Teacher page: limits and the derivative as a gradient

The one thing to do with the animation

Ask for the gradient at P before you press anything. Someone will give you the gradient of PQ. Accept it, write it on the board, then shrink h and ask again. Then again.

By the fourth answer the class has produced the idea of a limit themselves, and you have never used the word. Introduce the word afterwards as a name for the thing they just did, which is the right order.

Why this curve

y = x² at P(1, 1) gives a chord gradient of exactly 2 + h. Every row of the table is therefore exact, not rounded, so a student can see 3, 2.5, 2.1, 2.01 and spot the pattern without arithmetic noise. Pick a messier curve and the pattern hides.

The answers

Table rows	h = 1 gives 3, h = 0.5 gives 2.5, h = 0.1 gives 2.1, h = 0.01 gives 2.01. The limit is 2.
1. Estimate the gradient	4. The values 4.3, 4.1, 4.02, 4.001 are closing on 4.
2. Meaning of C′(100) = 9	B. At 100 chairs, one more adds about 9 baht. For reference C(100) = 900 and the true jump C(101) − C(100) is 9.02.
3. Balloon rate at r = 3	113.1 cm³ per cm, from 4π × 9 = 36π.

Where the marks go

1 markReading a limit from a table. Nearly free once they know that is the task. The guide does not ask for analytic limits on this course, so do not teach algebraic cancelling here: it is time spent on something that cannot be set.

1 markInterpreting a derivative in context with units. This is the one they lose, every year.

1 markChoosing the right derivative when the letters change. dV/dr and dV/dh are different questions about the same solid.

What each wrong answer tells you

They say	What it means
4.001 (Q1)	They have copied the last row rather than extrapolated. Ask what comes after 4.001 in the pattern.
4.3 (Q1)	They read the first row. Worth a word about why the biggest h gives the worst estimate.
"Costs 9 baht" (Q2)	The one that matters. Confusing a rate with a total. Put C(100) = 900 next to it on the board and the contrast does the teaching.
"Rising by 9 per cent" (Q2)	They have no sense that a derivative carries units. Go straight to "top units over bottom units".
36 (Q3)	Stopped before multiplying by pi. Arithmetic, not calculus.
1017.9 (Q3)	Used the volume formula instead of its derivative. They have not registered that dV/dr is a different function.

Other things they will say

"Why can't h just be 0?" The best question in the lesson. Let them try it: the run is zero and the division is undefined. That is exactly why the two-point method fails at a single point and why a limit is needed. Do not wave it away.

"So dy/dx is a fraction?" Treat it as one piece of notation at this level. It behaves like a fraction later, and arguing about it now costs more than it buys.

"The balloon grows at a constant rate." Common, and worth catching. dV/dr = 4πr² depends on r, so the same balloon at r = 6 grows four times as fast as at r = 3.

A possible order

	What is happening
1	Gradient of a straight line, from two points. Then ask for the gradient of a curve at a point and let the difficulty surface.
2	The animation. Predict, shrink, ask again. Build the table on the board alongside the one on screen.
3	Name it: limit, tangent, derivative. Notation table.
4	Rate of change. The chairs example, then the balloon. Insist on a sentence with units each time.
5	Questions 1 to 3. Question 2 is the one to take to the whole class.
6	Back to "why can't h be zero". Finish on the idea, not on an exercise.

Two things not to say

Do not say "the derivative is the gradient" and leave it there. Half of 5.1 is rate of change, and the marks are in the interpretation. A class taught only the gradient picture reads dC/dx as a slope on a graph nobody drew.

Do not introduce the power rule today, however much they want it. It arrives in 5.3 with a reason attached, and that page checks it against a real chord so the two ideas stay connected. Give them the rule now and the limit becomes a thing they sat through.

Questions to set

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.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

Chord gradientFor y = x² at x = 1, find the chord gradient with h = 0.1 and with h = 0.01.
2.1 and 2.01. Each is ((1 + h)² − 1) / h, which simplifies to 2 + h.
State the limitWhat do those chord gradients approach, and what is that number called?
2. It is the gradient of the tangent at x = 1, which is the derivative there.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

Do it for a cubeFor y = x³ at x = 2, find the chord gradient with h = 0.1, then state the derivative.
12.61 with h = 0.1. The derivative is 3x² = 12, and 12.61 is closing on it from above.
Read a rate in contextA stone's distance fallen is s = 5t² metres. Find its average speed from t = 2 to t = 2.1, and say what the limit of that would be.
(5(2.1)² − 5(2)²) / 0.1 = 20.5 m/s. The limit as the interval shrinks is the instantaneous speed, 20 m/s.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Why h cannot be zeroExplain why the chord gradient formula cannot simply be evaluated at h = 0.
At h = 0 it is 0/0, which is undefined. The whole point of a limit is to say what the expression approaches without ever reaching the value it cannot be given.
Algebra beats the tableShow algebraically that the chord gradient of y = x² at x = 1 is exactly 2 + h, and say why that is stronger than a table of values.
((1+h)² − 1)/h = (1 + 2h + h² − 1)/h = (2h + h²)/h = 2 + h. A table suggests the limit; the algebra proves it, and shows the error is exactly h.

Practicalities

Works on a phone, and the slider is easier than a drag on a touchscreen. No library is loaded from anywhere, so it runs behind a school firewall and offline once the page is open. Nothing a student types is saved or sent.