Math Teachers Pls Help! Intro to Calculus teaching?

Question

I was hoping for help for the math teachers that frequent this place. I am doing my 2nd prac teaching soon and I have to intro to calculus, basically develop the concept of the definition of a derivative (first principles)

What is the best order to teach this to a year 11 mathematics class (not sure if they are advanced or not).

I was thinking of starting off one lesson with limits & continuity, and then another one on secant/gradient and end up with the formula - how many lessons does it normally take to get to the formula? Any ideas or info would be much appreicated. thanks, Laura

Answer 1

I assume they've already had limits. The following is an outline for how I do it. This is usually one to two days.

1. Tell the class this is the single most important class of the year. Calculus is divided into three parts: limits (which provide the basis), differentiation (today's class), and a future topic (integration, but I don't give it a name). The rest is all an application of these three.

2. Supply motivation briefly (for those kids who like to see a destination). What is velocity, rate of flow? More generally, -> how do you measure or define rate of change? <-

3. Plop a line on the board and as a reminder, ask how much y changes when x changes by some amount. Of course, this is the slope, m.

4. Now plop a circle on the board, draw a tangent and ask for properties of the tangent. Specifically looking for the fact that the tangent only intersects at one point locally.

5. Now draw a quadratic and suggest that this is a more general idea, that smooth curves always have a tangent, and that's how we'll define the slope of a curve at any instant. This is all very well and good, but how do we find such slope?

6. Hilight a point (z,f(z)) and show a line to its left intersecting the curve at x=z-z1 and x=z+z1. Clearly that line is not tangent since it's intersecting at two points. Now show a second line intersecting the curve at x=z-z2 and z=z+z2 where z2
7. And now challenge them to see that to get the slope at the actual point, take the limit as Δz goes to 0. Easy sentence, hard concept.

8. Work an example with ax^2 at x=3. Generalize for x^2 at z. place this result to the left in a 3rd position. Work example with x^3 at z and put into 4th position at left of board.

9. come back to the line, y=x. No surprise, we get m. Of course you have to explain that this tangent intersects the line everywhere except instead of just at one point. Place this result in the 2nd position.

10. What about the slope of y=1? Zero. Put this result into a 1st position.

11. Now generalize to x^n. Put this into the 5th position. Emphasize that this one needs to be memorized cold.

12. Now explain that all that is needed is (f(z+Δz)-f(z))/Δz. This might come earlier so that the 2Δz in the denominator doesn't become ingrained.

13. Teachers always make a big deal out of proving that constants can be pulled out and that sums distribute, but at this point, I don't think the students really care. They've got far more going on trying to absorb a fundamental idea, so I'd rather just see this quickly thrown in. Oh yea, class, by the way, for ax^n at z, it's just anz^(n-1). Well, perhaps that's a little too cavalier.

14. Throw away the z. We call this concept, the (instantaneous) change in y with respect to x. The derivative. This concept of the slope is the change in y with respect to the change in x needs to be hammered home. So far, they're getting the concept of the slope of a curve, but they are not yet connecting it with the rate of change. That's the part that needs to be linked in their mind.

Emphasize that this concept of the derivative is fundamental both conceptually and from a mechanical standpoint, and for the next half year they will and breathe derivatives. There is not very much to memorize - most functions they encounter will have easy derivatives. The emphasis is on being able to work with this widely applicable idea.

15. Millions of exmples that the students work / homework. Specifically, initial examples are finding slopes so that the mechanics get ingrained, and then rate of change examples (e.g. velocity type of questions).

Answer 2

This is one possibility; it may not even be the best. I always use to start with a practical example which the students should be familiar with, namely a car accelerating. Draw a graph of distance against time. Pose the question what speed is the car doing after two seconds. We could take its position at 1.9 seconds and at 2.1 seconds and work out the speed by difference/0.2. However, this is only the average speed over that interval, not AT two seconds. A better result would be using positions at 1.99 and 2.01 seconds. From this you can develop the idea of instantaneous rate of change.

I would do a couple of examples of first principles differentiation, say y = x^2 and y = x^3 and then get them to do one themselves, say x^4. From the pattern you can develop the probable nature of the formula for x^n. Proving it will depend on whether the students know the binomial expansion for
(x + dx)^n.

How many lessons will it take? Depends on the ability of the students. Thankfully my teaching which started over 40 years ago was when advanced level students could be relied on to like maths and have real ability at it. Good luck!, I wouldn't like to be starting now.