Visual Introduction to the Definition of the Derivative

One of the new teachers I will be working with this year will be teaching Calculus, and this has got me thinking about some of the more successful strategies that I used the last time that I taught this course. A bunch of ideas come to mind, but one that sticks out from the others is the visual intro to definition of derivative that I stumbled upon the second year that I taught Calculus.  The first time I taught the course students struggled with this concept.  Using dynamic geometry software made all the difference for helping students learn this concept the next time I taught it.

To demonstrate, say you want to use the defintion of derviative to find the value of the derivative of f(x)=x^2 at x=-2.  Use the diagram below:

I've added a couple of objects to my diagram to help out with this explanation.  First, we note that we are really looking to find the slope of the tangent line of the function f(x)=x^2 at x=-2.  I don't tell students that my software can simply graph this tangent line and give me the slope, nor do I tell them that there are formulas to find the derivative of a function that we can use to find the slope of the tangent line.  That all comes in time.  For now my fixed point, a moving point attached to f(x), and a secant line between these points is all we need.

The next step is a reminder of what a tangent line is.  I let students discuss with a partner and then tell the class what we are looking at.  If the diagram above with a secant line is provided, I can ask students what to do to the moving point to make my dashed line become a tangent line.  I start dragging the moving point towards the fixed points and ask students to tell me when to stop.  If they tell me to stop too soon, I ask them if we can move the blue point closer to the fixed point.  We are visually developing the idea of using a limit to find the slope of a tangent line by physically moving one point as close to another point as possible.  Eventually we put the moving point on the fixed point.

Students notice right away that the secant line is no longer present.  Since you can't turn a secant line into a tangent line using a fixed point and moving point, we will do the best that we can.  And the best that we can do is to move the moving point as close to the fixed point as possible, and use these coordinates to estimate the slope of the tangent line.  I ask students to write this symbolically.  We introduce the following notation:

Fixed Point: (a, f(a))
Moving Point: (x, f(x))

Then the slope of the secant line is

We practice finding the value of the derivative by dragging the moving point as close to the fixed point as possible, and then finding the slope of the secant line.  When we introduce the definition of the derivative of a function at a given value, we use similar language.  Students can tell you the following about the slope of the tangent line with little help:

The slope of the tangent line can be found by taking the limit of the slope of the secant line as the moving point approaches the fixed point.  When we have had this conversation, it is not such a big leap to 

With the visual introduction, students can interpret the parts of this formula and feel empowered when the formula yields the same slope as our estimations did. 

My school district has open enrollment for AP and honors courses, and because of this the Calculus AB classes are very heterogeneous.  There is a nice split between students that have taken all honors math courses, students that are choosing AP Calculus AB as their first honors math course, and students that have never taken a honors/AP course in any subject.  Because of this I chose to keep the functions simple when we were finding derivatives using the definition.  This allowed for a greater focus on the conceptual understanding of derivative, and I believe that the visual introduction using dynamic geometry software and the extra conversations around the topic helped students ace this topic on the following test.  

You can use this premade presentation to introduce the definition of derivative in your own class.  This tool allows you to enter a function as well as coordinates for your fixed point and moving point.  A smaller version of the tool is below, and you can try this out by dragging the moving point.  

You can also use this presentation when talking about when a function is not differentiable, because it is easy to see that the slope of the tangent line will not approach the same value on either side of the cusp.

#July2014Challenge: Coordinate Algebra and Transformations

I was recently reviewing workshop materials from a transformational geometry session and was directed to the Georgia Deparment of Education website for a unit called Algebraic transformations. This is the fifth unit in their first year (9th grade) course titled Coordinate Algebra.  I found the activity/assessment below on page 109 (answer key on page 103).   The task was to draw some shapes on a coordinate grid, and transform them using 8 different coordinate rules.   Below is a list of similar problems.

What would happen to your shape if you transformed it using the given rule?
1.  (-x,y)    
2.  (x,-y)
3.  (x+2,y)
4.  (x, y-3)
5.  (x+1,y-4)
6.  (2x,2y)
7.  (-x,-y)
8.  (3x-2,y+1)

In case you've never seen these rules before, the process is that you take your preimage point (x,y), and substitute the values for x and y into the given coordinate rule to find the image point.  If your set of preimage points creates a shape, then it is easier to see the transformation given by the coordinate rule since your image will form the transformed shape.

