Transcript Lesson 2 Rational and Irrational Numbers Notes

Lesson 2 Operations with
Rational and Irrational Numbers
NCSCOS Obj.: 1.01; 1.02
Objective
TLW State the coordinate of a point on a
number line
TLW Graph integers on a number line.
TLW Add and Subtract Rational numbers.
TLW Multiply and Divide Rational numbers
The Number Line
-5
0
5
Integers = {…, -2, -1, 0, 1, 2, …}
Whole Numbers = {0, 1, 2, …}
Natural Numbers = {1, 2, 3, …}
Addition Rule
1) When the signs are the same,
ADD and keep the sign.
(-2) + (-4) = -6
2) When the signs are different,
SUBTRACT and use the sign of the
larger number.
(-2) + 4 = 2
2 + (-4) = -2
Karaoke Time!
Addition Rule: Sung to the tune of
“Row, row, row, your boat”
Same signs add and keep,
different signs subtract,
keep the sign of the higher number,
then it will be exact!
-1 + 3 = ?
1.
2.
3.
4.
-4
-2
2
4
Answer Now
-6 + (-3) = ?
1.
2.
3.
4.
-9
-3
3
9
Answer Now
The additive inverses (or opposites)
of two numbers add to equal zero.
Example: The additive inverse of 3 is
-3
Proof: 3 + (-3) = 0
We will use the additive inverses for
subtraction problems.
What’s the difference between
7 - 3 and 7 + (-3) ?
7 - 3 = 4 and 7 + (-3) = 4
The only difference is that 7 - 3 is a
subtraction problem and 7 + (-3) is
an addition problem.
“SUBTRACTING IS THE SAME
AS ADDING THE OPPOSITE.”
(Keep-change-change)
When subtracting, change the subtraction to
adding the opposite (keep-change-change)
and then follow your addition rule.
Example #1:
- 4 - (-7)
- 4 + (+7)
Diff. Signs --> Subtract and use larger sign.
3
Example #2:
-3-7
- 3 + (-7)
Same Signs --> Add and keep the sign.
-10
Okay, here’s one with a variable!
Example #3: 11b - (-2b)
11b + (+2b)
Same Signs --> Add and keep the sign.
13b
1.
2.
3.
4.
Which is equivalent to
-12 – (-3)?
12 + 3
-12 + 3
-12 - 3
12 - 3
Answer Now
7 – (-2) = ?
1.
2.
3.
4.
-9
-5
5
9
Answer Now
Review
1) If the problem is addition, follow
your addition rule.
2) If the problem is subtraction, change
subtraction to adding the opposite
(keep-change-change) and then follow
the addition rule.
Absolute Value
of a number is the distance from
zero.
Distance can NEVER be negative!
The symbol is |a|, where a is any
number.
Examples
7 = 7
10 = 10
-100 = 100
5 - 8 = -3= 3
|7| – |-2| = ?
1.
2.
3.
4.
-9
-5
5
9
Answer Now
|-4 – (-3)| = ?
1.
2.
3.
4.
-1
1
7
Purple
Answer Now
Find the sum.
1) -2.304 + (-0.26)
Line up the decimals and add (same signs).
-2.564
5 3
2) 
8  4 
5 6 

8  8 
Get a common denominator
and subtract.
1

8
Find the difference.
3)
5  3 
 
9  5 
Change subtraction to
adding the opposite.
5  3 
 
9  5
Get a common denominator.
25 27 

45 45 
Subtract and keep sign of
the larger number.
2
45
Find the difference.
1 6
4) 
2 7
1  6 
 
2  7 
7 12 

14  14 
5
14
Change subtraction to
adding the opposite.
Get a common
denominator and subtract.
5) Solve 6.32 – y if y = -3.42
Substitute for y: 6.32 - (-3.42)
6.32 + 3.42
9.74
1.
2.
3.
4.
-3.2
-2.8
2.8
3.2
Find the solution
6.5 – 9.3 = ?
Answer Now
Find the solution
1.
2.
3.
4.
5
.
12
1
7
3
7
11
12
2  1 
   ?
3  4 
Answer Now
A rational number is a number
that can be written as a fraction.
How can these be written as a fraction?
3
3=
1
11
3
2  
4
4
Inequality Symbols
Symbol
<
>
≠
≤
≥
=
Meaning
is less than
is greater than
is not equal to
is less than or equal to
is greater than or equal to
is equal to
Ordering Rational Numbers
2 ways to order from least to greatest
1. Get a common denominator
2. Change the fractions to decimals
(numerator  demoninator)
Which rational number is bigger?
4
3
or
11 8
1) Get a common denominator.
32 33

