ONE STEP EQUATIONS WITH INTEGERS
INTRODUCTION
The objective for this lesson on One-Step Equations with Integers is, the student will
solve one-step equations with integers in mathematical and real-word situations.
The skills students should have in order to help them in this lesson include, integer
operations and writing and evaluating expressions.
We will have four essential questions that will be guiding our lesson. Number one, why
do we use variables in solving equations? Number two, what is the goal when solving
equations? Number three, in an addition equation, how do we isolate the variable? And
number four, in a multiplication equation, how do we isolate the variable?
Begin by completing the warm-up for this lesson, on addition and subtraction of integers,
to prepare for our lesson on one-step equations with integers.
SOLVE PROBLEM PART ONE – INTRODUCTION
The SOLVE problem for this lesson is, Jennifer is saving her money to buy a new CD.
The CD is on sale this week only. The total cost of the CD, including tax, is thirteen
dollars and ninety nine cents. Jennifer has ten dollars and twenty nine cents saved. How
much more money does she need?
In Step S, we Study the Problem. First we need to identify where the question is located
within the problem and underline the question. The question for this problem is, how
much more money does she need?
Now that we have identified the question, we need to put this question in our own words
in the form of a statement. This problem is asking me to find the amount of money
Jennifer still needs.
During this lesson we will learn how to solve one-step equations with integers to
complete this SOLVE problem at the end of the lesson.
ADDITION EQUATIONS – CONCRETE MODELS
We’re now going to be looking at concrete model for addition equations.
Place one yellow unit tile on the left side of the scale. Identify the value of one yellow
tile. Positive one.
Is this scale balanced? Explain your thinking. No, because the value on one side is
positive one and the value on the other side is zero.
What can we do to balance the scale? Place a yellow unit tile on the right side.
What does it mean for the scale to be balanced? Both sides need to be equal in value.
Place one red unit tile on the left side of the scale.
Identify the value of one red tile. Negative one.
Is the scale balanced? No
What can we do to balance the scale? Place one red tile on the right side of the scale.
Practice balancing the scale by placing different values of yellow or red tiles on the left
side of the scale and identify what color and number of tiles will balance the scale.
Place one yellow tile and one red tile on the balance scale.
What is the value of the yellow? Positive one
What is the value of the red? Negative one
What is the value of the red and yellow tiles when they are combined? The value is zero.
What do we call the yellow and red tiles together? A zero pair
Place two yellow tiles on the left side and two yellow tiles on the right side of the scale.
Now place a zero pair on the left side.
Does adding the zero pair to the left change the value on the left? Explain your thinking.
No, the value does not change because the additive identity property for addition tells us
that we can add zero to any number and it will not change the sum or total value.
Let’s look at the equation c plus three is equal to five.
When modeling equations with integers there are two things that we must focus on.
What is the first? First we need to isolate the variable.
What does it mean to be isolated? To be alone or by yourself
What is the variable in our equation? The letter c
Explain the meaning of the word variable. A variable is a symbol that represents an
unknown value or number. Variables are usually written as letters. What is our second
focus? Keep the equation balanced.
What does the equal sign mean? If two things are equal, they have the same value.
Explain what this means when solving an equation. Whatever is on one side of the
equation must equal what is on the other side. Whatever you do to one side of the
equation, you must do to the other side in order to keep the equation balanced.
Place a cup on the scale on the left.
How can we model adding three, or positive three to our cup? Place three yellow tiles on
the left side.
What needs to be placed on the right side of the scale to represent five? Five yellow tiles
What are the two goals when working with equations? Isolate the variable and keep the
equation balanced so whatever you do to one side you must do to the other.
How can we isolate the variable in our equation? How could you use the idea of opposite
operations to isolate the variable? To isolate the variable, we perform the opposite
operation. If the equation is an addition equation, we will subtract. If the equation is a
subtraction equation, we will add.
Let’s subtract, or take away, three yellow tiles by removing them from the scale.
Is the variable now isolated, or alone? Yes
Is the equation still balanced? Explain your thinking. No, because the same operation
must be performed on both sides.
How can we balance the equation? Take away three yellow tiles from the right side.
Is the equation balanced now? Justify your answer. Yes, because we have performed the
same operation on both sides of the equation.
What is the value of cup or c? c is equal to two
Now let’s go back and check the problem using our tiles.
Go back to the original equation. c plus three equals five.
