Which two lengths tell the area of the triangle?

What this sequence teaches

The area of a triangle needs two lengths at right angles to each other. Find the right angle: the two lengths are the line that makes it and the side it meets. Every other length on the diagram is left out. The response to a triangle labelled 9, 8 and 10 is “8, 9” — not 36, and not ½ × 9 × 8. Picking the two lengths is where most area errors actually happen, so it is pulled out here to be practised on its own and seen on every whiteboard.

The teaching sequence

Example 1 — a triangle labelled 9, 8 and 10. The plainest case: the dashed line drops inside the triangle and meets the side of 9 at a right angle. The 10 is longer than both, so “take the two biggest” fails from the start.

Example 2 — the top corner slides sideways. The top corner slides parallel to the bottom side until the dashed line falls outside the triangle. The 8 and the 9 don’t move; the slant side changes from 10 to 17, and the answer doesn’t change at all. This is the example that teaches the right angle can sit outside the triangle, and that a length can change without the answer changing.

Two examples are enough. Turning the triangle, a dashed line from a different corner, and a labelled extension are all left to the testing sequence, where students have to produce the answer rather than watch it.

Running the sequence

• Keep it fast and uninterrupted — well under a minute. The testing sequence is where students take over.
• Reveal follows the rule in order. First the right-angle mark pulses and the line that makes it traces; its length flies into the answer. A short pause, then the side it meets traces and its length flies in. Last, the lengths that weren’t used dim: they are still there, just not needed.
• Moving to Example 2 is deliberately slow. First the answer dims. Then the 10 fades out, and the top corner slides across, taking the dashed line with it; as the right angle passes the end of the side, a dashed extension grows to meet it. Finally the 17 fades in and pulses once.
• Replay plays the change again from Example 1. Before Reveal it plays just the change to the question; after Reveal, the change to the question and then the reveal. On Example 1, Replay runs the reveal again.
• On Example 2, Example 1’s answer stays on screen, dimmed and labelled “Previous answer”. Ask: “How little can you change your answer?” Reveal traces the two lengths again, and the answer simply brightens. Nothing in it moves.
• Halfway through the slide there is an instant when the dashed line lies exactly along a side and the triangle is right-angled. Nothing needs saying about it now; the expansion sequence comes back to it.

The rule and the script

The rule, exactly as Show rule displays it: For the following examples: find the right angle. The two lengths that tell the area are the line that makes the right angle and the side it meets. Leave the other lengths out.

• Example 1 — “I found the right angle. The line that makes it is 8, and the side it meets is 9. The 10 doesn’t tell the area.”
• Example 2 — “The dashed line is outside the triangle now, but it still makes a right angle with the side: 8 and 9. The 17 doesn’t tell the area.”

When Why? is open, the page steps through the answer: the right-angle mark pulses and the line that makes it is underlined in the diagram and in the answer, then the side it meets is, and the lengths that were left out stay dimmed. Colour is never the only cue, so it works for colour-blind students and on a washed-out projector. Whether students see the rule and the reasons is your call.

Why these conventions?

Why not “base” and “height”? The words carry a picture — the base at the bottom, the height upright — that is exactly the misconception the testing sequence sets out to catch. The two lengths just need to be at right angles to each other, and either can be written first.

Why no ½, and no working out? Putting the lengths into the formula is the next step, and working out the area is the one after that. Each is a separate step and is never asked for here.

Why no units? They add nothing to the choice of lengths. Mixed units would bring in a conversion, which is a calculation, and this atom has none.

Why are the triangles drawn accurately? Every diagram is drawn to scale from its labels, so what students see never contradicts what they read. Every triangle is possible.

Where this atom sits

Before it: recognising a right-angle mark, and knowing that a side can be extended with a dashed line.

After it: setting up the area, A = ½(8)(9); then working it out; later, the same choice for parallelograms and trapeziums, where the two lengths are again at right angles.

