In the next two or three classes, students examine how the graphs of quadratics are affected by changes in a, h, and k. I particularly like this section of the unit as it is very visual, and I find that students who often struggle with algebra really enjoy these lessons.
This year, rather than use graphing calculators for these lessons, I have been using desmos.com on Chromebooks. This use of technology requires some mindfulness for me about setting up class and having students close Chromebooks during discussion periods (as some of them are unable to resist the lure of the computer screen!). I also have students link desmos to their school gmail account so they can save the graphs they are working on. This comes in handy at the end of the unit when students highlight their best work in a portfolio project.
Because of this use of technology, I build a little extra time into the lesson for students to get set up on desmos.
I start class today by letting students know that we are going to start looking at different families of quadratics. Today's task will focus on looking at the simplest of those families y=ax^2. I like a series of lessons in the IMP Year 2 curriculum that focus on a first, then k, and finally h, but have been modifying the task Transformer's Shift y's to cover the same material. To start with looking at a (rather than h and k) first, I follow questions 7 & 8 first. Students choose 4 different values for a and then create graphs to see the affect that a has on the parabola.
The bulk of class today will be spent with students working on their Chromebooks to investigate how changing the in y=ax^{2 }changes their graph. Students should begin the task by choosing different values for a and seeing the changes in their parabolas. For Part (a) especially, I like to have students compare each change to the original quadratic y = x^2, rather than having them list all 4 or 5 in the same screen. I think it's easier for them to notice the changes when they are comparing two graphs, rather than 4. I ask students to write down the changes they are noticing. After they explore 4 or 5 different quadratics, they will try to summarize the affects of the parameter, a.
As students work, here are some things that I will watch for:
Differentiation: As mentioned above, I find it helpful to guide students who need more support to always compare their changed equation to y=x^{2}. I also ask students to sketch both graphs and compare them, making notes about what they notice.
Teacher's Note: If some students finish before others, have them try to recreate these graphs using desmos. I find students get really excited about this activity and you can make it a friendly competition to see who can get their graphs closest to the originals.
When all students have compared at least four graphs, I will bring the whole group together for a discussion. I plan to ask students, “With each change in a, what has changed in the graph?” I will record my students' observations on the board.
As I scribe, I am listening for a statement like “as a gets smaller the parabola gets narrower.” I tend to caution students about oversimplifying to say that as a gets bigger the parabola gets narrower. Although this is true for positive numbers, it does not hold true for negative numbers. I ask students how they can more accurately describe what's happening. Try to get them to articulate the differences when a < 0 and when 0<a<1, along with a > 1.
If there is time, I will ask students to share the equations they used to recreate the designs in Question 3. But, I will certainly complete today's lesson by asking students to complete an Exit Ticket. Here is today's prompt:
Write about one thing you have learned today about quadratics in the y = ax^{2 }family.
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