Trigonometry Function Family TN

Teacher Notes

Mathematics can be beautiful, but the students who might appreciate the proof of the irrationality of root 2 are perhaps, and unfortunately, in a fairly small minority.  However, students really do appreciate the beauty of the graphs that they are able to produce in this activity. It is exciting that they can use the mathematics that they already know to produce some beautiful images like the picture on the left. This activity gets students to produce families of trigonometric functions using dynamic geometry software. By exploring the effect of changing parameters they really get a deep understanding of the properties of the main transformations: translations and stretches. Get ready for "Oos and Ahhs"!  

Solutions

I really hope that the activity engages the teacher to want to create the files for themselves; however, here is a solution to save a bit of time. Of course there are alternative solutions to these, which itself, creates a lovely opportunity for discussion! You can download a zip folder of all the Geogebra files that I used to create these images, but I encourage you to have a go at creating them for yourself.

How?

The following is some practical advice about how the activity might be run and how to prepare.

Resources 

Students need access to computers with Geogebra.  There is an online version that can be used without installing the software, but given the usefulness of this dynamic geometry software it is well worth getting the software installed on the computers that the students use. The help video on the main page of this activity gives help in how to use sliders in Geogebra to create these graphs.

Time needs

This activity could well be completed within 90 minutes depending of course on the previous experience of the students on work on transforming functions. I used about 30 minutes in class to introduce the idea and ensure that the students were comfortable using Geogebra and then asked them to complete it for a homework assignment.

Starting and finishing

  • The beautiful graphs provide an ideal starting point for this activity. A video of the first dynamic graph is provided to inspire the students.
  • To finish off students could try to create their own family of functions.  In fact, thinking carefully about what functions could be produced adds another level of difficulty so this could provide a great extension activity.  

Records

Students should take a screenshot of each of the graphs as a record of their achievements. They should also note down the function and the parameters that they used to produce each graph.  A ready made file has been made for this purpose.

What to expect?

The following paragraph outlines some of the possible sticking points that teachers might want to be prepared for.

  • To use Geogebra students will need to ensure that they have java enabled on their computer.
  • It is very important to use degree symbol to plot functions in Geogebra in degrees.  f(x)=sin(x) will plot the function in radians whilst f(x)=sin(x°) will plot the function in degrees.
  • It is important to make use of spaces to denote the multiplication of a function by a parameter, e.g. f(x)=asin(x°) will plot the arcsin(x), whilst f(x)=a sin(x°) will plot f(x)=a*sin(x°).
  • For the above example, it is important to define the parameter a before it is used in a sequence.  This can be done with a slider.
  • The rainbow effect colours on the graphs do add a lot to the appearance of the graphs and the motivation for the students to reproduce them.  However, students will need to be guided away from spending too much time playing with this effect.  Graphs made up of black cuvres are perfectly fine of course!
  • All of the above tips are included in the help video for the students on the main page.

Good Luck!

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