Download Grade 9 Mathematics Unit #1 – Number Sense Sub

Grade 9 Mathematics
Unit #1 – Number Sense
Sub-Unit #1 – Rational Numbers
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“I Can”
 Create a number line to order integers
 Identify integers
 Add integers
Subtracting Integers
 Subtract Integers
Multiplication & Division  Multiply Integers
with Integers
 Divide Integers
Order of Operations with  Solve problems involving order of operations
Integers
Identifying and
 Identify Rational Numbers
Representing the
 Identify Irrational Numbers
Number Systems
 Relate Whole Numbers, Rational Numbers and Integers
Describe a strategy that can create an irrational number
from a rational number
Ordering, Adding and
 Create a number line to place decimals in order
Subtracting Decimals
 Determine a decimal number between two given
numbers
 Add decimals
 Subtract decimals
Multiplication and
 Multiply decimals
Division of Decimals
 Divide decimals
Ordering Fractions
 Create a number line to order fractions
 Determine a fraction between two given numbers
Addition and Subtraction  Add Fractions
of Fractions
 Subtract Fractions
Multiplication and
 Multiply Fractions
Division of Fractions
 Divide Fractions
Order of Operations with  Perform the correct order of operations for decimals.
Rational Numbers
 Perform the correct order of operations for fractions.
Quiz #1 – Rational
 Demonstrate your understanding of above topics
Numbers
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Lesson Topic
1
Ordering & Adding
Integers
Lesson #1: Ordering & Adding Integers
Integers are used in everyday life, when we deal with weather, finances, sports, geography and
science.
Key Vocabulary: Define the following using the glossary in “Math Makes Sense 9”
Integer
Whole Number
Natural Number
Positive Integer
Negative Integer
Zero Pair
Opposite Integer
Zero Property
Distributive Property
Commutative Property
Product
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Order of Operations
2
Quotient
Ordering Integers
Create a Number Line to Order the Following Integers: -3, 4, -9, 5, -6, 0.
Step 1: Determine the lowest integer and the highest integer.
Step 2: Draw a straight line for your number line. Use a straight edge and be sure to put arrows
on both ends.
Step 3: Plot your lowest number to the far left hand side of your line; plot your highest number
to the far right hand side of your line. Mark equal intervals between to indicate each integer
between the two.
Step 4: Label each interval.
Step 5: Place the numbers onto the number line by indicating its place with a dot or a line and
writing the number ABOVE the number line.
Possibilities when adding Integers:
Money Method:
Think about the numbers in the addition statement as money – positive numbers is money you
have, while negative numbers is money you owe.
This would change the equation “3 + (-4)” to “You have $3 and owe $4”. What would you
have?
(
e) (
)
)
d)
(
f) (
)
)
(
)
3
c)
b)
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Use this to add the following:
a) ( ) ( )
g) ( )
( )
h) ( )
(
)
Ordering and Adding Integers Assignment:
Order the following integers on a the number line below:
Steps:
1. Sketch a number line and include markings for every integer
2. Place dots and label the numbers you are ordering on the number line
a) -3, 5, 6, -1, -7, 0
b) 2, -3, 4, -5, 6, -7
Assignment:
(-2) + (-4) =
(-1) + (+8) =
(+5) + (-7) =
(+9) + (+3) =
(+2) + (-7) =
(-2) + (+9) =
(-2) + (+1) =
(+6) + (-8) =
(+2) + (-6) =
(-3) + (-8) =
(-6) + (-8) =
(0) + (-2) =
(-3) + (+6) =
(-8) + (+4) =
(0) + (-7) =
(-8) + (-1) =
(+5) + (+1) =
(+6) + (+48) =
(+90) + (-14) =
(-57) + (-89) =
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(-7) + (+2) =
4
Add the following integers:
(-71) + (+47) =
(-47) + (-61) =
(-75) + (+78) =
(-53) + (+57) =
(-25) + (-31) =
(-46) + (-3) =
(+95) + (-58) =
5
(+72) + (-92) =
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(+73) + (-55) =
Lesson #2: Subtracting Integers
Subtracting Integers
Money Method, continued, with “Same, Change, Change”
Subtracting Integers requires the changing of signs within the mathematical statement. One
way to remember to do this properly is “Same, Change, Change”. What this means is to leave
the first numbers sign the same, no matter what it is, and change the second two.
