ADD INTEGERS
INTRODUCTION
The objective for this lesson on Add Integers is, the student will explore addition with integers
and apply this understanding to solve problems in mathematical and real world situations.
The skills students should have in order to help them in this lesson include, addition and
subtraction of whole numbers and use of the number line.
We will have three essential questions that will be guiding our lesson. Number one, explain how
to add integers that have the same signs. Number two, explain how to add integers that have
different signs. And number three, describe real-life situations where you would need to use
negative integers.
Begin by completing the warm-up on addition and subtraction to prepare for the add integers
lesson.
SOLVE PROBLEM – INTRODUCTION
Jackie gets paid five dollars and fifty cents an hour for forty hours of work per week. She
deposited one hundred thirty five dollars into her checking account. Her balance before the
deposit was negative thirty dollars. What was the balance of Jackie’s account after the deposit?
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, what was the balance
of Jackie’s account after the deposit?
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 in the account after
the deposit.
During this lesson we will learn how to add integers to complete this SOLVE problem at the end
of the lesson.
ADD INTEGERS – ADDING OPPOSITES
We have yellow tiles and we have red tiles to represent our integers. The yellow tiles are
positive values and the red tiles are negative values.
How do you represent positive four with yellow unit tiles? That would be four of the yellow unit
tiles.
How do you represent positive two with yellow unit tiles? Two unit tiles.
How could you represent negative three with red unit tiles? We would use three red unit tiles.
How do you represent negative five with red unit tiles? We would use five red unit tiles.
Daniel earns four dollars walking his neighbor’s dog. He pays his brother four dollars he owes
him from last week.
What does four dollars represent? It represents how much he earns and how much he owes his
brother.
Explain the meaning of the word earn. Earn will be a positive value because it is money you get
by doing something.
Explain the meaning of the word owes. Owes will be a negative value because it is money that
you will have to pay someone.
How could you use the unit tiles to represent the value that Daniel earned? Earning is a positive
amount so we can represent positive four with the four yellow tiles.
How could you use the unit tiles to represent the amount Daniel pays his brother? That would be
four red unit tiles.
How would you represent the one yellow tile using the number line? Move one to the right
because we start with zero and move one is the positive direction.
If we now represent the red integer tile on the number line, what value would we use? Negative
one because we start with zero and go one space in the negative direction.
How could we describe the values of positive one and negative one? They are opposites.
Explain your answer. Both positive and negative one are the same distance from zero but in
opposite directions.
Explain the relationship between the yellow and red tile. They represent opposite values.
What is the value of one yellow and one red tile? Zero. Justify your answer. If we start at zero,
add one and then take it away we have zero.
What color are the tiles that represent the money earned? Yellow
What color are the tiles that represent the money he paid his brother? Red
Circle each set of tiles that represents a sum of zero.
What direction is positive on the number line? To the right.
What direction is negative on the number line? To the left.
What direction do we move on the number line to model the amount that Daniel earned? To the
right or positive
What direction do we move on the number line to model the amount Daniel paid to his brother?
To the left or negative
What is the combined value of the positive four and negative four? Zero
What direction is positive on the vertical number line? Up
What direction is negative on the vertical number line? Down
Which direction do we move on the number line to model the amount Daniel earned? Up or
positive
Which direction do we move on the number line to model the amount Daniel paid to his brother?
Down or negative
What is the combined value of the positive four and negative four? Zero
Conclusion? When we add a number and its opposite, what is the sum? Zero
This is known as the additive inverse property.
ADDING INTEGERS WITH LIKE SIGNS – POSITIVE VALUES
Three plus two equals; what is the problem asking me to find? The sum of positive three and
positive two.
How can we model addition in the problem using unit tiles? Three unit tiles and two unit tiles.
When we add items, we push them together. What is the total number of tiles? Five
What do you notice about the color of the tiles? They are all yellow.
Draw a picture of the model in the box below Problem one. Use Y’s to represent the tiles.
Explain the color of the tiles. They are all the same color.
Explain what we did with the tiles. We pushed them together to add them.
What is the sum? Five yellows equals five.
Three plus two equals five. Now let’s use our number line as a model.
Explain how to model the first value of positive three. Draw a line in the positive direction – to
the right from zero to the three.
Explain how to model adding the positive two. Draw a line two spaces in the positive direction
starting from the three.
What is the sum of three plus two? Five
Now let’s look at the vertical number line.
Explain how to model the first value of positive three. Draw a line in the positive direction – up
from the zero to the three.
Explain how to model adding the positive two. Draw a line starting at three and moving two
spaces in the positive direction, which is up.
What is the sum of three plus two? Five
ADDING INTEGERS WITH LIKE SIGNS – NEGATIVE VALUES
Negative three plus negative two equals; what is the problem asking me to find? The sum of
negative three and negative two.
How can we model the problem using unit tiles? Three red tiles and two red tiles.
When we add items, we push them together. What is the total number of tiles? Five
What do you notice about the color of the tiles? They are all red.
Draw a picture of the model in the box below Problem one. Use R’s to represent the tiles.
