Download Grade 8 Math Study Guide

Grade 8 Math Study Guide
Exponents – the raised number used in a power to show the number of
repeated multiplications.
2
Example – in 4 , the exponent is 2 and the base is 4. The example is said to
be written in exponential form.
T o multiply powers with the same base, add the exponents.
4 2
6
3 x 3 =3
To divide powers with the same base, subtract the exponents.
5
2
3
4 -:- 4 = 4
Scientific Notation – a short form notation that involves decimals and
powers of ten.
5
2
Example – 849 000 = 8.49 x 10 743 = 7.43 x 10
In scientific notation, the decimal is always after the first number.
-3
Example – 0.0046 = 4.6 x 10
Percent – a fraction or ratio in which the denominator is 100.
Example - 1/25 = 4/100 To find the decimal
Also remember: N - numerator
I - inside
D - denominator
0 - outside
Square Roots – The square root of a number is a number that multiplies
itself to give the number.
Example: 8 x 8 = 64, so 8 is the square root of 64.
Integers – integers are numbers in a sequence. -3, -2, -1, 0, 1, 2, 3 …..
Examples: Adding integers: (+7) + (+3) = +10
(- 3) + (-8) = -11
(+5) + (-9) = -4
(-3) + (+12) = +9
Subtracting integers: (+7) – (+3) = +4
(-3) – (-8) = +5
(+2) – (+4) = -2
Multiplying integers: Pos x Pos = Positive
Neg x Pos = Negative
Pos x Neg = Negative
Neg x Neg = Positive
Dividing integers: Pos/Pos = Positive
Neg/Pos = Negative
Pos/Neg = Negative
Neg/Neg = Positive
Order of Operations – The rules to be followed when simplifying
expressions.
2
B – brackets
Example: 2 + 6 x (3 + 2) x 3 + 4
E – exponents
2
D – division
= 2 + 6 x (5) x 3 + 4
M – multiplication
= 2 + 6 x 25 x 3 + 4
A – addition
= 2 + 150 x 3 + 4
S – subtraction
= 2 + 450 + 4
= 456
Fractions – a number that describes part of a whole or part of a group.
Always express fractions in lowest terms!
Equivalent Fraction – fractions that represent the same fraction of the whole.
Example: ½ = 2/4
Adding Fractions: the denominators need to be the same in order to add
fractions.
Example: 1/3 + 1/3 = 2/3
¼ + ½ = ¼ + 2/4 = ¾
When adding mixed numbers, you should change each mixed number to an
improper fraction, then make sure the denominators are the same, and add.
Subtracting Fractions: ¾ - ¼ = 2/4 = ½ (lowest terms)
Same rules as adding fractions apply – the denominators must be the same.
Multiplying Fractions: To multiply fractions, you need to multiply the
numerators together and the denominators together.
Example: ¾ x 2/5 = 6/20 = 3/10 (lowest terms)
Reciprocals: 2 numbers whose product is 1.
Example: 2/3 – the reciprocal is 3/2
6 – the reciprocal is 1/6
2 ½ - change to an improper fraction 5/2 – reciprocal is 2/5
Dividing Fractions: To divide by a fraction, multiply by its reciprocal.
Example: ¾ -:- 2/5 = ¾ x 5/2 =15/8 = 1 7/8
2 2/3 -:- 1 ½ = 8/3 -:- 3/2 = 8/3 x 2/3 = 16/9 = 1 7/9
Greatest Common Factors (GCF): factors are numbers multiplied to give a
specific product.
Example: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Lowest Common Multiples (LCM): multiples are repeated additions within
a group.
Example: Find the lowest common multiple of:
4: 4, 8, 12, 16, 20, …..
8: 8, 16, 24, 32, …..
The LCM of 4 and 8 is 8.
Percents: A percent is a fraction or ratio in which the denominator is 100.
Example: 7% = 7/100
143.2% = 143.2/100 = 1.432
5 ½ % = 5.5% = 5.5/100 = 0.055
Express as a percent: 0.4 = 40/100 = 40%
3:8 = 3/8 = 3 -:- 8 = .375 x 100 = 37.5%
Percent of a number: To calculate 50% of 30, change the % to a decimal,
and “of” means multiply.
