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
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.