Explain the color of the tiles. They are all the same color.
Explain what we did with the tiles. We pushed them together to add them.
What is the sum? Five reds equals negative five.
Explain how to model the first value of negative three on our horizontal number line. Draw a
line in the negative direction to the left from zero to negative three.
Explain how to model adding the negative two. Draw a line two spaces in the negative direction
to the left, starting from the negative three.
What is the sum of negative three plus negative two? Negative five
Now let’s use our vertical number line.
Explain how to model the first value of negative three. Draw a line in the negative direction –
down from the zero to the negative three.
Explain how to model adding the negative two. Draw a line from the negative three two spaces
in the negative direction, which is down.
What is the sum of negative three plus negative two? Negative five
ADDING INTEGERS WITH LIKE SIGNS – CONCLUSIONS
Problem one: three plus two equals five
Problem two: negative three plus negative two is equal to negative five
What do you notice about Problem one and Problem two? In Problem one both values were
positive and in Problem two both values were negative.
In Problem one when you added two positive integers, what was the sum? It was a positive five.
In Problem two when you added two negative integers, what was the sum? It was a negative
five.
What is your conclusion? When we add two positive integers the sum will be positive. When
you add two negative integers the sum will be negative.
Justify your answer using a number line model. When we add two positive values, we start at
zero and because both values are positive we are moving in the positive direction so our answer
must be positive.
When we add two negative values, we start at zero and because both values are negative we are
moving in the negative direction so our answer must be negative.
ADDING INTEGERS WITH UNLIKE SIGNS
Three plus a negative two is equal to – What is the problem asking me to find? The sum of
positive three and negative two.
How can we model the problem using unit tiles? Three yellow tiles and two red tiles.
When we added in Problems one and two, what did we do with our tiles? We pushed them
together.
What do you notice about the color of the tiles? They are not all the same color.
Draw a picture of the model in the box below Problem three.
When we were adding opposites at the beginning of the lesson, what did we do when we had
yellow and red tiles? We created zero pairs.
Can we make any zero pairs with the yellow and red tiles? Yes. Circle the zero pairs. After
removing the zero pairs, what do we have left? One yellow tile.
Use Y’s and R’s to represent the tiles. Explain the color of the letters. Red tiles are negative and
yellow tiles are positive. Three Y plus two R.
Do we have zero pairs that we can combine? Yes
What operation did we use to combine the tiles? Addition
Explain what we did with the Y’s and R’s. We added them.
Explain how to model the first value of positive three. Draw a line in the positive direction – to
the right from the zero to the three.
Explain how to model adding the negative two. Draw a line from the three two spaces in the
negative direction – to the left.
What is the sum of three plus negative two? One
Let’s go to our vertical number line model.
Explain how to model the first value of positive three. Draw a line in the positive direction – up
from the zero to the three.
Explain how to model adding the negative two. Draw a line from the three two spaces in the
negative direction, which is down.
What is the sum of three plus negative two? One
Negative three plus positive two. What is the problem asking me to find? The sum of negative
three and positive two.
How can we model the problem using unit tiles? Three red tiles and two yellow tiles.
When we added in Problems one and two, what did we do with our tiles? We pushed them
together.
What do you notice about the color of the tiles? They are not all the same color.
Draw a picture of the model in the box below Problem three. When we were adding opposites at
the beginning of the lesson, what did we do when we had yellow and red tiles? We created zero
pairs.
Can we make any zero pairs with the yellow and red tiles? Yes, Circle the zero pairs. After
removing the zero pairs what do we have left? One red tile.
Use Y’s and R’s to represent the tiles. Explain the color of the letters. Red tiles are negative and
yellow tiles are positive.
We have three R’s plus two Y’s.
Do we have any zero pairs that we can combine? Yes
What operation did we use to combine the tiles? Addition
Explain what we did with the Y’s and R’s. We added them.
What is the sum? One red which equals negative one.
Negative three plus two is equal to negative one.
Now let’s look at our horizontal number line model.
Explain how to model the first value of negative three. Draw a line in the negative direction – to
the left from the zero to the negative three.
Explain how to model adding the positive two. Draw a line from the negative three two spaces
in the positive direction – to the right.
What is the sum of negative three plus two? Negative one
Now let’s go to our vertical number line.
Explain how to model the first value of negative three. Draw a line in the negative direction –
down from the zero to the negative three.
Explain how to model adding the positive two. Draw a line form the negative three two spaces
in the positive direction, which is up.
What is the sum of negative three plus two? Negative one
What did you notice about Problem three and Problem four? In both problems one integer was
positive and one was negative.
In Problem three when you added the two integers, what was the sum? It was a positive one.
In Problem four when you added the two integers, what was the sum? It was a negative one.
ADDING INTEGERS WITH UNLIKE SIGNS – EXPLORING RULES AND THE
COMMUTATIVE PROPERTY
Problem three is three plus negative two equals one. When we added the two values, where did
we start on the number line? Zero
Which direction did we move in first? To the right to represent the positive three.