50% = .50
x 30
15.0 Therefore, 50% of 30 is 15.
12% of 25 .12
x 25
3.00 Therefore, 12% of 25 is 3.
Discount and Sale Prices:
A shirt sells for $39.95. What is the sale price with a discount of 25%?
The discount is 25% of $39.95
39.95
X .25
9.9875 The discount is $9.99
Regular Price 39.95
25% Discount - 9.99
Sale Price $ 29.96
GST - 7%: Example – A bike costs $320.00. Calculate the total selling
price, including GST.
7% of 320 = .07 x 320 = 22.40
Bike $320.00
GST + 22.40
Total $342.40
Commission: Example – Carol works in an electronics store. She earns 4%
commission on her total sales. One day, Carol sold $3420.00 worth of
goods. What was her commission that day?
4% of 3420 = .04 x 3420 = 136.80
Carol’s commission was $136.80.
Finding the percent: Writing a decimal as a percent
0.8 = 8/10 = 80/100 = 80%
1.67 = 167/100 =167%
0.007 = .007 x 100 = .7 = 0.7%
Writing a fraction or ratio as a percent:
4 = 80 = 80%
5 100
90
72 = 90 -:- 72 x 100 = 125%
Example: 21 is what % of 24?
21/24
= 21 -:- 24
= .875 x 100
= 87.5%
100% of a number:
Example: If 7% of a number is 8.4, what is the number?
7% of the number is 8.4
1% of the number is 8.4 -:- 7
100% of the number is 8.4 -:- 7 x 100
= 840 -:- 7
= 120
If 125% of a number is 15, what is the number:
125% of the number is 15
1% of the number is 15 -:- 125
100% of the number is 15 -:- 125 x 100
= .12 x 100
= 12
Simple Interest: The formula to calculate interest is I = Prt
Interest = principal x rate of interest x time in years
Example: Mark bought a $500.00 Canada Savings Bond that paid 6.5%
interest a year for 5 years.
Interest = 500 x .065 x 5 = $162.50
Math Study Guide
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Equivalent ratios and proportions: Ratios are a comparison of numbers.
4:5 or 4/5 or 4 to 5 (these are all the same) Remember to express in lowest
terms!
Example: Find the missing term in each proportion.
5/6 = b/18 b = 15
Rate: A rate is a comparison of 2 numbers with different units.
Example: 60 km./hour is a rate.
Example: Express as a unit rate. $2.40 for 6 bagels
.40 for 1 bagel
2
2
2
Measurement: Pythagorean Theorem a + b = c
The area of the square drawn on the hypotenuse of a right angle triangle is
equal to the sum of the areas of the squares drawn on the other 2 sides.
Perimeter: Perimeter is the distance around a polygon. To find the
perimeter, find the sum of the lengths of all the sides.
Perimeter = length + length + width + width or 2L + 2W
Circumference of a circle: Circumference is the perimeter of a circle.
Circumference = Pi x diameter (Pi = 3.14)
Example: The diameter of a ferris wheel is 55 m. What is its
circumference?
C = Pi x d
= 3.14 x 55
= 172.7
The circumference is 172.7 m.
Area of a Rectangle: Area is the number of square units needs to cover a
surface. Area = length x width. Area is always indicated in units squared.
Area of a Square: Area = length x width
Area of a Parallelogram: Area = base x height
Area of a Triangle: Area of ½ base x height
2
Area of a Circle: Area = Pi x r
Example: The radius of a dartboard is 23 cm. What area of the wall is
covered by the dartboard?
2
Area = Pi x r
2
= 3.14 x 23
= 3.14 x 529
= 1661.06
The area of the wall covered by the dartboard is
2
1661.06 cm
Area of Composite Figures: Composite figures are made up of 2 or more
distinct regions. To calculate the area of a composite figure, determine the
area of each region and add them together.
Surface Area and Volume: When calculating the surface area of polyhedra,
you must calculate the area of each face and add them together.
Example: If you have a cube (6 faces) with dimensions of 12 cm x 12 cm x
12 cm., you need to calculate the area of each face and multiply x 6.
- the area of each face is 12 x 12 = 144 cm.
2
- Therefore the surface area will be 144 x 6 = 864
Volume of Prisms: The volume of a prism is the area of the base multiplied
by the height of the prism. Volume is always indicated in units cubed.
Example: Area of base = ½ b x h
Volume = area of base x height
Surface Area and Volume of a Cylinder:
The volume of a cylinder is calculated like the volume of a prism.
Volume = area of base x height of the cylinder.
Example: If you had a cylinder whose height was 10 cm. and diameter
was 6 cm., what would the cylinder’s volume be?
2
Area of the base: Area = Pi x r
2
= 3.14 x 3
= 3.14 x 9
2
= 28.26 cm
Volume = area of base x height of cylinder
2
Volume = 28.26cm x 10 cm
3
= 282.6 cm
Geometry:
Opposite Angles – the equal angles formed by 2 intersecting lines.
Supplementary angle – two angles whose sum is 180 degrees.
Complementary angle – two angles whose sum is 90 degrees.
Perpendicular lines – intersect to form right angles.
Parallel lines – lines in the same plane that do not intersect.
Transversal – a line or line segment that crosses 2 or more lines.
Algebra:
Variable – a letter or symbol used to represent a number.
Example: y + 4 , y = 2
4y , y = 3
=2+4
= 4 (3)
=6
= 12
8 – 2n, n = -1
= 8 – 2 (-1)
=8+2
= 10
a + b , a = -2, b = -3
= (-2) + (-3)
= -5
Writing Equations:
Write an equation for each sentence:
(1) 5 more than a number is 10 ***** n + 5 = 10
(2) 5 times a number is 30 ***** 5(y) = 30
Solving Equations:
Example: 3a + 13 = 37
3a = 37 – 13
3a = 24
a = 24 -:- 3
a = 8
n
5 = 4
n = 4x5
n = 20
Collecting Like Terms:
Like terms: terms such as 4x, 2x, 3x
2
Unlike terms: terms such as “2y”, “3z”, “4y , with different variables.
Example: a + b + a + b
= 2a + 2b
4y + 2z – 2y – 3z
= 2y – z
The Distributive Property:
Expand: 2(y + 5)
-3 (3m – 2n)
= 2y + 10
= - 9m + 6n
3 (5y + 2z + 4)
= 15y + 6z + 12
3 (y – 1) = y + 5
3y – 3 = y + 5
3y – y = 5 + 3
2y = 8
y = 8 -:- 2
y = 4
4y – 7 = 3 + 10
4y – 7 = 13
4y = 13 + 7
4y = 20
y = 20 -:- 4
y = 5
Mean, Median, Mode and Range:
Mean: the sum of the numbers divided by the number of numbers in a set.
Median: the middle number in a set of numbers arranged in order. If there
is an even number of numbers, the median is the average of the two middle
numbers.
Mode: The number that occurs most frequently in a set of data.
In 1, 2, 2, 5, 5, 5… the mode is 5.
Range: The difference between the highest and lowest numbers in a set.
Bar graphs: A graph that uses bars to represent data visually. Always
include a title and label both the horizontal and vertical axis.
Broken line graphs: A graph that represents data with line segments joined
end to end. Include a title, draw and label horizontal and vertical axis. Plot
the data and join the points with straight line segments.
Circle graphs: A graph that uses sectors of a circle to show how data is
divided into parts.
- after you have collected your data, write each number as a percent of the
total.
- Then calculate each percent of a circle (360 degrees)
- Draw a circle and measure each angle with a protractor. Label each
sector with a name and a percent. Give the graph a title.
Pictographs: A graph that uses pictures or symbols to display data.
- always include a title, column headings and a key which explains what
the symbols represent.
Probability: The ratio of the number of ways an outcome can occur to the
total number of possible outcomes.
The probability of an event:
P = number of favourable outcomes
total number of possible outcomes