Properties of Parabolas Day 2 of 2

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Objective

SWBAT identify how changes to parameters k and h affect the graph of a quadratic function.

Big Idea

Students get excited to explore how different families of quadratic equations change the shape and position of the resulting parabolas.

Opening

5 minutes

Today, I begin class by reminding students that in our last class, we looked at how the a in y = ax2 affects the shape of the resulting parabola. Then, I have students work on the Properties of Parabolas Day 2 Warm Up individually. After a few minutes I bring the group together to discuss.  Since this exercise should be review, we will go over it quickly. My goal is to remind students of the work they recently completed and prepare them to build on their knowledge in today's lesson.

Investigation

50 minutes

Today’s class has two sub sections. In the first section, students work individually to explore the effects of k on y = x2 + k. Students usually need between 10 and 15 minutes for this task. I like to set a timer to keep the class on track so we can cover both sections for today.

I usually start by explaining to students that we've looked at the affects that "a" can have on a quadratic, and now we'll explore another family of quadratics:  y = x2 + k.  

Depending on the class and how well they work independently, I either ask students to choose 4 different values for k and graph them all in desmos, or follow along with the Transformers task (Questions #4 and #5).

As students work, here are some things I watch for:

  • I remind students to use y = x2 as the equation they will compare everything else to.  
  • Sometimes, I have to prompt students to make a connection to the new vertex.  I might ask them, “What are the coordinates of the vertex of y = x2.  How does that compare to the vertex of y = x2 + 4?”
  • I repeatedly use the phrase “with each change in k, what has changed in the graph?”to help students articulate what they are seeing.
  • Other more general questions for working 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?

 

After 10 to 15 minutes of work on this task, I bring the class together to discuss their findings.  I will also be asking them to make predictions about adding "a" back into the equation.  I follow the pdf of the smartboard file (Properties of Parabolas Day 2 Discussion Questions) to guide the discussion.  Next, I show students that they can now reintroduce a back into the equation. I ask them to make predictions about a few equations that now have a and k in them. I try to elicit predictions from different students and have another student confirm or critique based on desmos.  I find that students get excited about this portion of the lesson!  They are eager to share their predictions based on what they have learned.

Next, I follow the same format for the variable "h".  (or Question 6 in Transformers). Now students will be looking at the h in y = (x-h)2.  Again, I have students work individually for 10 to 15 minutes and then bring the class together for a whole group discussion. 

As students work, here are some things I watch for:

  • Students may struggle with the negative sign when h itself is negative.  I try to be sure students understand that when h is negative y = (x – (-h))2 is the same as y = (x+h)2.  They may confuse which was the graph shifts based on when h is greater than or less than 0. I try to help them get clarity on this piece.
  • Again, I may have to ask specific questions to get students to articulate what is happening to the coordinates of the vertex as h changes.

After 10 to 15 minutes, I bring the class together again and follow the Properties of Parabolas Day 2 Discussion Questions discussion questions to guide the whole group share out.  I show students that all three families can be generalized together as y = a(x - h)2 + k.  Next, I tell students they will be thinking about this combined equation for homework and making predictions about what happens to the graphs of such equations.

 

DIFFERENTIATION:  I try guide students who need more support to access this activity to always compare their changed equation to y=x2.  They can sketch what both graphs look like and compare them, keeping track of what they notice.

Students who are ready for an extension activity can work on replicating fun designs of quadratics. Sometimes I'll ask students to share out their coolest designs or ask them to try to replicate a particular shape, like a spider.

 

Closing + Homework

5 minutes

Students have learned about two additional families of quadratic graphs today.  I try to give them time to reflect on their learning.  I usually assign an exit ticket with a prompt like: “Which family of quadratics do you think makes the most interesting graphs and why?” or “What are the effects of k and h on a parabola and how will you remember them?”

Homework: I assign students four equations to make predictions about for tonight’s homework. These equations are listed at the bottom of the Properties of Parabolas Day 2 Discussion Questions.  I let students know they should not use a calculator to make their predictions.  Instead, they will be combining their knowledge of a, k, and h and their effects on the graphs.

 

 

 

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