## Building Height

- Lesson

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.

Put students in pairs and pass out the following to each pair: a protractor, a small weight or washer, a piece of string or dental floss 4–6 inches long, a drinking straw, and 2 or 3 short pieces of tape. Explain that they will be measuring the height of a tall building, such as the school building. Ask students to brainstorm how to measure objects that are taller than a person using only the materials provided. List the ideas on the board and discuss which ideas are practical and reasonable ways to measure the building.

Draw an isosceles right triangle on the board with one leg drawn on the bottom horizontally. Ask students questions to introduce the properties of an isosceles right triangle. Here are some questions you may choose to ask the class:

- What does "isosceles" mean?
[Two sides of the triangle are congruent.]

- What does "right" mean?
[One angle of the triangle has a measure of 90°.]

- Can you identify the congruent sides and right angle?
[Have a student volunteer label the appropriate parts of the triangle.]

- Do we know anything about the other angles of the triangle? Can we figure out their measures?
[They are both 45°.]

Relate these properties back to the challenge of measuring the height of a building with the given materials, and ask for new or revised ideas on how to accomplish the task. Finally, introduce the idea of using a clinometer if it has not already been suggested. Pass out one Building Height Activity Sheet to each pair of students. Read through the directions on the first page and ensure that students understand how a clinometer works.

Building Height Activity Sheet

Help each pair of students create their clinometer. Direct them to follow the steps provided on the Clinometer Construction Overhead. Create a clinometer along with students, step-by-step, at the front of the room.

Clinometer Construction Overhead

Before going outside, read through the activity sheet again and address any student questions. Once outside, direct students to a specific area of the school building. Decide on a common definition of “height of the school” for students to measure, which will vary depending on the architecture of the building being measured. Allow students to find the height of the building following the instructions on the activity sheet. Once students are finished taking their measurements, go back to the classroom and collect data from each team. Each team should put their measurement for the height of the building on the board, and then record the data from the board onto the chart on page 2 of the activity sheet.

Lead a whole class discussion of the results, using the following questions as a guideline:

- Were the answers from the groups the same or different? Why do you think this is?
[There will most likely be variation among the answers from different teams. Explanations for this will vary, but possible answers could include accuracy in measurement, difficulty in reading the clinometer accurately, or precision in measurement (such as rounding error).]

- Does a person’s height affect the measurement of the building height?
[No, measuring eye-level height accounts for a possible difference. A person 2 inches taller than another will stand 2 inches closer to the building when taking the measurement.]

Direct teams to complete the Questions 3–5 on the activity sheet. Circulate among the pairs and provide assistance as needed. You may want to give hints for Questions 4 and 5, but do not provide direct answers. Once most pairs have finished, ask volunteers to put up their calculations for the mean, median, and mode, showing their work. Then have a class discussion about Questions 4 and 5. Use the following as a guide:

- What measure of central tendency (mean, median, or mode) best represents the "average" height? (Question 4)
[Answers will vary, but students should choose either the mean or the median. If there is an outlier, median would be best. If there are no outliers, mean would be the best value to report.]

- How could you improve the accuracy of the measurements? (Question 5)
[Answers will vary. Students may suggest standing still and using a more precise protractor.]

### Reference

Maxwell, Sheryl A. 2006. Measuring tremendous trees: Discovery in action. Mathematics Teaching in the Middle School 12:132–41.

- 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

**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). - Have students test the accuracy of a clinometer by measuring the height of something that they can directly measure with measuring tape (ex: height of a door).

**Questions for Students**

1. 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.]

2. Why are the answers for the height of the building different?

[Not all pairs measured with the same accuracy or precision.]

3. 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.]

4. 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.]

5. 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.]

6. 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.]

7. 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.]

**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?

### 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.

### NCTM Standards and Expectations

- Understand relationships among units and convert from one unit to another within the same system.

- Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

- Find, use, and interpret measures of center and spread, including mean and interquartile range.

### Common Core State Standards – Mathematics

Grade 6, Stats & Probability

- CCSS.Math.Content.6.SP.A.3

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Grade 6, Stats & Probability

- CCSS.Math.Content.6.SP.A.1

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, ''How old am I?'' is not a statistical question, but ''How old are the students in my school?'' is a statistical question because one anticipates variability in students' ages.