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 task requires determining the equation of linear function of the form y = 2ax, which happens to be the derivative of . Of course you will not introduce the term derivative at this year level. You are just planting the seed for this important concept which students will encounter later.
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 http://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: