2.1 Combining Opposite Numbers
Number Types
By now, you probably know what fractions and whole numbers are. The set of Whole
Numbers includes all positive numbers and zero, without fractions or decimals.
The set of Whole Numbers includes the Natural Numbers and zero. The set of Natural
Numbers is sometimes referred to as the set of counting numbers, because they are all the
numbers you would use to count whole things (fingers, dogs, rocks, trees, and so on).
Another group of numbers is the ______________________. The Integers are all the Whole
Numbers and their negatives. Again, no decimals or fractions, just {...-4, -3, -2, -1, 0, 1, 2, 3, 4,
...}. Whole numbers are known as a subset of integers (they are within the set of Integers).
The last set of numbers this book will address are rational numbers. A rational number is
any number that can be written as a _____________________________. This includes
terminating decimals (decimals that end, like 0.5) and repeating decimals (decimals that
repeat in a pattern, 0.5858585858..., also written as: _____________________). All whole
numbers and integers can also be written as fractions. For example, __________________.
Opposite numbers are the same number with opposite signs. All opposite numbers are called
additive inverses. All additive inverses have a sum of zero. If you were to place the number
'a' and its opposite, '-a', on a number line, you would see they are the same distance away
from zero.
Both 3 and -3 are 3 units away from zero, making them additive inverses and opposites. The
distance away from zero is also called the __________________________ of a number.
Absolute value is always positive because distance is always positive. Absolute value is
marked with vertical lines on either side of the number.
|−12|=12
|4.5|=4.5
Dance Moves
Juan really wants to impress the girls at the next school dance, so he decides to take dance
lessons. He notices that dancing is just a series of steps; move two steps forward, two steps
back, one to the left, one to the right, etc. At the school dance, Juan creates his own series
of moves. Here is a list of the steps:
● 3 steps to the left
● 2 steps to the right
● 4 steps to the left
● 3 steps to the right
Then he repeats. If his starting position is zero on the number line, where does he end up
after one cycle? Use the number line and Juan's feet to help you towards your answer.
Remember this!
● ________________numbers are positive numbers without fractional or decimal parts,
including zero.
● _________________are whole numbers that are positive and ____________________.
Whole numbers are a part of the set of integers.
● ______________________numbers are all numbers that can be written as a fraction,
including whole numbers, integers, fractions, terminating, decimals, and repeating
decimals.
● The absolute value of a number is its _____________________from zero on the number
line. Distance is always positive, therefore, so is the absolute value.
● Two numbers are opposites of each other, or additive inverses, if their sum is zero.
2.2 Modeling the Addition of Integers
The answer for an addition problem can be ________________or _________________. The
answer can also be zero if the integers are opposites. When adding integers, start at the
first integer on the number line. If the second integer is positive, move that many units to
the right and where you land is the answer. If the second integer is negative, move that
many units to the left and where you land is the answer.
Discussion Question
Start at 0. Describe two dance steps that will bring you back to where you started.
The commutative property of addition: For any two numbers, the order in which you add
them does not matter;
a+b=b+a
Since the order in which you add the three numbers does not matter, so do the
combination that is easiest for you. So, rather than do
9+(−3)+2,
you could do
9+2+(−3)
In this case, parentheses are used around the negative number to separate it from the plus
sign, so they do not change this problem in any way.
Remember this!
When adding integers, start at the first integer (on the number line) and move to the right
or left, depending on the sign of the second integer to find the answer.
Subtracting Integers
Subtraction is defined as taking away a value from another. As you saw in the previous
lessons, adding a negative number is subtraction. On a number line, adding an integer
meant moving units in the direction of the sign of that number. So, adding -4 would mean
moving 4 units to the left. With subtraction, it is the opposite. Subtracting -4 would mean
moving 4 units to the right. Therefore, subtraction can also be defined as adding the
opposite.
2−(−4)=2+4
When doing subtraction problems, change the problem to adding the opposite before
starting. KEEP, CHANGE, CHANGE!
