Then, around 1st grade, you learned about zero, and you started calling them whole numbers.

0, 1, 2, 3, 0, 1, 2, 3, ……

The first kind of rational number you learned was called a natural number.

1, 2, 3, 1, 2, 3, ……

Around 4th grade, you start hearing about

negative numbers, and you start calling them

integersintegers.

-2, -1, 0, 1, -2, -1, 0, 1, 2…2…

If a number doesn’t fall into any of those categories – but you can write it as a fraction, it’s just called a

rational numberrational number

If you can’t

write the number

as a fraction, it’s an

irrational number.

These are all the real numbers.

4.561…4.561…

ππ18

CLASSIFYING RATIONAL NUMBERSCLASSIFYING RATIONAL NUMBERS

N , W , Z , QN , W , Z , Q

QQ

QQ

Z , QZ , Q

Z , QZ , Q

QQ

W , Z , QW , Z , Q

QQ

N , W , Z , QN , W , Z , Q

N , W , Z , QN , W , Z , Q

QQ

QQ

N , W , Z , QN , W , Z , Q

QQ

Z , QZ , Q

irrationalrational

integer

whole

natural

irra

tional

rational

integer

whole

natural

200

0-

1.08-

11200

0-

1.08-

11200

0-

1.08-

11200

0-

1.08-

11

irrationalrational

integer

whole

natural

1 π-

871 π-

871 π

-87

1 π-

87

irrationalrational

integer

whole

natural

1

4 14471

4 14471

4 14471

4 1447

irrationalrational

integer

whole

natural

ABSOLUTE VALUEABSOLUTE VALUE

http://vn2.me/AQKD

For each value, write it opposite, then its absolute value.

Watch this:

opposite absolute value

1) -3 +3 3

2) 4 -4 4

3) 15 -15 15

4) -7 +7 7

5) 0 0 0

6) 1 -1 1

7) -1 +1 1

8) -50 +50 50

9) 10 -10 10

10)-2.5 +2.5 2.5

ABSOLUTE VALUEABSOLUTE VALUE

1717 – – ││14│14│

––14 14

-│7│-│7│

– – 7 7

16 - │ 10 │16 - │ 10 │ 16 - 1016 - 10 66

– – 15 + 1615 + 16 ++11 11

88

4 – 194 – 19 ––1515

16 – 516 – 5 1111

– – ││2│+ 192│+ 19 – – 2 + 192 + 19 1717

Integer Addition - Number LineInteger Addition - Number Line

1. 1. ––33 + + 99It’s an addition It’s an addition

problem.problem.

The first integer, The first integer, -3-3, is the , is the location,location,

The second The second integer, integer, 99, is the , is the

movement.movement.

-3-3 + + 9 9 = 6= 6

2. 2. 22 ++ ––77It’s an addition It’s an addition

problem. problem.

The first integer, The first integer, 22, is the location,, is the location,

The second The second integer, integer, ––77, is the , is the

movement.movement.

22 + + ––7 7 = -= -55

3. 3. ––66 ++ ––44

= -10= -10

It’s an addition It’s an addition problem. problem.

The first integer, The first integer, -6-6, is the , is the location,location,

The second The second integer, integer, ––44, is the , is the

movement.movement.

-6-6 ++ ––44

1.1. ––33 + + 99 9 9 –– 33 66 ++66

2.2. 55 + + ––88 8 8 –– 55 33 ––33

3.3. ––2 2 + + ––66 2 2 ++ 6 6 88 ––88

1. Signs are DIFFERENT.2. SUBTRACT the absolute values.3. Write down that number. 4. The answer is POSITIVE because the bigger absolute value (9) is positive.

1. Signs are DIFFERENT.2. SUBTRACT the absolute values.3. Write down that number. 4. The answer is NEGATIVE because the bigger absolute value (-8) is negative.

1. Signs are SAME.2. ADD the absolute values.3. Write down that number. 4. The answer is NEGATIVE because both integers are negative.

Integer Addition - The RulesInteger Addition - The Rules

Integer Addition - PracticeInteger Addition - Practice

= = ––34 = 34 = ––15 = 15 = ––31 = 31 = ––26 = 26 = ––2828

= 1 = 6 = = 1 = 6 = ––7 = 9 = 7 = 9 = ––88

= = ––7 = 14 = 7 = 14 = ––12 = 25 = 12 = 25 = ––3131

= = ––11 11 = 6 = 6 = 65 = 65 ==––8080 = 41= 41

1. 1. 33 – 9 – 9It’s a It’s a

subtraction subtraction problem.problem.

The first integer, The first integer, ++33, is the , is the location,location,

...the second ...the second integer, integer, ++9, 9,

meansmeansmove 9 spaces.move 9 spaces.

33 – 9 – 9 = -6= -6

2. 2. ––22 –– ––77

––22 –– ––7 7 = 5= 5

3. 3. ––66 –– 33

= -9= -9––66 –– 33

Integer Subtraction - Number LineInteger Subtraction - Number Line

The subtraction The subtraction signsign

means movemeans moveto the left...to the left...

