The Praxis Mathematics Tests
Workshop PPT #2
Slide show created by Jolene M. Morris
ETS Praxis Website
Official website where you can
find more information about
the Praxis tests
www.ets.org/praxis/
NCTM Website
The National Council of Teachers of
Mathematics (NCTM) is the national
organization that sets the standards for what
mathematics concepts should be taught and
at what grade levels.
www.nctm.org
Jolene’s Website
Jolene’s website where you can find Praxis
resources such as Jolene’s 400 Praxis
flashcards in a format to import into an app
such as Flashcards To Go or Anki or Quizlet
www.JoleneMorris.com
Agenda (Slide 5)
6. Number Theory (hunter green)
32. Fractions (black)
52. Decimals (light blue)
63. Ratios, Proportions, & Percentages (dk blue)
83. Integers (red)
96. Geometry & Measurement (burgundy)
111. Data, Statistics, & Probability (pink)
126. Algebraic Reasoning (green)
Number Theory
Our Number System
Place Value
Ordering Large Numbers
• Write them above one another with the
place values lined up.
• Starting from the left, look for the largest
value.
• For example, if you are asked to order:
5,139 986,733 3,950 77,922
Rounding Numbers
• Rounding a number requires that you
understand place value.
• Look at the digit to the right of the place
being rounded.
• If that digit on the right is 5 or higher, add
1 to the place being rounded; otherwise,
leave the place being rounded as is.
• Change all places to the right of the place
being rounded to zeroes.
Comparison Symbols
< less than
> greater than
≤ less than or equal to
≥ greater than or equal to
= equal to
Each of these symbols can also be negated by
putting a slash mark through them, such as
not equal to: ≠
Estimation
Squared & Cubed Numbers
Divisibility Rules
2 = Even numbers (ending in 0, 2, 4, 6, and 8)
3 = If repeated sums of the digits result in 3, 6, or 9
4 = If the last two digits are divisible by 4
5 = If the last digit is 0 or 5
6 = If the number is divisible by both 2 and 3
8 = If the last three digits are divisible by 8
9 = If repeated sums of the digits result in 9
10 = If the last digit is 0
Prime Numbers
• Prime Numbers = Integers greater than 1 with
exactly 2 factors or divisors; numbers that are
evenly divisible by only 1 and themselves.
• The number 2 is the first prime and it is the only
even number that is prime. The number 1 is
neither prime nor composite. Memorize the
prime numbers 1-100:
• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Other Number Classifications
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Prime vs. Composite
Odd vs. Even
Denominate numbers
Consecutive integers
Cardinal numbers
Ordinal numbers
Number Line
• A number line is a straight line where each point
of that line corresponds to a real number.
• A line is made up of an infinite number of points
and there are an infinite amount of real numbers.
• Usually the line is marked off to show the
integers, including zero.
• A number line is generally written as a horizontal
line.
Expanded Notation
Interpreting Remainders
• Do you round up? Do you round down? Do you
use the remainder as part of the answer (or as
the entire answer)?
– How many boxes can be filled? (use only the quotient;
ignore the remainder)
– How many cans are needed to paint the wall? (round
the quotient to the next greater whole #)
– How many in the last box that isn’t completely full?
(use only the remainder)
Prime Factorization
Thus, the prime factorization of 120 = 2 x 2 x 2 x 3 x 5
GCF
12 = 2 × 2 × 3
18 = 2
×3×3
30 = 2
×3 ×5
GCF = 2
×3
The factors common to all three
numbers above are 2 and 3.
2 x 3 = 6 so 6 is the GCF of 12, 18, and
30.
LCM
Multiplying by Powers of 10
Dividing by Powers of 10
4 Ways to Indicate Multiplication
• Using a small “×”, such as 3 × 5. Note that the “×”
is not used for multiplication in algebra because it
might be confused with the variable “×”.