( ) would mean to leave the first number, -5, as itself. Change the subtraction
Ex)
sign to an addition, then change the positive three to negative 3.
( ) becomes
This means:
( ).
You would then apply the money method to solve, meaning you’d think “I owe $5 and I also
owe $3.
Ex) Subtract the following, using the method of “Same, Change, Change”.
(
)
(
)
b)
)
(
)
d) (
e) (
)
(
)
f)
)
6
c) (
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a)
Subtracting Integers Assignment:
Subtract the following integers:
(-7) - (+8) =
(-5) - (+3) =
(+2) - (-6) =
(+2) - (+8) =
(-8) - (-8) =
(-7) - (+1) =
(+1) - (+9) =
(0) - (+8) =
(+9) - (-6) =
(-2) - (-6) =
(+5) - (+3) =
(-1) - (+3) =
(+2) - (+2) =
(+7) - (-3) =
(-9) - (-9) =
(+2) - (0) =
(-6) - (+8) =
(-3) - (-7) =
(+5) - (+1) =
(-8) - (+7) =
(-1) - (-1) =
(+8) - (+2) =
(-4) - (+8) =
(+2) - (+3) =
(+3) - (-1) =
(+7) - (+7) =
(-4) - (+4) =
(-5) - (-8) =
(0) - (+4) =
(-9) - (+7) =
(+3) - (-5) =
(+9) - (-2) =
(-4) - (-9) =
(-8) - (+6) =
(-7) - (0) =
(+28) - (-51) =
(-94) - (-73) =
(-65) - (-41) =
(+52) - (+47) =
(+35) - (+68) =
(-74) - (-61) =
(+65) - (-59) =
(+2) - (+78) =
(-10) - (-12) =
(+33) - (-53) =
(+49) - (-52) =
(-61) - (+92) =
(-9) - (-91) =
(-47) - (+3) =
(+69) - (+39) =
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(-6) - (0) =
(-4) + (+1) =
(-6) + (+10) =
(+14) - (-4) =
(0) + (+15) =
(0) + (-9) =
(+1) + (+24) =
(+3) + (-14) =
(-24) + (+14) =
(+17) + (+14) =
(+23) + (+14) =
(+14) + (+9) =
(+21) + (+21) =
(-3) - (+4) =
(+5) - (+3) =
(-22) + (+20) =
(+18) - (+21) =
(+6) + (-25) =
(+17) + (+6) =
(+24) + (+11) =
(+21) - (+3) =
(+35) - (+24) =
(-5) + (+8) =
(-4) - (-9) =
(+25) - (+2) =
(-22) + (-22) =
(-44) - (-19) =
(+3) - (+10) =
(+22) - (+4) =
(+14) - (+9) =
(+5) (+6) + (-13) =
(-7) (-4) + (+1)
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Find the sum or difference for each question.
(+14) - (+9) =
(+6) + (-13) =
8
Mixing Adding and Subtracting Integers:
Lesson #3: Multiplication & Division with Integers
When multiplying or dividing integers you treat multiply or divide the numbers as though they
are natural numbers, then use the properties below to determine the sign of the product or a
quotient.
Properties for Multiplication of two Integers:
(
(
(
(
Solve for the following products:
a) ( )( )
b) ( )(
)
c) (
)( )
d) (
)(
)(
)(
)(
)(
)
)
)
)
)
When two or more numbers are multiplied the result is called the product.