About the testing sequence

Six items in a designed order. The first is a reset: a new triangle, unlike either teaching example, so students have to use the rule from scratch rather than adapt Example 2’s answer. After that, some items change exactly one thing from the item before, and a second and third reset start again from nothing. Show the item, students write on mini-whiteboards, then reveal. Go through them in order and reveal each answer before moving on. There is no scoring — the whiteboards are the data.

What each item is diagnosing

1. 15 and 12, with 18 as the extra side. Reset. A tall triangle. “18 and 15” means the student took the two biggest.
2. The same triangle, turned until the 18 lies along the bottom. Nothing about the triangle has changed, so nothing about the answer should. An 18 on the board means the student chose the side at the bottom. Ask: “Did you change your answer? Why not?”
3. The same triangle, in the same position, with the dashed line redrawn from another corner onto the 18. Now the answer changes: 10 and 18. A board still showing 15 and 12 means the student recognised the triangle and repeated themselves; 15 and 18 means the old dashed line was carried over. Items 2 and 3 are the key pair: a student who picks “the side at the bottom” gets Item 2 wrong and Item 3 right; a student who follows the right angle gets both.
4. 8 and 5: the right angle is outside the triangle. Reset. The triangle is turned so no side is flat. The 10, which was on the answer to Item 3, is a slant side here: a 10 on a board means the student edited their last answer.
5. The same triangle, with the dashed extension labelled 6. The answer doesn’t change. 11 (5 + 6) or 6 means the dashed extension was taken as part of the side; “8 and 6” means “the two dashed lines”.
6. 9 and 14, with the 15 standing upright. Reset. A 15 means the student chose the upright line rather than the right angle.

Common confusions

Two stand out. The first is the upright line: many students have only ever seen triangles sitting on a flat side, so the length they want has come to mean “the vertical one”. Items 2, 3 and 6 are there to catch it. The second is the dashed extension: the line that makes the right angle and the extension are both dashed, so “take the dashed lines” feels like a rule. Item 5 is there for it: the rule says the side it meets, and the extension isn’t a side.

Discussion prompts

• “Say the rule, then use it on this one.”
• “Point to the right angle. Which line makes it? Which side does it meet?”
• “Same triangle as last time. Why did the answer change this time and not last time?”

About the expansion sequence

Seven items that show how far the rule reaches. Every one uses the same rule on something new, and each adds one thing; don’t skip items or run them out of order. The first item is one change from the last test item, so the card opens on familiar ground. The decimals, fractions and letters need students who read them comfortably.

What each step adds

1. Decimals. The same drawing as Test Item 6, labelled 1.4, 0.9 and 1.5. Lengths are copied exactly as written.
2. Every side labelled. Two extra lengths, all close together: 7, 6, 6.5 and 7.5. The longest side lies along the bottom and doesn’t tell the area.
3. A very thin triangle. The side is only 3, the dashed line lands far away along the extension, and all three sides are labelled: 3, 25 and 26, with the dashed line 24. “The two biggest” and “the two closest together” both fail.
4. A right-angled triangle, with no dashed line. The right angle is at a corner, so the two sides that make it tell the area. The reveal turns one of them dashed while it traces, then lets it turn solid again. The 13 is the longest side and doesn’t tell the area.
5. A side in two parts, 6 and 7, never labelled as a whole. The answer is “9, (6 + 7)”. The sum is written, not worked out: this atom never calculates.
6. Fractions: 1½ and 5½.
7. Letters. With no numbers there’s no shortcut left: the answer is n and p, and only the rule gets there.

Where classes stall

• The right-angled triangle. Students look for a dashed line and there isn’t one. Ask: “Where’s the right angle? What meets there?”
• The side in two parts. Some write 6 or 7, taking the part nearest the dashed line. Others write 13, which is right but did a sum this step doesn’t ask for; accept it, and point at the brackets.
• The letters. Some students freeze without numbers. The rule never used the numbers, only the right angle.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “Where’s the right angle when there’s no dashed line?”
• “Why is the answer (6 + 7), and not 6?”