A major understanding in today's lesson is that at any step in solving an equation the solution set should remain consistent. This requires students to slow down and be more precise than usual in their thinking (MP6). In slides 2 and 3 of Solving and Justifying Equations, students verify what they had learned in a previous lesson showing that a value substituted in for the variable will make the equation true after a term has been added to each side or each side has been multiplied by a non-zero constant.
(1) I will allow students to experiment with adding/subtracting a term of their own choosing. I want to remind students that the term that is added or subtracted could be a variable. Also, I remind students that the emphasis is not on isolating the variable, rather it is on verifying the equivalence of the two expressions after manipulating each side of the equation.
(2) "Multiplying" by a non-zero constant could also mean multiplying by a fraction with 1 in the numerator which would be the same as dividing by the denominator of that fraction. You also have the option of changing the statement above to multiply or divide by a non-zero constant depending on the terminology you choose to use with your students.
While students are working, I will try to pinpoint a few students who chose to manipulate the problem in an interesting way. I then have a handful of students (3-5) put their work on the board. I intentionally pick different approaches such as adding variables, integers, subtracting integers, etc. (Adding two different integers will not really show the variance in possibility).
Following up on the Opening, I will ask students to do a Think-Pair-Share around this question. I want to listen to student conversations to determine if any students realize that the equation has no solution. Once students have had an opportunity to discuss, I do a Non-Verbal Cue to determine the thoughts of the class (Thumbs up for agree, thumbs down for disagree). If most of the students agree, I ask them to find the solution. It is important to allow those who agree and disagree to voice their opinions and let the others critique their reasoning (MP3).
Teacher's Note: I will guide the class to the understanding that all of the rules that they have been learning about and using dealing with solving equations rely on the premise that the original equation actually has a solution. By following the properties of algebra (commutative, associative and distributive) and by adding a term to each side or multiplying by a non-zero constant they will get a true statement if the original equation is true. If the original equation does not have a solution then none of the following equations will be true.
In this section, I will have students justify each step in solving the equation. I put the reasons up and have students verify their work. The equation stops one line early for a reason...here we want to emphasize that, again, each line of this equation should have the same solution set (each line is a true statement). The last line 20=5x, while simpler than the original equation is still an equation which has a solution (x = 4) I have students find the final solution. On the next page, students should verify that each line of the equation has the same solution by substituting 4 in for x and showing that the equation is true. I have each student verify two lines for themselves and use the repeated reasoning to show that it would be true for each line (MP8).
*If you would like to approach this in a more organized way you could also assign a specific line or two to certain groups. This way you can bring the whole class back together and have groups report back on what they found showing that x=4 is the solution for each line.
On the third page, students will be working on a guided practice of solving equations and justifying each step of the process. In each partnership, I have one student solve and have the other justify the reason for each step (one leads the other follows). Then students can switch roles on each subsequent question. I ask students not to verify answers at this point as they will be doing so in the independent practice portion. The emphasis here is the justification of each line of the equation's solution.
In this section, students get some algebraic practice. Students do not need to justify the reason for each step in this section but they will be verifying their solution by substituting their solution set into the original equation to verify that it works there as well. As an extension, you could also encourage students to substitute their solution into another line of the solution as well. This step of substitution is once again attempting to ingrain the mathematical practice of being precise with their solutions (MP6).
I really want to encourage students that the emphasis is on the process not the product. With this in mind I will often put the answers to the questions up before students start working. This helps them to confirm that it is not about getting an answer but more importantly the process that you use to get there.
This closure is more about pre-assessing than about assessing the day's content. Through informal observation, I have determined a very good sense about which students are able to solve, justify and verify solutions. This question will assess students thinking about a future lesson dealing with common misconceptions in equation solving. The idea that the two solutions do not make the original equation true should get students thinking as to the possible reasons why. It is a common misconception (even among higher level math students) that a variable can be multiplied in or divided out from each side of an equation. A future lesson will address this more comprehensively.