Completing the Square Day 1

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Objective

SWBAT use the technique of completing the square to solve a quadratic equation for all values of x.

Big Idea

Completing the square can serve as a valuable technique when factoring a polynomial over the integers is not possible.

Warm Up

5 minutes

This Warm Up serves as a review of the previous lesson.  Students get some practice solving quadratic equations by taking square roots.  I also want students to solve the equations by factoring as they should be fluent with both methods.  I encourage students to write their solutions in simplest radical form for the second question.  

Teaching Note: If possible, I like to have students graph as many of these equations they have time for using technology. Using an online program like desmos.com, students can see very quickly what the roots of the equation are by tapping on them (e.g., desmos_roots.png). Students could also use a graphing calculator and calculate the roots using the function for calculating zeros to get a decimal approximation.  In either case, I will make a point to remind students that what they are actually doing when solving a quadratic equation is finding the zeros of the function.

Launch

10 minutes

I will use a really straight forward launch to the lesson. Students will have little trouble filling out the table on Complete_the Square_Day1.  The main focus is on the structure of the new terms they are writing (MP7).  Once students have had a chance to fill out the chart individually, I have them Think-Pair-Share with their partners about their observations.  Then, I encourage students to share to make a record of them on the board as they share with the class.  As students are sharing, I try to focus their comments on the patterns in the tables (MP3) (MP8). I expect to hear things like:

  1. The middle term in the answer (the b term) is double the constant term in the binomial being squared.
  2. The last term in the answer (c term) is the square of the constant term in the binomial being squared.
  3. All of the terms have an x^2
  4. The last term in the answer (c term) is always positive.

All of these ideas are leading towards the students coming up with the important formula that is used for completing the square (divide the b term by 2 and square the answer).  I don't want to give students this information, I want them to come up with it.  Together, we will talk more about this during the next segment of the lesson.  So for now, I just jot their answers down in a place where they can be left for the entire class and referred back to.  

Lastly, I explain to students that all of the trinomials on the sheet are called "perfect square trinomials."  I ask them why they think they may be called this?  I have them turn and talk around this idea.

Direct

20 minutes

Slide 3 (first slide of pdf)

Students solved a question similar to this (completing_the_square_day1_direct.pdf) in the last lesson.  Let them try this individually.  I have students take note of the perfect square that is in the problem (x+3)^2.  Put most of the emphasis on solving by taking square roots.  Depending on time, I might not even have students solve this equation by factoring.  If you choose to have them do both it is to make the point that these equations can be solved in multiple ways.

NOTE: Again, as stated in the warm up, if time permits, have students graph these equations using technology to find the zeros graphically as well as algebraically.

Slide 4

For Slide 4 and slide 5 I make sure to have students first solve by factoring and solving.  This should go relatively quickly.  I then choose a student to post the answer on the board.  I leave this answer posted while solving by completing the square.

Make sure students still have their tables available that they used in the launch.  I ask students to look at the left hand side of this equation and think about how they could turn it into a perfect square trinomial.  Then I show the equation like this complete the square setup.png.  Since students have their tables they can just find this trinomial and see how it should be written.  Some students may also be noticing the pattern that will help them find the (c value) constant value in the trinomial.  The box on the right hand side is used to balance the equation.  If we add 4 to the left we must also add a 4 on the right.  Then I have students write the trinomial as a binomial squared (from the table).  Lastly, they can use the same technique for solving that they used in slide 3.  They can take the square root of both sides. 

 

Slide 5

I will put this equation up and write it in a similar way (using the boxes) as in the previous problem.  This trinomial does not appear on the chart anywhere so at this point students need to develop a method for finding the constant term.  I pause from the problem and post the following trinomials from the table on the board:

x^2+12x+36

x^2+16x+64

x^2+18x+81

x^2-18x+81

 Now I ask students to find the connection between the middle term and the last term.  Have students think about this individually and then turn and talk with a partner after they have had time to think.  This is a crucial part of the lesson so don't rush it.  Give all students time to make meaning of the pattern in the numbers (MP2).  Once students see the connection, have several students share out the idea so that all students can hear it several time in different ways.  You can also refer back to some of the ideas gathered in the launch as a way to solidify that the middle term is divided in half and then squared to get the last term.  Put in the +36 and let students finish this example.

Slide 6

Students can work on these examples independently.  Have students post the answers on the board.

Slide 7

Let students try to solve this equation by factoring.  They should notice relatively quickly that they cannot factor this trinomial over the integers and so now the utility of completing the square becomes obvious.  Let students work through this example with their partner.

Slide 8

One more example to let students practice the procedure.

Closure

5 minutes

I let students work on Completing_the_Square_Day 1_Ticket individually. I encourage students to annotate the work as if they were analyzing a text.  I intend for them to identify errors and make notes to explain how to fix the error.  If students finish this early, I will ask them to solve the problem correctly on the reverse of the page. Then, check their result by graphing the equation using technology.