Common content, so this runs for both courses.
Ask for the angle of a 1 in 10 road before you say anything about gradients. The room will say ten. Several will say it as though it were a definition. Write the guesses up.
Then step the figure to the last view, where a 1 in 10 ramp and a 10° ramp sit on the same run. The 10° one rises nearly twice as far. The sentence to land is that the confusion always makes a slope sound gentler than it is, which is why it survives: nobody is ever surprised by a hill being easier than expected.
Say out loud that the vertical scale in the figure is stretched six times, and why: at true scale a 1 in 20 ramp is twenty pixels of rise in four hundred and you cannot see it. The angle readout is not stretched. Students who are not told this will tell you the figure is wrong, and they will be half right.
| Question | Answer |
|---|---|
| 1. 1 in 5 as an angle | tan⁻¹(0.2) = 11.3°. |
| 2. Gradient of 2x + 3y = 12 | y = −⅔x + 4, so −0.67. |
| 3. Perpendicular gradient | 1.5, and −⅔ × 1.5 = −1. |
| 4. Run for a 0.75 m rise at 1 in 12 | 0.75 × 12 = 9.00 m. |
| 5. The 1 in 10 against 10° claim | B. 5.7° against 10°, so nearly twice as steep. |
1 markThe rearrangement to y = mx + c, written out.
1 markThe gradient, with its sign.
1 markThe intercept, or the second gradient.
The sign is the mark most often dropped here. A general form with both coefficients positive always has a negative gradient, and that is not where the eye goes. Insist on the rearranging line even from students who can do it in their heads, because it is the line that earns the first mark when the arithmetic afterwards fails.
| They wrote | What happened |
|---|---|
| 5° on question 1 | Read the number off the sign. The expected answer, and the lesson. |
| 78.7° | Took tan⁻¹(5) instead of tan⁻¹(0.2). Rise over run, inverted. |
| 0.2 | Gave the gradient when the angle was asked for. Worth naming as its own habit. |
| +0.67 | On question 2, lost the sign moving 2x across. |
| 2 | On question 2, read the coefficient of x straight off the general form. |
| −1.5 | On question 2, divided the wrong way round: −3/2 instead of −2/3. |
| 4 | On question 2, gave the y-intercept. |
| −1.5 on question 3 | Applied "make it negative" to a gradient that was already negative. The reason to teach the product test instead. |
| 0.0625 | On question 4, divided 0.75 by 12 rather than multiplying. |
| 9.03 | On question 4, gave the slope length rather than the horizontal run. Three centimetres, and a mark. |
"Why is the gradient negative when both numbers are positive?" Because y is on the same side as x. Put x = 0 and x = 3 into 2x + 3y = 12 and read the two y values off: 4 and 2. It goes down.
"Which form should I use?" The one the question hands you, until you need a different one. Point-gradient is the one they under-use: it needs one point and a gradient and most questions give exactly that.
"Is 1 in 12 the same as 8%?" Yes, and that is worth a minute. A percentage gradient is the ratio times a hundred, so the three languages for one quantity are a ratio, a percentage and an angle, and only the first two are the same thing.
| Stage | What to do |
|---|---|
| Demonstrate | Fit a line to the two points (0, 4) and (6, 0), which is the line on this page, and read the gradient back as −0.667. Then point at the r² of 1 and say plainly that it means nothing here: any two points lie on a line exactly, so a perfect fit to two points is arithmetic, not evidence. |
| Where they stick | The Casio reports y = ax + b, so its gradient is a. Students who learned y = mx + c reach for b and write the intercept down as the gradient. Make them say which letter is which before they use it once. |
| The check | Before anything is typed, ask whether the gradient will be positive or negative. If they can look at 2x + 3y = 12 and say negative, the rest is arithmetic. If they cannot, the machine will not help them. |
Degrees mode matters here only for the ramp conversions, which are an aside rather than the sub-topic. It is still the right lesson to establish the habit in, because nothing is lost if it goes wrong.
| Step | What |
|---|---|
| 1 | "A road sign says 1 in 10. What angle is that?" Collect answers. Expect ten. |
| 2 | Step the figure to the comparison. Let the two ramps settle it. |
| 3 | Gradient as rise over run, and the tangent as the bridge to angles. Then leave angles alone. |
| 4 | The three forms, built by rearranging one line rather than learned as three facts. |
| 5 | Parallel and perpendicular, with the product test as the check. |
| 6 | The ramp question, which is the one that looks like a real design brief. |
Do not say "flip it and make it negative". Students apply it to a gradient that is already negative and produce −1.5 from −⅔. Say "the two gradients multiply to −1", which is a test as well as an instruction and cannot misfire.
Do not let "slope" and "angle" be used interchangeably in your own speech. This sub-topic is the one place the words are genuinely different and the confusion is imported from outside the room. If you say "a ten degree gradient" once, it will come back in an exam.