Labelling the sides of a right-angled triangle

What this sequence teaches

Before any trigonometry, students must name the three sides of a right-angled triangle relative to one marked angle. The hypotenuse is across from the right angle; the opposite is across from the marked angle; the adjacent is the side that’s left. The response to a triangle labelled 8, 15 and 17 is “H 17, O 8, A 15”: no ratios, no working. Mislabelling is where most right-angled trigonometry errors start, so it is pulled out here to be practised on its own and seen on every whiteboard.

The teaching sequence

Example 1 — an 8-15-17 triangle in the textbook view. The right angle is bottom-right and the marked angle bottom-left. The plainest case.

Example 2 — only the mark moves, to the other corner. The hypotenuse stays 17; the opposite and adjacent swap. This one change breaks three habits at once: “the upright side is the opposite” (the opposite now lies along the bottom), “the opposite is the short one” (it is now 15), and “each side has its label” (the labels belong to the angle, not the triangle).

Example 3 — the triangle turns until the hypotenuse lies flat along the bottom. The answer doesn’t change. Until now every hypotenuse has slanted across the top, which invites “the hypotenuse is the slanted side”; this example removes that before testing asks for it.

Running the sequence

• Keep it fast and uninterrupted — about thirty seconds. The testing sequence is where students take over.
• Reveal follows the rule in order. The right-angle mark thickens, a dashed arrow goes across from it, that side traces and its length flies to H. A pause; the same for the marked angle, and its side’s length flies to O. A pause; the side that’s left traces and its length flies to A.
• Moving to Example 2 is deliberately slow: the answer dims, the old mark fades out, then the new mark appears and pulses once. Example 1’s answer stays on screen, dimmed and labelled “Previous answer”. Ask: “How little can you change your answer?” Reveal keeps H 17 in place while the 15 and the 8 swap places, arcing past each other.
• Moving to Example 3 is a slow turn. On Reveal every value brightens where it is; nothing moves.
• Replay plays everything since the last example: before Reveal, the change to the question; after Reveal, the change and then the reveal.

The rule and the script

The rule, exactly as Show rule displays it: For the following examples: find the right angle. The side across from it is the hypotenuse. Find the marked angle. The side across from it is the opposite. The side that’s left is the adjacent.

• Example 1 — “The side across from the right angle is the hypotenuse: 17. The side across from the marked angle is the opposite: 8. The side that’s left is the adjacent: 15.”
• Example 2 — “The marked angle has moved. The hypotenuse is still 17. The side across from the marked angle is now the 15, so that’s the opposite, and the 8 is left: the adjacent.”
• Example 3 — “The triangle has turned, but nothing about it has changed. The hypotenuse lies flat now, and it is still across from the right angle. The answer stays the same.”

When Why? is open, the page steps through the three stages: the angle, its side and its value in the answer turn blue, underline and pulse. Colour is never the only cue. Whether students see the rule and the reasons is your call.

Why these conventions?

• Why “across from”, not “opposite”? If the hypotenuse were “opposite the right angle”, two sides would be “opposite” something — which is where “O 17” comes from. Here the word opposite only ever names the answer.
• Why “the side that’s left”, not “next to the angle”? The hypotenuse is next to the marked angle too. Finding the hypotenuse first, then taking what’s left, stops it being labelled adjacent.
• Why H, O, A in that order? It is the order of the rule, and the adjacent can only be found last. On whiteboards any order is fine, as long as each letter sits with its value.
• Why lengths on the sides? Students write the answer in seconds without redrawing the triangle, and it is the form trigonometry questions take.
• Why no size on the marked angle at first? The size changes nothing; a whole-number triangle’s angles aren’t whole numbers; and a number invites “does it matter?”. Degrees and θ arrive in the expansion sequence.
• Why are the triangles drawn accurately? Every diagram is drawn to scale from its labels, so what students see never contradicts what they read.