Diving Depths
Fatima is learning how to scuba dive. She is experimenting with different depths as she
swims in the ocean. If zero is the surface of the water, she starts 5 feet below that and then
dives another 12 feet further. How far below the surface is she?
When subtracting integers, remember to add the opposite. To help you organize
subtraction with the different cases, see the table below.
Discussion Question
Why is subtracting a negative integer the same as adding a positive integer?
Banking
Gina has a savings account with $23.64 in it. She makes a withdrawal of $15.67 and then a
deposit of $6.78. How much money is in her account?
Remember this!
● When subtracting integers, fractions, or decimals, always change the problem to
adding the opposite of the second number. Then, approach like addition.
● When adding, if the signs of the numbers are the same, add the values and then
give the answer the same sign as the numbers you added.
● When adding, if the signs of the numbers are different, subtract the values, and
then give the answer the sign of the larger number
Properties of Addition
In this lesson, you will combine multiple addition and subtraction expressions and solve.
To help solve these types of problems, there are two properties that can be used. The first is
the Commutative Property of Addition, which states that the order in which you add two
rational numbers ____________________. So, for any two rational numbers a and b,
a+b=b+a
The second property is the Associative Property of Addition. This property states that for
any three rational numbers, you may pick the two that you want to add together first. You
do not have to add them in the order in which they appear in the problem. So, in this case,
for any three rational numbers; a, b and c,
a+b+c=(a+b)+c=a+(b+c)
Notice the grouping symbols (parentheses) here. They indicate that it does not matter
which two numbers you add together first. Pick the two that are easiest for you, then add
the third. Other grouping symbols are braces, { }, brackets, [ ], and even absolute value, | |.
This property also works for more than three rational numbers.
Switch the Order
How many different ways can you add 7+4+(−6)+(−3)?
Multi-Step Multiplication and Division
Properties of Multiplication
Just like addition, multiplication has a commutative and associative property. The
Commutative Property of Multiplication says that the order in which you multiply any two
rational numbers, order does not matter. So, for any two rational numbers a and b,
a×b=b×a ab=ba a⋅b=b⋅a
Notice the three different ways to write the multiplication of two numbers. In addition,
you could put parentheses around a and b,
(a)(b)=(b)(a)
It is important to note that if there is no sign between two variables (letters) or a variable
and number, you can assume that the two numbers are being ______________________.
The Associative Property of Multiplication states that for any three rational numbers, you
may pick the two that you want to multiply together first. You do not have to multiply them
in the order in which they appear in the problem. So, in this case, for any three rational
numbers: a, b and c,
a×b×c=a×(b×c)=(a×b)×c abc=a(bc)=(ab)c a⋅b⋅c=a⋅(b⋅c)=(a⋅b)⋅c
Just like with the Associative Property of Addition, the Associative Property of
Multiplication can use any of the grouping symbols, including absolute value. This
property also works for more than three rational numbers.
Switch the Order
How many different ways can you multiply -8⋅2⋅-5⋅2?
The Commutative and Associative Properties work for multiplication like they do for
addition. There are two more properties to explore here: Identity and Inverse.
Recall that for addition the additive identity was ________because any number you add to
_________ is still that number. What number can you multiply by any number and the
answer is still that number?
a⋅1=a
The Identity Property of Multiplication states that for any number, a, when multiplied by 1,
the answer is still a.
The Inverse Property of Multiplication is similar. What number times a is equal to 1?
You can figure this out by working backwards from the identity property. Remember how
when dividing by a fraction, you can flip the fraction and multiply by the reciprocal
instead? Well, for any number a, you can multiply by its reciprocal to obtain 1.
________________________
It is important to note that the sign of the reciprocal of a number is the same as the sign of
the number. Reciprocals are not opposites.
Recall that the Associative and Commutative Properties do not work with division. For the
last question in the interactive, you need to perform the indicated operations in order
from left to right. Additionally, this interactive used multiplying by an inverse
interchangeably with dividing by the number. Familiarize yourself with using both
multiplying by fractions and dividing by a whole number. Depending on the problem, one
might be easier than the other.