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Intro to Mathematics
Nathan Frey
Arithmetic Concepts
Number systems, Base ten system,
Order of operations, Fractions,
Decimals, Percents, Ratios, Rates and
Proportions
Number Systems
Base 10 (decimal) system
Order of Operations
• PEMDAS
• Parentheses – any grouping symbol including
brackets, absolute value, fraction bar, etc.
• Exponents – also includes radicals like square
roots, cube roots
• Multiplication/Division – left to right
• Addition/Subtraction – left to right
Decimals
• Adding and Subtracting
Decimals
• Multiplying and Dividing
Least Common Multiple and Greatest
Common Factor
• Find the GCF and LCM of 60 and 42
Least Common Multiple/Denominator
(LCM/LCD)
Least Common Multiple/Denominator
(LCM/LCD)
Greatest Common Factor (GCF)
Greatest Common Factor (GCF)
Fractions
• Adding/Subtracting – need a common
denominator
Multiply Fractions
• Change all mixed numbers to improper
fractions, multiply numerators, multiply
denominators, and reduce the fraction
Divide Fractions
• Rewrite division as multiplication by the
reciprocal (flip the second fraction)
Converting Fractions, Decimals and
Percents
• To convert a fraction into a decimal do the division.
4/5 means 4 divided by 5
• To convert a decimal to fraction read the decimal to
the correct place value .25 means 25 hundredths
• To convert a decimal to percent multiply by 100 (move
decimal 2 places to the right and add percent sign)
• Percent % - means out of 100. Change a percent to a
fraction over 100.
• To convert a percent to a decimal divide by 100 (move
decimal 2 places to the left and drop the percent sign)
Common conversions
=‘[-p\
Ratio
(p. 174)
• A ratio is a comparison of two quantities using
the same units
• Example: In a sample group of patients there
are 15 men and 20 women. What is the ratio
of men to women? 15/20, 15:20, 15 to 20 or
simplified ¾, 3:4, 3 to 4
• What is the ratio of women to men?
• Practice on p. 176
Rates (p. 178)
• Rate is a comparison between two quantities
using different units written as a fraction.
• Example: You travel 250 miles in 4 hours. The
rate is 250 miles/4 hours or simplified 125
miles/2 hours
• Unit rate is when the denominator is 1
• 250 miles/4 hours = 62.5 miles/hour
• Practice p. 180
Proportion (p. 182)
• A proportion is an expression of equality of
two ratios or rates
• A proportion is true if both sides are equal
when written in simplified form
• 3/6=1/2 is true 6/10=10/15 is not true
• Solve a proportion by cross-multiplying
• Ex 1: 9/6 = 3/n
Ex: 5/7 = x/20
• Practice p. 186
Metric System (p. 372)
Converting metric units by using unit
multipliers
Setting up percent problems
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Translate keywords into math language
“of” means multiply
“is” means =
“what” tells us to use a variable
If you see “what percent” use p and remember to
convert answer to percent
• If you see “what number” use n and remember
your answer should just be a number
• Change percents into decimals
Percent Problems
• Translate as an equation. Use decimals.
Percent of Change
• Difference divided by the original amount
Simple Interest (p. 248)
• Amount of Interest=Principal x Annual Interest Rate x
Time (in years)
• A=Prt
• Maturity Value = Principal + Interest
• Monthly Payment = Maturity Value/Length of loan
(in months)
• Ex. 1: Find the maturity value of 5 year $1000 CD
with 5% simple interest.
• Ex. 2: Find the monthly payment for a 2 year loan of
$1000 with 7% interest.
Proportion problems
• When setting up a proportion, make sure that
you set up both sides comparing the same
way
• Apples to oranges = apples to oranges
• Not apples to oranges and oranges to apples
p
Geometry
Types of angles, properties of triangles,
types of polygons, solids, coordinate
geometry, congruence and similarity,
perimeter, area, and volume
Intro to Geometry
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Point - 0 dimensions
Line – 1 dimension
Line Segment – part of a line with 2 endpoints
Ray – part of a line with 1 endpoint
Plane and plane figures – 2d
Space and solids – 3d
Parallel lines
Intersecting lines
Types of Angles
• Angle – formed by two rays with common endpoint, measured in
degrees
• Right angle – 90 degrees
• Perpendicular lines – two lines that form a rt <
• Acute angle – less than 90 degrees
• Obtuse angle – more than 90 degrees
• Straight angle – 180 degrees
• Complementary angles – add up to 90
• Supplementary angles – add up to 180
• Vertical angles – formed by intersecting lines, across from each
other
• Adjacent angles – have a common endpoint and common side
Angles formed by parallel lines
• How many angles are formed from two
parallel lines and a transversal?
