Lesson 1.1 — Day 1 of 2
What Is a Definition?
Focus: the structure of good definitions; undefined terms; why every system must start somewhere
This lesson has two jobs: to make students feel, from the inside, why shared definitions are necessary and why some terms cannot be defined. The shoe task does the first half. If you’re lucky, the classroom discussion will do the second half for you.
The shoe task comes from Math Medic, whose free geometry materials are a valuable resource. The setup is simple: bring a collection of footwear to class — dress shoes, boots, sneakers, sandals, slippers, a doll shoe, an ankle bracelet, a thick sock — anything that might or might not meet whatever definition students come up with. Groups write a definition. You test it against the objects. They revise. Repeat.
Math Medic Geometry Materials — Math Medic (mathmedic.com)
The shoe task and the card task in Lesson 1.1 Day 2 are both sourced from Math Medic’s free geometry curriculum. Their materials are well-designed, freely available, and worth exploring alongside this course.
What makes this task work is that students are genuinely invested in their definitions — they argue, they poke holes in each other’s language, they revise. Encourage the debate. The point of the class is to construct convincing arguments and to respectfully find the flaws in others’. This task gives them a low-stakes arena to practice both.
The circular definition moment may or may not happen organically. In my first year teaching this lesson, it happened perfectly by chance. I took a word from one group’s definition and asked another student to define it. From their definition I took another word. We passed back and forth until one student said their word meant “to create” — and when I asked the next student what “create” meant, they said “to make.” Back to the first student: “what does ‘make’ mean?” “Well… to create.” The room recognized the loop immediately.
The philosophical payoff
The circular definition moment is not just a fun classroom trick. It is the clearest possible illustration of why every axiomatic system must begin with terms accepted without definition. If you define every word, you eventually either loop or regress infinitely. The only way out is to choose a small number of terms so fundamental, so widely agreed upon, that we accept them and build everything else from there.
Euclid actually did attempt to define his foundational terms. He wrote things like ‘a point is that which has no part’ and ‘a line is a length without breadth.’ These are so spare they function more as descriptions than genuine definitions — they tell you almost nothing useful while creating the appearance of having said something. Modern geometry recognized this and made the situation explicit: point, line, and plane are declared undefined terms, accepted as primitive notions on which everything else is built.
Downloads for Lesson 1.1 Day 1
Lesson Plan
Download the full Lesson 1.1 Day 1 plan (tasks, timing, and teacher notes).
CYU Problems
Download the Check Your Understanding problems for Lesson 1.1 Day 1. (best reserved for day 2)
Student Note Sheets
Download the student note sheets for Lesson 1.1 Day 1. May need to be finished day 2