James Bialasik SWEET HOME SENIOR HIGH SCHOOL, AMHERST, NY
Algebra I : Unit #9 - Understanding Equations : Lesson #12

CAUTION: Equation Solving Ahead

Objective: SWBAT identify incorrect procedures when writing equivalent equations.
Standards: HSA-REI.A.1 HSA-REI.B.3 HSA-REI.B.4b MP1 MP2 MP3 MP7
Subject(s): Math
60 minutes
1 Opening - 5 minutes

This opening will allow students to practice the work from previous lessons dealing with both solving an equation and justifying each step.  I want to make sure that students also verify that the solution they obtain will  make the initial equation true.  To accomplish this, I have students try this opening activity by themselves and then follow-up with a partner discussion about the solution they obtained (see Fostering Student Ownership of Learning for more information about how I use student conversations to support learning).

2 Launch - 5 minutes

In this section we will revisit the closing activity from the previous lesson.  I will have students read the scenario and work once again and do a think-pair-share to determine whether they agree or disagree with John's solution (MP3).  Each partnership should come to  a consensus as to whether or not they agree with the solution and the moves that John made.  I then ask students to do a non-verbal cue so that I can check each partnerships thinking about John's solution.  I have students who both agree and disagree with the solution state their case and allow others to support or refute the arguments.  In the end, I guide students to the understanding that if the solutions do not make the original equations true then some mathematically incorrect process was used during the solution. 

If the fact that John multiplied by a variable does not come out right away as the incorrect procedure, I assign each partnership a line of the equation to check the solutions {0,3}.  The solutions will make each line true except the first one showing that multiplying line 1 by x was the incorrect step.

The second slide of caution_equation_solving_launch is a rather obvious attempt to show that the statement "anything you do to one side you do to the other" can lead to mathematically incorrect statements.  I have students look at the work and they should realize rather quickly that John "got rid of" the denominator on each side thus leaving him with an untrue equation based on the original.  Lastly, I have students find the correct solution by multiplying each side by a non-zero constant.

3 Investigation - 25 minutes

This investigation is challenging and will require students to use several mathematical practices (MP1, MP2, & MP3).  Students will need to work with their partners to understand the algebraic steps that are taken in each question and reason abstractly about the new solution sets to the equations and whether or not they make the original equation true.  Students will also continue to follow "John" on his journey of following the rule, "Whatever you do to one side you do to the other."  In each question they will be asked to critique his reasoning.

The investigation is meant for students to explore common misconceptions among algebra students when solving equations.  By allowing the students to discover why these misconceptions lead to false mathematical statements they will be less likely to make the same mistakes themselves.

Teacher's Note: When students complete question 4b and 4e, the students do not need to simplify each side of the equation.  In other words, after multiplying by x they can leave the equation as x(x-3)=5x.  When the multiply by x-1 the result would be (x-1)(x)(x-3)=5x(x-1).  While students could use the distributive property to simplify both sides of this equation doing so is outside of the scope of the investigation.  The emphasis is on verifying the solution set and doing so is more obvious when the equation is written as it is above.  That being said, simplification of the above equation could be used as an extension.

4 Closing - 5 minutes

This closing will be a summary of the this lesson and previous lessons.  I ask the students to respond in writing to the following prompt:

What moves have we learned will not change the solution set of an equation?

What moves did change the solution set of an equation?

Once students have a couple of minutes to answer the above questions, I call everyone back together so that we can summarize as a class.  We close with the idea that you can try to use other moves on both sides of an equation and they could give you possible solutions but you will need to check those solutions in the original equation to ensure they result in a true equation.