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Integers

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Jennifer Ysgnesenauon

on 1 March 2014

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Transcript of Integers

Multiplying Integers
One rule to remember when multiplying integers is that whenever the signs are the same, the answer to the equation will always be positive. When the signs are different, the answer to the equation will always be negative.
Subtracting Integers with Like Signs
When subtracting integers with like signs, it's important to notice that the answer to the equation will either be positive or negative. Also, just because you are subtracting something from a number, it doesn't mean that the number will always be decreasing or getting smaller.
Adding Integers with Like Signs
When adding integers with like signs, without using a diagram to figure it out, it is to add them normally and remember what sign the answer should be.
Something to reconsider while doing the sum is that whenever you are adding a
positive
with a
positive
number you will always get a
positive
answer. Whenever you are adding a
negative
number with a
negative
number, the answer will always be
negative
.
(+4)+(+7)= +11
Adding Integers with Unlike Signs
(-8)+(+3)= -5
=8-3
=5
The bigger number is the 8 which has a negative sign to it so the answer will be -5.

When adding integers with unlike signs, remember that the answer can be either positive or negative. To figure this out, you can take the bigger number (not including the sign of the number) then subtract it from the smaller number. The sign for the answer belongs to the bigger number so what ever sign was with the bigger number is the sign to the answer.
Positive Integer
Negative Integer
When adding with integer chips use them to represent each number. So if you need 4 positive chips, you put that down and you need 7 more positive chips so you put that down. Then, you count how many positive chips you have in total to figure it out.
(-6)+(-3)= -9
When adding integers that are negative use the same process as adding positive numbers. If you need 6 negative chips, then you put that many down, then you put 3 negative chips down and then at the end, you count to see how many you have all together.
However, if you are multiplying with more than 2 numbers, you must break the equation down. That means break it down to multiplying 2 numbers at a time. Then the rule will vary because you can't identify the answer's sign right away. You have to break it down and figure out the sign when you are left with just 2 numbers for the equation to be able to compare them and then use the rule above.
Terms to Know
Positive
numbers:
numbers that are above zero
Negative
numbers:
numbers that are below zero
Zero Pairs:
pair of either positive or negative numbers that have a value of zero

Legend:
Integer Chips
Integer Chips
(-6)+(+4)= -2
Integer Chips
Mental Math
If you want to be able to do the math in your head, you have to remember a few things. If you are adding a positive with another positive number, then your answer has to be positive as well. If you are adding a negative number with another negative number, then your answer has to be negative as well.
Then, to figure the math out, all you need to do is add normally like there wasn't even the signs there. So ignore the signs and deal with those later.
(+4)+(+5)= +9
Positive
number
Positive
number
Positive
number
(-8)+(-1)= -9
Negative
number
Negative
number
Negative
number
Using a Number line
Example: (-6)+(+7)= ?
So the answer to the equation would be +1. So all that you have to do is take your starting number (-6) and move 7 numbers to the right because you are adding positive 7 to negative 6.
+1
+2
+3
+4
+5
+6
+7
Number line
(+2)+(+4)=?
+1
+2
+3
+4
(-5)+(-3)=?
-1
-2
-3
The answer
The answer
*Remember
When adding integers with like signs using a number line, adding positive numbers, you should be counting to the right, while adding negative numbers, you should be counting to the left side of the number line.
Adding
positive
numbers
Adding
negative
numbers
Count to the right
Count to the left
One negative chip with one positive chip cancel each other out and create something called a "zero pair". So the value of the two chips is zero. Because the value is zero, you cancel those out and see what chips are left to figure out the answer.
The rest of the numbers are canceled out and all that's left are the two negative chips.
Using Integers with Exponents
B
rackets
E
xponents
D
ivision
M

uliplication
A

ddition
S
ubtraction
Using "BEDMAS" means that you are doing the order of operation in that order.
Integers with More Than 2 Terms in the Equation
(+3)x(-8)x(-1)= +24
(-24)x(-1)= +24
At first glance, when you look at the equation you would think that the answer would be negative because all the signs are different, but you need to break the equation down and then compare the signs when the equation is down to 2 terms. When the rule of multiplying and dividing integers is when the signs are unlike the answer is negative and when the signs are like, the answer is positive can only be used when there are 2 terms. You can only compare the equations when there are only 2 terms.
Like signs = positive answer
I
n
t
e
g
e
r
s

