Going Around in Circles

Many years ago, while I was on the board of directors of the Ontario Association for Mathematics Education, we worked on the GUM project.  I was part of the grade 11 (I think) team and we were looking for a good introduction to sinusoidal graphs activity and some good evaluation questions to end the unit.  We did not reinvent the wheel (no pun intended - it's coming) but used the "Wagon Wheel" activity, also known as "Hula Hoop" and "Going Around in Circles". I have no idea to whom credit for this activity should go, but thanks nonetheless!

The idea is for students to work through this activity BEFORE they have seen the graphs of y = sin(x) or y = cos(x). Students should work in groups.  Each group needs some masking tape, a metre stick and a motion sensor (CBR2/GoMotion Sensor). 

The set up goes as follows: create a 1 m diameter circle on the floor using the metre stick (place on floor, use tape to mark centre and each end, rotate 90 degrees, mark each end then do 45 degree turns to get something that resembles a circle).  The alternative is to actually use hula hoops but I find that they don't travel well and are easy to trip over.  One student is the walker and one is in charge of the motion sensor but all have to predict the graph before starting (and they should switch jobs as they go through the activity).

Some advice - have the walker hold something like a large folder or whiteboard which will serve as the target for the motion sensor.  I have also found that the data is much better if the person holding the motion sensor rotates it to follow the motion of the walker.

Unfortunately I have not had the opportunity to use this activity for many years (I just haven't taught the course where this is introduced).  So here are some shots of the very old handout I have.

1. They get a feel for the shape of the graph - you will get all kinds of predictions before they do their first set of data collection. Then the starting position changes - this is a phase shift on the graph, but you will hear lots of different words to describe it as they have not learned the mathematical vocabulary yet.

2. Next, we go faster.  This is fun to watch!  And results in a period change.

3. The following questions have them thinking about changing the amplitude of the graph, again, without using the word amplitude.

4. Changing direction creates a reflection.

 5.  I am not sure why this is not up with #3...

6.  Changing the distance will creates a vertical shift.

The is an optional #7 which is to walk in the shape of an ellipse.  I recommend that the walker holds the motion sensor pointed at a wall to get good data for this.  You can do the whole activity this way, but I find it best for each member of the group to have a job.

By the end of the activity students understand how each of the parameters affects the graph without having seen a single equation.  This way, they have a foundation upon which to build when they see these parameters in an equation.  It's time well spent and it's fun!