How did we model adding the negative two? By moving to the left.
When we move to the right on the number line, what operation are we modeling? Addition
When we move to the left on the number line, what operation are we modeling? Subtraction
If we look at the number values without the signs, what operation did we use to determine the
answer in this problem? Subtraction
How was the sign determined? The sign of the sum is the sign of the value with the greater
absolute value.
When we added the two values in Problem four, where did we start on the number line? At zero
Which direction did we move first? To the left to represent the negative three.
How did we model adding the positive two? By moving to the right.
When we move to the right on the number line, what operation are we modeling? Addition
When we move to the left on the number line, what operation are we modeling? Subtraction
If we look at the number values without the signs, what operation did we use to determine the
answer in this problem? Three and two, subtraction
How was the sign determined? The sign of the sum is the sign of the value with the greater
absolute value.
Would the sum change if the order of the addends was switched? Was the sum the same for both
problems? Yes
Did the order of the addends change the sum? No
What property are we representing? The commutative property of addition.
Problem four: Negative three plus two is to negative one and two plus negative three is equal to
negative one.
Would the sum change if the order of the addends was switched? Was the sum the same for both
problems? Yes
Did the order of the addends change the sum? No
What property are we representing? The commutative property of addition.
INTEGER RULES
In your graphic organizer we have the integer rules for addition:
Two integers with the same signs:
Two integers with different signs:
Take a look back at the problems we completed during the lesson.
Completing the sentences in the box, let’s identify the rules for integer addition.
When we have integers with the same signs we add the two values and keep the sign.
When we have two integers with different signs we subtract the two values and use the sign of
the number with the larger absolute value.
ADDING INTEGERS FOLDABLE
Fold the paper hamburger style. Fold the paper hamburger style one more time. Now unfold the
paper.
Next, complete a hot dog fold. Unfold the paper and you should see four equal sections,
separated by a fold for a total of eight sections.
Now, cut each of these folds on the front flap only, you see the dotted line. On the outside of
each flap write the operation names. Addition, Subtraction, Multiplication, and Division
Inside on the right side of the first flap, write the rules for adding integers. On the left side, write
examples for adding integers.
SOLVE PROBLEM – COMPLETION
Going to go back to the SOLVE problem from the beginning of the lesson. The question was,
Jackie gets paid five dollars and fifty cents an hour for forty hours of work per week. She
deposited one hundred thirty five dollars into her checking account. Her balance before the
deposit was negative thirty dollars. What was the balance of Jackie’s account after the deposit?
In the S Step, we underlined the question and we complete this statement. This problem is
asking me the find the amount of money in the account after the deposit.
In the O Step, we Organize the Facts. First we have to identify the facts. We go back and read
through the problem again and make a vertical line or strike mark after each fact. Jackie gets
paid five dollars and fifty cents an hour for forty hours of work per week./ She deposited one
hundred thirty five dollars into her checking account./ Her balance before the deposit was
negative thirty dollars./ What was the balance of Jackie’s account after the deposit?
After we identify the facts, then we need to eliminate the unnecessary facts. We go back to the
problem again. Jackie gets paid five dollars and fifty cents an hour for forty hours of work per
week. That information is not necessary to determine the balance of her account, so we can
strike through that fact as unnecessary. She deposited one hundred thirty five dollars into her
checking account. That fact is necessary. Her balance before the deposit was negative thirty
dollars. That is also a necessary fact. What was the balance of Jackie’s account after the
deposit?
The next part of O is to list the necessary facts. She deposited one hundred thirty five dollars and
she had a previous balance of negative thirty dollars.
STOP! What are we trying to find? The amount of money in the account after the deposit.
What does this mean? The word deposit tells us that we will add the amount that she deposited.
Discuss: What was the value of her account before the deposit. Negative thirty dollars
What does this mean? We will be adding a positive value to a negative value. What operation
will we use? Subtraction. Explain your thinking. When we are adding integers with two
different signs, we subtract the values and use the sign of the number with the greater absolute
value.
L, Line Up a Plan. Write in words what your plan of action will be. Subtract the number that
represents the value of the previous balance from the deposit. Use the sign of the number with
the greater absolute value.
Choose an operation or operations. Subtraction
V, Verify Your Plan with Action. Estimate your answer. About one hundred dollars.
Carry out your plan. One hundred thirty five minus thirty is equal to one hundred five or one
hundred five dollars.
E, Examine Your Results
Does your answer make sense? Compare your answer to the question. Yes, because we are
looking for an amount.
Is your answer reasonable? Compare your answer to the estimate. Yes, it is close to our
estimate of about one hundred dollars.
Is your answer accurate? Check your work. Yes
Write your answer in a complete sentence. Jackie has a balance of one hundred five dollars in
her account.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, explain how to add integers that have the same sign. Add the values and
keep the same sign.
Number two, explain how to add integers that have different signs. Subtract the two values and
take the sign of the number with the larger absolute value.
And number three, describe real-life situations where you would need to use negative integers.
Some examples would be, bank statements, temperature rising and falling, borrowing money,
and elevations.