Unit 5
Quadrilaterals
Unit 5: Quadrilaterals asks students to see every quadrilateral proof as a triangle proof in disguise. The unit opens with the Birthday Cake Problem and a bucket sort that let the quadrilateral family emerge from constraints and student-created categories. From there, students prove parallelogram properties, add constraints to get rectangles, rhombi, and squares, and then tackle trapezoids and kites—all by drawing diagonals and using congruence tools from Units 3 and 4. A closing coordinate day re-proves the same results with slope, distance, and midpoint, highlighting hierarchy, inheritance, and the idea that two proof systems can agree on the same geometric truths.
For the full story of how this unit is sequenced, what changed from Year 1, read the complete Unit 4 overview and navigation instrument below.
Unit 5 Lessons
Lesson 5.1 – The Birthday Cake Problem and the Quadrilateral Family
Students meet quadrilaterals as solutions to a constraint puzzle, not as a list of names. The bucket sort then turns their informal categories into a formal hierarchy, tying directly back to Unit 2’s work on conditionals and converses.
Lesson 5.3 – Special Parallelograms: Rectangles, Rhombi, and Squares
Students add one constraint at a time to build rectangles, rhombi, and squares from parallelograms, proving each new diagonal property from earlier tools. The square’s properties then fall out entirely by inheritance, making the hierarchy feel like a genuine time‑saver, not a chart to memorize.
Lesson 5.4 – Trapezoids and Kites
Students step outside the parallelogram family to prove properties of isosceles trapezoids and kites using auxiliary constructions and triangle congruence. Symmetry becomes a guide for where equal parts should live, while the actual arguments stay grounded in the same congruence tools as earlier units.
Lesson 5.5 – Coordinate Proof Day
Students re‑prove quadrilateral properties using slope, distance, and midpoint, turning earlier algebra tools into a second proof system. The emphasis is on writing complete classification arguments and seeing that coordinate and Euclidean proofs agree on the same geometric truths.
Lesson 5.6 – Practice Day
Students work through a mixed set of problems across all five navigation‑instrument rows, choosing levels honestly and moving up as they are ready. The day is about consolidating proof, classification, and coordinate skills before the assessment, with the teacher using common errors to guide a short closing debrief.
Unit 5 Assessment
I’m happy to share my Unit 5 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.