Properties for Division of two Integers:
This …
Solve for the following quotients:
a) ( ) ( )
(
( )
( )
( )
( )
( )
( )
c) (
)
( )
d) (
)
(
)
)
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)
Means the same as this:
( ) ( )
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b) (
Symbolically…
( )
( )
( )
( )
( )
( )
( )
( )
e)
f)
(
)
g)
( )
(
)
h)
( )
(
)
(
)
(
)
(
)
Multiplication & Division of Integers Assignment:
Find each product.
(-6)x0 =
7x3 =
6x(-10) =
(-3)x(-5) =
8x(-2) =
(-4)x(-10) =
10x(-3) =
3x5 =
9x(-4) =
10x4 =
10x(-4) =
5x9 =
0x(-10) =
11x11 =
2x3 =
(-4)x(-12) =
(-4)x(-6) =
(-10)x(-2) =
3x12 =
4x7 =
3x(-3) =
Find each quotient.
(-40) ÷ (-10) =
(-55) ÷5 =
63÷9 =
(-63) ÷ (-9) =
(-8) ÷2 =
(-90) ÷ (-10) =
36÷ (-6) =
72÷12 =
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96÷(-12) =
33÷11 =
49÷7 =
(-100) ÷10 =
35÷5 =
(-25) ÷5 =
48÷ (-4) =
20÷10 =
(-24) ÷ (-12) =
(-96) ÷8 =
60÷ (-5) =
(-30) ÷5 =
14÷2 =
(-14) ÷ (-7) =
(-16) ÷2 =
(-110) ÷10 =
(-66) ÷11 =
(-63) ÷9 =
80÷ (-10) =
(-36) ÷ (-12) =
18÷9 =
18÷ (-2) =
16 ÷ (-4) =
9 - (-11) =
7+6=
8 × (-10) =
4 - (-6) =
(-3) + 11 =
(-2) + 5 =
20 ÷ (-2) =
(-8) × 4 =
(-9) × 6 =
(-1) × (-10) =
3 + (-11) =
1 × (-7) =
1 - (-9) =
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7 + (-8) =
11
Perform the indicated operation and simplify.
Lesson #4: Order of Operations of Integers
In lessons #1 and #2 you learned how to add, subtract, multiply and divide integers. Now you
will learn to extend your ability to evaluate expressions containing positive and negative
integers.
Before learning to evaluate using the order of operations you must know how to apply an
exponent to an integer. An exponent is a base (integer) raised to a specific power.
For example is an exponent. The base is the number 3 and the power is the number 2. This
means to multiply the base by itself two times.
Evaluate the following:
Exponential Form
Expanded Form
Simplified
When applying the order of operations the acronym BEDMAS can be used to remember the
order in which to perform the operations.
1. First any expression within Brackets must be performed. Brackets may be inside of
brackets, in which case the innermost brackets must be calculated first.
2. Secondly any Exponents must be calculated.
3. Thirdly any Division or Multiplication operations must be performed. These should be
done as they appear from left to right.
4. Lastly and Addition or Subtraction operations must be performed. These also should be
done as they appear from left to right.
(
Example 1: Evaluate the expression(
)
Apply BEDMAS
You must begin by solving within the brackets.
Use subtraction of integer rules to subtract these.
Find the solution by adding the integers.
)
(
).
12
)
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Evaluate the expression (
(
) (
)
( ) ( )
Evaluate the expression (
( )
( )( )
( )
( )( )
(
)
)
( )( ).
Apply BEDMAS.
Insert all invisible 1’s to make the negatives simpler. Since
there are no calculations within brackets you get to move on to
( )( )( )
evaluating the exponent. ( )
Multiply and Divide from left to right. DO NOT CONFUSE THE
SUBTRACTION SIGN BETWEEN AS A MULTIPLICATION!
Use rules of subtracting integers to make this into a simpler
expression.
Evaluate.
Example 2: Evaluate the expression
( )
a) ( )
Evaluate the expression
(
)
( )
[
(
[
[
)
(
(
[
(
(
b)
( )
(
)
(
)
(
)
)
)
)]
)]
]
]
.
