TWO STEP EQUATIONS WITH INTEGER
INTRODUCTION
The objective for this lesson on Two Step Equations with Integers is, the student will solve twostep equations with integers in mathematical and real-world situations. The skills students
should in order to help them in this lesson include, one-step equations, integer operations, and
order of operations.
We will have four essential questions that will be guiding our lesson. Number one, why do we
use variables in equations? Explain your thinking. Number two, which operation do we undo
first in two-step equations? Justify your answer. Number three, what is the inverse operation for
multiplication? And number four; what are the two concepts that we need to focus on when
solving equations? Begin by completing the warm-up, on solving one-step equations to prepare
for the lesson on Two-Step Equations with Integers.
SOLVE PROBLEM – INTRODUCTION
SOLVE problem for this lesson is, Jeremy has a collection of fifty-eight baseball cards. His dad
gives him a new card every month. He puts all the cards in a binder. He completely fills nine
pages of the binder and has four cards left over. How many cards are on each full page?
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 many cards are on
each full page?
Now that we’ve 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 cards on each full page.
During this lesson we will learn how to solve two step equations with integers to complete the
SOLVE problem at the end of the lesson.
TWO-STEP EQUATIONS WITH MULTIPLICATION AND ADDITION - CONCRETE
Let’s look at the equation two c plus two is equal to six.
How is this equation different from equations solved in the past? The equation has two
operations or two steps to solve.
There are two things that we need to remember whenever we are solving any equation.
What is the first concept we need to remember? Explain your answer. Isolate the variable. This
means that we need to have the variable alone on one side of the equals sign.
What is the second concept we need to remember? We need to keep the equation balanced.
That means that whatever operation we apply to the values on the left side of the equation, we
must apply to the right side of the equation.
What operation should we perform first on the equation? When solving equations with more
than one step, we should “undo” the problem by using the order of operations in reverse. This
means that when solving the equation, we will start with the value that is not attached to the
variable and add or subtract that value before we multiply or divide by the value attached to the
variable.
Let’s model the equation using cups and tiles.
How can we represent two c? If c equals one cup, then we could represent two c using two cups.
What else is on the left side of the equation? Adding two
How can that value be represented? Two yellow tiles
What should be placed on the right side of the scale? Six yellow tiles to represent the positive
six
What is our first goal when solving equations? Isolate the variable.
Remember that we need to perform the opposite operation and because this is a two-step
equation, the order of operations will be applied in the reverse order. Addition or subtraction,
then multiplication or division
What operation will we apply first? Explain your thinking. Subtraction, because it is the
opposite of addition and it is the first step towards isolating the variable.
How can we model subtraction of the tiles? Remove them from the balance scale.
Is our equation balanced? why or why not? It is not balanced until we remove two yellow tiles
from the right side of the equation.
What is the next step to solve the equation? Isolate the variable using division.
Why do we use division? We have a multiplication equation and we use the opposite operation
to isolate the variable.
Let’s divide the cups into two separate groups. Then, we’ll divide the four yellow tiles into two
equal groups.
Is the equation balanced? Yes, because we have performed the same operation to both sides of
the equation.
What is the value of c? c is equal to two.
Now let’s go back to the original equation on the scale to check the problem using tiles. Two c
plus two is equal to six.
Substitute two yellow tiles for each cup. What is the value on the left side of the balance scale?
Six yellow tiles which equals positive six.
What is the value on the right side of the balance scale? Six yellow tiles which equals positive
six.
Our equation is balanced.
TWO-STEP EQUATIONS WITH MULTIPLICATION AND ADDITION – PICTORIAL
Let’s represent the equation two c plus two equals six pictorially.
How can we model subtraction of the two using the pictorial model? We can cross out the two
Y’s on the left side of the equation.
What do we need to do to balance the equation? Cross out two Y’s on the right side of the
equation.
What is the next step? Divide the two c into two equal groups.
If we divide the left side of the equation by two what must we do to balance the equation?
Divide the four Y’s on the right side of the equation into two equal groups. There are two Y’s,
or yellows, in each group so the value of c is two.
Let’s check the equation by substituting the value of c, which is two Y’s into the original
equation.
Let’s represent the equation two c plus a negative two equals negative six pictorially.
How can we model subtraction of the two using the pictorial model? We can cross out two R’s
on the left side of the equation.
What do we need to do to balance the equation? Cross out two R’s on the right side of the
equation.
What is the next step? Divide the two c into two groups.
If we divide the left side of the equation by two, what must we do to balance the equation?
Divide the four R’s on the right side of the equation into two equal groups.
There are two R’s, or reds, in each group so the value of c is equal to negative two.
Let’s check the equation by substituting the value of c, which is two R’s into the original
equation.
And our equation is balanced.