• Using a small, raised dot, such as 3 • 5
• Using parenthesis, such as (3)(5) or 3(5) or (3)5
• Using no symbol, such as 3y meaning 3 times y
Properties of Operations
Order of Operations
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Simplify inside parentheses or grouping symbols
Simplify any expressions with exponents
Perform multiplication & division from left to right
Perform addition & subtraction from left to right
PEMDAS
Please Excuse My Dear Aunt Sally
Properties of Zero
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Division by zero is undefined
Zero is neither positive nor negative
Zero is the additive identity
Any number multiplied by zero equals zero
Zero is used as the universal place holder
Zero is neither prime nor composite
Zero has no multiplicative inverse
A number with an exponent of zero equals 1
Zero factorial equals 1
Properties of One
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One is the multiplicative identity
Any number multiplied by one equals the number
One is neither prime nor composite
A number with exponent of 1 equals the number
The number one raised to any power equals one
Any nonzero number divided by itself equals one
Identity Property
Number Theory
ANY QUESTIONS?
Fractions
What is a Fraction?
• A fraction represents equal-sized
parts of a whole.
• The top number is called the
numerator
• The bottom number , the
denominator, is a denominate
number (measurement of size)
• The line between numerator &
denominator indicates division and
is called a vinculum.
Proper & Improper Fractions
• A proper fraction has a numerator smaller than the
denominator and indicates a fraction less than one
whole: 3/8 1/4 14/15 4/5
• An improper fraction has a numerator larger than (or
equal to) the denominator and indicates a fraction that
is equal to one or more than one whole: 3/2 7/4 16/15
15/
99/
15
5
• An improper fraction can be changed into a whole
number or a mixed number by dividing the
denominator into the numerator.
Common Fraction
Mixed Number
• A mixed number is a whole number and a
proper fraction combined. Mixed numbers may
also be called mixed fractions.
• This graphic shows two whole pizzas and a
fraction of 3 pieces out of 4  2 3/4
Greatest Common Factor
• The GCF is the largest number that is a factor of all the
numbers.
• One way to find the GCF is to write the prime
factorization of each of the numbers above each other.
Then “bring down” those factors that are in common
and multiply them:
Find the GCF of 12, 18, 30:
12 = 2 × 2 × 3
18 = 2
× 3 × 3
30 = 2
× 3
× 5
GCF = 2
× 3
= 6
LCD
• The Lowest Common Denominator is
the smallest number that is a multiple
of all the denominators.
• One way to find the LCD is to count by
each of the denominators and find the
first number that is a multiple of all.
• Another way to find the LCD is to write
a prime factorization of each
denominator, and then “bring down”
one of each factor and multiply.
Equivalent Fractions
• Fractions that simplify to the same simple
fraction.
• When two fractions are equivalent, their cross
products are equal :
3 x 6 = 18
2 x 9 = 18
Simplify Fractions
Mixed Number
• A mixed number indicates the whole amounts
and the parts (a fraction).
• To convert a mixed number to an improper
fraction, multiply the whole number by how
many parts are in a whole, and then add the
remaining parts.
For example, 6 2/3 means there are 6 whole amounts of 3/3 (6
x 3 = 18) so there are 18/3.
Adding the remaining 2/3 results in a total of 20/3 in 6 2/3.
Improper Fractions
• An improper fraction shows a total number of
parts, but in those parts is at least one whole.
• To convert an improper fraction to a mixed
number, divide the numerator by how many parts
are in a whole. The quotient becomes the whole
number and the remainder becomes the
numerator of the fraction part of the mixed
number.
For example, 23/3 equals 23 divided by 3, which is 7 r 2 or 7 2/3
Why Find Common Denominators?
3 7-footers
+ 2 6-footers
5 13-footers
Why Change to Common
Denominators?
Why Change to Common
Denominators?
3 apples
+ 4 oranges
7 ??????
Add Fractions &
Mixed Numbers
• Write the two fractions/mixed numbers
vertically above each other (lining up place
value)
• Change the fractions to a common
denominator.
• Add the numerators only.
• Put that sum over the common denominator.
• Simplify the answer.
Subtract Fractions &
Mixed Numbers
• Write the two fractions/mixed numbers vertically
above each other (lining up place value)
• Change the fractions to a common denominator.
• Subtract the numerators only (careful to regroup
one whole (2/2, 3/3, 4/4, etc.) if you need to
borrow).
• Put that difference over the common denominator.
• Simplify the answer.
Multiply Fractions&
Mixed Numbers
Reciprocal
• A reciprocal is a fraction where the numerator
and denominator have been switched.
