SUBTRACT INTEGERS
INTRODUCTION
The objective for this lesson on Subtract Integers is, the student will explore subtraction 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, the number line, and addition of integers.
We will have three essential questions that will be guiding our lesson. Number one, explain how
to subtract integers using algebra tiles. Number two, describe the process we can use to take
away the second number in the subtraction problem if there are not enough algebra tiles.
Number three, will the following problems have the same solution? Four minus six and four plus
negative six. Justify your answer.
Begin by completing the warm-up on integer addition to prepare for the lesson on subtracting
integers.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, the highest elevation in North America is Mt. McKinley,
which has an elevation of twenty thousand three hundred twenty feet above sea level. The
lowest elevation in North American is Death Valley, which has an elevation of two hundred
eighty two feet below sea level. What is the difference between these two elevations?
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 is the difference
between these two elevations?
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 difference between the elevations.
During this lesson we will learn how to subtract integers to complete this SOLVE problem at the
end of the lesson.
REVIEWING ABSOLUTE VALUE
What is the range of number values on the number line? Negative five through positive five
Identify zero on the number line.
Describe the change as we move from zero to the right on the number line.
Describe the change as we move from zero to the left on the number line.
Identify the number of spaces from the zero in the positive direction. Three spaces
Identify the position on the positive side of the number line. Three
Identify the number of spaces from the zero in the negative direction. Three spaces
Identify the position on the negative side of the number line. Negative three
If the number of spaces from zero in both directions is three, explain why one of the points is
three and the other is a negative three. When we move to the left we are moving in the negative
direction; when we move to the right we are moving in the positive direction. The distance
moved is three spaces for both.
Identify what this concept is called. Absolute value
Explain what absolute value means. Absolute value is the distance a number is from zero on the
number line.
What is the absolute value of three? Three
What is the absolute value of negative three? Three
Let’s look at a vertical number line.
What is the value of the absolute value of negative four? It is four spaces from zero. So the
value is four.
What is the value of the absolute value of positive four? It is also four spaces from zero. So the
value is four.
The absolute value of a number is always a positive value because the absolute value represents
the distance from zero on the number line and a distance cannot be a negative value.
SUBTRACTING TWO POSITIVE VALUES
Four minus one
What is the problem asking us to find? The difference of positive four and positive one.
Explain how we can model the problem using unit tiles. Place four yellow tiles on the space.
What are some terms we can use to describe subtraction? Take away, remove, cross out
What do we want to take away from the four yellow tiles? One yellow tile
Let’s show how to take away one yellow tile. What do we have left after we take the one yellow
tile away? Three yellow tiles
What do you notice about the color of the tiles? They are all yellow.
Let’s draw a picture of the four yellow tiles in the box below Problem One and talk about how to
show subtraction. Cross out one yellow tile with an “X.”
We can also model using Ys.
How can we model taking away on Y? Cross out one Y.
Explain the color of the tiles. They are all the same color.
What operation did we use? Subtraction
What is the difference? Three yellows which equals three
Four minus one equals three.
Let’s look at the horizontal number line.
Explain how to model the first value of positive four. Draw a line in the positive direction – to
the right from the zero to the four.
Explain how to model subtracting the one. Draw a line starting at the four and then moving one
space in the negative direction.
What is the difference of four and one? Three
Four minus one equals three.
Now let’s look at the vertical number line.
Explain how to model the first value of positive four. Draw a line in the positive direction up
from the zero to the four.
Explain how to model subtracting one. Draw a line from the four one space in the negative
direction which is down.
What is the difference of four minus one? Three
Describe the relationship between subtraction and addition. They are opposite operations.
Compare the horizontal number line for Problem One. The number lines look the same.
How can you represent the number line model above with addition? Four plus negative one is
equal to three.
What conclusion can you make based upon the horizontal number line models? Subtraction is
the same as adding the opposite.
How can we write the two problems? Four minus one is equal to three corresponds to four plus a
negative one which is also equal to three.
SUBTRACTING TWO NEGATIVE VALUES
Negative four minus a negative one
What is the problem asking us to find? The difference of negative four and negative one.
Explain how we can model the problem using unit tiles. Place four red tiles on the space.
What are some terms we can use to describe subtraction? Take away, remove, cross out
What do we want to take away from the four red tiles? One red tile
Let’s show how to take away one red tile.
What do we have left after we take the one red tile away? Three red tiles
What do you notice about the color of the tiles? They are all red.
Let’s draw a picture of the four red tiles in the box below Problem Two and talk about how to
show subtraction. Cross out one red tile with an “X.”
We can also model using R’s.
How can we model taking away one R? By crossing out one R.
Explain the color of the tiles. They are all the same color.
What operation did we use? Subtraction
What is the difference? Three reds which is equal to negative three.
Negative four minus a negative one is equal to negative three.
Now let’s take a look at our model on the horizontal number line.
