Lesson 1: Solving Equations with Variables on Both Sides

-When you see variable terms on both sides of the equal sign, what mathematical property allows you to move an entire term to the opposite side to begin isolating the variable?
-If an equation contains parentheses and variables on both sides, why is it usually most effective to apply the distributive property before trying to collect like terms?
-Once you have variable terms on both sides, how do you decide which side should become the "variable side" and which should become the "constant side" to make the equation easiest to solve?

Solution
Coefficient
Constant
Variable
Distributive Property
Infinitely Many
No Solution
Inverse

8.EE.C.7 Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Target D.
-Analyze and solve linear equations and pairs of simultaneous linear equations.
-The student solves linear equations in one variable with rational coefficients, including equations with solutions that require expanding expressions using the distributive property and collecting like terms.

Description

In this lesson, students will progress from solving basic two-step equations to mastering multi-step linear equations that feature variables on both sides of the equal sign. Students will practice using the distributive property and collecting like terms to simplify complex expressions before isolating the variable. Through this process, they will learn to identify equations that result in a single value, as well as special cases that result in no solution or infinitely many solutions.

Purpose

The purpose of this lesson is to build students' algebraic fluency by teaching them how to maintain equality while performing multiple transformations on an equation. By learning to consolidate variables, students develop a deeper understanding of the structure of expressions and the logical reasoning required to solve real-world mathematical situations. This serves as a foundational skill for higher-level algebra, including solving systems of equations and analyzing linear functions.

DOK Level: 2 (Basic Application of Skills and Concepts)

The Cell Phone Plan Dilemma:

Compare two different monthly cell phone plans—one with a high flat fee but low data cost, and another with no flat fee but a higher data cost.

Business Break-Even Points:

Discuss how a small business owner (like someone selling custom t-shirts) calculates how many items they need to sell so that their total costs (expenses) equal their total earnings (revenue).

The Fitness Club Comparison:

Compare the cost of two different gyms over several months to find out exactly when the total price paid for both would be exactly the same.

Inverse Operation Errors: Students may try to combine variable terms from opposite sides of the equation by adding them together instead of using the inverse operation (e.g., adding 3x to both sides when it should be subtracted).

The "Vanishing" Variable: When variable terms cancel out (like in a No Solution or Infinitely Many Solutions problem), students may get confused or think they made a mistake because the "x" is gone.

Sign Flipping with Distribution: Students often forget to distribute a negative sign to the second term inside parentheses (e.g., -2(x - 5) becoming -2x - 10 instead of -2x + 10).

Concrete Modeling: Provide students with algebra tiles or a physical balance scale to visualize how "removing" a variable from one side requires the same action on the other to maintain equality.

Scaffolded Templates: Give students a graphic organizer that uses a "river" line drawn through the equal sign to help them visually separate the two sides of the equation and track inverse operations.

Choice Boards: Allow students to choose between solving standard algebraic equations or applying the concept to a real-world scenario, such as comparing the two cell phone plans discussed in the connections section.

Self-Checking "I Have, Who Has" Game: Use a card game where the solution to one student’s equation is the starting point for another, providing immediate feedback and peer support.

Error Analysis Tasks: For advanced learners, provide completed "solved" equations that contain common misconceptions (like distribution errors) and challenge them to find and correct the mistakes.

-Exit Tickets

-Quiz

-Check-Ins

-Performance Task

Manipulatives

Worksheets

Google Forms

Textbook