In my math course we are presently working on graphs of different functions.  Many students are able to do this by picking values for x (the independent variable) substituting that value for x in the given equation and obtaining a value for y (the dependent variable).  They then take the x they picked and y they obtained and plot the point (x,y).  If you do this enough times you get a series of points that when joined should yield the graph of the function in question.

This approach though comfortable does have its drawbacks.  One is that we may pick points that do not show the salient features of the graph.  In simpler terms, across the street from my apartment there is another building and to the right of it a large house.  If I look out my kitchen window I only see the building.  I might not realize that the house is even there because of where I am looking.  The same thing happens when you graph.  The points you plot might not allow us to see the whole thing.  Next, this method takes time.

I was hoping, therefore, to teach students about the use of transformations in graphing.  Here I know the salient features of a curve. Say I know that y=x^2 will yield a parabola and that off the top of my head I can easily graph that parabola using 3-5 points which I know highlight the salient features of the graph…I then am presented with y=x^2+1 and I know that this graph is the same as the prior except that each point is shifted up 1 making it easy to graph.  Basically I need to know what the “parent graphs” look like and the rules for shifting them up, down, left, right, flipping them over the axes and so on.

Well telling students the rules is no fun 🙂  Additionally it might increase students’ understanding if they found out the rules themselves thus being engaged with the material (as opposed to listening to me recite the rules).  To this end the class is broken up into groups.  Each group graphs 5-6 functions all of which highlight a certain transformation.  Graphing y=x^2 followed by y=x^2+1, y=x^2+2, y=x^2+3 and so on allows students to see that adding a positive constant (call it k) to the equation shifts the graph up k units.  The graphs are up on chart paper and hung across the front of the room once all groups have completed them.  In the past I then asked the class what they notice.  They might say “all of the graphs in that poster moved left” and from that we construct the transformation rules.

This time, in an effort to incorporate writing into the class (and to do my writing seminar homework) I did things slightly differently.  After all the graphs were hung up students were asked to pick two of the posters and write about (1) the transformation observed in each   (2) develop a rule in words for what is occurring in each (3) develop an equation whose graph could be sketched by utilizing this rule.  They were told to write for five minutes though I waited until most pencils had stopped moving before telling them to wrap it up.  During the writing exercise a few students called me over to make sure they were “doing this right” which I really did not expect.  Really we were writing to get us thinking about the transformations and be ready for a whole class discussion on the topic.  While there are ways to write rules that were mathematically inaccurate I was not going to grade this and was just hoping that they were engaged with the material and able to contribute later to the discussion.  I expect that the concern arose from my lack of explanation as to what the writing was for and a concern about being right that many students carry with them.  Perhaps if we do this more often they will be less concerned with doing it right and more able to get their thoughts down on paper.

Another thing that I noticed was that when we started the discussion many more students than usual contributed.  Some students who normally do not speak up in class did so even if they did so as a response to what another had said.  There were a lot of “you can use what he said in posted 4 as well except they go left instead of right” which was nice.  I think (only speculation now) that the writing gave everyone a chance to think about the graphs.  Had I just asked in class aloud right away despite even a long pause only the “quicker” and “more confident” students might have answered.  I think the writing also gave those that struggle a bit the opportunity to put their thoughts in order…to rehearse a bit what they might say in the discussion which may have given them the confidence to speak up.  I really like this lesson and have (without the writing) done it before in other classes.  I am curious as to how the students perceived the writing piece and even the whole idea of coming up with the rules rather than being told them.  I am wishing I had asked them to write about the lesson as a whole at the end of class just to get some feedback for myself about it.  I usually do this at the end of the course (evaluation surveys that I design) but never did so for a single lesson or activity.  Maybe the way to go is to ask them to reflect on the lesson next time we meet and to collect those to see what it is they thought about all this.