TWO-STEP EQUATIONS WITH MULTIPLICATION AND ADDITION USING ZERO
PAIRS – CONCRETE
Let’s look at the equation two c plus two equals negative six
Let’s create a model with cups and tiles.
What is our first step in solving this equation? Subtract two yellow tiles from the left side.
Why? It is our first step as we isolate the variable.
Is the 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 two yellow tiles from the right
side.
Is it possible to remove two yellow tiles? Why or why not? No, because all the tiles are red.
How could we subtract two yellow tiles from the right side of the equation? Is there a way that
zero pairs could help us?
How can we “create the possibility” of taking away two yellow unit tiles? By using zero pairs.
How can we use zero pairs to help us create the possibility? One red plus one yellow is a zero
pair, and its value is zero. So we can add or take away zero pairs from either side of the equation
without changing the value.
Add one zero pair (one red plus one yellow) to the right side. What is the value of the right side
of the equation? It is still negative six, because when we added zero, we did not change the
value.
Have we created the possibility of taking away two yellow unit tiles? No
What can we do? Add another zero pair.
What is the value of the right side of the equation? It is still negative six, because when we
added zero, we did not change the value.
Have we created the possibility of removing positive two or two yellow tiles? Yes
Is the equation now balanced? Yes, because the same operations have been performed on both
sides of the equation.
Is the variable isolated? No
What operation will need to be performed to isolate the variable? Division, because it is a
multiplication problem.
Let’s divide the two cups by separating them into two separate groups. Now divide the eight red
tiles into two equal groups.
What is the value of c? c is equal to negative four.
Now let’s go back to the original equation to check the solution.
Substitute four red tiles for each c. Remove the zero pairs on the left side of the scale. What is
the value on the left? Six red tiles. What is the value on the right? Six red tiles.
Because six red is equal to six red, the equation is balanced and we know the solution is correct.
TWO-STEP EQUATIONS WITH MULTIPLICATION AND ADDITION – USING ZERO
PAIRS
Pictorial model
Let’s model the equation two c plus two is equal to negative six.
What is the first step in isolating the variable? Cross out two yellows on the left to model
subtraction.
Can we cross out two Ys, two yellows on the right side of the equation? No
Is there a strategy we can use to create the possibility? Yes, with zero pairs.
Let’s add zero pairs.
Is it possible to take away two Ys or two yellows? No. We need to add another zero pair.
Can we take away two Ys or two yellows? Yes!
Now, we can write the two cups separately and divide the eight R’s on the right side of the
equation into two equal groups.
What is the value of c? c equals four reds
Let’s check the equation by subtracting the value of c, four R’s into the original equation. Six
R’s are equal to six R’s, so our equation is balanced and our solution is correct.
SOLVING TWO-STEP MULTIPLICATION AND ADDITION EQUATIONS
Algebraic Model
What operation do we use first in solving the equation? Subtraction
What will we subtract? Subtract two from both sides of the equation.
Is the variable isolated? No
What operation must you use to isolate the variable? Division, because it is a multiplication
problem.
Is the equation now balanced? Yes
Is the variable isolated? Yes
Check the problem by substituting the value of c back into the original equation. Six is equal to
six, our equation balances and our solution is correct.
TWO-STEP EQUATIONS WITH MULTIPLICATION AND SUBTRACTION- PICTORIAL
We will not model the equations with multiplication and subtraction with tiles. We can model
them pictorially by changing the subtraction equations to addition equations.
Three minus five is the same as three plus a negative five. Therefore, two c minus two equals six
is the same as two c plus a negative two equals six.
What is the first step in isolating the variable? Cross out two reds on the left to model
subtraction.
Can we cross out two R’s or two reds on the right side of the equation? No
Is there a strategy we can use to create the possibility? We can create the possibility with zero
pairs.
Let’s add zero pairs.
Is it possible to take away two R’s or two reds? No. We need to add another zero pair.
Can we take away two R’s or two red? Yes!
Is the equation now balanced? Yes.
Is the variable isolated? No
What operation do we have to use to isolate the variable? Division, because it is a multiplication
problem.
Now, we can write the two cups separately and divide the eight Ys on the right side of the
equation into two equal groups.
What is the value of c? c equals four yellows
Let’s check the equation by substituting the value of c of four Ys into the original equation. Six
Y is equal to six Y. Our equation balances, so our solution is correct.
SOLVING TWO-STEP EQUATIONS WITH MULTIPLICATION AND SUBTRACTION –
ABSTRACT
Let’s represent the equation two c minus two equals six by changing it to an addition equation:
two c plus negative two equals six
What is our first step? Subtract the negative two.
What do you need to do to balance the equation?
Subtract a negative two from the right side of the equation. Two c is equal to eight
What is the next step in solving the equation? We need to isolate the variable.
What operation must be used to isolate the variable? Division, because it is a multiplication
problem.
