Illuminations: Building Height

# Building Height

 Students will use a clinometer (a measuring device built from a protractor) and isosceles right triangles to find the height of a building. The class will compare measurements, talk about the variation in their results, and select the best measure of central tendency to report the most accurate height.

### Learning Objectives

 Students will: Find the measurement of a tall object Compare measurements and discuss errors Find mean, median, and mode of a set of data Select the best "average" measure to report

### Materials

 Protractors (preferably those with a center hole at 0°) String or dental floss Scissors Small weights, such as washers or coins Drinking straws (cut to the same length as the base of protractor) Tape Tape measure Clinometer Construction Overhead Building Height Activity Sheet

### Questions for Students

 Does it matter which person acted as measurer? Why or why not? [No, the building height would still come out the same because eye-level height was used in the calculation.] Why are the answers for the height of the building different? [Not all pairs measured with the same accuracy or precision.] Is there a more accurate way to measure the height of a building? How? [Yes. For example, a person could stand on the roof and use a tape measure against the building.] Why didn’t we use that method? [Answers will vary, but in the example of measuring from the roof student may not have access to the roof or a suitably long tape measure. Mention safety as a consideration as well.] In what kinds of situations would being able to measure the heights of objects this way be useful, or even profitable? [Answers will vary. For example, where the top of the building is not accessible, where the side of the structure is not perpendicular to the ground level (such as a hill), etc.] Can you think of an instance when mode would be the best measure of central tendency? [When measuring objects, the best measure of central tendency is mean or median. Mode is more useful with discrete sets, such as the "average" number of children in a family.] Why is median “better” than mean when there are outliers? [Median is calculated based on the number of elements in a set, while mean is calculated based on the values of those elements. A single outlier does not have a big impact on a median because it is still only 1 element of the set. However, a single outlier does change the mean significantly because its value is very different from the other values.]

### Assessment Options

 Have students write a letter to the school administration presenting the height they determined, including details on how the measurement was performed and why they believe it to be accurate. Have students write a journal entry about the process of measuring the height of objects using isosceles triangles. Have students write a journal entry about why individual measurements may be different. Students should address how reporting the mean or median of all measurements can be more reliable by accounting for the variation in measurements. Allow students to measure the height of another object on their own.

### Extensions

 Allow students to come up with other ways to measure building and try them. On a sunny day, they can measure their shadow and the shadow of the building, and use a proportion to find the height of the building. Students could also measure the height of other objects using their clinometers and tangent ratios. For example, using tangent x = opposite/adjacent, students could measure a common angle (30°, 45°, or 60°) substituting it for x, their distance from the object (adjacent), and solve for the height (opposite).

### Teacher Reflection

 Did students have sufficient understanding of how to use the clinometer before going outside? If not, how can you provide better direction? Did students work well in their pairs? How else could you group them? Did students have sufficient prior knowledge of isosceles right triangles? Did students understand the role of the triangle in calculating the building height? Did students understand that measurements will turn out the same no matter who measures an object?

### NCTM Standards and Expectations

 Data Analysis & Probability 6-8Find, use, and interpret measures of center and spread, including mean and interquartile range. Measurement 6-8Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision. Understand relationships among units and convert from one unit to another within the same system.

### References

 Maxwell, Sheryl A. 2006. Measuring tremendous trees: Discovery in action. Mathematics Teaching in the Middle School 12:132–41.
 This lesson was prepared by Katie Hendrickson as part of the Illuminations Summer Institute.

1 period

### NCTM Resources

 More and Better Mathematics for All Students
 © 2000 National Council of Teachers of Mathematics Use of this Web site constitutes acceptance of the Terms of Use The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.