Teach & case
Help a near-peer reason through the idea — when you explain it, you understand it best — then read the real decision it comes from.
Help Renard prove it
In a minute you'll be the teacher. Get ready to teach Renard: proof.
A diagram that looks right isn't a proof. A proof uses the givens and theorems, step by step.
What you've shown you can teach
A case · you decide
Renard and the triangle that looks right
Renard — a student who measured the base angles in a careful drawing and wants to call the result proved
Renard drew an isosceles triangle carefully, put a protractor on both base angles, and got 52° twice. "See — the base angles are equal, proved." But a drawing is one picture of one case, drawn by hand. Does measuring it settle the question for every isosceles triangle?
A diagram can mislead in two ways: it can be drawn slightly inaccurately, and it can accidentally show a SPECIAL case that isn't true in general. "They look equal" and "they measure 52° in this drawing" are observations about the picture, not the theorem.
A proof is a chain from the GIVENS, through definitions and theorems, to the claim — so it holds for every figure satisfying the hypotheses, not just the one on the page. That's the difference between "convincing" and "certain."
Renard's protractor reads 52° for both base angles. Is "the base angles of an isosceles triangle are equal" proved?
A step in Renard's write-up says "these two segments are equal (you can see it in the figure)." Is that a valid step?
Renard's whole move is turning "it looks true" into "here's why it must be" — the picture starts the conjecture, and the givens finish the proof.
When two ideas meet: 2 cases with two people
A case · you decide
Castell and Vireo: moved, but the same shape
Castell & Vireo — a student who moves shapes around, and a friend who asks when two shapes are the same
Castell rotates triangle ABC 90° about the origin, then enlarges a copy by a factor of 2.
Vireo measures side AB before and after each move.
Which statement is true?
Castell moved the shapes. Vireo checked what the moves kept.
A case · you decide
Talise and Osgood: the slope of a tangent
Talise & Osgood — a student who works in coordinates, and a friend who knows the circle theorems
A circle centred at the origin has radius 5 and passes through (3, 4). Talise needs the tangent line at that point.
Osgood reminds her that the tangent meets the radius at a right angle.
The radius from (0, 0) to (3, 4) has slope 4/3.
What is the slope of the tangent at (3, 4)?
Osgood's theorem gave the angle. Talise's coordinates gave the number.