Unit 6
Similarity and Scaling
This unit weaves together proportional reasoning, triangle similarity, and indirect measurement to show how geometry can measure the world and explain why “bigger” is not just bigger. Start by exploring the Unit 6 Navigation Instrument and Unit Overview to see how the lessons connect, what students will build, and where the key moments of productive struggle live.
Unit 6 Lessons
Lesson 6.1 – Similarity in Light and Shadow
Use Mount Rushmore and a flashlight shadow investigation to surface students’ informal ideas about similarity. Students reason proportionally, see scaling from a point in action, and describe what stays the same and what changes when a figure is scaled — all before any formal definitions appear.
Lesson 6.2 – Dilation and Scale Factor
Formalize dilation and similarity using a shadow diagram instead of physical materials. Students define center and scale factor, compare scale-factor and proportion methods side by side, and write precise similarity statements that encode correct correspondence.
Lesson 6.3 – When Is Similarity Guaranteed?
Investigate which conditions actually guarantee triangle similarity by hunting for counterexamples. Students test and refine conjectures until AA, SAS∼, and SSS∼ emerge as postulates, and see why failing to find a counterexample is strong evidence but not a proof.
Lesson 6.4 – Measuring the Inaccessible
Use similar triangles to measure heights that can’t be reached directly. Through Thales’ pyramid story, outdoor shadow work, and mirror and line-of-sight methods, students justify similarity with AA before writing any proportions and see geometry as a genuine tool for “measuring the world.”
Lesson 6.5 – When Size Changes, What Changes With It?
Explore how perimeter, area, and volume scale when a figure grows by a factor of k. Students discover the square–cube law, connect it to biological constraints like animal size and heat loss, and see how similarity explains why nature cannot simply scale organisms up.
Lesson 6.6 – Review
Consolidate the unit’s tools — scale factor, proportions, similarity shortcuts, and indirect measurement — in problems that still demand reasoning, not rote rehearsal. Assessment items separate calculation from justification so students must name the similarity step that underpins each proportion they write.
Unit 6 Assessment
I’m happy to share my Unit 6 assessments and grading rubrics for teacher use, but I don’t want them publicly available on the internet. If you’d like a copy, please email me from your school account at jay@thinkingthroughgeometry.net and let me know which school and state you teach in.