In this activity, students explore the approximately proportional relationship between the height of an object and the length of its shadow by looking at a photo of three people, a lamppost, and their shadows taken on a sunny day. They use what they know about proportional relationships and the length of a shadow to find the height of an object that is difficult to measure directly.

The given measurements are real measurements rounded to the nearest inch, and therefore, the values in the table are not in a perfectly proportional relationship. The quotient of each shadow length and its corresponding object’s height is around, but not exactly, . Students engage in quantitative reasoning by exploring relationships with real-world measurements (MP2).

The goal of this discussion is for students to recognize a relationship that is approximately proportional and consider reasons why the relationship is not exactly proportional. Ask students to brainstorm possible reasons why the measurements in the table are not exactly proportional, highlighting these possible factors:

• The measurements were rounded, making them inexact.

• The ground might not be perfectly level. 

• One of the people or the lamppost is at a slightly different angle to the ground.

• Anything that makes the real world differ slightly from a mathematical model.

If time allows, or if any students created a graph of the associated heights and shadow lengths, consider sharing this graph and asking how it could be used to determine the height of the lamppost. (This graph and the length of the shadow could be used to find the corresponding height.)

The purpose of this activity is for students to write a mathematical justification for the proportional relationship between heights and shadow lengths in the photo (MP3). A version of the photo is provided with some line segments drawn to strongly suggest an argument based on similar triangles. 

The sun is so far away relative to its size that the rays that reach Earth are extremely close to parallel. Share this information with the students or ask them to think about the shadows of the pens in the Warm-up.

Some students may struggle with deciding what is important to include in their explanation. Prompt them to start by thinking about what they want to show. For example, in this activity, we want to show that there is a proportional relationship between the side lengths in some triangles. Then ask: “What types of triangles have sides that are in proportion?” “How do you know when triangles are similar triangles?” “Which pairs of angles do you know are congruent?” and “Why are they congruent?”

The goal of this discussion is for students to share and refine their explanations. Invite 2–3 students to share their thinking. Ask other students to restate, support, refine, or disagree with their statements by asking questions such as  “How do you know that the triangles are similar?” and “Which pairs of angles are congruent?”

The goal of this optional activity is to solve a real-world problem (MP4) by finding the height of some tall object. This activity should be done on a sunny day and tried out ahead of time to ensure that the shadows created in your part of the world at the time your class takes place are cooperative! 

Find a tall object, or let students choose a tall object outside, that all students will find the height of (for example, a flagpole, a building, or a tree). The object should be on level ground, perpendicular to the ground, and tall enough that its height can’t easily be measured directly. Students will work outside with tape measures (or other measuring devices) to use what they’ve learned in this lesson to determine the height of the tall object.

The purpose of this discussion is for students to share their strategies for determining the height of a tall object. Invite 2–3 groups to share their thinking. Here are some questions for discussion:

• “What object did you use that had a height that you could measure with the tools available?” (A person, a fence, a street sign, etc.)

• “How do you know the relationship between the object you measured is proportional to the height of the tall object you couldn’t measure?” (Since the sun’s rays are approximately parallel to the ground and the objects are perpendicular to the ground, we know that the triangles created will be similar and their side lengths will be proportional.)