Lesson 6.4
Measuring the Inaccessible
Focus: Thales at the pyramid — shadow method — mirror method — line-of-sight method — two days
This is a two-day lesson. Day 1 covers the Thales pyramid task and the outdoor shadow investigation. Day 2 opens with the mirror task (the hypnotist problem) and closes with formal note-making and CYU problems. The two days connect naturally: Day 1 surfaces methods involving shadows and line of sight; Day 2 surfaces a third method using reflection; the notes (on day 2) consolidate all three with their assumptions and similarity justifications.
A standing expectation that must be made explicit in this lesson: before applying the properties of similar triangles — before writing a proportion, before computing a missing length — students must first justify that the triangles are similar. Which shortcut applies? What conditions satisfy it? This step was skipped by students in the first year, and the consolidation notes now ask for it explicitly in each method. Hold the line on this at the boards throughout the lesson. A student who writes a proportion without identifying the similar triangles and naming the shortcut has done the second step without the first.
Day 1 opening: the Thales task works well as a three-act math task. Give the story (Thales in Egypt, asked to measure the pyramid, no modern tools), then ask groups what information they would need. We have one group that figured out Thales’ actual method — waiting until his shadow equals his height — they understood something beautiful: sometimes the most elegant approach is to choose the right moment rather than the most complex calculation. In the first year, this was the only group that got Thales’ original method. After praising the insight, I gave them the extension: they’re in a rush and can’t wait. Most other groups were already solving it this way.
The ‘what if it’s cloudy’ extension: all groups struggled with this. The shadow method requires sunlight. The mirror method requires a reflective surface and a clear line of sight. When both are unavailable, groups need the line-of-sight method: one person lies on the ground and aligns the top of the height of something they know with the top of the distant object, while another person marks where the alignment occurs. The similar triangles here involve the known height, their distance to the base of the object, and the object’s height. Students find this method difficult to visualize. Modeling it physically — lying on the floor — is an appropriate accommodation when groups are genuinely stuck. Pictures demonstrating this method are included in the student note sheets for consolidation.
The note consolidation covers all three indirect measurement methods. For each: state the method, state the assumptions required, identify the similar triangles, name the similarity shortcut (AA for all three), and set up the proportion. This is where the justification expectation must be made explicit.
Shadow method: both objects are vertical (perpendicular to the ground); sunlight rays are parallel. By AA: corresponding angles equal (both right angles at the base, and the sun’s rays create equal angles with both objects). Proportion: shadow of person / height of person = shadow of object / height of object.
Mirror method: both objects are vertical; angle of incidence equals angle of reflection (derived in the task or Matt Chadbourne’s Desmos activity). By AA: right angles at the base and the reflection angles. Proportion: distance of person from mirror / eye height of person = distance of object from mirror / height of object.
Line-of-sight method: both objects are vertical; the observer’s line of sight from ground level connects the top of the near object to the top of the far object. By AA: right angles at the base and the shared angle at the observer’s eye position. Proportion: near height / near distance = far height / far distance.