SOLVING EQUATIONS WITH ONE VARIABLE
WITH EMPHASIS ON PROPERTIES OF RECIPROCALS AND OPPOSITES
Objective:
Students will be able to solve two-step equation by using the properties of reciprocals and opposites.
CA Standards:
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.
5.0 Students solve multistep problems, including word problems, involving linear equations and linear
inequalities in one variable and provide justification for each step.
Concept Map
Prior Skills:
Order of operation
Reciprocals
Opposites
Variables
Combining like terms
Solving equations in one variable:
Undoing the order of operations
Adding opposites to both sides of an equation
Combining like terms
Multiplying both sides of an equation with reciprocal
Solving linear equations
and inequalities
Real life
connection
Solving equations with
multiple variables
STUDENT ADVANCED ORGANIZER
Solving Equations with one variable
Adding the Opposite
x28
+(-2) +(-2)
x6
Multiplying by Reciprocal
x 3  4
+ 3 +3
x7
2 x  12
1 2
1
( )( ) x  12( )
2 1
2
x6
Solving Multi-step Equations
2 x  3  15
- 3 -3
2 x  12
1
1
( )2 x  12( )
2
2
x6
1
x23
6
+2 +2
1
x5
6
1
(6) x  5(6)
6
x  30
x
6
3
x
(3)  6(3)
3
x  18
Back Map: Prior Knowledge
Grade 7: Algebra and Functions
4.0 Students solve simple linear equations over the rational numbers.
Prior Experiences or Misconceptions
Real-Life Experiences
Cell Phone usage
Hourly wages
Driving (mph)
Misconceptions
2x + 2 = 8
 Students will add two to both sides and yet end up with 2x = 10
 Students will multiply both sides by ½ first, but will fail to distribute to the + 2
 Students will combine the 2x and the 2 and will write 4x = 8
 Students will subtract 8 from both sides and then write 2x = – 6 instead of 2x – 6 = 0
x/3 + 2 = 8
 Students will multiply both sides by 1/3 instead of by 3
Introduction:
Opening Word Problem
Mary received a free cell phone over the summer. The flat fee for using the cell phone each month is $15.99
plus 5¢ per minute she uses the phone. If her first bill was $33.04, how many minutes did she use the phone that
month. (Answer: 341 minutes)
1.
2.
3.
4.
5.
Have students quietly read the opening word problem.
Teacher will invite students to talk about their own phone bills
Have students make a prediction of possible solution.
(This problem will be revisited during closure.)
Hand out Student Advanced Organizer.
Day 1 Lesson Activity:
Ask students what the solution to x  2  8 might be. Instruct students to share their reasoning with a partner.
Call on a couple of students to share their process with the class.
Using graphic organizer, teacher model solving the equations.
Example: x  3  5
x
=
+3
2
+(-3)
5
Have students work on two more examples using additive inverse in pairs. Teacher selects two pair to present
their findings.
Repeat process for solving equations using reciprocals.
Students complete a worksheet allowing them to practice single step equations. For each problem students will
identify the property used.
Questions:
What is the opposite of 0?
What is the reciprocal of –2?
How else could you write x/3?
What is the coefficient of x?
Day 2 Lesson Activity:
Warm up: Review single step equations – Q: What is the difference between opposites and reciprocals? Provide
examples.
Using graphic organizer approach, teacher models solving two-step equation.
Students practice five problems in pairs and teacher selects pairs to share their findings.
Questions:
 Can you multiply by the reciprocal before adding the opposite to both sides? What possible error could
this lead to?
 What would you do if the equation was: 0x +2 = 2?
Closure:
Revisit the word problem used in the introduction. Students will derive the equation to solve the problem in
pairs and teacher will select pairs to share their results. Students will then be instructed to solve the equation and
compare their predictions with the solution.
Questions:
What is the purpose of using an equation to solve a problem rather than just guessing or performing multiple
steps of adding? What is the advantage of using an equation? Do you agree? If not, what is your reason for
disagreeing?