The discriminant of a quadratic equation, ax2 + bx + c = 0 is D = b2 – 4ac. If D>0, the quadratic equation has two distinct roots; if D<0, then the equation has no real roots; and, if D=0, the we have two equal roots. Let’s apply it in the following problem. Continue reading “Application of the Discriminant”
Tag: derivative
8 Different Ways to Think of the Derivative
In his paper The Transition to Formal Thinking in Mathematics, David Tall presents W.P. Thurston’s seven different ways to think of the derivative:
- Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
- Symbolic: the derivative of x^n is nx^n?1, the derivative of sin(x) is cos(x), the derivative of f ? g is f ? ? g ? g? , etc.
- Logical: f ?(x) = d if and only if for every ? there is a ? such that when 0 <|?x|< ? , then
- Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
- Rate: the instantaneous speed of f (t) , when t is time.
- Approximation: The derivative of a function is the best linear approximation to the function near a point.
- Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power. (Thurston, 1994.)
David Tall argued that the list excluded the global concept of local straightness so he added added the eighth that he believes that other 7 can be built.
8. Embodied: the (changing) slope of the graph itself.
In the same paper, David Tall presents a learning framework for derivative based on his Three Worlds of Mathematics Framework.
Teaching maximum area problem with GeoGebra
Remember that classic maximum area problem? Here’s a version of it: Pam wishes to fence off a rectangular vegetable garden in her backyard. She has 18 meters of fencing wire which she will use to fence three sides of the garden with the existing fence forming the fourth side. Calculate the maximum area she can enclose.
This problem is usually given as an application problem and is solved algebraically. For example if x is one of the two equal sides to be fenced then the area is the function f(x) = x (18-2x). The maximum area can be found by graphing or by inspection. If students have done a bit of calculus already then they can use the first derivate to solve the problem. But with free technology such as GeoGebra, there should be no excuse not to make the teaching of this topic less abstract especially for Year 9 or 10 students. It need not be at the end of the chapter on quadratic but as an introductory lesson for quadratics. Here’s a GeoGebra applet I made which can be used to teach this topic more visually and conceptually. Below is the image of the applet. I did not embed the applet here because it takes a while to load. Click maximum_area_problem worksheet to explore.
Here’s my suggested teaching approach using this applet. Students need to be given a bit of time exploring it before asking them the following questions:
- Pam wishes to fence off a rectangular vegetable garden in her backyard. She found fencing wires stored in their garage which she will use to fence three sides of the garden with the existing fence forming the fourth side. How long is Pam’s fencing wire? What are some of the sizes of gardens Pam can have with the fencing wires?
- If you were Pam, what garden size will you choose? Why?
- What do the coordinates of P represent? How about the path of P, what information can we get from it?
- As the length of BD changes so does the length of the other two sides. What equation will describe the relationship between the length of BD and EF? between BD and DE? between BD and area BDEF.
- What equation of function will run through the path of P? Type it in the input bar to check.
- What does the tip of the graph tell you about the area of the garden?
Feel free to use the comments sections for other questions and suggestions for teaching this topic. How to teach the derivative function without really trying is a good sequel to this lesson. More lessons in Math Lessons in Mathematics for Teaching.
Teaching the derivative function without really trying
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:
