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Math 8
Unit 1 Study Guide
Area
Area is the number of unit squares ( 1 cm  1 cm , 1 m  1 m , etc.) needed to cover an object or
surface. The unit for area is the unit used in the original measurements squared. For example, if
the original measurements were made using cm, then the unit for the area of the object will be
cm2.
Area of a Square
To determine the area of a square simply find the square of one of its sides or:
Area  s 2
where “s” is the length of a side for the square
Example
Determine the area for the following square:
3 cm
3 cm
3 cm
3 cm
Solution
To determine the area of the square, simply substitute 3 cm for “s” in the formula for the area of
a square or Area  s 2 and simplify:
Area  s 2
 (3 cm) 2
 9 cm 2
Substitute 3 cm for s.
simplify
Please note that this is the area of the figure so
the units of measurement must be squared.
Therefore the area of the given square is 9 cm 2 .
Area of a Rectangle
To determine the area of the rectangle, we simply multiply the measurement of the rectangle’s
base by its height:
Area  base  height
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Example
Determine the area for the following rectangle:
5 cm
12 cm
Solution
To determine the area of the given rectangle, simply substitute 12 cm for the base and 5 cm for
the height in the formula for the area of a rectangle or Area  base  height :
Area  base  height
Substitute 12 cm for the base.
 12 cm  5 cm
 60 cm 2
Please note that this is the area of the figure so
the units of measurement must be squared.
Therefore the area of the given rectangle is 60 cm 2 .
Squares
To square a number, just multiply it by itself.
“Squared” is often written with a little superscript 2.
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Squares of Negative Numbers
You can also square negative numbers.
Square Roots
The square root of a number is simply the value that when multiplied by itself gives us that
number. Looking at it another way, the square root goes the opposite way of squaring the
number.
The Square Root Symbol
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Perfect Squares
The perfect squares are the squares of whole numbers.
Example
What is the square root of 9?
Solution
The square root of 9 is simply the value that when multiplied by itself is 9. From an earlier math
course we know that 3 multiplied by itself is 9. As a result, we can conclude that the square root
of 9 is 3.
The symbol “
” indicates that we must determine the square root of the number that is
underneath it. For example 16 means that we must determine the square root of 16.
Therefore 16  4 .
The following is a list of the perfect square roots that can be easily determined from the
multiplication tables:
because 1  1  1
1 1
42
9 3
16  4
25  5
36  6
49  7
64  8
81  9
121  11
144  12
100  10
because 12  12  144
Estimating the Square Root of a Number
It is easy to work out the square root of a perfect square, but it is really hard to work out other
square roots. We may instead estimate the square root of a number that falls between 2 perfect
squares.
Example
Simplify 21
Solution
Since 21 is not a perfect square, the square root can be estimated using the perfect squares listed
above.
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Recall the perfect squares from above:
1
Notice that
4
9
16
25
36
49
21 is between the perfect squares of 16 and
64
72
81
81
100
100
25 or:
21
1
9
4
16
25
36
49
64
72
Since 21 is about halfway between 16 and 25, it can be assumed that 21 will also be between
16 and 25 . Since 16  4 and 25  5 , it can be concluded that 21 is between the values
of 4 and 5 or approximately 4.5.
Therefore
21  4.5
Perimeter
Perimeter is the distance around a polygon, a closed figure that is composed of 3 or more line
segments. To determine the perimeter of a polygon simply find the sum of its sides.
Example
Determine the perimeter of the following square:
3 cm
3 cm
3 cm
3 cm
Solution
To determine the perimeter of this square, simply substitute 3 cm for “s “in the formula
perimeter  4  s and simplify:
perimeter  4  s
 4(3 cm)
Substitute 3 cm for s.
simplify
 12 cm
Therefore the perimeter, or distance around the given square is 12 cm.
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The Number Line
The number line is a line labeled with positive and negative numbers in increasing order from
left to right, that extends in both directions. The number line shown below is just a small piece of
the number line from -4 to 4.
For any two different places on the number line, the number on the right is greater than the
number on the left.
Example
4 > -2, 1 > -0.5, -2 > -4, and 0 > -15
Absolute Value of Positive and Negative Numbers
The number of units a number is from zero on the number line. The absolute value of a number
is always a positive number (or zero). We specify the absolute value of a number n by writing n
in between two vertical bars: |n|.
Example
|6| = 6
|-0.004| = 0.004
|0| = 0
|3.44| = 3.44
|-3.44| = 3.44
|-10000.9| = 10000.9
Adding Positive and Negative Numbers
When adding numbers of the same sign, we add their absolute values, and give the result the
same sign.
Example
8.5 + (-17) = ?
The absolute values of 8.5 and -17 are 8.5 and 17. Subtracting the smaller from the larger gives
17 - 8.5 = 8.5, and since the larger absolute value was 17, we give the result the same sign as -17,
so 8.5 + (-17) = -8.5.
Subtracting Positive and Negative Numbers
Subtracting a number is the same as adding its opposite.
Examples
In the following examples, we convert the subtracted number to its opposite, and add the two
numbers.
7 - 4.4 = 7 + (-4.4) = 2.6
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22.7 - (-5) = 22.7 + (5) = 27.7
-8.9 - 1.7 = -8.9 + (-1.7) = -10.6
-6 - (-100.6) = -6 + (100.6) = 94.6
Note that the result of subtracting two numbers can be positive or negative, or 0.
Multiplication of Integers
To multiply integers simply follow the same method that was used to add and subtract negative
integers in Math 7.
 If the rational numbers have the same signs (both are positive or both are negative) then
their product will be positive.
 If the rational numbers have different signs (one positive and one negative) then their
product will be negative.
Example
Simplify:
3.0  0.2
Solution
Since the integers have the same positive sign, their product will be positive:
3.0  0.2
= 0.6
The signs are the same
therefore the answer will be
positive
Division of Integers
To divide integers simply follow the same method that was used to multiply negative integers in
Math 7.
 If the rational numbers have the same signs (both are positive or both are negative) then
their product will be positive.
 If the rational numbers have different signs (one positive and one negative) then their
product will be negative.
Example
Simplify:
 0.125   0.25
Solution
Since the rational numbers have the same sign (both negative), their quotient will be positive:
 0.125   0.25
Divide the decimal using a calculator.
 0.5
The signs are the same therefore
the answer will be positive.
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Order of Operations
Think about the things that you do to get ready to leave the house. When you dress in the
morning, you use some order of operations that are not mathematical, but the way you dress is an
order operations. You need to put your socks on before you put your shoes on for example.
In mathematics, we use the following convention for the order of operations:
Brackets
Exponents
Division (or)
Multiplication
Addition (or)
Subtraction
If you look at the first letters of the terms above you will see that they form the word BEDMAS.
This is an acronym that you can use to help you remember the order of operations.
Example
Simplify:
2×3+4÷2–5
Think
B
E
D or
M
A or
S
Ask yourself
Are there any operations inside brackets?
Are there any exponents?
Are there any multiplications or divisions?
The multiplication appears first so it is done before the
division
Are there any additions or subtractions?
The addition appears first so it is done before the
subtraction
In this case
No
No
Yes, 2 × 3
And 4 ÷ 2
6 + 2 then 8 - 5
Solution
2×3+4÷2–5
1st
2nd
=6+2–5
3rd
=8–5=3
Fractions
A fraction is the ratio of two whole numbers, or to put it simply, one whole number divided by
another whole number. We write a fraction like this:
3
4
numerator
denominator
The top part of the fraction is called the numerator and the bottom part is called the denominator.
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A fraction can also be thought of as a part or portion of a whole. Here are some whole shapes
and some ½ and 1/3 shapes (or fractions of a whole):
Proper and Improper Fractions
Normally the numerator (top of fraction) is smaller than the denominator (bottom of the fraction)
in which case the fraction is called a “Proper Fraction”. If the numerator is equal to or larger
than the denominator, then the fraction is called an “Improper Fraction”.
Here are some examples of improper fractions:
Mixed Numbers
We can also have mixed numbers. A mixed number is the combination of a whole number and a
fraction. We could convert an improper fraction into a mixed number to write it correctly. Here
are some examples of mixed numbers:
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Converting an Improper Fraction to a Mixed Fraction
An improper fraction can be changed to the mixed number by dividing the numerator by the
denominator. This gives a quotient and a remainder. The remainder is placed over the divisor.
Example
7
1
3
2
2
Converting an Improper Fraction to a Mixed Fraction
An improper fraction can be converted to a mixed fraction by multiplying the whole number of the mixed
fraction by the denominator and then adding the numerator of the fraction part.
Example
2
2 8

3 3
Reducing Fractions
To write a fraction correctly, we always reduce it to its lowest terms. To do this, we have to look
for the greatest common divisor or GCD. The greatest common divisor is the largest natural
number that the numerator or the denominator can be divided by with no remainders.
Examples
Adding and Subtracting Fractions
In order to add fractions, the fractions must have the same common denominator. To do this,
determine the lowest common multiple of the denominators and then use it to find equivalent
fractions that contain the lowest common multiple in the denominator.
Once the fractions have the same common denominator, add the numerators of the fractions
together and place this value in the numerator of a fraction with the lowest common denominator
as the denominator.
Don’t forget that when adding fractions, add the numerators only and place this value in a
fraction over the lowest common denominator in the denominator.
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It should be noted that that a whole numbers can be written as a fraction with the value of the
3
whole number in the numerator with a denominator of 1. For example 3 
1
Example
1 2
Add

4 4
Solution
To add fractions, the fractions must have the same common denominator. Since both
1
2
and
have a
4
4
denominator of 4, their sum can be determined by adding the numerators only and placing this value over
the lowest common denominator:
1 2 3
 
4 4 4
Therefore
Add the numerators together only
The denominator always stays the same
1 2 3
  .
4 4 4
Multiplying Fractions
To multiply fractions, multiply the numerators together and place this value in the numerator of a
fraction with the product of the denominators in the denominator or:
a c ac
 
b d bd
where a, b, c, and d are constants
Please note that the product of fractions must always be written in lowest terms.
Example
Simplify
1 2

4 3
Solution
To multiply fractions, multiply the numerators together and place this value in the numerator of a
fraction with the product of the denominators in the denominator:
1 2 1 2
 
4 3 43
2

12
multiply the numerators
multiply the denominators
simplify
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Don’t forget to write the product in lowest terms:
1 2 2
 
4 3 12
1

6
Therefore
Write the fraction in lowest terms by dividing
both the numerator and denominator by 2.
1 2 1
 
4 3 6
Reciprocals
The reciprocal of a fraction can easily be determined by simply switching the numerator and the
denominator. In other words, the numerator of the fraction becomes the denominator, and the
denominator becomes the numerator.
2
, simply place the numerator or 2 as the
3
denominator and the denominator or 3 as the numerator:
For example, to determine the reciprocal of
2
3
Original Fraction
2
3
The numerator of the fraction will simply
become the denominator, and the
denominator will become the numerator
3
2
The reciprocal of 2
3
Dividing Fractions
To divide fractions simply multiply the first fraction or dividend by the reciprocal of the second
fraction or divisor.
Example
Simplify
2 1

9 3
Solution
To divide fractions simply multiply the first fraction or dividend
2
by the reciprocal of the
9
1
second fraction or divisor :
3
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2 1

9 3
2 3
 
9 1
23

9 1
6

9
2

3
Therefore
The reciprocal of 1 is 3
3
1
Multiply the dividend by the reciprocal of the divisor.
Simplify
Write the answer in lowest terms.
2 1 2
  .
9 3 3
Percents
The unit “percent” means out of 100. To calculate percent, divide the numerator (or portion of
the total) by the denominator (or the total) of the fraction and multiply by 100%.
Example
Max’s mark on the first assignment was
18
. What percent of the assignment did he answer
20
correctly?
Solution
To determine your percentage, all that you have to do is divide the numerator by the denominator
and then multiply the decimal by 100%.
Divide the numerator
by the denominator
18
 100%
20
 0.9  100%
Remove the % symbol
 90%
Therefore, Max correctly answered 90% of the assignment questions.
It should be noted that percents can be expressed in 3 different ways and they are as follows:
 Written using the symbol for per cent “%”(as in example 1)
 Written as a decimal
 Written as a fraction.
As a result, we must know how to a percent from one form to another:
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Changing a Percent to a Decimal
To change a percent to a decimal, simply remove the % and divide our number by 100 (or move
the decimal 2 places to the left). For example, to change 82% to a decimal we remove the % and
divide 82 by 100 (or move the decimal 2 places to the left):
Remove the % symbol
divide 82 by 100
82
100
 0.82
82% 
82%  0.82
or
move the decimal 2
places to the left
To change a decimal to a percent, simply work backward or insert a % and then multiplying the
decimal by 100 (or move the decimal 2 places to the right):
multiply 0.35 by 100
0.35  0.35  100%
 35%
Add the %
symbol
or
0.35  35%
Move the decimal 2
places to the left
Changing a Percent to a Fraction
To change a percent to a fraction, simply remove the %, write the number in the numerator of a
fraction with a denominator of 100, and re-write the fraction in lowest terms. For example, to
change 54% to a fraction, simply remove the %, write 54 as fraction with a denominator of 100,
and then re-write the fraction in lowest terms:
remove the %
54
100
27

50
54% 
Place the number over 100
Re-write the fraction into simplest
terms by dividing both the numerator
and denominator by 2
To change a fraction to a percent, simply work backward or insert a %, divide the numerator by
the denominator, and the then multiplying this value by 100:
add the % symbol
divide the numerator by
the denominator
1
 0.25  100%
4
 25%
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Please note that we use the same method to change percentages greater than 100% from one
form to another. The only difference being is that our fraction will be a mixed number and the
decimal will be greater than 1.
Sales Tax
Some word problems require multiple calculations using percents. In these problems, the
original value is increased or reduced by certain percentage and then this “new value” is
increased or reduced by a second percentage. The following are some of the situations where
multiple calculations of percents are required:
 the value of an investment (savings accounts, Canada Savings Bond, etc.) or loan (credit
cards, personal loans, etc.) that use compound interest
 the sale price of an item after more than 1 reduction in price
 the total price of an item including GST (Goods and Services Tax) and PST (Provincial
Sales Tax)
To solve these types of problems, calculate the value after the increase or decrease (using the
first percentage) and then use this value to calculate the value after the second increase or
decrease (using the second percentage).
Example
A shirt at a local department store has been marked down from its original price by 20%.
Today, the store is holding a sale where all merchandise is marked down 10% of the lowest
ticket price. If the shirt was originally priced at $25 what will be the cost of the shirt if we were
to buy it today?
Solution
The first step in solving this problem is to determine the cost of the shirt after the original
discount of 20%:
$25  20%
 $25  0.2
0.20 is 20% written in decimal form
 $5
simplify
Therefore the price of the shirt after the first discount is $25  $5  $20 .
To find today’s sale price of the shirt, simply reduce the price of the shirt after the first discount
by 10%:
$20  10%
0.10 is 10% written in decimal form
 $20  0.1
 $2
reduce the price of the shirt after
the first discount or $20 by 10%
simplify
Therefore the price of the shirt after the second discount is $20  $2  $18 .
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Ratios
Ratios are used to represent the relationship between a series of items without including units.
You see and use ratios on almost a daily basis. For example, ratios are used in the sports column
of the newspaper to compare wins to losses to ties, to identify the odds of winning a contest, and
even recipes. The following are some examples of ratios:
10:3
4 to 5
1:2:3
1
4
3 to 5 to 8
2 Term Ratios
A two-term ratio is a ratio that contains only 2 numbers. The following are some examples of
two-term ratios:
1
5
1:3
2 to 7
Please note: All 2-term ratios will be
expressed in one of these forms.
3 Term Ratios
A three-term ratio is a ratio that contains 3 numbers. The following are some examples of threeterm ratios:
2:5: 7
1: 3 : 8
5 to 2 to 1
Unlike two-term ratios however, a three-term ratio has only 2 different forms namely:
2:5: 7
or
2 to 5 to 7
It should be noted that 3 term ratios can also be written as a series of 2 term ratios. For example,
the ratio 2 : 5 : 7 can be broken down into the following 2 term ratios:
2:3
First term and middle
term of the original
ratio.
5:7
2:7
Middle term and
last term of the
original ratio.
First term and last
term of the
original ratio.
Writing a Ratio
To write a ratio, simply write down all of the numbers that are given in the order that they appear
in the question.
Example
John has 1 nickel, 2 dimes and 3 quarters in his pocket. What is the ratio of nickels to dimes to
quarters that John has in his pocket?
Solution
To answer this question, all that we have to do is write down the number of each coin that John
has in his pocket in the order that the questions asks (nickels to dimes to quarters). When you
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put the number of each coin that John has in the order that is asked, your ratio will look like the
following:
1:2:3
Number of nickels
Number of dimes
Number of quarters
Simplifying Ratios
Ratios, like fractions, must always be written in its simplest form. In other words, we must
factor or divide each term of the ratio by the greatest common factor. Don’t forget that the
greatest common factor is the largest number that will divide into each term of our ratio.
Example
Simplify 5 : 25 : 50
Solution
To simplify the given ratio, all that we have to do is divide each of the terms by the greatest
common factor. Looking at the terms, we see that the greatest common factor of the terms in the
ratio is 5. As a result, to simplify the given ration we must divide each term by 5:
Divide each term in
the ratio by 5
5 : 25 : 50
1 : 5 : 10
This is the ratio in simplified form.
Therefore the ratio 5 : 25 : 50 written in its simplified form is 1 : 5: 10.
Determining the Share in a Ratio
To determine the share or portion of the total number of items as determined by the ratio:
 Determine the number of shares by finding the sum of the terms in the ratio.
 Determine the value that every single share in the ratio represents by finding the quotient
of the total number of items by the number of shares.
 Multiply each term of the ratio by the number determined in step 2 to determine the
actual value that each term of the ratio represents.
Example
John, Sammy, and Jake share a restaurant in the ratio of 1:3:5. If the restaurant made a profit of
$5400 last week, what was the value of each person’s share?
Solution
The first step in solving this problem is to determine the total number of shares in the restaurant
by finding the sum of the terms in the ratio:
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1: 3 : 5  1 3  5
Find the of the terms in the ratio
9
Determine the value that every single share in the ratio represents by finding the quotient of the
total number of items ($5400) by the number of shares (9).
$5400  9  $600
The value of every share in the ratio is $600
Multiply each term of the ratio by the number determined in step 2 to determine the actual value
that each term of the ratio represents.
1:3:5
John
Jake
Sammy
John has 1 share in the restaurant, therefore the portion of the profit that he will receive is:
1 $600  $600 .
The number of shares that
John owns in the restaurant
The amount of profit that each
share in the restaurant receives
Sammy has 3 shares in the restaurant, therefore the amount of profit that he will receive
is 3  $600  $1800 .
Jake has 5 shares in the restaurant, therefore the amount of profit that he will receive
is 5  $600  $3000 .
Therefore, John earned $600, Sammy earned $1800, and Jake earned $3000 from their restaurant
last week.
Proportions
A proportion consists of equivalent ratios. The following are some examples of proportions:
3 15

1: 2  2 : 4
2:3  6:9
4 : 5  8 : 10
5 25
When asked to solve a proportion, determine the unknown value present in one of the ratios.
The unknown value in a proportion is represented by a letter (most commonly “x”) and is called
a variable. The following is an example of a proportion that contains an unknown value:
1 n

3 12
This is our unknown value that we
must determine
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Solving Problems Involving Proportions
To solve problems involving proportions, find the relationship between the equivalent ratios and
then use this characteristic to determine the unknown value.
Example
Solve for “n”:
1 n

3 12
Solution
To solve problems involving proportions simply find the relationship between the equivalent
ratios and then use this characteristic to determine the unknown value.
Looking at the given proportion we see that the denominator of the second ratio is simply the
denominator of the first ratio multiplied by 4. Since the ratios are equivalent, the unknown value
“n” can easily between determined by multiplying the numerator in the first ratio by 4:
4 
1
n

3
12
4 
As a result:
To get the second ratio of the proportion, the
numerator and denominator of the first ratio
were both multiplied by 4.
1 4  n
4n
simplify
Therefore the unknown value in the given proportion is 4.
Cross-multiplication
Cross multiplication is a way of solving certain equations. Let’s go back to proportions.
Remember that a proportion is an equation which states that two ratios are equal. Here is an
example of two equivalent ratios:
3 6

4 8
When the terms of a proportion are cross multiplied, the cross products are equal. Cross
multiplication is the multiplication of the numerator of the first ratio by the denominator of the
second ratio and the multiplication of the denominator of the first ratio by the numerator of the
second ratio. For example:
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3
4
6
8
If we cross multiply the 3 x 8 we get 24. And if we multiply 4 x 6 we also get 24. So we know
that these are equivalent expressions.
Using Cross-multiplication to find an Unknown Variable
If one term of a proportion is not known, cross multiplication can be used to find the value of the
unknown term.
Example
Find the unknown variable in the proportion below.
x
3

70 10
Solution
x
3

70 10
x(10)  3(70)
10 x  210
10 x 210

10
10
x  21
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Unit 2 Study Guide
What is an Expression?
Numerical relationships or patterns in mathematics are generally shown in the form of an
expression. An expression is a mathematical sentence that contains both letters and
numbers. The following are some examples of expressions:
x  3y  7
2n  m  4
4 g 1  h
It should be noted that a letter in an expression that is used to represent a number is called a variable.
Evaluating Expressions
To evaluate an expression, substitute or replace the desired values into the given expression for
the corresponding variable and simplify.
Please note to simplify an expression:



Eliminate the brackets.
Perform all of the multiplication and division parts of the question.
Perform all of the addition and subtraction parts of the question.
Creating Expressions
To create an expression from a given statement, always begin by looking for the key word(s) that
indicate the mathematical operation(s) (,,, or ) and numbers to be used. Please note that the
following table lists some of the key words that indicate the operation(s) that will be used in an
expression:
Addition ()
Sum
Plus
Increase
Subtraction ()
Minus
Difference
Decrease
Multiplication ()
Times
Product
Multiply
Division ()
Divide
Split
Quotient
Once the key word(s) have been identified, simply break up and translate each key word of the
statement into algebraic terms and write them in the form of an expression.
Creating (or deriving) an Equation from a Given Set of Data.
The following are the steps that should be followed when creating or deriving an equation from a
given set of data:
 Organize the data
Re-organize the data so that a relationship or pattern can easily be seen. Generally
speaking arrange them in increasing order for one of the variables in the table.
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 Look for a pattern
Look at the data to see if there is a relationship between the numbers in each column.
 Write an equation
Write an equation that describes the pattern that was identified in step 3.
 Verify the equation
Test the equation using the given set of data.
Like Terms
Terms that have the same variable are called like terms. The following are some examples of
like terms:
3x
1
x
4
 8x
2x
The following is another set of like terms:
 2a
3a
6a
Unlike terms have different variable parts or variables raised to different powers. The following
are some examples of unlike terms:
3 xy
1
x
4
2y
 8x 2
Another example of unlike terms:
2x
2x 2
4r
Like terms can be combined. For example you can add like terms simply by adding the numbers
that are in front of like terms (coefficients).
In an expression, the operations of addition or subtraction can only be performed on the like
terms. To combine the like terms in an expression, add or subtract the co-efficients of the like
terms followed by the like variable and its exponent. It should be noted that this process of
simplifying the like terms is called, “combining the like terms.”
The Distributive Property
To expand an expression with brackets means to remove the brackets by multiplying. This is
done using the Distributive property. Another way to describe using the distributive property
is to multiply through by what is outside of the brackets or multiply every term in the brackets by
what is outside of the brackets.
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To expand or simplify expressions similar to the following:
3( x  1)
 2(3  x)
4( x  5)
Multiply each of the terms inside the bracket by the term that is outside of the bracket.
Solving One Step Equations
When asked to solve an equation, you must determine the value of for the variable that satisfies
or makes the equation a true statement (the value on the left-hand side of the equal sign is equal
to the value on the right hand side). The following are some examples of true statements:
55
 3  3
20  20
To solve an equation, simply re-arrange the terms so that the variable is on one side of the equal
sign with a co-efficient of 1 and the constants on the other side. It should be noted that this
process is called isolating the variable. The following are some examples of what an equation
looks like after isolating the variable:
x2
z  15
y  4
To isolate the variable in an equation, simply perform the opposite operation on both sides of the
equation. For example:
 the opposite of addition is subtraction and vice versa
 the opposite of multiplication is division and vice versa
Please note that equations are balanced expressions. In other words, the information on the left
side of the equation (left of the = sign) will be equal to the right side of the equation. As a result,
in order to keep the equation balanced, when an operation is performed on one side of the
equation, the same operation must be performed on the other side of the equation.
Solving Two Step Equations
Some equations require 2 or more steps in order to be solved. These types of equations are
called 2-step equations. The following are some examples of 2-step equations:
3x  2  1
1
y  1  2
4
2y  3  y  4
3x  1  4
Notice that each of the above equations either contains a variable with a co-efficient other than 1
or more than 1 variable term and more than 1 constant.
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To solve these types of equations algebraically, simply perform the opposite operation(s) on both
sides of the equation to:
 Isolate the term(s) that contain the variable or re-arrange the equation so that the term(s)
containing the variable are on one side of the equal sign and the constants are on the other
side. After performing this step, the original is now in a form that is similar to the
equations that were discussed in section 3.02.
 Isolate the variable. In other words divide or multiply both sides of the equation by a
constant (number) in order to change the co-efficient in front of the variable to 1.
The 4 Step-Method for Problem Solving
When solving word problems it is a good idea to follow the 4-step method for problem solving.
The following are lists the steps for this method:
 Think about the problem
Write down in detail the actual question that the problem is asking. Once the actual
question has been determined, write down all of the additional information that is given
in the problem.
 Make a plan
Write a plan that can be followed in order to answer the problem.
 Solve the problem
Follow the plan listed in step 2 to solve the problem.
 Conclusion
Write a short statement to present the solution to the problem.
Some word problems will require the use of an equation in order to solve them. These problems
are easy to identify because they involve a relationship that contains at least one unknown value.
Creating a Table of Values
When we are given an algebraic expression, we can generate a list, or table of values by
replacing the variable with specific numbers. We are then able to do the corresponding
computation. For example, given the expression x2 – 1, if x = 1, then the expression equals zero
because 12 – 1 = 0.
If a relation is represented by the equation y = 2x + 1, you can write a table of values as:
1
2
3
4
5
6
7
x
3
5
7
9
11
13
15
y
A related pair of x and y values is called an ordered pair. Some ordered pairs for the relation
above are:
(1, 3), (2, 5), (3, 7), (4, 9), (5, 11), (6, 13), (7, 15), (x, y)
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Graphing Linear Relations
Once we have created a table of values from an expression and found some ordered pairs, we can
proceed to graph the expression. To graph the information from the chart above, we would plot
all the x values along the horizontal axis of a graph and plot all the y values along the vertical
axis.
