When Size Changes, What Changes With It?

This lesson is the conceptual culmination of the unit. Up to this point, students have used similarity to compare shapes, scale images, and measure inaccessible objects. Here, similarity extends to explain how different kinds of measurements — perimeter, area, and volume — scale differently when a figure grows. That mismatch is not a mathematical curiosity. It constrains what can exist in the natural world.

 

The thinking task asks groups to create a scaled version of a simple figure (square, rectangle, triangle, or cube) and determine what happens to side lengths, perimeter, area, and volume. No formulas are provided. Students must reason from counting, geometry, or physical models. The group-work phase surfaces the pattern; the consolidation names it and explains why it must be true.

 

Watch for two common false patterns in student work. First: some groups use a side length equal to their scaling factor (e.g., start with a square of side 2, then scale by 2). This leads to a conjecture that area scales by x⁴ rather than x², because they are conflating the scale factor with the original side length. The pattern holds for everything they test — which is why it is a productive false conjecture. Try a case where the scale factor does not equal the side length and see what happens. Second: testing only k = 2. As one student argued, with k = 2 you cannot tell whether area is doubling or squaring. Test k = 3. Both observations demonstrate exactly the kind of reasoning Units 1 and 2 built.

 

The biological connections — elephant ears as radiators, insect size limits and atmospheric oxygen, why large animals need disproportionately thick legs — are presented as a guided discussion after groups share their scaling patterns. The goal is not to teach biology but to show students that the geometric relationships they just discovered predict real constraints in the world. Geometry limits what can exist.

 

The Kleiber’s Law thread is worth following. The geometry predicts metabolic rate should scale as mass²ᐟ³. The empirical value is mass³ᐟ⁴. The discrepancy is not a failure — it is evidence of additional structure (fractal branching in biological networks) that traditional geometry alone does not capture. This mirrors the course’s broader epistemological thread: mathematical prediction combined with empirical observation is how knowledge grows. The model’s failure to fully predict Kleiber’s Law is itself a scientific finding that leads to new discoveries.

 

The Neptune story (Le Verrier and Adams predicting Neptune’s position because Uranus deviated from prediction) is available as a second example of the same pattern: discrepancy between mathematical prediction and observation leads to discovery. Use it if your class is engaged and has time.