Unit 1- Foundations of Geometry
This unit will look familiar to anyone who has taught or taken a traditional geometry course. Definitions, notation, distance and midpoint, area and perimeter, angle relationships. In many ways it mirrors the opening chapter of most geometry textbooks. For a full description of how and why the unit is sequenced as it is please download the Navigation Instrument and Unit Overview.
Unit 1 Lessons
Lesson 1.1 Day 1 – What Is a Definition?
Students build and test their own definition of “shoe,” discovering why good definitions need both classification and differentiation. The circular‑definition moment motivates the need for undefined terms and sets up point, line, ray, segment, and plane as the primitive building blocks of the course.
Lesson 1.1 Day 2 – Notation and the Foundational Objects
Through a card‑task, students must communicate using only symbols and words, They finish the foundational vocabulary and learn to distinguish line, ray, and segment notation, experiencing why precise symbols matter before they ever see a formal proof.
Lesson 1.2 – Distance and Midpoint in the Plane
Starting from a Google Maps puzzle and a taxicab‑geometry story, students reason their way to the distance formula and midpoint formula. They see the hidden right triangle inside the coordinate plane and connect distance back to the Pythagorean theorem and forward to proportional reasoning for similarity.
Lesson 1.3 – Polygons, Perimeter, and Area in the Plane
A security‑camera task brings back the work on definitions as students sort convex and concave polygons by what they have in common. They use the distance formula for slanted sides, derive the triangle area formula from a rectangle, and begin to see perimeter and area as connected ideas on the coordinate grid.
Unit 1 Differentiation Day – Cumulative Practice and Navigation Work
nStudents use the navigation instrument to choose problem sets that target their own gaps across Lessons 1.1–1.3. The day is built around spaced retrieval: revisiting distance, midpoint, slanted‑side perimeter, and foundational vocabulary so that honest self‑assessment leads directly to focused practice.
Lesson 1.4 – Angle Relationships and the First Proof
Students move from naming angles precisely to discovering linear pairs and vertical angles. Measuring leads to a conjecture; then they build a genuine deductive argument that vertical angles must be congruent. This is framed explicitly as their first proof, grounded in definitions they helped construct.
Optional Review Day – Targeted Practice and Honest Self‑Assessment
If used, this day gives students another low‑stakes chance to work with the navigation instrument before the unit assessment. They identify a specific weak row, work on aligned problems, and update their self‑assessment, reinforcing the habit of using the instrument as a living document rather than a form to fill out.
Unit 1 Assessment
I’m happy to share my Unit 1 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.