It is also important to label both the horizontal axis and the vertical axis and write the equation
of the relation on the graph. Every graph also needs to have a scale on it. The points may lie in
a straight line, if so the expression is linear.
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Unit 3 Study Guide
Right Triangles
A right triangle is just a triangle that contains a right or 90 0 angle. The following are some
examples of right triangles:
Notice that each right triangles has a right
angle (as indicated by the little boxes)
Labeling Triangles
The angles of triangle are always labeled using capital letters:
A
C
B
The sides of a triangle are always labeled with the small letter of the angle that is opposite it. For
example, the side opposite angle “A” will be “a”:
A
c
b
C
a
B
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The Pythagorean Theorem
Around 500 BC, a Greek philosopher and mathematician named Pythagoras noticed that there
was a relationship between the length of the longest side in a right-angle triangle, which he
called the hypotenuse, and the 2 smaller sides. The relationship that he noticed was that the
length of the hypotenuse squared is equal to the sum of the squares of the other 2 sides or:
h 2  a 2  b2
This formula is called the
Pythagorean Theorem.
The sum of the squares of the other 2
sides (a and b in no particular order)
hypotenuse (h) squared
Please note that the hypotenuse will always be the side that is directly opposite the right angle.
The Pythagorean Theorem is a very useful relationship because it allows us to determine the
length of any side in a right triangle when the other 2 sides are given.
Example
Determine the length of “h” in the following right-angled triangle:
h
4
3
Solution
Side “h” is the side that is directly opposite the right angle of the triangle. Therefore side “h” is
the hypotenuse of the triangle.
To determine the length of side “h”, use the Pythagorean Theorem:
The Pythagorean Theorem
Eliminate the squares
h2  a2  b2
h  (3)  (4)
2
2
2
Substitute the lengths of the 2 smallest sides of the
triangle for “a” and “b” and then solve for “h”.
h 2  9  16
h 2  25
h 2  25
To determine the length of side “h”, find the
square root on both sides of the equation.
h5
Therefore, the length of side “h” in the given triangle is 5 units.
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Nets
A net is a 2-dimesional (flat) diagram or pattern of a 3-dimesional object. In other words, it is as
if the 3-dimensional object is unfolded (like a cardboard box) and laid flat on the page. Please
note that all net diagrams will in fact be composite figures. The following are some examples of
nets:
A flattened cardboard box
Top
Front
Back
Side
Side
Bottom
A flattened prism shaped tent
side
front
back
bottom
side
The Surface Area of a Prism
Surface area is the total area of all of the surfaces (sides) of a 3 dimensional object. To
determine the surface area of a prism, calculate the area of each of its surfaces or sides and then
find their sum.
The Area of a Triangle
The following is a review of Grade 7 material. In order to determine the area of a triangle, we
must first identify the triangle’s base and height. The following diagram illustrates the base and
height of a triangle:
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The height of a triangle is the length of the
perpendicular line (at a right angle) that connects
the base of the triangle to the top of the triangle.
The base of a triangle will always be the side of the triangle that has
a perpendicular line (at a right angle) off of it.
Once the base and height have been identified, the area of the triangle can be easily determined
using the following formula:
Area 
1
base  height
2
The Surface Area of a Right Triangular Prism
Surface area is the total area of all of the surfaces (sides) of a 3 dimensional object. To
determine the surface area of a right triangular prism, calculate the area of each of its surfaces or
sides and then find their sum.
The Volume of a Right Rectangular Prism
The volume of object is the number of unit cubes that will fit into the object. The following are
some examples of the unit cubes that we will use in this section:
Notice that the unit cubes are 1
unit long by 1 unit wide by 1
unit high.
1m
1cm
1cm
1cm
1m
1m
The unit for volume is the unit used in the original measurements cubed or with an exponent of
3. For example, if the dimensions of the object were measured in cm, the volume of the object
would be expressed in cm 3 . Similarly, if the original measurements were made using meters,
the volume of the object would be expressed in m 3 .
To determine the volume of a rectangular or triangular prism, multiply the area of the base of the
prism by its height or:
Volume  area of the base of the prism  height of the prism
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The Volume of a Right Triangular Prism
To determine the volume of a right triangular prism, we still multiply the area of the base of the
prism by its height, however we use one of the triangular sides as the base of the prism.
Therefore the formula will be:
Volume  area of the base of the prism  height of the prism
= (1/2 base x height) x height of the prism
The Diameter and Radius of a Circle
The diameter of a circle is the length of the line segment that connects two points on a circle and
passing through the center of the circle:
diameter
The radius of a circle is the length of the line segment that connects the center of a circle to any
point on the circle:
radius
Please note that the radius of the circle is in fact half of the diameter. Similarly the diameter is
twice the radius.
Circumference
The perimeter of a circle is called the circumference. To calculate the circumference of a circle,
substitute the diameter of the circle into the following formula and simplify:
circumference    diameter
This symbol is called Pi (pronounce pie)
Pi has a value of 3.14
and
Area of a Circle
To determine the area of a circle simply substitute the measure of its radius into the following
formula and simplify:
Area    radius 2
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The Surface Area of a Cylinder
A cylinder is similar in shape to a rectangular prism; the only difference being is that the ends (or
base) of a cylinder are circles rather than rectangles. The following are some examples of
cylinders:
The Volume of a Cylinder
To determine the volume of a cylinder, we again use the method we used to find the volume of a
prism. We still multiply the area of the base of the cylinder by its height, however the base of a
cylinder is a circle. Therefore the formula will be:
Volume  area of the base of the cylinder  height of the cylinder
 (r 2 )  height of the cylinder
Transformations
Symmetries create patterns that help us organize our world conceptually. Symmetric patterns
occur in nature, and are invented by artists, craftspeople, musicians, choreographers, and
mathematicians.
Plane symmetry involves moving all points around the plane so that their positions relative to
each other remain the same, although their absolute positions may change. Symmetries preserve
