Solving Equations

This opening is designed to get students thinking about the fact that when the properties of arithmetic (commutative, distributive and associative) are performed on one or both sides of an equation the equivalence of that equation is maintained.  This idea will also be extended to include addition (or subtraction) of a constant or variable to each side of an equation and multiplication by a non-zero constant.

Slides 2 and 3

Start with having students verify that each equation in fact has a solution of x=2.  Students can then visually inspect the equations to determine how the first equation is related to the second equation.  You may want to prompt students to think about the properties (commutative, associative, and distributive) from previous lessons when analyzing the two equations.  Students will need to look at the structure of each equation (MP7) to determine the change that was made. 

At the start of this investigation I explain to students that they will be exploring how the equivalence of an equation can be maintained while performing certain operations to each side of the equation.  The investigation tasks take the emphasis off of actually performing the manipulations by trying to focus student attention on the fact that in each subsequent part, the original equation has the same solutions.  

In grade 8, students learned how to solve linear equations. We drill deeper in this course, focusing more attention on the idea of equivalence or the skill of manipulating equations to rewrite them in a an equivalent form.  This lesson will help students deepen their understanding of equation solving, because they will see that as long as operations are performed correctly to each side equivalence is maintained.

Take your time with this portion of the lesson as there are some important points to sum up.  First, have students do a think-pair-share around the following prompt:

Based on the work you have done so far in class today and what you have learned in previous classes, what are you allowed to do in order to maintain equality (that is, to make a new equation that has the same solution)?

Once students have had an opportunity to think and share with their partner, begin to make a list from the entire class.  I guide the discussion (don't tell the students) towards:

1. Using the properties (commutative, associative, and distributive) to re-write the expressions on one or both sides of an equation.
2. Add or subtract a term or multiply or divide by a non-zero constant**

I ask, "Is x = 3 an equation?  What is its solution? What about x^2 = 25?  What is its solution?" This is almost so obvious it is more difficult than it needs to be.  Of course the solution is 3 or +5, -5 respectively.

** A future lesson will include an investigation dealing with what happens to an equation when multiplying by a variable.  If students ask about this situation it is up to the teacher's discretion how far to take the conversation at this point.

Now students can do some practice with solving equations.  When facilitating this portion of the lesson, I use questioning to keep students focused on why they are able to perform each step in solving an equation (are the using a property, adding/subtracting, multiplying by a non-zero constant).*

Teacher's Note: The process of combining or collecting like terms is not listed above due to the fact that it is a natural algebraic step.  If two like terms are next to each other they should be combined into one term.

Exercise 2 again takes the emphasis off of "getting an answer" and puts it back on the process of equation solving.  Students will see through experimentation and repeated reasoning (MP8) that there is not one right way to solve an equation (which will actually take some of the pressure off for a few students).  One of the strengths of algebra is that it enables many different paths to the same solution.

For the Solving_Equations Ticket Out, eight minutes may be cutting it a little close. While the students will not be solving any of the equations, they will consider the structure of each equation to determine which equations have the same solutions (MP7).  This will give me a good understanding about which students are able to recognize the properties in context.  As an extension, students could explain for each equation why it is equivalent to the other based on the algebraic manipulation that is performed.