Integer Chips
(-6)-(-2)= -4
Here using a diagram, because you have six chips and you need to get rid of 2, you take away 2 chips to be left with 4 negative chips.
(+2)-(+7)= -5
When ever you are subtracting a larger number from a smaller number, you have to use something called a "zero pair" because you can't take away something from nothing and because the value of a "zero pair" is zero, you won't be disrupting the equation. In the example, you need to take away positive 7 from positive 2, but you only have positive 2. You can't just add five more positive chips because that would be adding 5 to 2 making it positive 7 subtracted from positive 7 and it would no longer be (+2)-(+7). You have to add zero pairs because you can't change the equation and by adding zero to it, you won't be changing it. You need to add 5 zero pairs to be able to take positive 7 away. Then, you still have positive 2 take away positive 7 which leaves you with negative 5.
Using a Number Line
(-5)-(-2)= -3
A Different Way
(-2)-(-4)
=(-2)+(+4)
= +2
Change the operation to addition
Reverse the sign
Another way to figure out integers is by reversing the operation and the sign of the second number. By doing this, it is the equivalent of the other equation, but you are just looking at it, in a different perspective.
(+3)-(+8)
=(+3)+(-8)
= -5
Operation reversed
Sign reversed
I think that this way of figuring out subtraction is easier because addition seems to be an easier operation compared to subtraction. As well, once you have reversed the operations, you can figure out the answer to the question any way that you find the easiest, if that means using integer chips or using a number line.
Examples
(+7)x(+6)= +42
Both signs are the same
So the answer will be positive
(+3)x(-8)= -24
Both signs are different
So the answer will be negative
When looking at the equation, you can first identify the signs to see if they are the same or different so you know if the answer will be either positive or negative.
(-4)x(-9)= positive number
Both signs are the same
Even not knowing what the real answer might be, you know part of what it would be by looking at the signs and because the signs are both the same, the answer will be positive.
(-5)x(-2)x(+4)
(+10)x(+4)
= +40
To figure out if the answer to the equation is positive or negative at glance, is only effective when you are comparing two integers. When the equation contains more than two integers, you have to break down the equation until you are left with two integers to compare.
If say you are looking at the example above, you would notice that the signs in the equation are different. Then, you would think that the answer should be negative because the signs are different, but in fact, the answer is positive. But, once you break down the equation to 2 integers, you will see that the signs are both the same so the answer will be positive rather than negative.
(-3)x(-7)x(-6)
(+21)x(-6)
= -126
As well, the example to the side shows that you think that the answer would be positive because all the signs are negative and they are all the same. But, once it is broken down to 2 integers, you will see that it changed to different signs making the answer to the equation negative.
Like
signs=
Positive
answer
Unlike
signs=
Negative
answer
Yellow
= Positive
Red
= Negative
(+3)-(+6)= -3
When using exponents with integers, remember to simplify the exponents first before doing anything else. Once you have simplified the exponents, you can just add normally after that. If the equation contains more than one operation, then that is when "BEDMAS" comes in. It tells you the order you must do the equation in to be correct. Just remember that division and multiplication are technically in the same spot as well as addition and subtraction.
Rule: Keep, Change, Change
Subtracting Integers with Unlike Signs
When subtracting integers with unlike signs, it is important to remember that the answer could be either negative or positive.
Bigger
Positive
- Smaller
Positive
=
Positive
Answer
Smaller
Positive
- Bigger
Positive
=
Negative
Answer
Bigger
Negative
- Smaller
Negative
=
Negative
Answer
Smaller
Negative
- Bigger
Negative
=
Positive
Answer
e.g. (+4)-(+2)= +2
e.g. (+3)-(+5)= -2
e.g. (-8)-(-5)= -3
e.g. (-2)-(-6)= +4
(-3)-(+5)= -8
(-4)-(+1)= -5
(+6)-(-5)= +11
(+7)-(-9)= +16
Positive
Number -
Negative
Number=
Positive
Answer
Negative
Number -
Positive
Number=
Negative
Answer
Example:
Example:
Rule:
Dividing Integers
Dividing integers follows the same rules as multiplying integers.
Rule: Like signs=
Positive
Answer