Apply BEDMAS. Beware of “Invisible Brackets” that
exist around the entire numerator and entire
denominator.
Insert “Invisible Brackets” and “Invisible 1’s”
Perform any multiplications within Brackets (once
inside the brackets BEDMAS must be applied again)
Add and subtract within the top brackets.
Divide the integers.
(
(
)
)
b)
(
)
(
)
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a)
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Example 3: Evaluate the expression
– 5 + (-5)
x (3 + (-1))
4. -3
+ 6 + (-9)
5. 5
÷ (-1)4
6. -10
7. 1
x 1+(-9)
8. -1
9. [(10
11. (6
÷ (-10))7]2
- (-10)) ÷ (4 ÷ (-2))
13. (-2)
÷ 2 ÷ (1 ÷ (-1))
10.
– 2 ÷ (-2)
- (-9 + (-3))
2 x (-6) ÷ ((-6) x 2)
12. (2+1)
14. 9
x 6 ÷ (-9)
- (10 ÷ (-2) - 5)
14
3. -8
2. 9
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Order of Operations of Integers Assignment:
Perform the operations in the correct order.
1. 2 x 5 - 7
17. -6
- (-10 + (-1 - (-5)) x 5)
19. 8
÷ (-2 – 4 + 8 + (-4) + (-6))
20. 9
÷ (10+6 - (1 - (-6))) + (-4)
16. 1
x (-2 + 3) x (-9 + 7)
18. (10
- 8) x (-4) ÷ 2 x 4
15
x (-9 + (-9) + 22)
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15. 1
Lesson #5: Identifying and Representing the Number Systems
Numbers are grouped into systems called the Number Systems.
The Natural Numbers: The Natural Numbers (N) are the simplest counting numbers we know
of. They are never negative, nor are they decimals or fractions. They begin at the number 1
and increase as though you are counting objects.
Ex) 1, 2, 3, 4, … are Natural Numbers
The Whole Numbers: The Whole Numbers (W) expand upon the natural numbers with the
inclusion of the number zero. Whole numbers are also never negative, nor are they decimals or
fractions. They begin at the number zero and increase as though you are counting.
Ex) 0, 1, 2, 3, 4, … are Whole Numbers
The Integers: Integers (I) are whole numbers along with their negative opposites. When
simplified Integers will never be fractions (with a denominator other than 1) or a decimal.
Ex) …, -3, -2, -1, 0, 1, 2, 3, … are Integers
The Rational Numbers: Rational Numbers are any number that can be written in the form ,
where a and b are Integers and
.
 Rational Numbers also include decimals that repeat or terminate (since they can be
written as fractions)
 Powers and Square roots may be rational numbers if their standard form is a rational
number.
The Irrational Numbers: Irrational numbers are any numbers that are not rational. That is to
say any non-terminating, non-repeating decimals, any imperfect square (etc.) roots.
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All of the above numbers can be grouped into the Real Number System. That is to say,
numbers that exist and can be plotted on a number line. Non-real (or Imaginary) numbers do
exist but will not be taught at this level.
Venn Diagram of the Number Systems:
Writing Rational Numbers in different forms:
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, etc. Any rational number that reduces to 5 is the same value.
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Many forms of rational numbers look different but represent the same number. For example,
the number 5 can be represented by each of the following:
Dealing with negative in Rational Numbers:
By convention, the negative symbol on a fraction is written in one of two places: either
attached to the numerator of the fraction:
, or out front of the fraction:
. This means
these two fractions have the same value, even though they look different.
Many times people will leave the negative sign with the denominator. This doesn’t change the
value, but doesn’t meet criteria of what is considered appropriate.
Thus:
. When a rational number has the same value in the numerator and the
same value in the denominator and the same overall sign as another they are equivalent.
Ex) Write 3 equivalent fractions to each of the following:
a)
b)
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Assignment: Page 101 #5, 6; page 103 #26, 27:
Lesson #6: Ordering, Adding & Subtracting Decimals
When ordering decimal numbers we can use a similar strategy to ordering integers but extend
the strategy to the right side of the decimal.
