Properties of Parabolas Day 1 of 2

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Objective

SWBAT identify how the parameter "a" in y = a*x^2 affects the graph of a parabola.

Big Idea

Students use technology to explore the relationship between changes in the parameter a in y = a*x^2 and the corresponding changes in the graph.

Opening

5 minutes

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.

Investigation

45 minutes

The bulk of class today will be spent with students working on their Chromebooks to investigate how changing the in y=ax2 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:

  • I find students like to use the words “bigger” and “smaller” to describe the changes they see in their graphs.  I encourage them to be more specific and try to elicit words like wider and narrower so they are using more descriptive language.
  • Students may skip values for "a" that are between 0 and 1 because they can’t “find it” in their viewing window.  If I observe this, I will prompt them to figure out how to zoom out far enough to be able to see it. Desmos is definitely easier to work with here than graphing calculators. Once the idea of zooming is present in the room, I will ask students to predict where they graph may be, before reminding them to zoom. 
  • Students may try to enter all the equations at once. This will make it very difficult for them to figure out which changes in the parabolas match with which changes in the equation.  I encourage my students to compare new equations with the original (y=1x2), and, maybe with three parabolas at time at most.
  • I try to repeatedly use the phrase “with each change in a, what has changed in the graph?” to help students articulate what they are seeing in precise mathematical language.
  • Some other guiding questions I may use with students during this investigation:
    • What would happen if….?
    • Will the parabola be the same if we use different numbers?
    • What patterns are you noticing?
    • What is the same? What is different?

 

Differentiation: As mentioned above, I find it helpful to guide students who need more support to always compare their changed equation to y=x2.  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.

 

Discussion + Closing

10 minutes

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 = axfamily. 

Citations

  1. Transformers: Shifty y's is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

    http://www.mathematicsvisionproject.org/secondary-mathematics-ii.html