New mathematical ideas are usually built on another mathematical idea or ideas. Because of this, the teaching of mathematics if it is to make sense to students, should reflect this ‘building on’ process. Students should be able to see how the new idea is connected to what they already know. Good teaching of mathematics also demand that this new knowledge be useful and connected to the mathematics that students will encounter later.
Here is an example of a lesson that teaches the idea of derivative without really teaching it yet. This means that you can introduce this in Year 9 or 10 in their lesson about graphs of second degree function. The only requirement is that they understand the function of the form 
The lesson uses the applet below. Of course, much of the success of the lesson will still be in questions you will asked after students initial exploration of the applet. You can find my proposed questions for discussion below the applet. [iframe https://math4teaching.com/wp-content/uploads/2012/02/Deriving_function_from_ax_2.html 750 620]
Questions for discussion
- You can move point A but not point B. Point B moves with A. What does this imply?
- What do you notice about the position of B in relation to the position of A?
- What is the path (locus) of point B? Right click it and choose TRACE then move A to verify your conjecture.
- What is the same and what is different about the coordinates of points A and B?
- To what does the coordinates of B depends on?
- What is the equation of the line traced by B?
- Refresh the applet then use the slider to change the equation of the graph, say a=3. What is the equation of the line traced by B this time?
- What do you think will be the equation of the path of B if the graph is
By the end of this lesson students should have the intuitive notion of derivative and can find its equation given the function
There are actually 8 ways to think of the derivative. If you want to know more about Calculus, here’s a good reference:
Your ideas do not teach the derivative – either by trying or not trying. A student seeing this will not understand anything about the derivative because the line t(x)=2ax is not the same tangent line as your graph implies, by lining up the x coordinate of t(x)=2ax with f(x)=ax^2. Why so? Because x changes as you drag point A. It is completely non-remarkable that the x-values are aligned. The derivative relates to the slope of the tangent line.
There is so much nonsense on this page, that it’s difficult to know where one can start:
https://math4teaching.com/8-different-ways-of-thinking-about-the-derivative/
1. There is no such thing as an infinitesimal – it exists only in the dysfunctional mind.
2. By “Logical”, you mean “epsilonics”, which has nothing to do with the derivative.
The calculus does not require limits at all. In fact, limits have no place in calculus.
3. There is no such thing as an instantaneous rate, only a slope at a given point. Such a slope may not even be a rate of any kind. Usually, a rate implies a time difference.
To talk about volume of sphere changing as radius changes is not a rate.
4. “Approximation” is the nonsense inherited from Isaac Newton and propagated by fools like you.
5. “microscopic”? – Now that’s one to have a good laugh about!
I am not surprised that so many American students can’t do mathematics, let alone calculus!.
Thank you for your humour all along your text + for your authorised comment on Isaac Newton’s math contribution.
We may also be kind toward Erlina who tries to dispend an intelligent and human way of approaching mathematics. This task is millions time more difficult than the US and many other nations’s easy ways which consist in giving kids pre-swallowed formula and rules to absorb.
A last point: Not sure, but, as far as I know, Erlina’s living not in the U.S.A but in the far Pacific (I can’t remember the name of this island now. Sorry. Search me!). Hence she wouldn’t be responsible for the pathetic way US children are taught mathematics (I agree with you @ this).
The alarming worst that I experienced is the series of exercises given to students by the online org Kaplan (mostly calculus). Nothing’s due to be explained by the student, not even justified. They tend to reduce the game to a yes or no. Win or lose, as in casino games. Thus reducing the student to a kind of robot-teller.
And Kaplan is the US leader, major in online adult education !