When we order integers:
 Positive integers are larger than negative integers
 Ie: 5 is larger than -5
 When comparing positive integers:
o The more place values to the left of the decimal the larger the number is
 Ie: 0.3 is larger than 0.003
o If two numbers have the same number of places, the first digit from the
left that is larger will determine which value is larger.
 Ie: 0.0035 is larger than 0.0032
 When comparing negative integers:
o Use opposite rules as to positive integers
o The fewer place values to the left of the decimal, the larger the number
 Ie: -0.003 is larger than 0.3
o If two numbers have the same number of places, the first digits from the
left that is larger will determine which value is smaller.
 Ie: -0.0032 is larger than -0.0035
Remember! When a decimal repeats the number that repeat are designated with a line above
them:
Ie: 0.27272727272727…. = ̅̅̅̅ and ̅ = 0.999999999999999999999…
Ex) Compare the following decimals with <, >, =
a) 3.25 3.52
b) -52.3
52.5
c) 0.3000
0.3
Adding and Subtracting Decimals:
When adding or subtracting decimals it is very important that you are adding values together
for the same place value, which means you must line up the decimal when adding numbers
with decimals.
Ex)
First: Line up the decimals:
Page
Ex 2)
19
Then add/subtract columns as they appear from right to left.
Assignment:
Comparing Decimals
1. Compare each pair of decimals using a <, > or = sign.
22.2
22.58
16
11.96
12.83
-21.7 -21.44
18.16 18.78
-2.39
-2.87
11.68 11.84
13.7
13.68
4.04
4.47
-5.79
-4.82
6.87
7.05
4.55
5.3
19.76
19.68
10.97
10.53
-24.48
24.03
8.15
22.04
22.56
15.17
14.53
-23.63
22.84
8.82
8.75
23.22
22.76
-10.22
2.29
3.27
16.06
8.03
-11
-11.75
-10:32
2. Order the following sets of decimals on a number line.
a) 3.42, 4.3, 4.32, 3.44
b) -3.42, -4.3, -4.32, -3.44
3. Order each set of decimals.
0.62, 0.38, 0.35, 0.49, 0.1, 0.21, 0.54, 0.6, 0.51, 0.28
Least to Greatest
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0.23, 0.2, 0.77, 0.49, 0.74, 0.91, 0.65, 0.23, 0.03, 0.83
Greatest to Least
1.42, 1.41, 1.18, 1.34, 1.44, 1.52, 1.61, 1.03, 1.26, 1.56
Least to Greatest
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4. Add or subtract the following decimals.
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5. Add or subtract the following as indicated. Use loose leaf to line up decimal
places.
Lesson #7: Multiplying and Dividing Decimals
**Remember rules for Multiplication and Division of Integers:
 Positive x Positive = Positive

 Negative x Negative = Positive

 Positive x Negative = Negative

 Negative x Positive = Negative

Positive ÷ Positive = Positive
Negative ÷ Negative = Positive
Positive ÷ Negative = Negative
Negative ÷ Positive = Negative
In other words: if the signs are the same the product or quotient will be positive; if the signs are
different the product or quotient will be negative.
Multiplication with Decimals:
When multiplying decimals, multiply them as you would whole numbers (ignore the decimal
place). Then count the total number of decimal places in both multipliers (the numbers you
are multiplying); this number is the total number of decimal places in the product.
Ex) a) Multiply
First: Set it up vertically:
Then perform multiplication.
Count decimal places and place decimal.
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Dividing with Decimals:
Set up the division as a fraction. Multiply numerator and denominator in your fraction (which is
both divisor and dividend within the fraction) by multiples of 10 until there is no more decimal
places. Then divide numerator by denominator.
Ex) a) Divide
Set up fraction:
24
b)
Multiply by 10’s:
Divide.
b)
c)
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25
Assignment:
Part A:
Part B:
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26
Part C:
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27
Lesson #8: Ordering Fractions
Fractions Review:
The denominator goes in da bottom. The denominator also determines the number of pieces
one whole is divided or cut into. The numerator goes on the top and determines how many
pieces there are.