distances, angles, sizes, and shapes.
In this lesson we will look at 3 types of transformations. They are rotations, reflections and
translations:
1. For example, rotation by 90 degrees about a fixed point is an example of plane
symmetry.
2. Another basic type of symmetry is a reflection. The reflection of a figure in the plane
about a line moves its reflected image to where it would appear if you viewed it using a
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mirror placed on the line. Another way to make a reflection is to fold a piece of paper and
trace the figure onto the other side of the fold.
3. A third type of symmetry is translation. Translating an object means moving it without
rotating or reflecting it. You can describe a translation by stating how far it moves an
object, and in what direction.
Rotation
To rotate an object means to turn it around. Every rotation has a center and an angle.
Translation
To translation an object means to move it without rotating or reflecting it. Every translation has
a direction and a distance.
Reflection
To reflect an object means to produce its mirror image. Every reflection has a mirror line. A
reflection of an “r” is a backwards “R”.
Tessellations
The word “tessera” in latin means a small stone cube. They were used to make up ‘tessellata’ –
the mosaic pictures forming floors and tiles in Roman buildings. The term has become more
specialized and is used to refer to repeating shapes or patterns, which cover a surface without
gaps or overlaps. The shapes in a tessellation can be geometric like squares and triangles, or
shaped like animals and people.
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There are only 3 regular geometric shapes that will tessellate. “Regular” means that all sides and
internal angles are equal. These are the equilateral triangle, square and hexagon.
A tessellation of triangles
A tessellation of squares
A tessellation of hexagons
When you look at these three samples you can easily notice that the squares are lined up with
each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time,
they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they
cannot be formed by directly lining shapes up under each other - a slide (or a glide!) is involved.
Shape
triangle*
square*
pentagon
hexagon*
more than six sides
Angle measures in degrees
60
90
108
120
more than 120 degrees
Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle
must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and
you can show working tessellations for these figures. For all the others, the interior angles are not
exact divisors of 360 degrees, and therefore those figures cannot tile the plane.
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Unit 4 Study Guide
Line Graphs
Line graphs are used to show changes in data where one of the components is time. The
following is an example of a line graph:
Sales (in $1000)
Monthly Sales for the First Six Months of the Year
2001
7
6
5
4
3
2
1
0
Jan
Feb
Apr
May
Jun
Jul
Months
The following are the steps that are used to draw a broken line graph:
 Label (name) each axis and choose an appropriate scale. Please note that the graph itself
must also have a title.
 Plot the points on your graph that correspond to the data.
 Connect the dots using line segments (straight lines).
It should be noted that to create a multiple (more than 1) broken line graph simply follow the
same steps that were used for example 1 and plot each set of data independently of each other on
the same grid. Please note that each set of data in a multiple line graph should be drawn using
different colors and a legend should be include to indicate the color that is used to represent each
set of data on the graph. The following is an example of a multiple line graph:
Temperature (C)
The Average Daily Temperature For Calgary and
Toronto Last Week
15
legend
10
Toronto
Calgary
5
0
Mon
T ues
W ed
T hurs
Fri
Day of the Week
According to the legend, this line
represents the temperature in Calgary.
According to the legend, this line
represents the temperature in Toronto.
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Bar Graphs
Bar graphs are graphs that display data by using horizontal or vertical bars whose lengths are
proportional to the numbers they represent. The following are 2 examples of bar graphs:
The Favorite Colours of Math Students
25
Yellow
20
Purple
15
Colour Green
10
Blue
5
Red
0
Red
Blue
Green
Purple
Yellow
0
5
Colour
10
15
20
25
Number of Students
The following are the steps that we use in order to draw a bar graph:
 Label (name) each axis and choose an appropriate scale. Please note that the graph itself
must also have a title.
 Draw the bars that correspond to each piece of data.
It should be noted that to create a multiple (more than 1) bar graph simply follow the same steps
that were used for example 1 and plot each set of data independently of each other on the same
grid. Please note that each set of data in a multiple bar graph should be drawn using different
colors and a legend should be include to indicate the color that is used to represent each set of
data on the graph. The following is an example of a multiple bar graph:
The Number of Cases of Fruit Sold at a Local
Fruit stand in Calgary Last Week
Number of Cases
Sold
Number of Students
The Favorite Colours of Math Students
30
25
20
15
10
5
0
Mon
Tues
Wed
Thurs
Day
The blue bars represent the
number of oranges sold.
Fri
Oranges
Apples
legend
The purple bars represent
the number of apples sold.
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Circle Graphs
Circle graphs are graphs displaying data using the segments or portions of a circle. This type of
graph is used to show the amount (usually percentage) that each term in a set of data represents
with respect to the total amount of the data set. The following is an example of a circle graph:
Frozen Desert Consumption
Sherbert
3%
Ice Milk
24%
Ice Cream
63%
Yogurt
10%
The following are the steps that are used to draw circle (pie) graphs:
 Calculate the percentage of the total for the set of data that each term in the data set
represents.
 Determine the portion of the circle that will be designated for each term by multiplying the
decimal form of the percentages calculated in step 1 by 360 0 .
 Draw a circle with a line to represent the radius and then measure the number of degrees,
calculated in step 2 for the first term in the data set and draw a line to represent this angle
measurement.
 Complete step 3 for the remaining pieces of data.
 Give the graph a title and label each portion or wedge of the circle appropriately.
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Choosing an Appropriate Graph
Each type of graph has its advantages and disadvantages. Consult the table below when choosing
a graph. Each entry in the table has a link to an example of that graph. Just click on the
underlined name of the graph.
Type of Graph
Strengths
Limitations
Line graph
A line graph plots continuous
data as points and then joins
them with a line. Multiple data
sets can be graphed together,
but a key must be used.


