In this lesson, students examine the length of shadows of different objects. There appears to be a proportional relationship between the height of the object and the length of the shadow. Students use this relationship to predict the height of a lamppost given the length of its shadow. In order to justify the proportional relationship, students use the hypothesis that the rays of sunlight making the shadows are parallel, together with their knowledge of similar triangles. Finally, students go outside and make their own measurements of different objects and the lengths of their shadows and use this technique to estimate the height of a tall object.
This lesson involves modeling (MP4), not only because students interpret real-world data (both the given heights and shadow lengths and the measurements that they take themselves) but also because they need to make simplifying assumptions in order to justify why the relationship is proportional.
- Calculate the unknown heights of objects by using proportional reasoning and explain (orally) the solution method.
- Justify (orally) why the relationship between the height of objects and the length of their shadows cast by the sun is approximately proportional.
Let’s use shadows to find the heights of an object.
Before doing the last activity, conduct the experiment ahead of time to ensure that shadow lengths will be cooperative at the time your class takes place. Also, make preparations to take your class outside. They will need measuring devices (tape measures, yard sticks, rulers) as well as a way to record their measurements.
- I can model a real-world context with similar triangles to find the height of an unknown object.