Common content, so this runs for both courses.
Ask for the angle before you show them anything. Collect three numbers on the board, then run the three triangles. Most of the room will have said something in the thirties, and the correct answer is 26.6, so the reveal does the teaching rather than you.
The useful part is not that they were wrong. It is that every wrong answer was too big, and once that is on the board they have a check they can use for the rest of their lives: an angle to the base that comes out larger than you expected means you measured along an edge.
| Question | Answer |
|---|---|
| 1. Angle for a 9 by 12 by 8 box | Base diagonal 15, then tanθ = 8/15, so 28.1°. |
| 2. Its space diagonal | √(81 + 144 + 64) = √289 = 17 exactly. |
| 3. The student who answered 33.7° | B. That is tan⁻¹(8/12), so they used the 12 edge. |
1 markFinding the base diagonal, and showing it.
1 markA correct trigonometric statement using it.
1 markThe angle to the required accuracy.
A labelled diagram with the triangle picked out earns the first mark by itself, even when the arithmetic afterwards collapses. Students who refuse to draw lose a mark they were given for free.
| They wrote | What happened |
|---|---|
| 33.7° | Used the 12 edge. The commonest answer in the room. |
| 41.6° | Used the 9 edge. Same error, shorter edge, bigger angle. |
| 61.9° | Triangle inverted: they took tan⁻¹(15/8). Worth asking which side is opposite the angle. |
| 15 | On question 2, stopped at the base diagonal and never went up. |
| 29 | On question 2, added 9 + 12 + 8. They have the right idea and the wrong operation. |
"Why can't I just use the longest edge?" Because the triangle has to close. Put a finger on the corner under the far end of the cable and ask them which floor line reaches it.
"Is the formula with three squares a different method?" No, it is the same method with the first step substituted in. Derive it once in front of them and they stop treating it as a fourth thing to memorise.
| Stage | What to do |
|---|---|
| Demonstrate | Do the two steps as two separate calculator entries, writing the 10 on the board between them. Then do it again storing the base diagonal, so they see that rounding it to 10.0 and to 10 give the same answer here but would not always. |
| Where they stick | Leaving the machine in radians, which turns 26.6° into 0.46 and looks like a different kind of error. Also taking tan⁻¹ of a length rather than a ratio. |
| The check | Ask what tan⁻¹(0.5) should be before anybody presses a key. Anyone who cannot say "about 26 degrees" is not reading their own answer. |
Ask which machine each student has before the lesson, not during it. Degrees mode is the setting that ruins this whole topic, and the two machines show it in different corners of the screen.
| Step | What |
|---|---|
| 1 | Ask for the angle cold. Collect three guesses. Do not comment on them. |
| 2 | Run the three triangles. Stop on the second one and ask why it is worse, not just wrong. |
| 3 | Derive the two-step method, then the three-square shortcut from it. |
| 4 | Distance and midpoint, which are quick and give the room a win. |
| 5 | Solids, finishing on the sphere where 36π appears twice. |
Do not say "just use the 3D Pythagoras formula". It is the thing they will write down without understanding, and it does not help at all with the angle, which is what is actually examined.
Do not call the base diagonal obvious. It is the one line in the question that is not drawn on the solid, which is precisely why the whole room misses it.