Classification of Triangles
Warm-up
Give pairs six straws cut to lengths 3, 4, 5, 5, 5, and 9 cm and one challenge: "Build as many DIFFERENT triangles as you can. Record the side lengths of each." Within minutes someone tries 3-4-9 and discovers the sides won't close.
Two discoveries surface before any vocabulary: triangles come in genuinely different types, and not every trio of lengths even makes a triangle. Both drive today's classification work.
Explore
Classification museum: pairs sort a zoo of paper triangles two independent ways — by SIDES (all equal / two equal / none equal) and by ANGLES (all acute / one right / one obtuse). Then the cross-classification grid: can a triangle be both isosceles AND right? (Build one: legs 5-5 and the right angle between them ✓.) Equilateral and obtuse? (Every attempt fails — equilateral forces all angles to 60°.)
The grid with possible/impossible verdicts, each backed by a build or an argument, is the product. Impossible cells need a WHY, not just a shrug.
Formalize
Formalize the two independent classification axes and the naming convention (side-name + angle-name):
The tick-mark and angle-arc notation carries the classification: matching tick marks claim equal sides; the square corner claims 90°. Reading and writing these marks is the literacy skill — a triangle with marks is a set of CLAIMS, checkable with ruler and protractor.
Practice
Practice: classify eight triangles by both axes from their marks (not by measuring); draw a triangle for each possible grid cell; two "impossible or possible?" challenges with one-sentence justifications.
Exit ticket: "A triangle has angles 90° and 45°. Classify it completely." (Third angle 45° → isosceles right triangle — the angles force the sides.)
Exit ticket
Practice: classify eight triangles by both axes from their marks (not by measuring); draw a triangle for each possible grid cell; two "impossible or possible?" challenges with one-sentence justifications.
Exit ticket: "A triangle has angles 90° and 45°. Classify it completely." (Third angle 45° → isosceles right triangle — the angles force the sides.)
The triangle: two sides carry matching single tick marks; the angle between those two sides carries a square corner mark.
Step 1: Read the side claims: two sides equal → isosceles (at least).
Step 2: Read the angle claim: the marked angle is 90° → right triangle.
Step 3: Full name: right isosceles triangle.
Step 4: Deduce the remaining angles as a bonus: the two base angles must be equal (isosceles) and share → 45° each. The marks determined EVERYTHING — no protractor touched the page. That's the power of notation read correctly.
The trios: (a) 3, 4, 5 (b) 3, 4, 9 (c) 4, 4, 8.
Step 1: The principle from the warm-up, now stated: the two shorter sides must together OUT-REACH the longest side, or the arms can't close.
Step 2: Test (a): ✓ — builds (the famous right triangle, in fact).
Step 3: Test (b): ✗ — the short sides fall flat before touching.
Step 4: Test (c) — the edge case: . The arms just barely reach, collapsing FLAT into a line segment: a degenerate "triangle" with zero area. Verdict: no honest triangle. Equality fails; the inequality must be strict.
Step 5: Name it for the future: the Triangle Inequality — met here as a straw fact, returning in Grade 9 as a theorem.
The challenge: build or refute a triangle that is both equilateral and obtuse.
Step 1: Attempt honestly: draw an obtuse angle (say 120°) and try to complete an equilateral triangle from it. The two arms must be equal — fine — but closing the third side always produces a LONGER side opposite the fat angle.
Step 2: The argument that settles it: equilateral triangles have all sides equal, which forces all angles equal, and three equal angles summing to 180° must each be 60°. Sixty is acute — no room for an obtuse angle, ever.
Step 3: Contrast with isosceles obtuse: 120°-30°-30° with two equal sides builds happily ✓ — the impossibility was specific to equilateral, not to "pointy shapes with equal-ish sides."
Step 4: The takeaway about mathematical truth: one failed attempt proves nothing (maybe we drew badly), but the angle-sum argument closes every possible attempt at once. Builds show possibility; arguments show impossibility.