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Alternate Interior
Alternate Exterior
Same-Side Interior
Corresponding
Vertical
Adjacent
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Alternate Interior
3 & 6, 4 & 5
Alternate Exterior
1 & 8, 2 & 7
Same-Side Interior
4 & 6, 3 & 5
Corresponding
1 & 5, 3 & 7, 2 & 6, 4 & 8
Vertical
1 & 4, 2 & 3, 5 & 8, 6 & 7
Adjacent
1 & 2, 2 & 4, 1 & 3, 3 & 4, 5 & 6, 6 & 8, 8 & 7, 7 &5
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Triangles
Base – one side of a triangle
Height – perpendicular to the base
Angles add up to 180
Right Triangle – has a right angle (90 degrees)
Hypotenuse – longest side of a right triangle
Acute Triangle – all angles less than 90 degrees
Obtuse Triangle – has one angle more than 90
degrees
• Equilateral – all three sides congruent (same length)
• Isosceles – two sides congruent
• Scalene – no sides congruent
Quadrilaterals
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Parallelogram – two pairs of parallel lines
Rectangle – a parallelogram with 4 right angles
Rhombus – a parallelogram with 4 equal sides
Square – both a rectangle and rhombus
Trapezoid – only one pair of parallel sides
Isosceles Trapezoid – a trapezoid with congruent
sides
• Kite – has two pair of consecutive sides that are
congruent
Solids
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Prism
Pyramids
Cylinder
Cones
Sphere
TEST 1 MATERIAL ENDS
Midpoint Formula
• Given two points the midpoint is halfway
between them and is found by taking the
average of the x’s and the average of the y’s
S,m,x,mcll x xz
Pythagorean Theorem
• For right triangles
Polygons
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Triangle – 3 sides
Quadrilateral - 4
Pentagon - 5
Hexagon - 6
Heptagon -7
Octagon - 8
Nonagon - 9
Decagon -10
Dodecagon -12
Perimeter
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Perimeter is distance around a polygon
Shortcut formulas:
Rectangle P = 2L + 2W
Square P = 4s
Circumference is the name given to the length
around a circle
• C = 2πr
Area
• Area is how much space a shape contains
measured in square units
• Rectangle A = L x W
• Square A = s2
• Parallelogram A = b x h
• Triangle A = ½ b x h
• Circle A = πr2
Perimeter and Area of composite
figures
Congruent Triangles
• Congruent triangles have all the same angles,
and all the same side lengths. Congruent
means they are exactly the same.
• SSS
• SAS
• ASA
• AAS
Volume
• Volume is the measure of space in an
enclosed surface.
