Post #4

 In the spring, I will have the opportunity to teach Calculus for a few weeks to my Honors Precalculus class. I'm excited to teach mathematics at such a high level! I had great teachers who used visuals to help me understand what calculus actually tells us. I developed a great appreciation for Calculus and went on to study math because of it! So, my goal when creating this activity was to establish a good visual to help my students make sense of instantaneous rate of change.

New York State Standards do not cover Calculus. Even so, I want my students to use what they know about slope to create visuals on a graph that represent the instantaneous rate of change at specific values. I want them to understand that a derivative is a formula that gives you the instantaneous rates of change (slopes of the tangent line) at specific points.

I graphed the following function on Desmos. I chose this function because it has a few simple curves where students could see how the slope changes.

I also entered the derivative of the function f(x) at a specific point c. I created a slider for c. This is to show students that the derivative acts like a formula that will give the slope of a line that is tangent to the graph at all different points. That slope changes at different points.

When students change the value of c, the formula (derivative) in line 4 gives a value (on the right side). Here, when c is 0, the slope of the tangent line is -1.

The next step is for students to see what the tangent line looks like on the graph. I give them slope-intercept form of the linear function (y=mx+b) and have them change the slope value (m) to the value from line 4, using a slider. Here, they would make m=-1.

Then, the b slider is there so they can move the line with the correct slope around on the plane until it is tangent to the graph of f(x) at the point c. Here, when b is 0, the line tangent to f(x) at c.

Here's another example:

When c=-1, the slope of the tangent line is 2.

After adjusting the m and b sliders, the linear function again shows the slope of f(x) at the exact moment that x = c = -1!

Reflection:

I'm excited to return to this activity in the spring and put it into action! I think that visual learners will benefit from something like this. Moreover, I think that this activity encapsulates why I love math- and might spark that same love in my students! I think it is powerful that I can create a general formula that can tell me so much about a function. I think my next step would be to connect this to velocity and acceleration in physics. By using derivatives, it seems we can discover how things work around us much more easily! Have you ever used visuals or manipulative to solidify abstract concepts for your students?