Mirror, mirror on the ground!

In this lesson, students will use their knowledge of similar triangles and indirect measurement to measure the heights of various objects.

A lesson plan for grade 8 Mathematics

By Denise Corbett

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Related topics

• Learn more about geometry, mathematics, and measurement.

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Learning outcomes

Students will learn to measure indirectly using similar triangles.

Teacher planning

Time required for lesson

2 hours

Materials/resources

• fifty feet of measuring tape per team of two students
• one mirror team
• data sheet
• calculator

Pre-activities

I review the concepts of similar triangles and proportions the day before we begin our measurements.

Activities

1. Determine a variety of objects on site for students to measure, such as flag pole, goal post, tree, etc. You should select the same number of items to measure as the number of groups you will have. The students do not have to go to every station, but if they do you will only have one group per station at a time without groups having to wait. Decide how many stations you want the students to go to. I found that four was all we could do in an hour. The teacher should also measure the height of these objects prior to the lesson using the same techniques as the students.
2. Divide students into groups of two. Choose students with significant height differences when possible.
3. Give each group a mirror, tape measure, and data sheet.
4. Tell each group the order in which they are to visit the different stations.
5. Demonstrate the correct procedures for measuring by measuring the height of a student. On the data sheet, the first example should be the set up for the teacher and student.
   1. Have the student stand straight and still and place the mirror several feet in front of the student.
   2. The teacher faces the student with the mirror between them and backs up until the teacher can see the top of the student’s head reflected in the center of the mirror.
   3. Another student then measures the distance from the center of the students feet to the center of the mirror and records this in the diagram on the data sheet. Then he measures the distance from the center of the mirror to the teacher and records this measurement.
   4. The teacher’s height will be listed on the data sheet as given information. The students at this time should set up their proportions and solve for the height of the student.
   5. After students have finished their calculations, measure the height of the student to compare results.
6. Explain that the students will need to follow the same procedures for each station. The students will use their own heights as given information. Students must draw their own diagrams for each object they measure and show all equations and calculations.
7. Also explain that the students need to measure each object twice, once with the first student using his height and then the other. The students will discover that since they are different heights they will have to stand in different spots to see the top of their object in the mirror, therefore the distances will be different. They will also discover that they will get the same results because the triangles they each form are also similar. You may need to prompt for this discovery.
8. Answer any questions the students may have and then send them to their respective objects to begin measuring.
9. Once back in the classroom, discuss the questions at the end of the data sheet. We also discuss the accuracy of this type of measurement.

Assessment

After completing this lesson, I give students a scenario in writing with no diagram. Students will have to draw and label the diagram independently and calculate the missing height.

Supplemental information

Comments

This lesson came about as a result of a workshop given on inquiry based learning.