Interactivate: 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

Standards Addressed:

Textbooks Aligned:

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

Key Terms

addition	The operation, or process, of calculating the sum of two numbers or quantities
additive inverse	The number that when added to the original number will result in a sum of zero
algorithm	Step-by-step procedure by which an operation can be carried out
constants	In math, things that do not change are called constants. The things that do change are called variables.
multiplication	The operation by which the product of two quantities is calculated. To multiply a number b by c is to add b to itself c times
multiplicative inverse	The number that when multiplied by the original number will result in a product of one
variables	In math, things that can change are called variables. The things that do not change are called constants.

Lesson Outline

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?

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.

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"?

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.

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.

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.

Interactivate

Solving Equations