Vertex Form to Standard Form

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Objective

SWBAT use an area model to square binomials. Students will be able to rewrite quadratic functions from vertex form into standard form.

Big Idea

Students apply the multiplication models they have recently learned to the task of rewriting quadratic functions from vertex form into standard form.

Opening

10 minutes

I begin class with a warm up activity where students are first asked to identify parameters "a," "b," and "c" for a quadratic and then multiply some simple binomials. This multiplying binomials piece is a review of what students have been working on and I have a few students come to the board to show their area models.  Students work up to Question #8 as part of the warm up for class. These warm up questions give students a chance to revisit and solidify what they learned in the last lesson and give me the opportunity to check for understanding.  From a quick look at the warm up, I can identify students who need targeted instruction in multiplying binomials, or who might need more scaffolding when it comes to factoring later in the unit. 

Next, I tell students that today’s task will focus on applying these new multiplication skills to change quadratic functions that are in vertex form into standard form. I also like to emphasize with students that it is important for them to be able to work fluidly between quadratics in different forms.  Today we will be working from the vertex to the standard form, and later in the unit, we will go in the opposite direction.

 

Investigation

35 minutes

I like to group students in homogeneous groups of three to four students for this activity. I hand out the Vertex to Standard Form ask and we read through the assignment together.   A good place to start is on Question #10 where students see a quadratic in vertex form (where k = 0).  I might do Questions 10 and 11 as a whole group and then let students start on Question #12 in their small groups.

Students first have to think about how "k" affects the number of x-intercepts a quadratic will have. I like how this portion of the task connects the vertex form further with x-intercepts which is where we are headed as a class. Next, students graph quadratics in vertex form. I emphasize that students should graph at least three points on both sides of the vertex.  Of course, students will start to remember about the symmetry of quadratics and graph accordingly.  Finally, I ask students to put each of those vertex form equations (the ones they just graphed) into standard form.

 Things I look for while students work:

  • Students who struggle to recognize the effects of k may need help sketching some basic graphs to see how many x-intercepts result with a change in k.
  • Some students in my class with inevitably have difficulty graphing the quadratics in vertex form. They may need prompting to remember how to identify the vertex and then a hint about using that as the starting point to the graph and looking at points on either side.  I encourage them to make a table in order to graph their points.
  • Sometimes students need reassurance they can use the area model even if the constant term is negative.  I try to help them construct a diagram that shows this.
  • Students may struggle with the order of operations in Question 20.  They may need reminding to square the binomial first and then use the Distributive Property to multiply each term in the resulting trinomial.  Finally, they will add the constants at the end.

 

 

Discussion & Closing

15 minutes

Discussion

I try to leave ample time to discuss this activity.  I have students who finish early share some of their graphs and standard form equations on the board. 

I always find that graphing quadratics is a good way to incorporate the SMP7: Look for and make use of structure.  In my discussion of both Questions 12 through 17 and 18 through 23 we can talk about why the graphs come out the way they do. Students should be able to see the effect of parameter k on the x-intercepts and of the x squared on the shape of the parabola.  I believe actually having students plot points helps in this understanding.

Next, we discuss how to turn these vertex form equations into standard form.  Some students may struggle with following the general rules of the order of operations.  I try to elicit from students what order we need to do things in and how to think about a negative sign out in front of the parenthesis. Students may need reminders about the distributive property, but most of the guidance and corrections can come from other students in the class, rather than me.

 

Reflection

Have students complete an Exit Ticket related to Reflection.  Ask them to complete the follow prompt on an index card:

Summarize what you have learned in writing:  How would you summarize the steps for going from Vertex Form to Standard Form?

 

 

 

Citations

  1. Structures of Expressions 2.2 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