Equations involving Factored Expressions

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Objective

SWBAT solve equations involving factored expressions by using the zero product property.

Big Idea

The zero product property is a powerful idea in mathematics that can be used to solve higher order equations.

Warm UP

5 minutes

I plan to have students work individually on this Warm-Up.  Many students will be able to solve these questions by inspection (without showing any work, just looking at the problem).  Once all students have finished,  I will have each pair of students do a Turn-and-Talk to discuss how they solved the two equations.  Afterward, I will allow a few students to share out their ideas.  

When students share their strategies, I encourage students to try to verbalize everything that went through their mind when they were solving the equations (MP3).  Such descriptions help others in the class get some insight into the thought processes required to interpret and solve these equations.  When a student finishes sharing, I will probably do a non-verbal cue (thumbs up/thumbs down) to see which students in the class shared a similar way of thinking.

 

Direct Instruction

15 minutes

I will use the factored_expressions_solve_direct presentation to share today's direct instruction.

We will begin with Slide 1. I let students do a Think-Pair-Share around this slide.  Many students will make the leap that the answers will be the same as those on the previous slide, but they won't be sure why.  As students are discussing their ideas in their partnerships, I listen in for students who are making some breakthroughs about the importance of the number zero in a factored equation.  In my experience, I have also noticed that students do not often see the left hand side of the equation as an expression involving multiplication.  Hopefully, the students will observe that the expressions are being multiplied together.  And, since the result is zero, one or both of the values must be equal to zero.

Slide 2 allows students to extend their hypothesis to an equation involving more expressions. I will let students work on this question with a partner. Again, I will listen for students who are recognizing that when any value is multiplied by zero, the resulting product must be zero.  

Before putting up slide 3, I plan to have students make a conjecture about what these two exercises have showed them (MP2).  I want them to discuss their ideas with a partner and then try to formalize them in writing (MP3).  I will give my students about 2-3 minutes to do this. Some students may get this exactly, but most will not.  My goal is to give students the opportunity to slow down their thinking; to interact with the content on a deeper level so that we can make progress later, if not now.

Depending on the confidence level of my class, I may discuss Slide 3 abstractly (as it is written). Alternatively, I may show some concrete examples by plugging in 0 and another number.  I may also go back to Slide 2 and show how "a" is really the first part of the expression (e.g. (x+10)) and "b" is really the second part of the expression (e.g. (x-5)).  If the opportunity affords itself, I will challenge my students by asking them to write the Zero Product Property for the expression in Slide 2 (if abc=0 then a=0 or b=0 or c=0 or a=b=c=0)

Slide 4 should help students see where the a and b "come from" in the Zero Product Property. I expect my students to come up with a solution for the first question on Slide 6.  Then, I will ask them to think with a partner about how to solve the second part of Slide 6 (MP1).  Eventually, I will have a student put their work under the document camera or on the board to explain their thinking behind finding the equation whose answers were given.

Slide 5 is another thinking question that students can work through with their partners (MP1). (Aside: I include this question because I have seen this incorrect extension of the Zero Product Property from some of the brightest calculus students who don't understand the concept of the property).  Rather than show students why this does not work, I want them to discover it for themselves.  If students are struggling with how to start, I will have them solve the two equations for x and then substitute their answers back in to see that the result is false.  You can then have students extend their thinking to the abstract if ab=1 is it true that a=1 and b=1 or are their other solutions (other solutions would be 2*.5, 4*.25, 2/5*5/2, etc.)

Practice

15 minutes

After the Direct Instruction, I will ask my students to work on the following problems on their own. Towards the end of class, I will allow them to compare answers with a partner. At the same time, I will ask several students to come up to the board to show and explain their work.  

If I discover that students are struggling with Question 6, I may give them the hint that they should use the distributive property to try to make that equation look like the others.  If I see the students misapplying the 'Zero Sum Property' I will remind students that the Zero Product Property only works with multiplication. So, if a + b = 0, we can't assume that a = 0 or b = 0.  It may be an interesting exercise to have students justify this as well.

Close

5 minutes

This ticket out the door asks students to put an abstract concept into their own words so that they can more deeply internalize its concept (MP3).  I will encourage my students to think about more than just the example that is given (x-5)(x+10)=0.  In order to fully explain the concept they may want to pull in other examples from class or make up some of their own.  I will entice my students to be thoughtful and creative by telling them that I will be on the lookout for the most interesting way to explain this idea to another student.