What is our value of c? Two
Substitute in two yellow tiles for the c.
What is the value on the left side of our equation? Five yellow tiles
What is the value on the right? Five yellow tiles
Is the equation balanced? Is our answer correct? Yes, because five yellow is equal to
five yellow, the equation is balanced and we know the answer is correct.
ADDITION EQUATIONS – PICTORIAL MODELS
What letter do we use to represent the cup? C. What letter do we use to represent the
positive three? The letter Y, to represent yellow.
What is our first step in solving the equation. Isolate the variable. How can we isolate
the variable? We can subtract the three Y’s by crossing them out. Have we isolated the
variable? Yes. Is the equation balanced? Explain your answer. No, because we only
subtracted the positive three from one side.
What do we need to do to balance the equation? Subtract three Y’s from the right side of
the equation because whatever operation you do to one side, you must do to the other.
What is the value of C? C is equal to two Y. C is equal to two yellows.
Let’s check the equation by substituting the value of C, which is two Y’s, into the
original equation. Two y plus three y is equal to five y.
Let’s look at the equation C plus three R’s is equal to five R’s.
Our first step is to isolate the variable. How can we isolate the variable? We can subtract
the three R’s by crossing them out. Have we isolated the variable? Yes. Is the equation
balanced? Explain your answer. No, because we only subtracted the negative three from
one side.
What do we need to do to balance the equation? Subtract three R’s from the right side of
the equation because whatever operation you do to one side, you must do to the other.
What is the value of C? C is equal to two reds. Let’s check the equation by substituting
the value of C into the original equation.
ADDITION EQUATIONS –USING ZERO PAIRS AT THE CONCRETE LEVEL
Let’s look at our equation c plus three is equal to negative five.
What is the variable? c
How can we model the variable? Place one cup on the left side of the balance scale.
How can we model adding three, or positive three to the cup? Three yellow tiles
What needs to be placed on the right side of the scale to represent the negative five? Five
red tiles.
How can we isolate the variable? Take away or subtract the three yellow tiles from the
left side of the scale.
Is the variable isolated or alone? Yes
Is our equation balanced? No, because the same operation must be performed on both
sides.
What do we need to do to keep the equation balanced? Subtract three yellow tiles from
the right side. Is it possible to take away, or subtract, the three yellow tiles from the right
side? No, because all the tiles are red.
What can we do to “create the possibility” of taking away three yellow tiles? We can use
zero pairs. Add one zero pair, one red and one yellow, to the right side.
What is the value of the right side? It is still negative five because we added zero, which
did not change the value.
Can we now take away three yellows? No. Add another zero pair to the right side. What
is the value of the right side? It is still negative five because we added zero, which did
not change the value.
Can we now take away three yellows? No
Add another zero pair to the right side. What is the value of the right side? It is still
negative five, because we added zero, which did not change the value.
Can we now take away three yellows? Yes
Is the variable now isolated? Yes
Is the equation now balanced? Explain your answer. Yes, because the same operation
has been performed on both sides of the equation.
What is the value of c in the equation? Negative eight, because all the tiles are red.
Let’s check the equation by going back to the original equation. Substitute eight red tiles
for the c.
What should we do with the tiles that are different colors on the same side of the equal
sign? We can make zero pairs and remove those from the equation without changing the
value.
After removing the three sets of zero pairs, identify the value on the left side. Negative
five
After removing the three sets of zero pairs, identify the value on the right side. Negative
five
The equation is balanced and therefore, the answer is correct.
PICTORIAL – ADDITION EQUATIONS – USING ZERO PAIRS
Let’s model the equation c plus three is equal to negative five.
Can we take three Y’s or three yellows from five R’s or five reds? No
How could we possibly take away the three Y’s? Create the possibility with zero pairs.
Add a zero pair to the right side. Is it possible to take away three Y’s? No
Add another zero pair to the right side. Is it possible to take away three Y’s? No
Add another zero pair to the right side. Is it possible to take away three Y’s? Yes!
How can we isolate the variable? We can subtract the three Y’s by crossing them out.
Is the variable now by itself? Yes
Is the equation now balanced? No
What do we need to do to balance the equation? Subtract three Y’s from the right side of
the equation because whatever operation you do to one side, you must do to the other.
What is the value of C? C is equal to eight reds.
Let’s check the equation by subtracting the value of C, which is eight R’s into the
original equation.
