Solving One-Step Equations Using Reasoning

In partners, students work together to solve the Think About It problem  (TAB.pdf)  in any way that they can.  

After a few minutes of talk time, I bring the class together and ask them how they represented this problem.  I want to validate strategies and make sure that students hear the different ways of approaching this problem.  I ask, "How many of you used a picture to represent this?" I follow up with, "Who wrote an equation?" I want to gauge the students' use of representations.  

I ask for a student who modeled using a picture to share his/her work, and I display it on the document camera.  The student who created the picture shares what (s)he was thinking, and then I open it up for feedback from other students (If no one drew a picture, ask 'what could a visual model look like for this problem?').

I have 1-2 other students share out their thinking to reach 15 marbles.  I ask the group to explain to me why 5 marbles would not make sense as an answer.

Then, to transition from the Think About It to the new material, I frame the lesson:  "Today we are going to be seeing equations like this which represent problems in our everyday lives with an unknown starting amount.  Using reasoning, we can get back to the number we started with, just like you did in the Think About It."

(The goal of this lesson is to have students reasoning about operations and relationships between numbers.  Some kids will start to talk about inverse relationships, and that's great. But, it is not the goal for students to start solving for variables using inverse operations yet.)

The questions I use to guide students through the Examples:

• What is a variable?   Today we will be solving for unknown amounts, or variables in equations.
• What is an equation? 
• For the first example, what do we know?  
• What is another way we can say this?  
• Should x be smaller than or greater than 10? 
•  What is 6 more than 10? 
•  Let’s check if 16 is my unknown number.  

I then walk students through the rest of the examples following the same line of reasoning.  For the division problems, I have students re-write them using a division symbol.  This strategy more easily allows them to see that you can re-write it as a multiplication problem to solve. 

Also, we discuss an exemplar written response:

I determined the value of x in 11 - x = 5 by first thinking about the equation as “I started with 11 and subtracted some number, and was left with 20.”  I know that this means that the number I started with has to be 5 less than 11.  11 – 5 = 6.  And 11 - 6 = 5.

Then, we move to the next section where students need to write an equation for the solution x = 4.  I make sure that we have examples shared out where the variables are in different locations (ex:  x - 3 = 1 and 8 - x = 4).

During Partner practice, students work together to solve for the variables  (Partner Practice.pdf).  They work in a Player/Coach format:

• the first student will serve as the 'coach' and will explain to the second student how to complete the first problem.  
• the second student asks questions to help with understanding
• once both students agree on the answer, it is recorded on both papers
• the roles switch for the second problem, and so on

As students work, I circulate, listen, and ask:

• How did you know that the variable equals this value in this equation?
• How did you check your work to know that you are correct?

 I'm looking for:

• Are students able to explain how they came up with their answer?
• Are students able to explain why their answer makes sense?
• Are students using substitution to check their answers? 

 After 10 minutes of partner practice time, I bring the class back together to talk about their work.  I ask for a student to show the work from a problem of his/her choice on the document camera.  The student talks through the thinking that went into the problem and explains how (s)he checked the answer.

I make sure that we talk about one of the problems that involves the variable divided by a constant ( n/4 = 12).  I want to be sure that we talk about 3 not being a reasonable answer, and reinforce the importance of using substitution to check the answer.

Students work on the independent practice problems (Independent Practice.pdf).  I make my way around the room, and make sure that students are solving for the variables and also proving their answer using substitution.

In this lesson, students who are able to quickly determine the value of the variable sometimes try to skip the proving step.  I keep an eye out to be sure they are providing a top quality answer, including the check.  They are building good habits for more complicated problems.  

After independent work time, I have students share out the equations they've written for question 2.  The entire class uses substitution to prove whether or not g = 5 is a solution for the shared equation.  I am sure to have multiple students share out for the same operation.  I want students to understand that there are an infinite number of equations that have 5 as a solution.

For a challenge, I ask students to create an equation using fractions that has g = 5 as a solution. 

Students work independently on the Exit Ticket to end the lesson.