Easy to draw and read
Shows data changes
over time
 Can be used to estimate
values between or
beyond data points

Bar graph
A bar graph displays discrete
data in separate columns. A
double bar graph can be used
to compare two data sets.
Categories are considered
unordered and can be
rearranged alphabetically, by
size, etc.


Easy to draw
Lengths of bars
compare data values
 Can easily compare two
or three data sets
 Scale can be used to
find the total

Pie chart
A circle graph (pie chart)
displays data as a percentage
of the whole. Each pie section
should have a label and
percentage. A total data
number should be included.





Pictograph
A pictograph uses an icon to
represent a quantity of data
values in order to decrease the
size of the graph. A key must
be used to explain the icon.



Shows parts of a whole
Shows percent of total
for each category
 Sizes of sectors
compare parts of the
whole
Does not show parts of a
whole
 Zig-zag pattern can be
difficult to interpret
May be difficult to read
depending on scale used
 Does not show percents of
the total for comparison
 Two sets of data in one
graph can be confusing
 Can only be used to show
discrete data
No exact numerical data
Hard to compare 2 data sets
Total unknown unless
specified
 Best for 3 to 7 categories
 Difficult to draw accurately
Easy to read
 Large number of symbols
Visually appealing
make it difficult to read
Handles large data sets
 Best for only 2-6 categories
easily using keyed icons  Does not show parts of a
whole
 Difficult to draw
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Misrepresenting Data
Graphs are a visual way of representing data. The information can be interpreted much quicker
than by a person reading through a data table. However, they can create false impressions by the
way they are drawn.
There are many different ways in which graphs can be drawn to misrepresent data.
Misinterpretations of the data may lead to incorrect conclusions or assumptions. A wise
consumer needs to become aware of the ways in which graphs can be misleading.
Here are some of the ways in which a graph can be drawn to misrepresent data:
1. Using an exaggerated or incorrect scale.
2. Starting the scale on the vertical axis of a bar graph at a number other than 0.
3. Using different size symbols in a pictograph or bars of different widths in a bar graph to
make one piece of data appear greater than another.
4. Treating a part of a graph differently to draw people’s attention to that piece of data.
5. Not including enough context to make the graph meaningful.
Probability
Probability is the likelihood (or chance) that a random event will occur. Probability is
commonly used is in the description of the odds of winning a contest. For example:
 The odds of winning a free bag of chips is 1 in every 4 bags.
 The odds of winning a new bicycle at a local sport store is 1 in 10 000 shoppers.
Probability is usually expressed in the form of a decimal. The following are some important
probabilities that you aware of:
 If the probability is 0, than the event is unlikely to occur.
 If the probability is 1, than the event is likely to occur.
 If the probability is 0.5, than there is a 50/50 chance that the event will occur.
To determine the probability of an event occurring we simply use the following formula:
number of favorable outcomes
P(event ) 
total number of outcomes
The number of times that the
desired event occurs.
The probability of
an event occurring.
The total number of possible
events that can occur. Please
note that is also called the
sample space.
Please note that probability is usually expressed as a fraction or a percentage.
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Equally Likely Events
If more than one event is equally likely to occur than the probability of each event occurring will
be:
1
, where n is the number of events.
n
A good example of this is the rolling of a die. When a die is rolled, there is an equal chance of
rolling a 1, 2, 3, 4, 5, or 6. Therefore the probability of rolling any number on the die will be will
be exactly the same or:
1 1

n 6
There are 6 possible events that can occur when
rolling a die namely rolling a 1, 2, 3, 4, 5, or 6.
Therefore the probability of rolling any number on a standard die is 1 .
6
The Probability of an Event Not Occurring
If the probability that a event A occurs is P( A) than the probability that event A does not occur would
be 1  P( A) .
Tree Diagrams
Determining the probability that an event occurs more than once (such as winning 2 bags of
chips in a row) or the probability that 2 events will occur consecutively or one right after the
other (rolling a six and drawing a diamond from a deck of cards) can be extremely complicated.
As a result, it is helpful to use a tool called a tree diagram to solve these types of problems.
A tree diagram is a type of graph that is used to show the possible outcomes for multiple or
consecutive events. The following is an example of a tree diagram:
The “trial” is in fact the
outcome of each event.
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Independent Events
If the outcome of one event does not affect the outcome of the other event then these events are
called independent events. The following are some examples of independent events:





Flipping a coin more than once
Flipping 2 coins
Drawing a card from a standard deck of 52 cards
Rolling 2 dice
Spinning a multiple colored spinner and flipping a coin.
The outcome (heads or tails) of flipping the first
coin does not affect the outcome (heads or tails)
of flipping the second coin.
The following is an example of events that are not independent:
 Drawing one card from a deck of 52 cards and then drawing a second card without replacing
the first.
The outcome (which card will be drawn out of the deck)
directly affects the outcome of the second draw because
there is 1 fewer card in the deck for the second draw.
To determine the probability that 2 independent events will occur, multiply the probability that
the first event occurs by the probability that the second event occurs.
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