88 88
2) or convert the fraction to a decimal.
0.363 < 0.375
4
3
<
11 8
Which rational number is bigger?
7 11
or
4 6
1) Get a common denominator
42 44
<
24 24
2) or convert to a decimal
1.75 < 1.83
7 11

4 6
Which symbol makes this true?
5
3
__
7
4
1. <
2. >
3. =
Answer Now
Which symbol makes this true?
-2
-1
__
4
9
1. <
2. >
3. =
Answer Now
Multiplying Rules
1) If the numbers have the same
signs then the product is positive.
(-7) • (-4) = 28
2) If the numbers have different
signs then the product is negative.
(-7) • 4 = -28
Examples
1) (3x)(-8y)
-24xy
2)
4 

20 
5
Write both numbers as a fraction.
Cross-cancel if possible.
 20 4 
 1 5 
Multiplying fractions:
80

5
top # • top #
= -16
Bottom # • bottom#
When multiplying two negative
numbers, the product is negative.
1. True
2. False
Answer Now
When multiplying a negative number
and a positive number, use the sign of
the larger number.
1. True
2. False
Answer Now
2
5




3) 

 5  8 
10
=
40
=
1
4
Multiply: (-3)(4)(-2)(-3)
1.
2.
3.
4.
72
-72
36
-36
Answer Now
an easy way to determine the sign of the answer
When you have an odd number of negatives,
the answer is negative.
When you have an even number of negatives,
the answer is positive.
4) (-2)(-8)(3)(-10)
Do you have an even or odd number of
negative signs?
3 negative signs -> Odd -> answer is negative
-480
Last one!
1
2



2
5)  65
 2 
 5 
Positive or negative answer?
Positive - even # of negative signs (4)
Write all numbers as fractions and multiply.
 1  6 52  2 
 2  1 1  5  1 
120
10
=12
What is the sign of the product of
(-3)(-4)(-5)(0)(-1)(-6)(-91)?
1.
2.
3.
4.
Positive
Negative
Zero
Huh?
Answer Now
Dividing Rules
1) If the numbers have the same
signs then the quotient is positive.
-32 ÷ (-8 )= 4
2) If the numbers have different
signs then the quotient is negative.
81 ÷ (-9) = -9
When dividing two negative
numbers, the quotient is positive.
1. True
2. False
Answer Now
When dividing a negative number and
a positive number, use the sign of the
larger number.
1. True
2. False
Answer Now
a
b
The reciprocal of
is
b
a
where a and b  0.
The reciprocal of a number is called its
multiplicative inverse.
A number multiplied by its
reciprocal/multiplicative inverse is
ALWAYS equal to 1.
Example #1
7
2
The reciprocal of
is
.
7
2
2 7
1
 
1
7 2
1
Example #2

1
The reciprocal of -3 is  .
3
1
3
1
1

3

1
1
Basically, you are flipping the fraction!
We will use the multiplicative inverses
for dividing fractions.
Which statement is false about
reciprocals?
1.
2.
3.
4.
Reciprocals are also called
additive inverses
A number and its reciprocal
have same signs
If you flip a number, you get
the reciprocal
The product of a number and
its reciprocal is 1
Answer Now
Examples
1)  3  5
4
8
When dividing fractions, change
division to multiplying by the reciprocal.
3
8
24



20
4
5
6

5
3 
2)  1  
 
5
 10 
1  10 

 

5  3 
10

15
2

3
1.
2.
3.
4.
18
-18
7
-7
What is the quotient of
-21 ÷ -3?
Answer Now
1 10

?
5
3
1.
2.
3.
4.
2
3
2
3
3
50
3
50 Answer Now
.
.
.
.