Is the equation now balanced? Yes, because five R’s is equal to five R’s.
Let’s model the equation c plus negative three is equal to five.
Can we take three R’s away from five Y’s? No
How could we possibly take away the three R’s? Create the possibility with zero pairs?
Add a zero pair to the right side. Is it possible to take away three R’s? No
Add another zero pair to the right side. Is it possible to take away three R’s? No
Add another zero pair to the right side. Is it possible to take away three R’s? Yes
How can we isolate the variable? We can subtract the three R’s by crossing them out.
Is the variable now by itself? Yes
Is the equation now balanced? No
What do we need to do to balance the equation? Subtract three R’s from the right side of
the equation because whatever operation you do to one side, you must do to the other.
What is the value of C? C is equal to eight yellows.
Let’s check the equation by substituting the value of C into the original equation. Is the
equation now balanced? Yes, because five Y’s or five yellows are equal to five yellows.
PICTORIAL – SUBTRACT EQUATIONS
We will not model subtraction with tiles, but we can model it pictorially by changing the
subtraction equations to addition equations. Remember that when subtracting, we can
add the opposite.
Therefore, three minus five is the same as three plus a negative five. We can represent
the equation c minus three is equals five pictorially by changing it to c plus negative three
is equal to five.
What is the first goal for solving our equations? To isolate the variable.
Can we take three R’s or reds away from the five yellows or five Y’s? No
How could we possibly take away the three R’s? Create the possibility with zero pairs.
Add a zero pair to the right side. Is it possible to take away three R’s? No
Add another zero pair to the right side. Is it possible to take away three R’s? No
Add another zero pair to the right side. Is it possible to take away three R’s? Yes
How can we isolate the variable? We can subtract the three R’s by crossing them out.
Is the variable now by itself? Yes
Is the equation now balanced? No
What do we need to do to balance the equation? Subtract three R’s from the right side of
the equation because whatever operation you do to one side, you must do to the other.
What is the value of C? C is equal to eight yellows
Let’s check the equation by substituting the value of C into the original equation. Is the
equation now balanced? Yes, because five Y’s is equal to five Y’s.
ABSTRACT – SOLVE ADDITION EQUATIONS
C plus three is equal to five
What is the first step in solving an equation? Isolate the variable
Explain how to isolate the variable. Perform the opposite operation.
What operation will we use in an addition equation to isolate the variable? Subtraction
Subtract three from the left side of the equation. Is the equation now balanced? No
What do we need to do to balance the equation? Subtract three from the right side of the
equation because whatever operation you do to one side, you must do to the other. C is
equal to two.
What is the value of C? c is equal to two.
Substitute the value of c back into the original problem to check. C plus three is equal to
five. We substitute in two the value c. Two plus three is equal to five. And five is equal
to five.
ABSTRACT – SOLVE SUBTRACTION EQUATIONS
Let’s look at the equation c minus three is equal to five
Remember that we can represent c minus three equals five by changing it to an addition
equation: c plus negative three equals five.
What is the first step in solving an equation? Isolate the variable.
Explain how to isolate the variable. Perform the opposite operation.
Because this is an addition equation, how can we isolate the variable? Subtract a
negative three. Subtract negative three from the left side of the equation.
Is the equation now balanced? No
What do we need to do to balance the equation? Subtract negative three from the right
side of the equation because whatever operation you do to one side, you must do to the
other.
What is the value of c? C is equal to eight.
Substitute the value of c back into the original problem to check. C minus three is equal
to five. Eight minus three equals five. Five is equal to five.
SOLVE PROBLEM PART ONE – COMPLETION
This concludes part one of our lesson and now we’re going to go back to the SOLVE
problem for part one.
The SOLVE problem was, Jennifer is saving her money to buy a new CD. The CD is on
sale this week only. The total cost of the CD, including tax, is thirteen dollars and ninetynine cents. Jennifer has ten dollars and twenty-nine cents saved. How much more
money does she need?
In the S Step, we Study the Problem. Underline the question and complete this statement.
This problem is asking me to find the amount of money Jennifer still needs.
In the O Step, we Organize the Facts. We start by identifying the facts. We go back and
read our SOLVE problem and mark each fact with a vertical line. Jennifer is saving her
money to buy a new CD./ The CD is on sale this week only./ The total cost of the CD,
including tax, is thirteen dollars and ninety nine cents./ Jennifer has ten dollars and
twenty nine cents saved./
After we identify the facts, then we eliminate any unnecessary facts. When we look at
our SOLVE problem we can identify one unnecessary fact, that the CD is on sale this
week only.