Explain how to model the first value of negative four. Draw a line in the negative direction – to
the left from the zero to the negative four.
Explain how to model subtracting the negative one. If we are subtracting the negative we are
moving in the opposite direction of negative one which would be one space to the right.
What is the difference of negative four and negative one? Negative three
Negative four minus negative one is equal to negative three.
Now let’s look at our example on the vertical number line.
Explain how to model the first value of negative four. Draw a line in the negative direction
down from the zero to the negative four.
Explain how to model subtracting negative one. Draw a line from the negative four one space up
because we are doing the opposite of subtracting one.
What is the difference of negative four and negative one? Negative three
Describe the relationship between subtraction and addition. They are opposite operations.
Compare the horizontal number lines for Problem Two. The number lines look the same.
How can you represent the number line model above with addition? Negative four plus one is
equal to negative three.
What conclusion can you make based upon the horizontal number line models? Subtraction is
the same as adding the opposite.
How can we write the two problems? Negative four minus negative one is equal to negative
three, corresponds to negative four plus one is equal to negative three.
SUBRACTING A NEGATIVE FROM A POSITIVE
Four minus a negative one
What is the problem asking us to find? The difference of positive four and negative one.
How can we model the problem using unit tiles? Place four yellow tiles on the space.
What are some terms we can use to describe subtraction? Take away, remove, cross out
What do we want to take away from the four yellow tiles? One red tile
Is it possible to take one red tile away from four yellow tiles? No
When we were adding integers, what did we have when we represented one red and one yellow?
A value of zero.
We called that a zero pair because the value is zero.
How could we use zero pairs to create the possibility of take away a red tile? What happens if
we add one zero pair? Does the value of the tiles change? Explain your thinking. No, the value
does not change because we can add or subtract the value of zero to a number without changing
the value.
What property does this relate to? The identity property of zero.
Have we created the possibility of taking away one red tile? Yes
What do we have left after we take the one red tile away? Five yellow tiles
What do you notice about the color of the tiles? They are all yellow.
Let’s draw a picture of the four yellow tiles in the box below Problem Three and talk about how
to show subtraction. Add a zero pair and then cross out one red tile with an “X.”
We can also model using Y’s.
How can we model taking away one R? Add a YR to represent the zero pair and then cross out
the R.
Explain the color of the tiles. They are all the same color.
What operation did we use? Subtraction
What is the difference? Five yellows which is equal to five.
Four minus a negative one is equal to five.
Let’s look at our example on the horizontal number line.
Explain how to model the first value of positive four. Draw a line in the positive direction – to
the right from the zero to the four.
Explain how to model subtracting the negative one. If we are subtracting the negative we are
moving in the opposite direction of negative one which would be one space to the right.
What is the difference of four and negative one? Positive five
Four minus a negative one is equal to positive five.
Now let’s look at our vertical number line.
Explain how to model the first value of positive four. Draw a line in the positive direction up
from the zero to the four.
Explain how to model subtracting a negative one. Draw a line from the positive four one space
up because we are doing the opposite of subtracting one.
What is the difference of positive four and negative one? Five
Describe the relationship between subtraction and addition. They are opposite operations.
Compare the horizontal number lines for Problem Three. The number lines look the same.
How can you represent the number line model above with addition? Four plus one is equal to
five.
What conclusion can you make based upon the horizontal number line models? Subtraction is
the same as adding the opposite.
How can we write the two problems? Four minus a negative one is equal to five corresponds to
four plus one is equal to five.
SUBTRACTING A POSITIVE FROM A NEGATIVE
Negative four minus one
What is the problem asking us to find? The difference of negative four and positive one.
How can we model the problem using unit tiles? Place four red tiles on the space.
What are some terms we can use to describe subtraction? Take away, remove, cross out
What do we want to take away from the four red tiles? One yellow tile
Is it possible to take one yellow tile away from four red tiles? No
When we were adding integers, what did we have when we represented one red and one yellow?
A value of zero. We called that a zero pair because the value is zero.
How could we use zero pairs to create the possibility of taking away a red tile?
What happens if we add one zero pair? Does the value of the tiles change? Explain your
thinking. No, the value does not change because we can add or subtract the value of zero to a
number without changing the value.
What property does this relate to? Identity property of zero
Have we created the possibility of taking away one yellow tile? Yes
What do we have left after we take the one yellow tile away? Five red tiles
What do you notice about the color of the tiles? They are all red
Let’s draw a picture of the four red tiles in the box below Problem Four and talk about how to
show subtraction. Add a zero pair and cross out one yellow tile with an “X.”
We can also model using R’s.
How can we model taking away one Y? Add a YR to represent the zero pair and then cross out
the Y.
Explain the color of the tiles. They are all the same color.
What operation did we use? Subtraction
What is the difference? Five reds is equal to negative five.
Negative four minus one is equal to negative five.