It’s a It’s a subtraction subtraction

problem.problem.The first integer, The first integer,

––22, is the , is the location,location,

The subtraction The subtraction signsign

means movemeans moveto the left...to the left......but, the ...but, the

second integer, second integer, ––

7, means7, meansreversereverse

direction, then direction, then move move 77 spaces. spaces.

It’s a It’s a subtraction subtraction

problem.problem.The first integer, The first integer,

––66, is the , is the location,location,

The subtraction The subtraction signsign

means movemeans moveto the left...to the left...

...the second ...the second integer, integer, ++3, 3,

meansmeansmove 3 spaces.move 3 spaces.

1.1. 4 – 104 – 10 44 + + ––1010 10 – 10 – 44 66 ––66

2.2. –– 7 – 17 – 1 – – 77 + + ––11

7 7 ++ 1 1 88 ––88

3.3. –– 1 – 1 – ––33 – – 11 + + ++33

33 –– 1 1 22 ++22

1. Change the subtraction to addition, then...

2. ...change the sign of the 2nd integer.3. Signs are DIFFERENT.4. SUBTRACT the absolute values.5. Write down that number. 6. The answer is NEGATIVE because the

bigger absolute value (––1010) is negative.

1. Change the subtraction to addition, then...

2. ...change the sign of the 2nd integer.3. Signs are SAME.4. ADD the absolute values.5. Write down that number. 6. The answer is NEGATIVE because

both integers are negative.

1. Change the subtraction to addition, then...

2. ...change the sign of the 2nd integer.3. Signs are DIFFERENT.4. SUBTRACT the absolute values.5. Write down that number. 6. The answer is POSITIVE because the

bigger absolute value (++33) is positive.

Integer Subtraction - The RulesInteger Subtraction - The Rules

Integer Subtraction - PracticeInteger Subtraction - Practice

= = ––85 = 85 = ––54 = 54 = ––65 = 65 = ––56 56 = = ––5959

= = – – 15 = 15 = – – 84 = 79 = 4 84 = 79 = 4 = = ––3737

= 60 = 93 = 98 = 97 = 60 = 93 = 98 = 97 = 36= 36

– – 16 – ( 16 – ( – –

95)95)

– – 5 – ( 5 – ( – –

9)9)

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

Integer MultiplicationInteger MultiplicationWatch this:

http://nlvm.usu.edu/en/nav/frames_asid_322_g_1_t_1.html?from=topic_t_1.html

1. –7(8) =

Why is this multiplication?There’s a number smashed next to

parenthesis.

When you multiply or divide integers, it’s easy:

Step 1: Multiply (or divide) the absolute values.

7 • 8

5656

Step 2: Now, look the signs:

• If they match...

it’s positive

• If they’re different...

it’s negative––

Integer DivisionInteger Division

2. 9

90

Why is this division? Fractions are

division.When you multiply or divide integers, it’s easy:

Step 1: Multiply (or divide) the absolute values.Step 2: Now, look the signs:• If they match...it’s positive

• If they’re different...it’s negative

= +10or

10

Integer Multiplication and Division - Integer Multiplication and Division - PracticePractice

= = ––48 = 48 = ––50 50 = 143 = 143

= 168 = = 168 = ––105 105 = = – – 2424

= = – – 70 = 70 70 = 70 = = – – 9696

=288 = 24 =288 = 24 =– 200=– 200

= = – – 21 = 21 = – – 11 11 = 17 = 17

= = – – 5 = 5 = – – 20 20 = 13 = 13

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

Integers OperationsIntegers Operations

44

--5050-8-8

11

-7-7

--1010--125125-3-3

-3-3

9696

11

-1-1

44

-9-9

-4-4

77

-5-5

33

-1-1

-3-3

Integers OperationsIntegers Operations

4. Samantha has $688.52 in her checking account. If she makes a withdrawal of $127.78, what will be the new balance?

5. Jared and Natalie were playing basketball. After playing for a long time, Jared was losing by six points. Then he scored ten points in a row, but Natalie scored the next five points. Then they had to stop playing. Who won and by how many points?

3. The temperature at 10 P.M. one evening was 7oC. At 4 A.M. the next morning the temperature was - 3oC . What was the change in temperature from 10 P.M to 4 A.M.?

2. Lina started the week with a checking account balance of $496. During the week, she wrote a check in the amount of $58.50, another check in the amount of $147.29, and then made a deposit in the amount of $180.00. What was her checkbook balance after this deposit?

6. Gertrude plays bingo every Tuesday. In the last 5 weeks, she has lost $7, lost $4, won $8, lost $9, and won $2. What was her average weekly gain or loss?

1. The change in elevation from the top to the bottom of the Grand Canyon is -1.83 km. A tour guide hikes down to the bottom every day for a week, but rides an ATV back up. For the week, what is the total change in elevation that he hiked?hiked?

-12.81 -12.81 kmkm

$470.2$470.211

--1010ooCC

$560.74$560.74

Natalie won by a Natalie won by a point.point.

$-2 or $2 $-2 or $2 lossloss