Fractions can be represented with a pie chart picture where a circle is cut into the number of
pieces indicated by the denominator and the number indicated in the numerator is shaded in.
Whole numbers are indicated by an entire filled in circle.
Ex) Represent the following fractions with a picture:
a)
b)
c)
d)
Equivalent Fractions are numbers which appear different but have the same value. They may have
different numerators and denominators, be improper and proper, etc. You can determine if they are
equivalent by reducing until the fraction is in lowest terms.
Ex) Determine if the two fractions are equivalent.
a)
b)
c)
b)
Page
a)
28
Ex 2) Write 2 fractions that are equivalent to each:
Lowest Common Denominator LCD) is the number which is the smallest multiple of two given different
denominators. This can be found easily be writing multiples of both denominators out and finding the
lowest one in common.
Ex) Find the LCD for the two given fractions:
a)
b)
Converting Decimals to Fractions:
Method 1:
The simplest way to convert a fraction is to divide the numerator by the denominator:
Ex) ;
Method 2:
Make an equivalent fraction with a base of power 10 (10, 100, 1000, etc.); then move the decimal place
appropriately:
Ex)
Since we are dividing by 100 we move the decimal place 2 times to the left.
Converting Decimal to Fractions:
Any decimal can be converted to a fraction by multiplying the number by a power of 10 to make the
decimal a whole number and putting that power of 10 as the denominator of the fraction.
Ex) 1.65 =
The fraction then must be fully reduced by dividing by common factors.
Ordering Fractions:
29
in order from greatest to least:
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Ex: Write the numbers
Method: Use common denominators:
(Do positive numbers first):
(And negative):
Assignment: (TO BE DONE IN ASSIGNMENT SECTION ON LOOSE LEAF!)
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30
Page 101 #10, 14, 15, 24, 25:
Lesson #9: Adding and Subtraction Fractions:
Use common denominators:
**REMEMBER: The denominator in a fraction determines the number of pieces one whole is
divided (cut) into. The numerator in a fraction determines how many pieces there are.
Since denominators determine the size of the piece, it is important to ensure the denominators
are equivalent or common.
Example 1: Add:
Sequence:
1. Make all fractions improper.
2. Make determine common denominator; multiply numerator and denominator by the
number required to get the denominator:
3. “Smoosh” them together on one denominator. (This step may be skipped when you have
gotten 5 correct in a row in your assignment)
4. Add/Subtract the numerators.
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Assignment: page 111 # 11; page 119 #9, 12, 13abc
31
Ex 2)
Lesson #10: Multiplication and Division of Fraction:
Multiplication of Fractions:
When multiplying fractions you simply multiply the numerator by the numerator and the
denominator by the denominator. Fractions must be in improper form and must be fully
reduced. You may choose to reduce before multiplying if you prefer.
Ex) Multiply the following:
a)
c) (
b)
)( )
d)
Division of Fractions:
When dividing fraction we multiply by the reciprocal. This is a fraction that has been inverted.
Ex) The reciprocal of is . Note: when reciprocating a negative number the negative stays with
the numerator out of convention. That is to say that the reciprocal of
is
. Fractions must
still be fully reduced.
Ex) Divide the following fractions:
b)
c)
d)
Assignment: Page 127 #4, 7; page 134 #4 and the questions on the following page:
Page
32
a)
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33
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34
Lesson #11: Order of Operations with Rational Numbers:
Recall the correct Order of Operations for mathematics using the acronym BEDMAS:
B–
E–
D–
M–
A–
S–
Use the order of operations to complete the following:
Ex 1)
)
]
Assignment: Page 140 #3ac, 4ac, 7ac, 8, 12acd
Quiz Review: Page 144 #1-5, 7-12, 14, 16, 19, 20, 23
35
[(
Page
Ex 2)