After we eliminate the unnecessary facts then we list the necessary facts. The CD cost is
thirteen dollars and ninety-nine cents, and Jennifer has ten dollars and twenty-nine cents.
In the L Step, we Line Up a Plan. Write in words what your plan of action will be. Write
an equation that I can use to solve the problem and then solve the equation.
Choose an operation or operations. Our operation is going to be subtraction.
In the V Step, Verify Your Plan with Action. Estimate your answer. Our estimate here is
about three dollars.
Carry our your plan. We write the equation, x, which is the unknown value of how much
more money she needs, plus the amount she has saved, ten dollars and twenty-nine cents
for the total amount of the CD, which is thirteen dollars and ninety-nine cents.
We solve it with subtraction, by subtracting the amount of ten dollars and twenty-nine
cents from each side of the equation to isolate our variable. The value of our variable x is
three dollars and seventy cents.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, because we
were looking for how much money Jennifer needs.
Is your answer reasonable? Compare your answer to the estimate. Yes, because it is
close to the estimate of about three dollars.
Is your answer accurate? Check your work. Yes
Write your answer in a complete sentence. Jennifer still needs three dollars and seventy
cents.
PART TWO OF OUR LESSON – ONE–STEP EQUATIONS WITH INTEGERS
The SOLVE problem for this part of the lesson is, Mr. Thompson’s math class is working
on activities with equations. There are twenty-eight students in his class, and he divides
the class into groups of four students. Each group must make a class presentation about
the equation they are assigned. How many groups will give a presentation?
In the S Step, we Study the Problem. First we need to identify where the question is
located within the problem and underline the question. The question for this SOLVE
problem is, how many groups will give a presentation?
Now that we have identified the question, we need to put this question in our own words
in the form of a statement. This problem is asking me to find the number of groups that
will give a presentation.
During this lesson we will learn how to solve one-step equations with integers using
multiplication and division to complete this SOLVE problem at the end of the lesson.
MULTIPLICATION EQUATIONS WITH A CONCRETE MODEL
What are the steps for solving equations? Isolate the variable and keep the equation
balanced.
When we were solving addition equations, what operation did we use? The opposite
operation, of subtraction
What operation do you think we will use when solving a multiplication equation? The
opposite operation of division
Let’s look at our equation, two c equals six.
If c is equal to one cup, how can we represent two c? With two cups
What do we need to place on the right side of the scale to represent the six? Six yellow
tiles because the six is positive.
How can we isolate the variable? Division, because it is a multiplication equation
We can divide the two cups by separating them into two separate groups. Have we
isolated the variable? yes
How can we balance the equation? Divide the six yellow tiles on the right side of the
equation into two groups.
Is the equation balanced now? Yes, because we have performed the same operation on
both sides of the equation.
What is the value of c? c is equal to three
How can we check the problem using tiles? Substitute three yellow tiles for the c.
Substitute thee yellow tiles for each cup, c.
What is the value on the left? Six yellow tiles
What is the value on the right? Six yellow tiles
Is the equation balanced? Explain your answer. Yes, because six yellow equals six
yellow, and the answer is correct.
MULTIPLICATION EQUATIONS – PICTORIAL REPRESENTATIONS
Two c equals six
Let’s start by representing the equation two c is equal to six.
We can model division of the two cups by writing two separate c’s. We can model
dividing the six Y’s on the right side of the equation into two groups. Remember that
whatever operation is done on one side must also be done on the other. There are three
Y’s, or yellows, in each group so the value of c is three.
Check the equation by substituting the value of c, which is three Ys into the original
equation.
Is the equation balanced? Yes, because six Y’s is equal to six Y’s.
Let’s look at the equation two c is equal to negative six.
Let’s start by representing the equation two c equals negative six.
We can model division of the two cups (two c) by writing two separate c’s.
We can model dividing the six R’s on the right side of the equation into two groups.
Remember that whatever operation is done on one side must also be done on the other.
There are three R’s, or reds, in each group so the value of c is negative three.
Check the equation by substituting the value of c, which is three R’s, into the original
equation.
