Why a correct gradient still produces a wrong line, and the picture that makes the missing step unforgettable.
Set the question, let them answer, then press the mistake button. Ask for the tangent to y = x² − 2x + 3 at x = 3. A good number of the class will write the point as (3, 3) and produce y = 4x − 9.
Press the button. The line appears, exactly parallel to the true tangent, sliding past the parabola and touching nothing at all. Solving x² − 2x + 3 = 4x − 9 gives x² − 6x + 12 = 0, whose discriminant is −12, so there is no contact anywhere. The correct tangent gives x² − 6x + 9 = 0, discriminant zero, touching once. That contrast is the lesson.
A tangent that touches nothing is absurd enough to remember, which is more than can be said for "remember to find the y value".
The vertical axis is compressed relative to the horizontal one, so the normal does not look perpendicular on screen. A parabola rising 13 units over 5 cannot be drawn to equal scales in a landscape figure. The caption says so on the student page rather than letting the picture mislead quietly, and it is worth a sentence in class: perpendicularity lives in the gradients, not in how it looks when the axes are scaled differently. Students who have only ever seen equal-axis diagrams find this genuinely confusing if nobody names it.
| Worked example | At x = 3: y = 6, dy/dx = 2x − 2 so m = 4. Tangent y = 4x − 6. |
| The normal | Gradient −¼, giving y = −0.25x + 6.75, or x + 4y = 27. |
| 1. Gradient of y = 5x² at x = 2 | 20. The derivative is 10x. The point, if they need it, is (2, 20), and the tangent is y = 20x − 20. |
| 2. Equation of the normal | B. y = −0.25x + 6.75. |
| 3. What went wrong | B. They used (3, 3) instead of (3, 6). |
1 markThe y value of the point, found from the curve. This is the one that is lost, and because the point feeds the rest of the question it usually takes the next mark with it.
1 markThe gradient from the derivative. Almost nobody loses this.
1 markThe equation, in the form the question asked for. If it says ax + by + c = 0, then a y = form is not finished.
1 markFor a normal, the negative reciprocal. Watch for sign-only and reciprocal-only errors; they are different mistakes and want different corrections.
| They write | What it means |
|---|---|
| y = 4x − 9 | Used x as y. The whole point of the page. Their calculus is fine; their reading is not. |
| Point (3, 4) | Used the gradient as the y coordinate. Rarer but it happens when all three numbers are in play at once. |
| Normal gradient −4 | Changed the sign without taking the reciprocal. Point at the two readouts moving in opposite directions. |
| Normal gradient ¼ | Took the reciprocal and forgot the sign. Perpendicular gradients multiply to minus 1, so test it: 4 times a quarter is 1, not minus 1. |
| 10 for Q1 | Derivative right, substitution missing or x = 1 used. |
| 40 for Q1 | Treated the derivative as 10x². The power was multiplied but not reduced. |
"Can a tangent cross the curve?" Yes, elsewhere. A tangent is only required to match the gradient at its point of contact. On a cubic a tangent will often cut the curve again further along, which surprises them and is worth thirty seconds.
"What if the gradient is zero?" Slide to x = 1 on the figure. The tangent is horizontal and the normal is vertical, so the normal has no gradient at all and its equation is x = 1. The page prints "undefined" rather than a number, deliberately.
"Is the normal ever useful?" Yes: reflection and refraction in physics, and the shortest distance from a point to a curve. Worth naming so it is not just a second thing to memorise.
| What is happening | |
|---|---|
| 1 | Equation of a line through a point with a given gradient. Pure revision, and the whole sub-topic depends on it. |
| 2 | Set the tangent question cold. Collect answers, including the wrong one, without correcting yet. |
| 3 | The mistake button. Let the absurdity do the work, then formalise the three steps. |
| 4 | The worked example in full, with the y value as its own numbered line. |
| 5 | Normals. Negative reciprocal, and the steep-shallow sanity check on the slider. |
| 6 | Questions 1 to 3. Question 3 is diagnostic: it tells you who has internalised the fix. |
| 7 | The final habit: substitute back and check the line passes through the point. |
Do not write the point as (3, y) and fill it in later. Students copy the layout, and a blank waiting to be filled is exactly the blank that stays empty under exam pressure. Work the y value out first, on its own line.
Do not say "the normal is just the negative reciprocal" as the whole instruction. Two distinct errors hide in that phrase, sign and reciprocal, and students who make one of them hear their own method in your sentence.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
The slider is easier than dragging on a phone. The mistake button pins the point at x = 3 so the demonstration is always the same, and moving the slider clears it. No external library, nothing stored, nothing sent.