Unlike Signs=
Negative
Answer
Once you have figured out what sign belongs to the answer, all you have to do is divide the answer regularly like you would do if there wasn't signs in front of the numbers.
Example: (-24)÷(+4)
= 24÷ 4
Because the signs are unlike, the answer will be negative
= 6
= -6
Integer Chips
(-6)-(+2)= -8
Because there weren't any positive chips to take away, you have to add zero pairs because you can't disrupt having 6 negative chips and by adding zero to it, you won't be. So if you need to take positive 2 away from negative 6, you need to put down 6 negative chips and then add 2 zero pairs because you need to take away 2 positive chips. So basically you add as many zero pairs (one positive and negative chip) until you are able to take away what you need. So once you take away 2 positive chips, you are left with negative 8 as your answer.
Using a Number Line
(-3)-(+4)= -7
= (-3)+(-4)= -7
(+7)-(-2)= +9
=(+7)+(+2)= +9
When using a number line with subtraction it is best to use when you are adding integers rather than when you are subtracting integers. So, if you convert the subtraction equation to an addition equation, then you can easily use the number line. So, change the operation from subtraction to addition, and change the sign of the second term to the opposite. Positive sign turns into a negative sign and vice versa.
Operation changes to addition
Sign changes to its opposite
Operation changes to addition
Sign changes to its opposite
A Different Way
When subtracting integers, you can do something different and convert the subtraction equation to an addition equation which is easier to figure out. You can do this by changing the operation sign to addition and then change the sign of the second term to its opposite. Negative sign changes to a positive sign. Positive sign changes to a negative sign. Then you keep the first term the same and don't change anything about it.
(+8)-(-6)= +14
=(+8)+(+6)
= +14
Change operation to addition
Change sign to negative
Keep it the same
Keep, Change, Change
And then, you have changed the equation so both signs are the same and you are now adding integers with like signs. You can now add the numbers with any other method like using a number line or integer chips.
When using a number line to figure out the equation. The direction of the arrow will vary. However when using a number line to figure out a subtraction equation is hard because you don't know what way to count most of the time. But, when using addition it is easier. So if you convert the subtraction equation to an addition equation using the "keep, change, change" rule that was mentioned earlier, it would make the number line so much easier because addition with a number line has rules on what way to count.
(+4)+(-5)= -1
The answer
*Remember that whenever you are adding a negative to a positive number, then you will be counting to the left. If you are adding a positive number to a negative number, then you will be counting to the right.
Negative
+
Positive
Positive
+
Negative
Count to the right
Count to the left
(+8)
2
Example:
+ (-6)
3
=(+64)+(-216)
= -152
Examples
(+7)
2
x(-3)
3
+(-5)
First Step: Exponents
= (+49)x(-27)+(-5)
Second Step: Multiplication
= (-1323)+(-5)
Third Step: Addition
= (-1328)
B
rackets
E
xponents
D
ivision
M