Is the equation balanced? Yes, because six R’s equals six R’s
SOLVE MULTIPLICATION EQUATIONS – USING THE ALGEBRAIC MODEL
Two c is equal to six
What is the first step in solving our equation? Isolate the variable
What operation will we use? Explain your thinking. This is a multiplication equation, so
we will use the opposite operation of division.
Divide the left side of the equation by two. Is our equation balanced? No
What do we need to do to balance the equation? Divide the right side of the equation by
two.
What is the value of c? c is equal to three
Substitute the value of c back into the original problem. Two c is equal to six; two times
three equals six; six is equal to six.
Do you think it is possible to model Problem Three? No, because you cannot represent
negative two with cups or pictorially.
Because the equation in Problem Three is a multiplication equation, you can isolate the
variable by dividing by negative two.
Divide the left side of the equation by negative two.
Is the equation balanced? No
What do you need to do to balance the equation? Divide the right side of the equation by
negative two, because whatever operation you do to the one side, you must do to the
other.
What is the value of c? c is equal to negative three.
Substitute the value of c back into the original problem to check. Negative two c is equal
to six; negative two times negative three is equal to six; six is equal to six.
SOLVE DIVISION EQUATIONS – USING AN ALGEBRAIC MODEL
Identify the format of the division equation. The problem is written as a fraction bar with
the fraction bar representing the division symbol.
We will not model with tiles or use a pictorial model for division equations. However,
we will follow the same steps as with multiplication equations to solve the division
equations.
What is the first goal when solving our equations? Isolate the variable. Explain how we
can isolate the variable. We can perform the opposite operation.
This is a division equation. What operation will we use to isolate the variable? The
opposite operation which is multiplication
What number do we need to multiply by? Two
When we multiply c over two by two the product is two c over two.
What is the value of two divided by two? One, because any number divided by itself is
one.
Multiply the left side by two. Is the equation now balanced? No.
What do we need to do to balance the equation? Multiply the right side of the equation
by two, because whatever operation you do to one side, you must do to the other.
Multiply the right side of the equation by two. What is the value of c? c is equal to six.
Substitute the value of c back into the original problem to check.
SOLVE PROBLEM PART TWO - COMPLETION
We are now going back to the SOLVE problem from the beginning of part two of the
lesson. The SOLVE problem was, Mr. Thompson’s math class is working on activities
with equations. There are twenty-eight students in his class, and he divides the class in to
groups of four students. Each group must make a class presentation about the equation
they are assigned. How many groups will give a presentation?
In the S Step, we Study the Problem. Underline the question and complete this statement.
This problem is asking me to find the number of groups that will give a presentation.
O, Organize the Facts. First we need to identify the facts. We go back and reread our
SOLVE problem and place a vertical mark at the end of each fact. Mr. Thompson’s math
class is working on activities with equations./ There are twenty eight students in his
class,/ and he divides the class into groups of four students./ Each group must make a
class presentation about the equation they are assigned./ How many groups will give a
presentation?
After we identify the facts, we eliminate the unnecessary facts. In this word problem we
do not need to know that each group must make a class presentation in order to determine
the number of groups, so that is an unnecessary fact.
After we eliminate the unnecessary facts, we list the necessary facts. There are twentyeight students in the class; and there will be four students in each group.
L, Line Up a Plan. Write in words what your plan of action will be. Write an equation
that we can use to solve the problem and then solve the equation.
Choose an operation or operations. Division
V, Verify Your Plan with Action. First we estimate your answer. Our estimate here is
less than ten.
Carry out your plan. We write the equation, four, which represents the number of
students in the group, times x, which is our unknown number of groups; will equal the
total number of students, twenty-eight. This is a multiplication equation, so we’ll solve
using the opposite operation of division. We divide both sides of our equation by four,
and find that the value of our variable x is seven.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, because we
were looking for how many groups would give a presentation.
Is you answer reasonable? Compare your answer to the estimate. Yes, because it is close
to the estimate of less than ten.
Is your answer accurate? Check your work. Yes
Write your answer in a complete sentence. There will be seven groups giving a
presentation.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, why do we use variable in solving equations? We use variables to
represent unknown values.
Number two, what are the goals when solving equations? Isolate the variable and keep
the equation balanced.
Number three, in an addition equation, how do we isolate the variable? Use the opposite
operation of subtraction.
And number four, in a multiplication equation, how do we isolate the variable? Use the
opposite operation of division.