overlapping, the area of all five is the same as the sum of the area of the three triangles.
First, you need to arbitrarily choose a unit with which to measure the perimeters. One way to do this is to choose the length of a leg of the smaller triangle as your unit. Whatever unit you choose, you will discover that the square has the smallest perimeter. The rectangle’s perimeter is slightly larger. The parallelogram, triangle, and trapezoid are tied for the largest perimeters.
Standard units provide a frame of reference that can always be relied on. Although we can say, “This one is three times longer than that one,” standard units provide everyone with an equivalent value for “that one.” With standard units, anyone measuring the same item could say, “This one is six times longer than 1 cm.”
Small triangle: Using the Pythagorean theorem to determine the length of the hypotenuse, we can write the following:
c2 = a2 + b2 = 12 + 12 = 2, or
c = √2
The length of the hypotenuse is √2, or approximately 1.414 units.
Medium triangle: The length of the legs is equal to the hypotenuse of the smaller triangle, or √2. So, to determine the hypotenuse, we can write the following:
c2 = a2 + b2 =√22 +√22 = 4
So the length c is equal to 2 units.
Remember, our unit is the length of one leg of the small triangle. So the perimeters are as follows:
4 •√2, or approximately 5.656 units
Rectangle that is not a square:
2 • 1 + 2 • 2 = 6 units
Parallelogram that is not a rectangle:
2 •√2 + 2 • 2, or approximately 6.828 units
2 + 2 + (2 •√2), or approximately 6.828 units
2 •√2 + 1 + 3, or approximately 6.828 units
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.