Comparing and Graphing Quadratic Functions in Different Forms

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Objective

SWBAT match paper dominoes when given different structures of Quadratic Equations to their graphs by complete missing information of the key features of each structure.

Big Idea

To go deeper, not wider! Identify misconceptions, and expert learners to teach other students.

Warm Up

10 minutes

In the Warm Up of this lesson, I draw three columns on the board.  In each column, I write a Quadratic Equation in one of the three forms:

1.  Standard Form

2.  Vertex Form     (Completing the Square Form)

3.  Intercept Form (Factored Form)

I hand each student a Warm Up of the three columns and instruct students to list any key features about the Parabola from each given function.  Once students have listed every feature that they can identify on their own, then we review the key features of each as a class.  Here is an example  of a complete Warm Up after the review.  I allow students to use the Warm Up as notes to guide them in the following Partner Activity, and also to keep in their notebooks for further reference if needed. Listed below are some of the key features for each form.

  • As you can see from the complete Warm Up, a is easily identified in all three of the different structures as positive or negative.  This positive or negative indicates if the Parabolas opens upward or downward.
  • In Standard Form, the constant (c) in the equation is always the y-intercept. 
  • In Vertex Form, the Vertex is easily identified, however, students need to be careful of the signs of each coordinate.
  • In Intercept Form, the x-intercepts can easily be identified, but again students need to pay close attention to the signs of each x-intercept.

Other features may be found from each structure with more work, and the use of formulas and symmetry.  Students may also be able to change between structures by multiplying or completing the square.  I have not taught students how to complete the square yet, so most students are not going to use this method.  I do however, introduce it to a few of the higher level groups in the next activity.

 

Partner Activity

30 minutes

The Partner Activity in this lesson, and the idea for the Warm Up come from a lesson called Representing Quadratic Functions Graphically from the Shell Mathematics Assessment Project at http://map.mathshell.org/download.php?fileid=1734 (last accessed 6-27-15).  I have also provided a full copy of the lesson here.

After reviewing the Warm Up with the students, I hand each set of table partners a set of paper dominoes that I have cut apart.  Each domino has two ends.  On one end of the domino is a graph of a Parabola, and on the other end of the domino is a set of Quadratic Equation(s).  The equations and the graph on the same domino are not matches.  The students have to match the graph side of the domino to equations on another domino.  Students then match the equations side of the given domino to a graph on the other end of another domino.  Students continue matching dominoes end to end until all of the matches are complete.

Table partners are already in homogeneous pairs based on their work thus far with the Quadratics Function Unit.  Each set of table partners receives a set of paper dominoes with labels A-J. All of the graphs are provided on the dominoes, but the Quadratic Equations have incomplete information. There is a location on the Equation side of the Domino for all three forms of each Quadratic Equation. They are listed in the order of Standard Form, Intercept Form, and Vertex Form.  With pieces of the equations missing, it requires students to be able to do some work to verify that a set of Equations matches a specific graph.  

I have students begin, as instructed in the original lesson with dominoes A, E, and H to demonstrate how to match the dominoes.  Students begin working as I walk around to monitor their progress. I encourage students to write the features on the dominoes as they are identified.  Once I have checked that a pair of students have matched dominoes in the order of A, H, and E respectively, I instruct them to continue matching the rest of the given dominoes.  I demonstrate the questions to ask students as they are working with the graph on domino A to the equations on domino H in the video below.

 

I provide adequate time, about 20 minutes for the collaborative Partner Activity of matching dominoes.  Not all of the students will match all of the dominoes, and that is okay.  The students should still be able to identify key features from different structures of the Quadratic Equation and how to use those features to match the Equations to the correct graph.  I have provided pages T-9 and T-10 from the teacher resources of the original lesson to show the correct order that the dominoes should be placed.  Some students place the dominoes in a rectangular shape or oval shape, and some students just have a few dominoes aligned, but haven't configured them into a shape.

As I am walking around questioning students, I also write down learners names and the pair of dominoes that I want them to share with the class. I do not tell the students that I have selected them to present at the end of the Activity.  I choose these students based on their correctness of identifying a match, difficulty of identifying a match, or maybe even based on a change that was made between matches using the key features. By using a variety of students to present for different reasons, it helps all of the students in the class reach the objective.  Again, the objective is for students to be able to identify key features in each form of a Quadratic Equation, and to be able to use these features to match it to a correct graph.  

After providing students with about 20 minutes of matching the dominoes, I call on the students that I have selected during my observations.  These students are purposely planned to show something about the objective that I feel is important.  For example, two equations may have the same zeros, but some other feature makes them have different graphs, like the sign of a. So I may have the students present the dominoes that are correct and incorrect.  

Exit Slip

10 minutes

In this Exit Slip, I am assessing the students ability to write equivalent Quadratic Equations for a given Parabola on the coordinate plane.  Students are to write the Quadratic equations in all three forms:

1.  Standard Form

2.  Intercept Form 

3.  Vertex Form

I instruct students to work the Exit Slip on their own and hand in before leaving class today.  If time becomes an issue, I will give the Exit Slip at the beginning of the next class period. I do not want to provide this Exit Slip as homework because I want to see each student's own ability.  

I do not instruct students on a specific method to write the equations.  Students may use key features from the graph, like zeros, y-intercept, the Vertex, or other points to write the equations. Students may also develop an equation from another form already known.  I do ask that students show their work, and explain their reasoning.