uliplication
A

ddition
S
ubtraction
*Remember to follow the order of operations ("BEDMAS") to get the right answer. Other rules in integers still apply when "BEDMAS" is occurring like multiplication rules and addition rules. Also, because exponents are multiplication, don't forget the rule of integers while doing them. Like signs=
positive
answer. Unlike signs=
negative
answer
Multiplication/Division
Examples
(+25)÷(-5)= (-5)
Both signs are different
So the answer is negative
(-35)÷(-7)= (+5)
Both signs are the same
So the answer is positive
When looking at a division equation, you are able to identify whether the answer is going to be positive or negative at first glance due to the signs in the equation. When looking at the first example, you can see that the signs are different which means the answer will be negative. While looking at the second example, you can see that both the signs are the same and that the answer will be positive due to that. So the easiest and one of the most important things to do first is identify the answer's sign before moving one to anything else.
After finding the answer's sign, all you have to do is divide regularly like there weren't signs in front of the numbers.
Rules
When you do exponents and the exponent is an even number, the answer is always going to be positive.
(-8)
2
x(-3)
= (+64)x(-3)
= -192
(-3)
4
x(+2)
= (+81)x(+2)
= +162
When you do exponents and the exponent is an odd number, the answer is always going to be negative.
(+4)
3
x(-9)
= (-256)x(-9)
= (+2304)
(-5)
5
x(-1)
= (-3125)x(-1)
= (+3125)
This is because when you do exponents, you have to include the rule of multiplying back into it. The rule that you use in multiplication is when the integers in the equation are the same, then the answer is positive. When the integers in the equation are different, then the answer is negative.
But, when you do integers and there are more than 2 terms, that's when the rule of multiplication and the having an odd and even number as an exponent takes place because you can't use the rule with more than 2 terms. You have to break the equation down to 2 terms to be able to compare them and determine the sign's equation.
(-6)
3
x(+4)
(-6)x(-6)x(-6)
= (+36)x(-6)
= -1296
The answer will always be negative because the signs in the end, when there are two terms will be different every time.
(+5)
2
x(-3)
(+5)x(+5)
= +25
The answer will always be positive because there will always be an even amount of the same integer to even each other out which means the sign at the end of all the breaking down will be the same as each other
The sign's answer will vary whenever there are more than 2 terms in the equation. At that point, you can no longer identify the answer's sign at glance. You can only identify it with 2 terms and when there are more than 2 terms, you need to break it down and figure it out until there are 2 terms left.
(-64)÷(+4)÷(-8)
-16
= (-16)÷(-8)
= (+2)
The answer seems to be negative knowing that all the signs are unlike.
However, whenever you break it down, and use the rule with 2 terms, you will soon find out that they are like signs.
Now, you can compare the signs to find out whether it is positive or negative and because the signs are like, the answer will be positive.
Integers
: whole numbers that are either positive or negative
1.
2.
1.
2.
Integer Chips
Arrays
e.g. (+3)x(+3)
Arrays can only be used when the 2 terms are positive if you need the the actual representation of the data.
What is an array?
An array is an orderly arrangement of data. Usually shown in rows or columns.
How can I use an array in multiplication?
This means that you are going to need 3 groups of 3 things.
When representing this in a visual, you first need to make 3 groups and make sure that each group has 3 chips inside of it. Then, you can count how many chips are in each group to figure out the multiplication equation.
* Remember, you can't use arrays when the 2 terms have unlike signs and when there are 2 like negative signs and get the right answer for the sign. However, you can use arrays just to figure out the multiplication equation without the signs there to interfere with it.
e.g. (-4)x(+4)
*Because the signs are unlike, I know that the answer will be negative.
You need 4 groups of 4
= 1
You can now count how many chips are in each group and use the rule about multiplying integers to figure out the sign of the answer.
Put 2 zero pairs to be able to take away 2 positive chips
Integer Chips
Array
While using an array to represent a division equation, remember that to be able to have an exact representation with the correct signs for the chips, can only be used when both terms are positive.
e.g. (+18)÷(+6)
One of the first things you should do is take out 18 chips. Then, you should start dividing the chips into 6 equal groups. When you have separated them into 6 equal groups, you need to count how many chips are in each group which is the answer to the equation.
*Remember
When dividing numbers with unlike signs and when the 2 terms are negative, can't be represented using arrays. However, it can be represented to figure out the division statement alone without the signs.
e.g. (+15)÷(-3)
Remember to still use the rule of division so you know what the sign of the answer should be. So because both signs are unlike, the answer will be negative.
= 1
Now, you must take out 15 chips and divide the chips into 3 equal groups. Once you have divided the chips in 3 groups, making sure that all the groups have the same amount of chips in it. You need to count the amount of chips in the group to figure out the answer to the problem. Don't forget to add the sign to the answer with the division rule.
(+12)+(+3)= +15
(-9)+(-5)= -14
Positive
+
Positive
=
Positive
Negative
+
Negative
=
Negative
*Whenever an equation doesn't show an order of operation, it means that it is an multiplication statement.
e.g. (+5)(-7)= -35
T
h
e

E
n
d

Tips
Sometimes, whenever a number doesn't have a sign in front of it, it will be a positive a number.
e.g. 6-(-4)x3
2
When ever there are actual brackets in the equation and it is not to organize the equation, they would be square brackets and not round brackets.
e.g. (-9)x[(+4)+(-1)]
(-3)
These brackets mean the brackets in "BEDMAS" so you will be doing the equation in the brackets first before anything else.
= (-9)x(-3)
= (-27)
As these brackets don't group an equation together, they are there to help classify negative numbers because the subtraction sign is the same as the negative sign. The positive sign is the same as the addition sign.
3 to the power of 4 equals
3
4
= 3x3x3x3
Exponents are multiplication. It's the number multiplied by it self the amount of times the exponent says to.
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