## Exploring Point Slope Form through Geogebra

In geometry we are studying parallel and perpendicular lines and the theorems that can be proven about them. In thinking about how to present the connection between algebra and geometry for this unit, I wanted to include an exploration of what exactly makes lines (and linear functions) so special: constant rate of change, or slope.

We did not get a chance to do this entire exploration in class, but I am expecting students to look at it at some point over the weekend. I know they have seen slope-intercept form, but point-slope form is convenient in many ways, especially in the way it applies the concept of slope between points located on a line.

## Using #Geogebra to Predict and then Verify

Last year’s class introducing logarithmic and exponential differentiation was a bust. I tried to include it as an application of implicit differentiation, but I knew afterwards then, and still believe now that doing so was an incredibly horrible idea. There’s no way students are going to ‘see’ an application of an abstract concept like implicit differentiation better…by using it in another abstract concept. I’ve accepted that, and vowed this year to do a much better job.

I also had a shocking moment yesterday when a Calculus student came to me after school and asked me ‘what is the derivative?’ We had started the unit with a conceptual development of the derivative using limits and average rate of change, and had since moved to applying differentiation rules, so we were deep in that process – power rule, quotient rule, product rule, chain rule…really the primary ‘rules’ section of any Calculus course. I was taken aback by the comment – had I really stopped emphasizing the definition of the derivative in our class activities? In a way, yes. We had been writing equations for tangent lines and graphing them, but we hadn’t seen the limit definition (which I’ve been impressed by students remembering) in a little while. This proved that not only did I need to do a better job with logs and exponential functions, but that a little conceptual basis in that process would be useful.

I always like using Geogebra as a tool to pre-load information I am about to give students – what is about to happen? What should my result look like when I do this on pencil and paper? The graphing capabilities make it really easy to do this and set this up – I created this file and made it look the way I wanted in a few minutes.

These were the instructions I gave students:

Sketch what you would expect the derivative of y = 2^x to look like. Then click the ‘Show Derivative Function’ to graph the actual derivative. How close were you?

How would you expect your sketch to change for the derivative of y = 3^x?

Graph and make a prediction of the graph of the derivative of y = 2^-x. Check and see how close you were using the Geogebra tool.

Can you adjust the slider value for a so that the derivative is the same as the function itself? Use the arrow keys to adjust the slider more precisely.

Go through this same process to sketch the derivative of y = ln(x) in a new Geogebra window. Create this by going to the ‘File’ menu and selecting ‘New Window’.

It was really great seeing students predicting what the derivative would be, and then using the applet to confirm what they thought. There were lots of good conversations about scale factors and reflections, and some of them pretty much nailed what the general forms were going to be. This made the algebraic derivation a piece of cake – they knew where it was headed.

I also sprung this on them:

I’ve been really getting into the idea of standard based grading, and have been doing a form of it through my quizzes for a while, but it is still a small component of the overall grade calculation. While their grades aren’t being calculated any differently at the moment, I shared that this list would make a really good tool as we prepare for the unit exam on derivatives next week, and most started going through on their own and deciding what they needed to work on.

I’m still getting caught up after a couple very busy weeks, but I really like how this group in Calculus has been developing and maturing as math students in only a couple months. Their questions are more directed: ‘I don’t understand this application of the chain rule’ compared to ‘I don’t get it’. Their written work is detailed and clear, making it easy to locate errors. As a group, they get along really well, and class periods are filled with moments of furious productivity and camaraderie as well as humor and smiles throughout.

It was raining hard all day. I watched some students walk into class, look outside at the afternoon sky, and sink into their chairs, clearly feeling a bit down. I told them it was perfect Calculus weather – why not sit inside and do some differentiation?

Probably not what they had in mind. By the end of class, everyone left the classroom looking much more positive than when they walked in, and at least feeling good about the work they had in front of them.

## Lens Ray Tracing in Geogebra

One of my students came to me today to ask about ray tracing in preparation for his SAT II tomorrow in Physics. What happened is a good example of what tends to be my thought process in using technology to do something different.

Step 1 – I looked through some of my old worksheets, which I haven’t used in a while since I haven’t taught physics since 2009. The material I was happy with back then suddenly didn’t work for me. Given the fact he was standing there (and that time was of the essence) I wasn’t about to make a whole new worksheet.

Step 2 – I started drawing things on the board. This started working out fine, but I realized that every drawing I made would have to be erased or redone or saved in some other format. The student, after all, was most interested in learning how to do it and getting some practice. We did a couple sketches for mirrors, but when we got to lenses, I realized there had to be a better way. The sign for me for technology to step in is when I find myself doing the same thing over and over again, so the next step was pretty obvious.

Step 3 – Geogebra to the rescue. This is a particularly sharp student, so I was pretty happy with just talking him through what I was doing and asking him questions as I put together a quick demo of how to do this. He was pretty impressed with how logical the concept of ray tracing was, and had read the basic procedure in the textbook, but actually seeing it happen made a big difference. As he was standing there, his questions pushed me to make the applet (to steal Darren Kuropatwa‘s term) “a little more awesome.”

He asked what happens when the object is inside the focus of the lens. This led to throwing in some simple logic to selectively display the rays to show the location of the image when it is virtual and real. He asked what the difference is for a diverging lens. I told the student that I didn’t know what would happen if I switched the primary and secondary foci in Geogebra, but we talked about why that would relate to a diverging lens. Sure enough, the image appeared virtual and upright in the applet.

Step 4 – I then adjusted it a bit to show a diverging lens when the primary focus was on the left side of the lens, cleaned up some things, added colors, and now I have this cool applet to use when I get to working on lenses in the spring.

I like when I can think on my toes and use a tool like Geogebra to make something that will really make a difference. When I do this activity in the spring, it would be cool to put this side by side with an actual lens and an object and have students compare what is happening in both cases.

Check out the applet here: http://www.geogebra.org/en/upload/files/weinbergmath/Lens_Ray_Tracing.html

You can direct download the Geogebra file from here but be aware that I made the mistake of creating it in the beta version of 4.2. At some point, I’ll do it in the stable version.

You can drag the head of the arrow around, as well as move the primary focus F_p around to change it into a diverging lens. Clearly there are limitations to this – drag the object to the right side of the lens, for example, but I think it’s pretty cool that Geogebra can show something like this after an hour or so of playing around.

Have fun!