Graphing Sine and Cosine

Recently in Trig, we've embarked on the task of graphing sine and cosine curves. Before this, we constructed the unit circle and students became familiar with the exact values of sin and cos at the special angles. To introduce the parent graphs, I used an activity that I found at Kate's blog last year (who I'm sure got it from somewhere else, but I don't know where).

To first see the "unwrapped" unit circle that is the sine (and later, cosine) function, students used yarn to mark intervals along the circumference of a unit circle, then used spaghetti to measure the y (later, x) value at each place. They then transfered these lengths of spaghetti to an x-y plane with x intervals of length matching the circumference intervals. This was a great picture for them of where the shape of the sine function comes from. One student even explained it (works much better in ASL), as though you've taken the bottom half of the unit circle and spun it around to make up the second half of the sine function. (my adaptation of worksheet)

Graphing sine and cosine with changes in period and amplitude came easily for my students, but when we started translating with horizontal and vertical shifts, the students were lost. The first day was a major failure on my part and I knew that I needed to have a new approach when I started the next day. I placed an open call for help on Twitter and was forwarded two GeoGebra applets (thanks Dave!). I quickly came up with a guided investigation to go along with the first applet.

We went to the computer lab, and had some success! Normally, when I've taken classes to the computer lab to use Geometer's Sketchpad or GeoGebra (or other programs), the students end up playing with the program, going through the motions, and they leave with little to no connection back to the paper and pencil world of the classroom. It's probably the way I present it, but they may understand the concept more indepth while using the computer program, without any transference back to the original idea or any application to the next topic.

This time, I required the students to answer the questions in a word doc, and I taught them how to use Print Screen to capture the image on screen and paste it into their document in order to later compare it to another graph. The next day, we continued by using the second applet (with some of the parameters changed) to walk us through the process of graphing sine functions with translations *and* changes to period/amplitude. We closed the day with an application of what the applet taught us about sine to help us graph cosine functions with translations. As a bonus, my boss happened to come and observe me explaining the second applet to the students. He just loves it when teachers use technology (and happens to observe me on days when I'm using it well...lucky me!)

Results: I know at least one of my (two) students benefitted from this process. We are still not at a point where they can graph the functions independently, but they have some strategies to help them, and a deeper understanding of what the numbers in the ugly looking equations mean and how changing them, changes the function. As their Algebra II teacher, I know I could have done a better job when we did function transformations. That probably adds to their confusion and weakness in this area. Something to think about for next time.

**update** I know I haven't even posted yet, so I feel like I shouldn't update, but I did see some good progress in my students today. Now we have a shared experience that I can link back to as they continue to develop their graphing/graph analysis skills.


  1. Glad it worked out. I have become a huge fan of GeoGebra over the last year. It's nice to create an applet that you can use to keep your students focused on the task at hand. I'm all for letting kids play with the math, but there's definitely a time and place.

  2. Dave,
    Wow! You're quick at commenting...must be the name. :-)

    I agree about the "letting kids play with math" comment. There is a time and a place, and when you want them to get something more foundational from an investigation, and all they do is play, it is a bit frustrating. My principal (a former math teacher) was wow-ed, too!

    I'm not skilled with GeoGebra yet, but I think I could get the hang of it if I sat down and played myself for a bit. Maybe spring break...