Solving Equations

Abstract

The following discussions and activities are designed to help students understand the concepts behind and methods of solving equations. This lesson is best implemented with students working in groups of 2-4.

Objectives

Upon completion of this lesson, students will:

• understand that there are multiple ways to solve an equation and get the same result
• appreciate the different ways to solve single-variable linear equations
• be able to classify processes as additive and multiplicative inverses

Student Prerequisites

• Arithmetic: Students must be able to:
  • add, subtract, multiply, and divide
• Algebraic: Students must be able to:
  • understand the concept of variables
  • manipulate variables and constants separately

Teacher Preparation

• access to a browser
• copies of the Worksheet

Lesson Outline

1. Focus and Review

   Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

   • Does anyone know what an additive inverse is?
     • What is the root word in "additive"?
     • So if we are adding something, what might it mean to take the "inverse"?
     • Based on that, what do you think an additive inverse is?
   • Does anyone know what a multiplicative inverse is?
     • What is the root word in "multiplicative"?
     • Since we already know what an inverse is, can anyone guess at what a multiplicative inverse might be?

2. Objectives

   Let the students know what it is they will be doing and learning today. Say something like this:

   • Today, we are going to find different ways to solve equations with a single variable. We will be working on computers for part of this lesson, but please do not turn on your computers until I tell you to do so.

3. Teacher Input

   Start with a complicated logic problem for students to solve step by step:

   • How much money does Bernard have if he has $5 more than Andrew, and Andrew has $10?
   • How much money does Cole have if he has twice as much as Bernard?
   • How much money do Dave, Ellen, and Fitzgerald have if you know the following things:
     • Dave has $2 more than twice as much as Ellen
     • Ellen has $8 less than half as much as Fitzgerald
     • Fitzgerald has $20 less than Cole

   Show that this problem can be represented by the set of equations below which can then be solved for each of the variables through substitution.

   • A = 10
   • B = A + 5
   • C = 2B
   • D = 2E + 2
   • E = 1/2F - 8
   • F = C - 20
   Ask students how they would solve this system of equations based on the logic problem they just solved.

   Based on their experiences solving the preceding logic problem, ask students the following questions:

   • What does it mean to solve an equation?
   • How should a simplified equation look? Where are the variables and where are the constants?
   • How can we move variables or constants from one side of the equation to the other?
   • What can we do if we have something like "2x" or "10x" and we just want "x"?

4. Guided Practice

   Navigate to Equation Solver and choose an equation to solve. For best results, choose a relatively difficult equation to ensure that there are numerous ways to solve it.

   • Ask students to guide you, step by step, to solve the equation.
     • Make sure students understand how to designate their steps as additive inverse or multiplicative inverse.
     • Point out the fact that Equation Solver always does the same thing to both sides of the equation - this is important for students to remember when solving equations without computerized help.
   • After solving the equation, ask students for other methods by which the equation can be solved.
   • Divide the students into groups and ask them to each develop as many different algorithms as possible to solve equations.
   • Have students briefly describe their algorithms to come up with a class-wide list of at least five different equation-solving methods.
     • Note: Even if students suggest algorithms that fail to work sometimes, they can still try them out to see why and how their algorithms don't work.

5. Independent Practice

   Have each group of students solve 5-10 Level 1, 2, and 3 equations on Equation Solver using the various algorithms developed as a class.

   • As they do so, have students complete the Worksheet, writing down the number of steps it takes them to solve each equation as compared with the recommended minimum number of steps.
   • Depending on the size and number of groups, either have each group solve some equations with every algorithm, or have each group solve equations using just one algorithm, and then compare between the groups.

6. Closure

   Have students compare the number of steps it took them to solve their equations and discuss which algorithms were easiest, fastest, or most effective. Ask the following questions:

   • Which algorithm solved the equations in the fewest steps for Level 1 problems? Level 2? Level 3? Overall?
   • Did all algorithms help you to arrive at the correct solution, or did some fail to solve the equation?
   • Were any of the algorithms particularly easy to use and remember?

   Discuss important considerations when solving equations:

   • Common misconceptions about whether it matters if you put variables on the left v. right side of equation
   • Multiplying or adding first
   • How to multiply a fraction through the use of reciprocals

Alternate Outline

This lesson can be rearranged in the following ways:

• Students with less experience solving equations may benefit from a teacher-presented algorithm for solving equations as a base for their own algorithms
• To combine this lesson with statistics, have each group tally the number of steps it took them compared with the optimum number of steps in order to numerically compare the efficacy of various algorithms

Suggested Follow-Up

To reinforce and practice solving equations, have students compete in Algebra Quiz or Algebra Four.