7th Grade Math and Pre-AP Math Integers 7th Grade Math and Pre-AP Math

Integers can be positive (+) or negative (-). What is an Integer? An integer is any whole number or its opposite. Integers are part of a group of numbers called “real numbers.” Integers can be positive (+) or negative (-). -3, 1, 104, -76 : all these are integers!

OK – so what’s an opposite? An opposite has the same absolute value but a different sign. The opposite of 5 is -5. The opposite of -7 is 7.

OK – so what’s absolute value? Absolute value tells how far a number is from zero. It tells how “big” the number is without considering the sign.

Absolute Value Absolute value is always positive. The symbol for absolute value is two vertical lines around a number.

Absolute Value |-5| = 5 |8| = 8 |-13| = 13 (Read: The absolute value of -5 is 5.) (Read: The absolute value of 8 is 8.) (Read: The absolute value of -13 is 13.)

IMPORTANT!!! If a number does not have a sign, it is considered POSITIVE. A number with a plus sign in front of it is also considered positive.

Are all numbers integers? NO!! Only whole numbers or the opposite of whole numbers (negatives). What can’t be an integer? Numbers that have fractions or numbers after a decimal are not integers.

Are all numbers integers? These numbers are not integers: 2/3 1.05 5 7/8 2.5 7/9 -- But they are still “real numbers.” You can add, subtract, multiply & divide them, but they are not integers.

Putting Integers in Order Let’s look at a number line. -6 -4 -2 0 2 4 6 Numbers get bigger (more positive) as you go to the right They get smaller (more negative) as you go to the left.

Why do we even need negative numbers? Negative numbers give us a way to talk about how much less than zero something is.

Why do we even need negative numbers? Like on Jeopardy when somebody has no money and then they lose $500. Their score is -500. Or like when it gets really cold outside, negative numbers tell us how far below zero it is.

Other Ways to Use Negatives Direction (positive for east, negative for west) Yardage lost in football Money (negative money is a debt) Altitude (negative altitude is below sea level) Time (negative time is before an event, positive time is after)

Using < and > Sometimes we need to use the “greater than” and “less than” symbols to compare two numbers. If you read the symbols from left to right, it’s easy to remember which symbol is which.

Using < and > If you come to the open end first, say “greater than.” If you come to the pointed end first, say “less than.”

Using < and > Practice with these: 8 > 3 -9 < -7 8 > 3 -9 < -7 3 < 8 |4| <|-6| -7 > -9 |-22|>|18|

Adding Integers When two numbers have the same sign, the sum will also have the same sign. Positive + Positive = Positive Negative + Negative = Negative

Adding Integers 8 + 3 = 11 (or +8 + +3 = +11) -7 + -10 = -17 3 + 19 = 22 -2 + -3 = -5

Adding Integers Adding integers with different signs is a little more complicated. You have to figure out which number has the greatest absolute value. (That just means which number is farther from zero.)

Adding Integers To add two numbers with different signs, find their difference (subtract) and use the sign of the number that has the largest absolute value.

Adding Integers -10 + 3 = -7 15 + -4 = 11 8 + -13 = -5

Subtracting Integers Instead of subtracting, we want to always talk about adding. We are going to change all our subtraction problems to addition.

Subtracting Integers In any subtraction problem: Change the minus to a plus and change the sign of the second number. (We could also just say “add the opposite.”)

Subtracting Integers 9 - +3 = 9 + -3 = 6 -15 - +6 = -15 + -6 = -21 7 - + 12 = 7 + -12 = -5 -6 - -8 = -6 + 8 = 2 13 - -4 = 13 + 4 = 17

Multiplying & Dividing Integers There are only two kinds of multiplying & dividing problems: Integers with the same sign Integers with different signs

Multiplying & Dividing Integers You have been multiplying & dividing numbers with the same sign for a long time! Multiplying or dividing two numbers with the same sign always makes a . . .

POSITIVE!!!

Multiplying & Dividing Integers 3 x 8 = 24 -3 x -8 = 24 50 / 2 = 25 -50 / -2 = 25

Multiplying & Dividing Integers There is only one possibility left for multiplying & dividing numbers with different signs. Multiplying & dividing numbers with different signs always makes a . . .

NEGATIVE!!!

Multiplying & Dividing Integers 45 / -9 = -5 -12 x 5 = -60 -5 x 9 = -45 60 / -5 = -12