Where this atom sits

Before it: recognising a right-angle mark and a marked angle. After it: which ratio links the two sides in play; then setting up the equation with every value in brackets, e.g. sin(θ) = (5)/(x), with no working out; then rearranging.

About the testing sequence

Six items in a designed order. The first is a reset: a new triangle, so students have to use the rule rather than adapt Example 3’s answer. After that, some items change exactly one thing from the item before, and two more resets start again from nothing. Show the item, students write on mini-whiteboards, then reveal. There is no scoring — the whiteboards are the data.

What each item is diagnosing

1. 9, 12 and 15, marked angle at the top (reset): H 15, O 12, A 9. “O 9” means the upright or the shorter side was taken as the opposite. “O 15” means the right angle was treated as the angle. “A 15” means the adjacent grabbed the hypotenuse.
2. The same triangle, turned until the hypotenuse stands upright. The answer doesn’t change. A changed board means the student relabelled by position; “H 12” or “H 9” means “the hypotenuse is the slanted side”. Ask: “Did you change your answer? Why not?”
3. The mark moves to the other corner: H 15, O 9, A 12. A board still showing O 12 means the labels were stuck to the triangle. Items 2 and 3 are the key pair: a student labelling by position gets one right and one wrong; a student following the angle gets both.
4. 20, 21 and 29, hypotenuse along the bottom (reset): H 29, O 20, A 21. “A 29” means “the bottom side is the adjacent”, or “the adjacent is the side next to the angle”.
5. The mark moves along the hypotenuse: H 29, O 21, A 20. Neither short side is upright and their lengths are almost equal, so no position or length cue is left. An unchanged board means the labels were stuck to the triangle; O and A the wrong way round usually means guessing between two similar lengths.
6. 5, 12 and 13, tilted so no side is flat or upright (reset). An error here means a position cue is still in use somewhere. Ask the student to say the rule, then point.

Common confusions

Three stand out. Position: “the opposite is the upright one, the adjacent is the bottom one” comes from textbook pictures; Items 2, 3, 4 and 6 catch it. The hypotenuse as adjacent: it really is next to the marked angle, which is why the rule finds it first. The right angle as the angle: students label everything relative to the right angle, and the board shows “O 15” or “O 29”.

Discussion prompts

• “Say the rule, then use it on this one.”
• “Point to the right angle. What’s across from it? Now point to the marked angle. What’s across from that?”
• “Same triangle as last time. Why did the answer change this time and not last time?”

About the expansion sequence

Eight items, each adding one new thing, with no further instruction. The marked angle in Items 1–5 is bigger than 45°, so when it gets a label (Items 4 and 5) the label sits right beside the angle it names; don’t skip items or run them out of order.

What each step adds

1. x on a shorter side. Copy exactly what is written: H 10, O x, A 6.
2. x moves to the hypotenuse. Students who have been finding the hypotenuse by picking the biggest number stall here; the right angle still decides.
3. One side has no length. It still gets its letter, written “A –”. Most real questions label only two sides, and students often skip the third.
4. The marked angle is 53°. Nothing changes. A student who writes the 53 somewhere hasn’t yet separated angles from sides.
5. θ instead of the number. The same marked angle under a new name, ready for the next atoms.
6. Two triangles sharing a side. Two right angles, one marked angle: label the triangle the marked angle is in. Boards with the 20 or the 16 used the wrong triangle; “9 + 16” labelled the outer triangle, which isn’t right-angled.
7. The mark moves to the top of the other triangle. The shared 12 goes from opposite to adjacent — the clearest demonstration on the page that a label belongs to the angle.
8. Letters, with the two shorter sides equal. Only the marked angle can separate O from A. The answer, H r, O q, A p, runs in reverse alphabetical order, so reading the letters in order fails.

Where classes stall

• Item 2, among students who never really used the right angle.
• Item 6, where two right angles make “find the right angle” feel ambiguous until students look for the marked angle first.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “There are two right angles. Which triangle is the marked angle in?”
• “The 12 was the opposite a moment ago. Why is it the adjacent now?”