Solving Linear Equations in Two Variables

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SWBAT solve problems involving two linear equations in two variables. SWBAT interpret the meaning of algebraic expressions.

Big Idea

This MAP Classroom Challenge provides students with an engaging and thought-provoking activity about notebooks, pens and cash registers to learn about solving equations in two variables!

Entry Ticket: Pre-Assessment - Notebooks and Pens

15 minutes

To begin the lesson, I ask students to complete a Pre-Assessment on their own. For this problem students work on critiquing the reasoning of two students on a problem about a store that sells notebooks and pens. I like this pre-assessment as students are asked to critique the reasoning of others, right in line with Math Practice 3. This lesson is set up nicely in that it provides a good integration and mix of content and math practice standards. Students get a lot of practice on interpreting and solving systems, and utilize the skill stressed in MP.3 to improve on their ability to demonstrate mastery of solving systems.  

Teacher's Note: Teachers can also assign this pre-assessment for a previous night's homework or for classwork in a previous class to have some time to assess student thinking prior to implementing the lesson.

I have included a Student Work Sample as a resource in this section. In this particular work sample. the student shows some good critiquing of the first person's thinking, but the prompt is incomplete. I find many of my students are not yet able to articulate their thinking well on this exercise and it many either need additional time to complete the task or do not completely finish the task in the allotted time frame.


Task Source: solving_linear_equations_complete.pdf accessed from the Mathematics Assessment Resource Service at




Individual Work: Cash Registers

15 minutes

The next problem involves students interpreting and solving a system of equations involving the number of quarters and dollar bills in a cash register. I like the task because it applies the concept of solving systems of equations to a concrete, real-life example for students. 

I usually teach this lesson during the latter half of my unit on systems of equations, so I expect many students to be able to explain the meaning of the two equations based on the context. I expect a variety of levels of thinking in terms of students' ability to solve the system of equations. A student work sample is included as a resource in this section.

I utilize this problem as a dip stick formative assessment to assess student understanding and ability to interpret and solve systems of equations.  




Collaborative Activity: Cash Registers

15 minutes

After working on the cash register problem independently, I ask students compare and contrast their strategies and solutions in pairs or in small groups. I want students to engage in a conversation with other classmates to:

  1. Communicate their own reasoning 
  2. Critique the reasoning of others

During this section of the class, I encourage students to share their thinking with classmates. I regularly take opportunities like this to emphasize Math Practice 3.  In general, the solving_linear_equations lesson from the Math Assessment Project does an excellent job of providing time, support and structure to give students an authentic opportunity to collaborate with one another. For a more detailed explanation of my thinking, watch Engaging students in reviewing students' work.

Collaborative Activity: Analysis of Sample Student Work

15 minutes

I like to close the exploration of the Cash Registers task by having students analyze the student work samples

 provided in the lesson materials. In this section of the lesson, students explore different ways to solve systems of linear equations by looking at fictional examples of student work. (It may also be possible to use examples from today's lesson, but I think that the activity is important enough to use hypothetical examples if necessary.)

One of the sample works involves a students who solves the problem by guessing and checking. Another student solves the system by graphing both linear equations and finding the intersection of the two, etc. Examples of Student work are included in resourse in:  Student Sample Work: Assessing Mia and Student Sample Work: Assessing Ava

During this section it is important to keep the primary intent of the lesson, namely strategies for solving systems, at the forefront of students' minds. While textbooks often chunk the idea of solving systems into discrete, almost unconnected mini-lessons (first let's learn about guess and check, now let's learn about solving systems by elimination, etc.), this activity provides students with an opportunity to make direct connections between strategies and gain a sense of the big picture goals of solving systems of equations. 

Exit Ticket and Homework: Revisiting the Cash Register Problem

10 minutes

To conclude the class, I congratulate students on all of their hard work and have them take out their original work on the Preassessment about notebooks and pens. For the last ten minutes of class (and for homework), I have students analyze their original thinking and make revisions based on the work they did in class today.

I also have students make note of what they changed and why. One method that helps at this point is to have students write any revisions in colored pencil. Students are more likely in a different color ink. It is also easier for me to review their work.