• Lesson
6-8
1

Students extend their knowledge of proportions to solving problems dealing with similarity. They measure the heights and shadows of familiar objects and use indirect measurement to find the heights of things that are much bigger in size, such as a flagpole, a school building, or a tree.

This lesson should be completed on a sunny day.

Give pairs or groups of students two meter sticks or yardsticks, string, and scissors. Have the students measure each other's height with the string and use the meter stick to measure the string. Students should record their individual data on the In Your Shadow activity sheet.

Within groups, students should be consistent when measuring, using either all metric measurements or all customary measurements.

Have students measure each other's shadow, following the same procedure as that for measuring height. Once again, record the data on the activity sheet.

Next, ask students to measure the shadow of something tall, such as a flagpole, a school building, or a tree. Once the data is collected on one or more objects, return to the classroom and post the data on the chalkboard or overhead. (You may assign different groups of students to measure different objects while outside.)

Ask the students how they might find the height of the tall object just by knowing the length of its shadow. Discuss what the word "similar" means in reference to geometry.

Next, help the students see how to use proportions to determine the heights of the tall objects that they measured. For example, show students how to use the measurement of the shadow of a tree along with a student's height and shadow measurements to figure out the height of the tree. The three pieces of known data can be used to set up a proportion with one unknown.

Work with the students on setting up proportions to solve for missing measures. Offer several examples using the data that the students collected in the experiment.

• Metersticks or Yardsticks
• String
• Scissors
• Outdoor objects to measure (flagpole, school building, tree)
• Calculators (Optional)

Assessments

1. At this stage of the unit, students should be able to do the following:
• Apply the use of proportional reasoning to multiple types of problems
• Understand the concept of similarity
• Set up and solve proportions
2. To assess whether the students are able to use indirect measurement, have them use their own heights and shadows (as measured earlier) to figure out the measures of the shadows of classroom objects (e.g., chalkboard, bulletin board, cabinets, etc.). Of course, this can be done only if the students know the actual heights of the classroom objects.

This activity reverses the previous process because now the students have the actual heights but don't know the measures of the shadows of those objects. This allows you to assess whether the students can apply indirect measurement in multiple settings.

Questions for Students

1. When do we need to use the concept of similarity and indirect measurement? Why is it helpful?
[When we don't know the height of a tall object, for example, we can use shadows and indirect measurement to help us determine the height.]
2. Why are we able to use shadows to make these predictions about heights?
[Assuming we make several measurements at the same time, the lengths of the shadows should be proportional, thereby allowing us to make predictions about heights of tall objects.]
3. Why are similar figures proportional?
[The ratios of the length and height of one object is equal to the ratio of the length and height of another object, when they are similar.]

Teacher Reflection

1. Are the students making the connection between ratio and proportion?
2. Are the students already familiar with the term "similar"?
3. If the location of the unknown changes due to the problem situation, are the students able to independently and correctly set up proportions?

### Learning Objectives

Students will:

• Find the heights of very tall objects by measuring shorter objects and their shadows
• Use similarity and indirect measurement
• Set up and solve proportions