integers 7 th grade math and pre-ap math. what is an integer? an integer is any whole number or its...

IntegersIntegers77thth Grade Math Grade Math

and Pre-AP Mathand Pre-AP Math

What is an Integer?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 OK – so what’s an opposite?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’sOK – so what’sabsolute value?absolute value?

Absolute value tells how far a number is

from zero.

It tells how “big” the number is without

considering the sign.

Absolute ValueAbsolute Value

Absolute value is always positive.always positive.

The symbol for absolute value is two vertical lines

around a number.

Absolute ValueAbsolute 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!!!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 Are all numbers integers?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.

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.

Are all numbers Are all numbers integers?integers?

Putting Integers in OrderPutting 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.

Negative numbers

give us a way to talk about

how much less than

zero something is.

Why do we even need Why do we even need negative numbers?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.

Why do we even need Why do we even need negative numbers?negative numbers?

Other Ways to Use NegativesOther 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 >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 >Using < and >

If you come to the open end first, say “greater than.”

If you come to the pointed end first, say “less than.”

Practice with these:

8 > 3 -9 < -7

3 < 8 |4| <|-6|

-7 > -9 |-22|>|18|

Using < and >Using < and >

Adding IntegersAdding IntegersWhen two numbers have the same sign, the sum will also have the same sign.

Positive + Positive = Positive

Negative + Negative = Negative

Adding IntegersAdding Integers

8 + 3 = 11 (or +8 + +3 = +11)

-7 + -10 = -17

3 + 19 = 22

-2 + -3 = -5

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 IntegersAdding Integers

Adding IntegersAdding IntegersTo add two numbers with different signs, find their difference (subtract) and

use the sign of the number that has the largest

absolute value.

Adding IntegersAdding Integers

-10 + 3 = -7

15 + -4 = 11

8 + -13 = -5

Subtracting IntegersSubtracting Integers

Instead of subtracting, we want to always talk about

adding. We are going to change all

our subtraction problems to addition.

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 IntegersSubtracting Integers

Subtracting IntegersSubtracting Integers9 - +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 Multiplying & Dividing IntegersIntegers

There are only two kinds of multiplying & dividing problems:

• Integers with the same sign• Integers with different

signs

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 . . .

Multiplying & Dividing Multiplying & Dividing IntegersIntegers

Multiplying & Dividing Multiplying & Dividing IntegersIntegers

3 x 8 = 24

-3 x -8 = 24

50 / 2 = 25

-50 / -2 = 25

There is only one possibility left for multiplying & dividing numbers with different signs.

Multiplying & dividing numbers with different signs always makes a . . .

Multiplying & Dividing Multiplying & Dividing IntegersIntegers

Multiplying & Dividing Multiplying & Dividing IntegersIntegers

45 / -9 = -5

-12 x 5 = -60

-5 x 9 = -45

60 / -5 = -12