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:

Grade 10

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Grade 6

Functions and Relationships

The student demonstrates algebraic thinking.

Grade 7

Functions and Relationships

The student demonstrates algebraic thinking.

Grade 8

Functions and Relationships

The student demonstrates algebraic thinking.

Grade 9

Functions and Relationships

The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.

The student demonstrates algebraic thinking.

Grade 6

Algebra and Functions

1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results

Mathematical Reasoning

1.0 Students make decisions about how to approach problems

2.0 Students use strategies, skills, and concepts in finding solutions

Grade 7

Algebra and Functions

1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs

4.0 Students solve simple linear equations and inequalities over the rational numbers

Mathematical Reasoning

1.0 Students make decisions about how to approach problems

2.0 Students use strategies, skills, and concepts in finding solutions

Grades 8-12

Algebra I

2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.

4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.

5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

Algebra

Arithmetic with Polynomials and Rational Expressions

Perform arithmetic operations on polynomials

Use polynomial identities to solve problems

Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning

Solve equations and inequalities in one variable

Seventh Grade

Expressions and Equations

Use properties of operations to generate equivalent expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Sixth Grade

Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.

Reason about and solve one-variable equations and inequalities.

Grades 6-8

Algebra

Understand patterns, relations, and functions

Numbers and Operations

Understand meanings of operations and how they relate to one another

Grades 9-12

Algebra

Understand patterns, relations, and functions

Numbers and Operations

Understand meanings of operations and how they relate to one another

Algebra I

Algebra

Competency Goal 4: The learner will use relations and functions to solve problems.

Grade 7

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 5: The learner will demonstrate an understanding of linear relations and fundamental algebraic concepts.

Grade 8

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Introductory Mathematics

Algebra

COMPETENCY GOAL 4: The learner will understand and use linear relations and functions.

7th Grade

Algebra

The student will demonstrate through the mathematical processes an understanding of proportional relationships.

8th grade

Algebra

The student will demonstrate through the mathematical processes an understanding of equations, inequalities, and linear functions.

Elementary Algebra

Elementary Algebra

Standard EA-4: The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.

8th Grade

Algebra

The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

Number and Operation

The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

Algebra I

Algebra

Students will describe, extend, analyze, and create a wide variety of patterns and functions using appropriate materials and representations in real world problem solving.

Numbers and Operations

Students will recognize, represent, model, and apply real numbers and operations verbally, physically, symbolically, and graphically.

Algebra I

Foundation for Functions

4. The student understands the importance of the skills required to
manipulate symbols in order to solve problems and uses the necessary algebraic skills required to
simplify algebraic expressions and solve equations and inequalities in problem situations.

Linear Functions

7. The student formulates equations and inequalities based on linear functions, uses
a variety of methods to solve them, and analyzes the solutions in terms of the situation.

6th Grade

Patterns, Functions, and Algebra

6.23b The student will solve one-step linear equations in one variable, involving whole number coefficients and positive rational solutions

7th Grade

Number and Number Sense

7.3 The student will identify and apply the following properties of operations with real numbers: the commutative and associative properties for addition and multiplication; the distributive property; the additive and multiplicative identity properties; the additive and multiplicative inverse properties; and the multiplicative property of zero.

Patterns, Functions, and Algebra

7.22a The student will solve one-step linear equations and inequalities in one variable with strategies involving inverse operations and integers, using concrete materials, pictorial representations, and paper and pencil

8th Grade

Patterns, Functions, and Algebra

8.15 The student will solve two-step equations and inequalities in one variable, using concrete materials, pictorial representations, and paper and pencil.

Secondary

Algebra I

A.01 The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and apply these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions.

A.03 The student will justify steps used in simplifying expressions and solving equations and inequalities. Justifications will include the use of concrete objects; pictorial representations; and the properties of real numbers, equality, and inequality.

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.