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Everything about Integers. Interesting Integers!. We can represent integers using red and yellow counters. Red tiles will represent negative integers, and yellow tiles will represent positive integers. Positive integer. Negative integer. Definition.

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Interesting integers
Interesting Integers!

  • We can represent integers using red and yellow counters.

  • Red tiles will represent negative integers, and yellow tiles will represent positive integers.

Positive integer

Negative integer


Definition
Definition

  • Positive number – a number greater than zero.

0

1

2

3

4

5

6


Definition1
Definition

  • Negative number – a number less than zero.

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6


Definition2
Definition

  • Opposite Numbers – numbers that are the same distance from zero in the opposite direction

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6


Definition3

7

-7

opposite

Definition

  • Integers – Integers are all the whole numbers, their opposites and zero.


What are Opposites?

  • Two integers the same distance from the origin, but on different sides of zero

  • Every positive integer has a negative integer an equal distance from the origin

  • Example: The opposite of 6 is -6

    • The opposite of -2 is 2


Interesting integers1
Interesting Integers!

  • If there are the same number of red tiles as yellow tiles, what number is represented?

    zero pair

  • It represents 0.


Definition4
Definition

  • Absolute Value – The distance a number is from zero.

  • Distance is always positive therefore, absolute value is always positive.

The absolute value of

9 or of –9 is 9.


What is Absolute Value?

  • Distance a number is from zero on a number line (always a positive number)

  • Indicated by two vertical lines | |

  • Every number has an absolute value

  • Opposites have the same absolute values since they are the same distance from zero

  • Example: |-8| = 8 and |8| = 8

  • |50| = 50 and |-50| = 50


Examples

7 =

7

10 =

10

-100 =

100

 - 8 =

8

y

 y =

w

 - w =


|7| – |-2| = ?

  • -9

  • -5

  • 5

  • 9


|4 + (-3)| = ?

  • -1

  • 1

  • 7

  • Purple


Greater than (>)

  • Any number that is to right of another

  • number on the number line.

6

Six is greater than negative one because six is at the right of negative one on the number line

>

-1

6

5

4

3

2

1

0

-1

-2


Less than (<)

  • Any number that is to the left of another

  • number on the number line.

Negative two is less than one because negative two is at the left of one on the number line

1

<

-2

6

5

4

3

2

1

0

-1

-2


Practice

Place the symbol(<,>,=) to make

the sentence true:

-9____8

-3____-3

0____-1

-12____12

-5____-7

<

=

>

<

>




Negative numbers are used to show debt
Negative Numbers Are Used to Show Debt

Let’s say your parents bought a car but

had to get a loan from the bank for $5,000.

When counting all their money they add

in -$5.000 to show they still owe the bank.


Hint

  • If you don’t see a negative or positive sign in front of a number it is positive.

9

+


Addition and subtraction

Interesting Integers!

ADDITION AND SUBTRACTION


Adding integers
ADDING INTEGERS

  • We can model integer addition with tiles.

  • Represent -2 with the fewest number of tiles

  • Represent +5 with the fewest number of tiles.


Adding integers1
ADDING INTEGERS

  • What number is represented by combining the 2 groups of tiles?

  • Write the number sentence that is illustrated.

    -2 + +5 = +3

+3


Adding integers2
ADDING INTEGERS

  • Use your red and yellow tiles to find each sum.

    -2 + -3 = ?

ANSWER

= -5

+

-2 + -3 = -5


Adding integers3
ADDING INTEGERS

ANSWER

-6 + +2 = ?

+

- 4

=

-6 + +2 = -4

-3 + +4 = ?