C is equal to four. Check the problem by substituting the value of c, which is two, back into the
original equation.
Our equation is balanced and so our solution of c equals four is correct.
SOLVING TWO-STEP DIVISION EQUATIONS – ALGEBRAIC MODEL
c divided by two plus one equals three
How is the division written in the equation? It is a fraction with the fraction bar as a 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 two operations are represented in Problem One? Division and Addition
What operations will we use to solve the problem? The opposite operations of subtraction and
multiplication.
What operation is first? Why? We subtract a one from both sides of the equation because we
are following the opposite of the order of operations.
c divided by two is equal to two
Is the variable isolated? No
What operation must be used to isolate the c? Multiplication
Why? Because we need to use the opposite operations to isolate the variable.
Multiply by two. What is the product? The product is two c over two or two c divided by two.
What is the value of two divided by two? One, because any number divided by itself is one.
What is the value of c? c is equal to four
Substitute the value of c back into the original problem to check. Four divided by two plus one
equals three. Two plus one equals three. Three is equal to three.
Our equation is balanced so our solution is correct.
WRITING AND SOLVING TWO-STEP EQUATIONS IN REAL-WORLD SITUATIONS
Melissa has a total of two hundred fifty dollars in her saving account. She started with one
hundred ten dollars and then earned the rest of her savings by babysitting. Melissa charges seven
dollars per hour. Write an equation to represent the situation and determine the number of hours,
represented by h, that she babysat last week.
What is the problem asking us to find? An equation to determine the number of hours she
babysat last week.
What variable will we use to represent the hours in the word problem? h
What should be on the left side of the equation? The amount she started with
What other information will be written on the left side of the equation? The variable times the
charge per hour
What will be written on the right side of the equation? The total in her account, two hundred
fifty dollars.
Let’s write the equation in Column Two. Seven h plus one hundred ten equals two hundred fifty
What two operations ae represented in Problem One? Multiplication and Addition
What operations will we use to solve the problem? Subtraction and Division
Let’s solve the problem. We first start by subtracting the value that is not attached to our
variable. We have to subtract one hundred ten from both sides to keep our equation balanced.
We now have a multiplication equation. Seven h equals one hundred forty
To solve multiplication we use the opposite operation of division. We have a value or a solution
of twenty which is twenty hours for the variable h.
Let’s check the solution. We substitute twenty back into our original equation to represent the
variable h. When we multiply and then add, we find our equation is balanced and so our solution
is correct.
SOLVE PROBLEM – COMPLETION
We are now going to go back to the SOLVE problem. The question was, Jeremy has a collection
of 58 baseball cards. His dad gives him a new card every month. He puts all the cards in a
binder. He completely fills nine pages of the binder and has four cards left over. How many
cards are on each full page?
In S step, we Study the Problem. Underline the question, and complete this statement. This
problem is asking me to find the number of cards on each full page.
O, Organize the facts. We first identify the facts. We go back and read our SOLVE problem
again, and make a vertical line after each fact. Jeremy has a collection of 58 baseball cards.| His
dad gives him a new card every month.| He puts all the cards in a binder.| He completely fills
nine pages of the binder and has four cards left over.| How many cards are on each full page?
After we identify the facts, we go back to the original problem and eliminate any unnecessary
facts. In this word problem, the fact that his dad gives him a new card every month does not help
us determine how many cards are in each full page, so it is an unnecessary fact. After we
eliminate the unnecessary facts, we list the necessary facts. There are fifty total cards, there are
nine full pages, and there are four cards left over.
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 and operation or operations.
Subtraction and Division.
V, Verify your Plan with Action.
Estimate you answer. About seven.
Carry out your plan. We write the equation nine x plus four equals fifty-eight. We first subtract
four from both sides, and then we divide our equation nine x equals fifty-four by nine on both
sides. The value of our variable x is six.
Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, because we were
looking for how many cards on a full page.
Is your answer reasonable? Compare your answer to the estimate. Yes, because it is close to the
estimate.
Is your answer accurate? Check your work. Yes.
Write your answer in a complete sentence. There are six cards on each full page.
CLOSURE
Let’s go back and discuss the essential questions from this lesson.
Our first question was, why do we use variables in equations? Explain your thinking. We use
variables to represent unknown values in mathematical and real-world situations. We can then
write an equation and solve for the value of the variable.
Number two, what operation or operations do we undo first in two-step equations? Justify your
answer. The first operation we apply in two-step equations is addition or subtraction. That is
because in order to isolate the variable we must first add or subtract the value that is not attached
to the variable.
Number three, what is the inverse operation for multiplication? The inverse operation for
multiplication is division.
And number four, what are the two concepts that we need to focus on when solving equations?
In solving equations, we need to isolate the variable and keep the equation balanced.