Teach the Cast
Some of MathCircle’s cast are still learning. When you teach them, YOU learn it best — catch what they get wrong, then quiz them.
Teach Tess Try-Small
Teach the connections
Show how the problem-solving moves connect
Tess Try-Small (specializing — try the smallest cases first) × Gemma General (generalizing — find the pattern and state the rule for all n)
Tess found the sums 1, 3, 6, 10 for the first 1, 2, 3, 4 numbers. What does Gemma DO with those four numbers?
Tess Try-Small then says: “You should generalize FIRST (guess the rule for n), then only check small cases if you have time.” — is that right?
Now they see it: Tess's baby cases aren't busywork — they're exactly the evidence Gemma turns into a formula.
Wendy Wonder (noticing & wondering — really look at the problem before charging in) × Hattie Hunch (conjecturing — make a bold, testable guess)
Hattie is about to guess a formula. Where should her guess come from?
Wendy Wonder then says: “Skip the noticing — a strong solver just jumps straight to a bold conjecture with no observation behind it.” — is that right?
Now they see it: a wild guess and a good conjecture differ by one thing — whether you noticed something first.
Hattie Hunch (conjecturing — make a bold, testable guess) × Cass Check (sense-checking — verify the answer actually makes sense)
Hattie guesses "the sum of the first n odd numbers is n²." What does Cass do next?
Hattie Hunch then says: “Once you've made a confident conjecture, checking it is a waste of time — confidence is enough.” — is that right?
Now they see it: the bold guess and the humble check belong together — that pairing is how guessing becomes math.
Capstone — crack "the sum of the first 100 odd numbers" as a team
The problem: what is 1 + 3 + 5 + 7 + … (the first 100 odd numbers)? Work it the math-circle way — each phase below needs ONE strategy. Bring in the right person for each.
First — really look at the problem and ask what's interesting about a running sum of odds.
Compute the sum of the first 1, 2, 3, 4 odd numbers to get some data.
The sums are 1, 4, 9, 16. Make a bold guess for the pattern.
State the rule for all n and why the sum of the first n odds is n².
Someone says the sum of the first n odds is 400. Work backwards to find n.
Check the whole thing: does the rule give 100² = 10000 for 100 odds, and is that sensible?
The whole circle, working together — because you taught each move and saw how they hand off to each other.
Your concept map