• Rectangular Prism V = L x W x H
• Cube V = s3
• Sphere V = (4/3)πr3
• Cylinder V = πr2h
• Cone V = (1/3) πr2h
Intro to Algebra
• Variable – letter standing for a number
• Constant- a known number
• Coefficient – number being multiplied by a
variable ex. 2x, the 2 is the coefficient
• Term – part of algebraic expression separated
by plus sign
• Like terms – have the same variables with the
same exponents
Properties
• Commutative property – a property of
addition or multiplication that allows one to
change the order
a+b=b+a
ab = ba
Ex:
1+2=2+1=3
2x4=4x2=8
Properties
• Associative property - a property of addition
or multiplication that allows one to change
the grouping
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Ex:
(1 + 2) + 3 = 1 + (2 + 3) = 6
(2 x 4) x 3 = 2 x (4 x 3) = 24
Properties
• Distributive property - a property of addition
and multiplication
a(b + c) = ab + ac
Ex:
2(3 + 5) = 6 + 10 = 16
2(3x – 4) = 6x - 8
Translating Verbal Expressions into
Algebraic Expressions
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Addition
Subtraction
Multiplication
Division
Raise to a power
TEST 2 MATERIAL ENDS
One step equations
Slope/Rate of Change
• Slope measures the steepness of a line
• Slope = rise/run
• Horizontal lines have 0 slope
• Vertical lines have undefined slope
Linear Equations
• y= mx + b
• m is the slope
• b is the y-intercept (where the graph crosses yaxis)
• Ex. 1: y = 2x - 3
• Ex. 2: y = -x + 2
• Ex. 3: y = ½ x
Solving Systems
• A system is two or more equations to solve
simultaneously
• Solve each equation for y
• Graph each equation
• Find the points of intersection
Descriptive Statistics
• Statistics concerns data and methods of
representing and analyzing data
Pictograph
Circle graph or Pie Chart
9%
27%
15%
18%
20%
11%
Scatterplot
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Series1
30
20
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Bar Graph
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Series1
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Line Graph
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Series1
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Histograms
Statistical Measures
• Arithmetic mean (average)
• Median – the middle number
– If the data set contains an even number of values,
take the mean of the two middle numbers
• Mode – the most frequent number
– There may be one mode, no mode, or more than
one mode
Box and Whisker Plot
• Find the median of the data
• Find the first quartile Q1 by taking the median of
all the values below the median
• Find the third quartile Q3 by taking the median of all
the values above the median
• The range of the data is the difference between the
highest number and the lowest number
• The interquartile range is the difference between Q3
and Q1
Box and Whisker Construction
21, 21, 23, 24, 35, 35, 36, 38, 40, 41, 46, 48
Counting techniques
• Multiplication principle for independent events: If one
step can be done in m different ways and the second
step can be done in n different ways, then together
they can be done in m x n ways.
• Example: If you have 6 shirts, 4 pants, and 2 pairs of
shoes, how many different outfits can you make?
• At a restaurant, you can choose one of 3 salads, one of
5 entrees, and one of 2 desserts. How many different
meals?
• A student ID is the students first initial, last initial, then
2 digits.
Permutations
• Permutations are ordered arrangements
• Factorial – multiply all the numbers up to and
including the number – n! = n x (n-1) x …x 3 x
2x1
• nPr – the number of ordered arrangements
from a group of n with r objects selected
Permutation Examples
• 3 people are running a race, how many different
ways can they place?
• 4 people are running a race, how many different
ways can they place?
• 5 people are running for president, vice
president, and secretary. How many different
possibilities?
• 6 people are running for president, vice
president, and secretary. How many different
possibilities?
Combinations
• Combinations are unordered arrangements
• nCr – the number of unordered arrangements
from a group of n with r objects selected
Combination Examples
• Five people are running for city council with three
to be elected. How many different possibilities?
• Six people are running for council with two to be
elected. How many different ways can this be
done?
• You decide to make a fruit salad using three of
the following: blueberries, bananas, strawberries,
peaches, grapes. How many different ways?
Probability
• Probability is the mathematics of chance and
measuring uncertainty
• Experimental vs. Theoretical probability
• Sample Space is the set of all possible
outcomes
• Event is one or more outcomes
• Probability
Probability
• Probability is a ratio between 0 and 1
• A probability of 0 means the event is impossible
• A probability of 1 means the event is certain to
happen
• Example 1: What is the probability of rolling a
number greater than 7 with one die?
• Example 2: What is the probability of rolling a
number less than 7 with one die?
• Example 3: What is the probability of rolling a
prime with one die?
Rolling two dice
• List all possible outcomes of rolling two dice
• How many ways can we get each number?
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Probability with 2 Dice
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P(even)
P(roll a 4)
P(roll a 7)
P(roll a number less than 5)
P(roll a number less than 4 or greater than 9)
P(roll a prime)
Flipping a coin
• A coin is flipped 2 times. How many different
ways can this be done?
• A coin is flipped 2 times what is the
probability of getting at least one head?
• A coin is flipped 4 times. How many different
ways can this be done?
• What is the probability of getting exactly two
heads?