Learning Math: Measurement
Fundamentals of Measurement Homework
Session 2, Homework
Imagine that you are playing the Between game with a partner. Player A picks a decimal number between 5 and 6 — say, 5.7 — crosses out the number 5, and writes down 5.7 in its place. Player B picks a number between 5.7 and 6, such as 5.9, crosses out the number 6, and writes down 5.9. Now Player A must pick a number between 5.7 and 5.9 — i.e., 5.8 — and replace Player A’s previous number (5.7) with it. Play continues for a total of 10 rounds (the above describes three rounds).
Imagine that the Between game was a measuring task where you were trying to become more accurate with each measurement. What does this game tell you about the nature of measurement?
Suppose that a scale of 1 in.:20 ft. was used for building a model train. What does this mean? If a railcar is 40 ft. long, how long is the scale model?
A science class wants to create insects that are larger than life. They found that a queen ant is 0.5 in. long. The large model they plan to create is 5 ft. long. What is the scale factor for the ant model?
Here is your car’s gas gauge:
- Suppose your full gas tank holds 16 gallons. Put an arrow on the gas gauge to show how much gas you would have left in your tank if you filled it up and then took a drive that used 6 gallons:
- Your gas tank said “Empty,” but you were low on cash. You used your last $4 to buy gas, paying $1.139/gallon. Suppose a full tank holds 14 gallons. Put an arrow on the gas gauge to show how much gas you had in your tank after your purchase:
- You filled up your tank this morning, then took a drive in the country to enjoy the fall colors. Your odometer said that you had gone 340 miles, and you have been averaging about 31 miles/gallon. If your gas gauge looked like this when you got home, how much does your gas tank hold when it is full?
Suppose that you had the following 10 measurements (in centimeters) of the same object:
- With these data, what would you give as the best approximation? Explain why you give that approximation. What would be the precision unit for your best estimate?
Think about finding the mean or an average data value of the set.
Suppose that you made five measurements in addition to the 10 listed above:
32.1 32.2 32.3 32.3 32.4
- What is your best approximation now?
Suppose that you made a total of 20 measurements — the 15 above and the following five:
32.0 32.9 32.4 32.2 32.1
- What is your best approximation now? Did it change?
- In general, what effect on a best approximation do you expect if the results of more and more measurements are reported?
Take a piece of paper and measure its length and width. What level of precision will you use to measure? What is the accuracy of your measure in terms of the relative error?
Remember, the level of precision depends on the measuring instrument and its unit of measure. Accuracy depends on relative error. You can record the accuracy as a percent by subtracting the relative error from 1 and writing the resulting decimal as a percent.
Problem H4 adapted from Lamon, Susan J. Teaching Fractions and Ratios for Understanding. pp. 121-123. © 1999 by Lawrence Erlbaum Publishers. Used with permission. All rights reserved.
The fact that this game does not reach an “ending” (with no moves available) shows that there can always be a more accurate measurement, and therefore the act of measuring can never be exact.
This means that all lengths in the model are 1 in. long for every 20 ft. in the original model. If the original railcar is 40 ft. long, we can set up the following proportion to calculate the model’s length:
Solving the equation for x would give us x = 2. So the model’s length will be 2 in.
The scale factor is 5 ft.:1/2 in. For easier calculations, we can convert the ratio into inches. Since 1 ft. = 12 in., the scale factor would be 60:1/2, or 120:1.
|a.||One way to do this is to partition the gas gauge into 16 equal parts by dividing it in half four times (2 • 2 • 2 • 2 = 16). You can then count 6 gallons. Another way is to note that 6/16 = 3/8, so partitioning the gauge into eighths is sufficient.
|b.||One way to solve this problem would be as follows: Filling up the entire tank would cost $1.139 • 14 = $15.94. Four dollars is one-fourth of $16, so it is just a little more than one-fourth of $15.94. The arrow should be just slightly above one-fourth full.
|c.||First you need to find out how much gas you used for 340 miles:
340 31 = 10.97
So approximately 11 gallons were used. Next, look at the needle position: It is approximately one-third of the way from “Empty” to the first quarter mark. Further subdividing the gauge into 12ths, we see that the needle is about 1/12 of the way from “Empty.” Since about 11 gallons were used, and 11/12 of the tank is empty, the tank therefore holds about 12 gallons when full.
|a.||The best approximation might be about 32.2 cm, which is in the center of the data set. The average (mean) of the 10 measurements is 32.17 cm. The precision unit appears to be tenth (or 1 mm), as all 10 measurements are recorded to the nearest 0.1 cm.|
|b.||The mean of the 15 measurements is 32.2 cm. Again, the precision unit might be 0.1 cm. Our approximation, however, is better now since we have more measurements.|
|c.||The mean is now 32.23 cm. Even though the measurement errors are pretty high, this gives us even more confidence that the actual measure is close to 32.2 cm.|
|d.||With more measurements we will get an increasingly better approximation.|
The level of precision depends entirely on the measuring instrument. If we use a measuring stick with 1-in. precision, the maximum error is 0.5 in.
The accuracy can be obtained by subtracting the relative error from 1 and writing the resulting decimal as a percent. For the longer side of the paper, the relative error is 0.5/11, or about 0.045 (so, the accuracy is 1 – 0.045 = 0.95, or 95%). The relative error for the shorter side is 0.5/8.5, or about 0.059 (so the accuracy is 1 – 0.059 = 0.94, or about 94%), which is less accurate. A measuring stick with more partitions will give more precise and more accurate measurements.
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.