(Link to worksheet file is in the middle of the post. TL;DR summary at the end.)
The textbook we use at our school (Glencoe 2005) has a lot of things that I don’t like about it (why am I teaching the Quadratic Formula before simplifying radicals?) but when I started looking through the chapter on quadratics I found something interesting. After the text introduces graphing a quadratic and identifying the axis of symmetry and vertex, they devoted two pages to this “Graphing Calculator Activity”. The idea was that students would explore families of quadratics by using a graphing calculator to graph the parent graph – y=x^2 – and various manipulations of the graph to see how changing values affected the graph.
I liked this idea, since my students have not had nearly enough time this year to explore graphs and their equations, but I don’t have graphing calculators in my room. Most of my math colleagues have class sets of calculators, but I’m new this year. (This is also the reason my students have not been able to explore equations and their graphs this year.) I haven’t taught them how to use graphing calculators and anytime I’ve needed to graph something in class, I’ve done it on the projector using Desmos.com. I’ve also encouraged them to use Desmos to check their work and even required them to use it to do their homework on Solving a Quadratic by Graphing. (Their graphing-by-hand skills are abysmal.)
A quick comment here: I have about 4 students with their own TI-83 or 84 graphing calculators, including a visually-impaired student whose calculator talks to her. I forgot what a GIANT PAIN IN THE ASS they are to use! Seriously – 2nd > TRACE > scroll down to ZERO or press 2 > scroll across to LEFT BOUND > scroll over to RIGHT BOUND > GUESS > then it gives you the answer Then you have to do it all over again for the second solution?! After spending the whole school year using Desmos (where you just click on the graph and it gives you the coordinates and it doesn’t cost $100), I’m just not doing it anymore.
Where was I? Graphing worksheet. Right.
So instead of using graphing calculators, I booked a day in the computer lab to use Desmos. I created a worksheet that uses all the same concepts from the activity in the textbook, only I edited my instructions a little so my students would understand what I meant. I informed the kids that we would be in the lab, and passed out this worksheet when they got to the lab with basically no explanation. (I probably should have clarified expectations better – see “The Bad” below.) Most of them finished it, and a few finished very early. (Also see below.) I refused to let them turn it in until I was happy with their explanations and observations, which was very good. I made them answer Question #3 the way it was supposed to be answered, and ensured that they were noticing the things they should have been noticing.
Download the Word document (.docx format) here:
Graphing Quadratic Functions Exploration
In general, I think my students noticed the effects that different changes to the equation can have on the graph. Their written explanations and observations tended to show a fairly solid understanding of this. Because of our weird schedule that week (standardized testing on different days for different grade levels), I didn’t get a good chance to discuss their findings as a class. I’m also not sure if they truly generalized their observations, or if they retained the information.
My students generally seemed to understand and enjoy the activity. Considering how often my class feels like I’m pulling teeth, I consider this a huge advantage. I will absolutely consider activities like this anytime I introduce graphing in the future.
After spending the whole class period graphing on Desmos, a few kids were fascinated by everything Desmos can do. A couple of students graphed a bunch of quadratic and linear functions on the same coordinate plane and used a variable to animate their graphs. This turned out really cool, and I made them save their work on their accounts. A few kids got to explore all the graphs that Desmos contains in the menu, including crazy polar graphs. Some even noticed the Staff Picks graphs on the Desmos homepage, opened them up, and started playing with them. This was so awesome – they kept calling me over to show me stuff on Desmos. I’ve never seen them interested in and exploring something that has to do with math before. At all.
I probably should have given my students a better idea of my expectations for the worksheet. I should have explained that I wanted them to make discoveries for themselves; that I hadn’t explicitly taught them the answers to the questions on the worksheet yet. I also should have explained that I was looking for them to use the vocabulary terms they learned in class (parabola and vertex, mainly).
Question #3 wasn’t phrased very well for my students. (In my defense, this is partly because they didn’t read the whole question. They tried to graph y=ax^2 and were confused when Desmos either wouldn’t graph it or immediately defined a=1 and there was no change to the graph from the parent graph.) I should have made it more obvious that I was looking for a general explanation of what happens when you substitute a number in for a.
I didn’t realize how quickly some students would finish, and I should have had an alternate activity prepared for when they did. It didn’t occur to me until my very last period of Algebra 1 that Function Carnival would be a perfect extension for this lesson.
I loved this activity. Almost every student was engaged almost every moment. I have a couple of improvements for next time, most of which were caused by my own inexperience (first-year teacher). I’ll be using similar activities to introduce graphing lessons a lot more next year.