ANSWER

+1

=

+

-3 + +4 = +1


Solve the problems
Solve the Problems

-8

  • -3 + -5 =

  • 4 + 7 =

  • (+3) + (+4) =

  • -6 + -7 =

  • 5 + 9 =

  • -9 + -9 =

11

7

-13

14

-18


Solve these problems
Solve These Problems

-2

  • 3 + -5 =

  • -4 + 7 =

  • (+3) + (-4) =

  • -6 + 7 =

  • 5 + -9 =

  • -9 + 9 =

3

-1

1

-4

0


Adding Integers

Number line

T-Method


Adding Integers

using the number line

- 3 + 2 =

-1

Negative three means

move 3 places to the left

Positive two means

move 2 places to the right

1

2

0

-3

-2

-1

The answer is:

-1


Adding Integers

using the number line

8 - 3 =

5

Positive eight means

move 8 places to the right

Negative three means

move 3 places to the left

4

5

6

3

1

7

8

2

8

0

7

5

6

1

2

3

The answer is:

5


Adding Integers

T- Method

T-Method


Use your red and yellow tiles to find each sum.

2 + 5 = ?

-

+

2

+

=

5

7

T-Method


Use your red and yellow tiles to find each sum.

-2 + (-5) = ?

-

+

2

+

=

5

7

T-Method


Use your red and yellow tiles to find each sum.

2 + (-5) = ?

+

-

5

2

2

+

=

3

T-Method


Use your red and yellow tiles to find each sum.

-2 + 5 = ?

-

+

2

2

5

+

=

3

T-Method


Subtracting Integers and

T- Method

T-Method


Subtracting integers
SUBTRACTING INTEGERS

  • We often think of subtraction as a “take away” operation.

  • Which diagram could be used to compute

    +3 - +5 = ?

+3

+3


Subtracting integers1
SUBTRACTING INTEGERS

  • This diagram also represents +3, and we can take away +5.

  • When we take 5 yellow tiles away, we have 2 red tiles left.

We can’t take away 5 yellow tiles from this diagram. There is not enough tiles to take away!!


Subtracting integers2
SUBTRACTING INTEGERS

  • Use your red and yellow tiles to model each subtraction problem.

-2 - -4 = ?

ANSWER


Subtracting integers3
SUBTRACTING INTEGERS

Now you can take away 4 red tiles.

2 yellow tiles are left, so the answer is…

This representation of -2 doesn’t have enough tiles to take away -4.

Now if you add 2 more reds tiles and 2 more yellow tiles (adding zero) you would have a total of 4 red tiles and the tiles still represent -2.

-2 - -4 = +2


Subtracting integers4
SUBTRACTING INTEGERS

Work this problem.

+3 - -5 = ?

ANSWER


Subtracting integers5
SUBTRACTING INTEGERS

  • Add enough red and yellow pairs so you can take away 5 red tiles.

  • Take away 5 red tiles, you have 8 yellow tiles left.

+3 - -5 = +8


Subtracting integers6
SUBTRACTING INTEGERS

Work this problem.

-3 - +2 = ?

ANSWER


Subtracting integers7
SUBTRACTING INTEGERS

  • Add two pairs of red and yellow tiles so you can take away 2 yellow tiles.

  • Take away 2 yellow tiles, you have 5 red tiles left.

-3 - +2 = -5


SUBTRACTING INTEGERS

  • -3 – ( -5) =

  • 4 - 7 =

  • (+3) - (+4) =

  • -6 – (-7) =

  • 5 - 9 =

  • -9 – (-9) =

2

-3

-1

1

-4

0


SUBTRACTING INTEGERS

  • 3 – (-5) =

  • -4 - 7 =

  • (+3) - (-4) =

  • -6 - 7 =

  • 5 – (-9)=

  • -9 - 9 =

8

-11

7

13

14

-18


In order to subtract two integers, you would need to re-write the problem as an addition problem.

Keep in mind that subtract means add the opposite.

-

-

2

2

(-5)

-5

=

=

5

+

T-Method


SUBTRACTING INTEGERS re-write the problem as an addition problem.

  • -8 – ( -2) =

  • 5- 4 =

  • (+7) - (+2) =

  • -8 – (-7) =

  • 8- 5 =

  • -10 – (-2) =

6

1

5

-1

3

-8


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