Explain how to model the first value of negative four. Draw a line in the negative direction
down from the zero to the negative four.
Explain how to model subtracting a positive one. Draw a line from the negative four one space
down.
What is the difference of negative four and positive one? Negative five
Negative four minus one is equal to negative five.
Describe the relationship between subtraction and addition. They are opposite operations.
Compare the horizontal number lines for Problem Four. The number lines look the same.
How can you represent the number line model above with addition? Negative four plus a
negative one is equal to negative five.
What conclusion can you make based upon the horizontal number line models? Subtraction is
the same as adding the opposite.
How can we write the two problems? Negative four minus one equals negative five corresponds
to negative four plus negative one is equal to negative five.
SUBTRACTING INTEGERS ANND CORRESPONDING ADDITION PROBLEMS
Four minus one
What is the difference of four minus one? Three
What is the corresponding addition problem using integer tiles? Four plus a negative one is
equal to three.
Complete Problems Two through Four.
What conclusion can you make about these problems? Subtracting an integer is the same as
adding the opposite value.
INTEGER RULES
Let’s look at the graphic organizer that has the integer rules.
Subtraction is the same as: Make two changes to the problem. After these two steps:
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 subtraction.
Subtraction is the same as adding the opposite.
Make two changes to the problem: One, change the subtraction sign to an addition sign.
Number two, change the sign of the second number to its opposite.
After these two steps: Follow the rules for addition of integers.
SUBTRACTING INTEGERS FOLDABLE
We have the four operations written on the front flaps of your foldable. Addition, Subtraction,
Multiplication and Division.
Inside, on the right side of the second flap, write the rules for subtracting integers. On the left
side, write examples for subtracting integers.
SOLVE PROBLEM – COMPLETION
Now going back to the SOLVE problem from the beginning of the lesson. The question was, the
highest elevation in North America is Mt. McKinley, which has an elevation of twenty thousand
three hundred twenty feet above sea level. The lowest elevation in North America is Death
Valley, which has an elevation of two hundred eighty two feet below sea level. What is the
difference between these two elevations?
S, Study the Problem. Underline the question and complete this statement. This problem is
asking me to find the difference between the elevations.
In Step O, we Organize the Facts. We first need to identify the facts. We go back to the original
problem and as we read it again we mark with a vertical mark or strike each fact. The highest
elevation in North America is Mt. McKinley,/ which has an elevation of twenty thousand three
hundred twenty feet above sea level./ The lowest elevation in North America is Death Valley,/
which has an elevation of two hundred eighty two feet below sea level./
After we identify the facts we eliminate any unnecessary facts. In this problem all the facts are
necessary.
So the last part of O is to list our necessary facts.
Mt. McKinley has an elevation of twenty thousand three hundred twenty feet above sea level.
Death Valley has an elevation of two hundred eighty two feet below sea level.
STOP!
What are we trying to find? The difference between the elevations.
What does this mean? The word difference tells us to subtract.
Discuss:
What integer can we use to represent the first elevation? Twenty thousand three hundred twenty
What integer can we use to represent the second elevation? Negative two hundred eighty two
What does this mean? We will be subtracting a negative value from a positive value.
Discuss:
How do we subtract integers? We add the opposite of the second value. Explain your thinking.
We change the subtraction sign to addition and then change the sign of the second value to its
opposite. We then follow rules for addition.
Step L, Line Up a Plan. Write in words what your plan of action will be. Change the subtraction
problem to an addition problem by adding the opposite. Add the absolute value of the elevation
below sea level to the elevation above sea level.
Choose an operation or operations. Write as a subtraction problem, then change to addition and
add the opposite value.
V, Verify Your Plan with Action. First we estimate your answer. About twenty one thousand
feet.
Carry out your plan. Twenty thousand three hundred twenty minus a negative two hundred
eighty two is equal to twenty thousand three hundred twenty plus two hundred eighty two which
is equal to twenty thousand six hundred two feet.
Examine Your Results
Does your answer make sense? Compare your answer to the question. Yes, because we are
looking for the difference in the two elevations.
Is your answer reasonable? Compare your answer to the estimate. Yes, it is close to our
estimate of about twenty one thousand feet.
Is your answer accurate? Check your work. Yes
Write your answer in a complete sentence. The difference in the two elevations is twenty
thousand six hundred two feet.
CLOSURE
Let’s go back and discuss the essential questions from the lesson.
Our first question was, explain how to subtract integers using algebra tiles. Represent the first
value with the red and yellow tiles. Take away the second value.
Question Two, describe the process we can use to take away the second number in the
subtraction problem if there are not enough algebra tiles. Create the possibility by using zero
pairs.
Number Three, will the following problems have the same solution? Four minus six and four
plus a negative six. Justify your answer. Yes, the two problems will have the same solution of
negative two because subtracting is the same as adding the opposite.