#July2014Challenge: Guess my graph

Can you guess my graph?

I have done this activity with countless students over the years, in all classes from Algebra 1 to Calculus AB.  I'll admit that it goes better when there is candy involved, but even without the candy engagement is high.  Students need to have a graphing calculator for this activity (Desmos or Geogebra are also fine).  I show them a graph using either my graphing calculator under the document camera, or a premade powerpoint with images from Geogebra/Desmos.  Students take a guess, input into the calculator, and view the graph to check their answer.  If it is wrong, they discuss with a partner or ask a question.  I generally circulate during this time to see how many students are getting the correct answer.  We have a quick class discussion at the end of each round to address misconceptions.

I like to use this activity for review of quadratic functions in factored form.  To start the activity, I would show the graph from above and walk students through the process of writing factors given roots.  The work that I show on the board might look like the diagram below.  We don't worry about the vertical stretch/shrink at this point in time.

I am not suggesting that students show this work.  I am simply trying to model my thinking for students, and I really want to make sure they understand that the roots of a quadratic function are points on the graph.  It is not by magic or coincidence that a factor of (x-4) makes a root of 4.  If I model my thinking for them now, they are more likely to use this language or reasoning pattern later when I ask them to explain their thinking.  I might expect them to write an abbreviated version of this on the classwork or homework, but for the guess my graph activity we are going to use a graphing calculator and our voices to practice with this reasoning.  I really liked the suggestion below for showing work when you find roots.  This is from a TMC14 presentation by Glenn Waddell, which I read about at Misscalcul8.

After we walk through the first problem together, I'll show a few more graphs of the form
 y=a(x-b)(x-c) where a=1.  Then I will give problems with a= -1, and a=1/2 so we can discuss vertical stretch/shrink as well as reflecting over the x-axis.  I let them know that the only a-values that I will pick are a= -2, -1, -1/2, 1/2, 1, or 2.  I save double/repeated roots for last.  This is a favorite, and many students are able to guess the equation for a graph with repeated roots without any help from me on the very first try.  This activity is also great for an introduction to cubic and quartic graphs.

Another version of guess my graph uses mini whiteboards or Wikki Stix.  In this version I tell you the equation, and you show me the graph.  I started using Wikki Stix because I knew it would appeal to my kinesthetic learners, and teachers share mini-whiteboards at my school so it isn't always an option to use them.  I hand each student 4 Wikki Stix.  They form the axes with two Wikki Stix, then they form the parabola y=x^2 with the third.  Then with the fourth stick students can form the correct graph.  Below is an appropriate graph for y=-1/3(x-2)(x-2).

We lose some of the specificity by using Wikki Stix instead of mini-whiteboards, but I kind of like this.  It means that students have to make some generalizations about how to graph quadratic functions.  To make distribution and collection easier, I ask students to make a twizzler (see below) with the Wikki Stix.  (Warning: some students in your next class will think this is a piece of candy.  Make sure they know it is a wax twizzler and not the real thing!)