Thinking Through Geometry - Free Lessons, Notes, and Tasks for Thinking Classrooms.
0.2 Teacher Commentary
The Palindromes task does several things that the first lesson did not. It rewards organization in a way that students can feel — groups that list palindromes in order notice the +110 / +11 pattern; groups that check cases randomly miss it entirely. That difference is worth naming explicitly during consolidation: how you arrange what you have found often determines what you are able to see.
The algebraic generalization — writing any four-digit palindrome as ABBA and attaching place values — is the mathematical heart of the lesson. It is also the first time in the course that students encounter the move from specific examples to a proof that covers all cases. The argument is accessible: 1001A + 110B = 11(91A + 10B), so every four-digit palindrome is divisible by 11, without exception, because the algebra says so. Walk through it carefully during consolidation and then ask: can you see any way this argument could fail? The answer is no — and that matters, because students are starting to feel the difference between inductive evidence and deductive certainty.
This lesson also includes two cultural activities that are as important as the mathematics: the group work rubric co-creation and the growth mindset video. The rubric matters because it gives students ownership over the norms they are being held to, and because the act of describing good and bad group work requires them to articulate what collaboration actually looks like. The video — Jo Boaler on growth mindset and mathematical learning — matters because many students arrive carrying the belief that mathematical ability is fixed, and addressing that belief early changes what they are willing to attempt.
On the conjectures in Period 6
In one section, students generated several conjectures beyond the main task: that palindromes with digit count divisible by 4 are always divisible by 11; that AAAA palindromes divided by 11 yield A0A; and — the most interesting one — that when B > A in an ABBA palindrome, the result of dividing by 11 is itself a palindrome.
These were not the intended direction of the lesson, but they were intellectually alive and worth honoring. The right move, which is also the harder move, is to celebrate the curiosity while redirecting toward the question: do we know any of these are always true, or do we have evidence that they might be? That distinction — between a conjecture supported by examples and an argument that covers all cases — is exactly what this course is building toward. Even if we did not arrive at proofs, we arrived at the question that proof is the answer to.
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