In case you want to check your answers:
1.  Reflect across the y-axis
2.  Reflect across the x-axis
3.  Shift (translate) right two units
4.  Shift (translate) down 3 units
5.  Shift (translate) right 1 unit, down 4 units
6.  Dilate from the origin by a scale factor of 2
7.  Rotate 180 degrees counterclockwise about the origin (or reflect across both axis)
8.  Horizontal stretch and shift, and shift up one unit.  (I am guessing this was an extension of what they learned, so not a precise description).

I am thrilled that the Common Core allows for more time spent on coordinate rules. These rules are somewhat intuitive, and easy to verify using a diagram.  Activities can easily involve multiple representations of knowledge.  This study of coordinate transformations is even included in the 8th grade standards:

8.G.A.3.  Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

My hope is that all of this study of coordinate transformations will help with the understanding of function transformations.  Many of the newer textbooks and resources still teach function transformations via observation.  You might see an introductory activity that asks students to graph each of the following functions and to notice the similarities and differences between each graph.

Students easily make the correct observations, and they can even memorize the rules for vertical translations in this case.  But once we teach students all of the rules for function transformations, it starts to make less sense if the introduction is based on observation instead of reason.  I'd rather review the rules for coordinate transformations each time I introduce a function transformation.  In this case we are transforming a set of points instead of just moving a graph around.  This will also help students see functions as a set of points instead just a graph with a shape.  In my work over the next year I will be looking for textbooks or online resources that utilize this method of teaching function transformations.  

I made the following tool to use when practicing coordinate transformations.  To use this tool, enter a coordinate rule for point B that depends on the coordinates of point A.  For example, if Point B= (-x(A), -y(A)), then point B will be rotated 180 degrees about the origin from point A.  You can use the tool below to try this out.  Once you've entered coordinates for point B, drag point A across the semicircle to see the transformation created by the trace of point B.


I'll admit that this tool isn't as polished as I would like it to be. I wanted to share anyway because this is a concept where the use of technology can enhance instruction and practice.  The updated version will be added to the Geogebra Tools list at the top of this blog when available.  For now you can access a download of this tool from the Geogebratube website, and from there you can open with Geogebra and resize the window and turn on the option to view coordinates, etc.  

Interactive Intro to Domain and Range

One of my favorite things about Common Core is the opportunity to learn new ways to introduce content. I have been teaching high school for 7 years, so I haven't had the chance to teach with too many books.  With common core and the impending adoption of new materials, I am finding new ways to teach almost weekly!

I had a chance this week to play around with domain and range using the TI Inspire.  I can't take credit for most of the ideas below, but this introduction is too good not to share!  Check out the Texas Instruments website for free worksheet downloads that go with domain and range presented in this manner, plus many other topics.  The worksheets go with TI calculator activities, but you can still get great ideas from them if you don't have access to the technology.

Drag the gray point in the presentation below over points A, B, C, D and E.  Watch as the domain and range values appear on the axes.  To try a different problem set, hit "clear domain and range", drag each point to a new location, and repeat the activity.

Can you arrange the points so that the domain is {-2,3,5} and the range is {1,2}?  This question has multiple solutions and will give students the opportunity to discuss the mathematics and check answers in real time. They are also practicing with multiple representations.



Students can find domain and range of a line segment below by dragging point C.  Again the domain and range values appear along the axes.  To try a new problem hit "clear trace" and drag the endpoints to a new location.

Can you find the line segment that has domain greater than or equal to -3 and less than 4, and range greater than -1 and less than or equal to 3?  Insert inequality statements into these problems to help students practice with the multiple representations used in this context.



For the presentation below drag point A to create a relation.  Can you graph the relationship with domain and range described below?  Is your relation a function?

To show domain and range for a specific function, enter the function into the input box below. Change the left and right limit of the function to make it piecewise, or to see what the domain and range are for the the function itself change the limits to values that are not on the screen below.  (So -6 and 8 will work).



Links you may find useful:

Geogebra Tools from Making Math Visual blog.
Geogebra Series II: Transforming a Graph With Sliders

Geogebra Series: Inverse Functions