Higher Level only. Standard Level does the same sectors in degrees, with θ/360, and the page at 3.4 deliberately tells them not to carry ½r²θ back there.
Go straight to the third view and leave it up. It draws 18.85 against 1080 inside a circle whose whole area is 113.1. The bar for 1080 is nine and a half times the circle. You do not have to say anything; someone will.
Then wind back to the one-radian view, where the arc and the radius are drawn as the same length. That is the definition, and it is the only thing in this sub-topic worth learning as a picture: a radian is the angle whose arc is one radius. Everything else is arithmetic from it. 2π radii fit round the circle, so a full turn is 2π.
The order matters. Definition first and the catastrophe second does not land, because they have no reason yet to care. Catastrophe first and they want the definition.
| Question | Answer |
|---|---|
| 1. 60° in radians | 60 × π/180 = π/3 = 1.047. |
| 2. Sector area, r = 6, θ = π/3 | ½ × 36 × π/3 = 6π = 18.85 m². |
| 3. Arc length, same sector | 6 × π/3 = 2π = 6.28 m. |
| 4. One radian in degrees | 180/π = 57.3°. |
| 5. The fastest check on an answer of 1080 | B. The whole circle is 113.1, so a sector of it cannot be 1080. One comparison, no working. |
Questions 2 and 3 are the same sector, and the two answers are 18.85 and 6.28. Both are multiples of π worth writing exactly: 6π and 2π. On this course the decimal is what gets marked, but the exact form is the fastest check that the angle went in as a fraction of π.
1 markThe conversion, or the angle correctly in radians.
1 markThe right formula, ½r²θ or rθ.
1 markThe answer, with units.
There is no mark for the mode the calculator was in, which is exactly why it costs so much. A student in degrees mode loses every subsequent mark on a question and has nothing on the page that looks wrong.
| They wrote | What happened |
|---|---|
| 3437.75 | On question 1, multiplied by 180/π instead of π/180. The conversion ran the wrong way. 3437.75 radians is 547 full turns, so the size alone condemns it. |
| 0.017 | On question 1, converted ONE degree. They found the right factor and forgot the 60. |
| 1080 | On question 2, put 60 into ½r²θ. The error this whole sub-topic exists to prevent. Hold it up against 113.1. |
| 37.70 | On question 2, left out the half. Note that 37.70 is ALSO the circumference of this circle, which is a coincidence of r = 6 and not a clue; say so if someone spots it. |
| 6.28 for the area | Gave the arc. They have both formulas and have not read the question. |
| 360 | On question 3, 6 × 60. The whole circumference is 37.70, so this is ten times round. |
| 6.283 | On question 4, a full turn rather than a single radian. They know 2π is the circle and have not divided. |
"Why not just use degrees for everything?" Because the calculus in Topic 5 needs radians: the derivative of sin x is cos x only when x is in radians, and in degrees it picks up a factor of π/180. That is the honest answer and it is worth giving, even though 3.7 contains no calculus. Radians are not a harder unit, they are the unit that makes the later formulas true.
"So π/3 means 60, I can just type 60?" No, and this is the sentence to catch. π/3 and 60° are the same ANGLE in two units, the way 1 inch and 2.54 cm are the same length. The formula ½r²θ is written for radians, so what goes in is 1.047.
"How do I know which mode I am in?" Look at the status line before the question, not after the answer. Both machines display it. Better still, test it: sin 30 is 0.5 in degrees and −0.988 in radians, so one keystroke tells you.
"Can I leave the answer as 6π?" On Applications, give the decimal, because the course is built around the calculator. Write 6π beside it if it helps you check.
| Stage | What to do |
|---|---|
| Demonstrate | Before any of the mathematics, put the machine in degrees and compute 0.5*36*π/3. It gives 18.85, because the expression has no trigonometric function in it and the mode is irrelevant. Make that point explicitly, or half the class will decide mode causes the 1080 error. It does not: 1080 comes from typing 60 where 1.047 belongs. Mode matters the moment a sin, cos or tan appears. |
| Where they stick | On the Casio, SET UP is per-application, so the Angle setting they change in Run-Matrix does not follow them into Graph. Have them set it in whichever application they are about to work in. On the Nspire, Document Settings is doc → Settings → Document Settings, not ctrl menu, which is the context menu; 5 2 from the home screen also gets there. |
| The check | Teach one habit: compute the whole circle first. πr² = 113.1 here, and every sector answer has to be under it. It costs one line and it catches the factor-of-57 error, the missing half, and a misread radius, all three. |
A student who writes the angle as a decimal immediately, 1.047, loses the exact form that would have let them check. Encourage π/3 on the page and 1.047 in the machine.
| Step | What |
|---|---|
| 1 | Figure, third view. 18.85 against 1080 against a circle of 113.1. Say nothing. |
| 2 | Collect what they think went wrong. Someone will say "the units". |
| 3 | One-radian view. Build the definition: arc equals radius. 2π of them fit. |
| 4 | Both conversions, derived from 180° = π, not memorised as two fractions. |
| 5 | ½r²θ and rθ as the degrees versions with the fraction folded in. Show the algebra once. |
| 6 | The whole-circle check, then the five questions. |
Do not say "radians are just another way of writing degrees". It is true and it is the sentence that produces 1080, because it suggests the two are interchangeable inside a formula. Say instead that each formula is written for one unit and tells you which.
Do not say "always work in radians". On this course the bearings in 3.3 and the triangles in 3.2 are in degrees, and a student who has been told to convert everything will convert a bearing. The rule is to read the formula.