• Multiplying any fraction by its reciprocal results
in an answer of 1. As such, a reciprocal is called
a multiplicative inverse.
NOTE: Since there is no rule on how to divide fractions, but
because multiplication is the inverse of division and a reciprocal is
the inverse of a fraction, you can divide fractions by multiplying by
the reciprocal of the divisor. Hence, the inverse of an inverse
results in the same answer as if you had divided.
Divide Fractions&
Mixed Numbers
• Change any mixed numbers and any whole numbers to
improper fractions with a “1” on the denominator.
• Write the two fractions horizontally beside each other.
• Write the reciprocal of the divisor (flip the second
fraction upside down) and change the operation to
multiplication.
• Expand each numerator and each denominator into a
prime factorization. “Cancel” any ones such as 3/3 or
5/5. Multiply what is left straight across.
Change Fractions to
Decimals & Percentages
• Change a fraction to a decimal by dividing the
denominator into the numerator. Keep dividing
until the decimal number repeats or
terminates. Draw a line (vinculum) above the
repeating portion.
• Change a fraction to a percent by first changing
it to a decimal as explained above, and then
moving the decimal point two places to the
right. Remember to append the percent
symbol.
Comparing & Ordering Fractions
• Cross Multiply (Means extremes property)
• More than two fractions: change them to common
denominators (or use reasoning, such as: are they more
than half)
Fractions
ANY QUESTIONS?
Decimals
Decimal Point
A decimal point is a period that indicates the
location of the one’s place – the decimal point
always comes to the right of the one’s place. If
there are no fractional decimal numbers to the right
of the decimal point, the decimal point doesn’t
have to be written. It is understood.
Comparing Decimal Numbers
Line up the decimal numbers according to
place value (as though you were going to add
them). Starting at the left-most place value,
compare the numbers in each place value to
find the largest, next largest, etc.
Addition
• Decimal numbers are added exactly the same as whole
numbers: line up the numbers by place value and add
each place value from the right to the left.
• When the decimal numbers are lined up by place value
properly, the decimal points in each number are also
lined up.
• Any number without a decimal point is lined up so the
ones place is right before the decimal point (there is an
understood decimal point after the one’s place).
• It may help to write zeros in empty places to facilitate
addition.
Subtraction
• Decimal numbers are subtracted exactly the same as
whole numbers: line up the numbers by place value and
subtract each place value from the right to the left.
• When the decimal numbers are lined up by place value
properly, the decimal points in each number are also
lined up.
• Any number without a decimal point is lined up so the
ones place is right before the decimal point (there is an
understood decimal point after the ones place). It may
help to write zeros in empty places to facilitate
subtraction.
Multiplication
• Decimal numbers are multiplied by temporarily ignoring
the decimal point. Multiply the two numbers as though
they were whole numbers. In the final product, place the
decimal point to signify the number of decimal places in
both numbers of the original problem.
• For example: 2.3 (one decimal place) x 1.456 (three
decimal places) is the same as 23 x 1456 with the answer
having four decimal places (1 + 3 from the original
problem)
Division
• Division is not defined for decimal numbers. In order to
divide by a decimal number, we change that divisor into a
whole number: First multiply each number by powers of
10 – multiply by whatever is necessary to make the
divisor a whole number. Then divide as you would with
whole numbers. Wherever the decimal point is in the
dividend, it floats directly up to that position in the
quotient (answer).
Multiplying by Powers of 10
Dividing by Powers of 10
Scientific Notation
Scientific Notation is a way to write very large or very small
numbers using powers of 10. To convert a number into
scientific notation, move the decimal point so the resulting
number is between 1 and 10. Then state the power of 10.
Because we use a Base 10 number system, an easy way to
know what power of 10 is needed, the exponent indicates
the number of decimal places the decimal point was moved.
The exponent is negative if the decimal point was moved to
the right; the exponent is positive if the decimal point was
moved to the left.
1234.5  1.2345 × 103
Decimals
ANY QUESTIONS?
Ratios, Proportions, &
Percentages
What is a Ratio?
• A ratio is a comparison of two numbers
using division.
• Write a ratio using a fraction bar, a colon,
or the word “to”
3:2
3/2
3 to 2
Fractions & Ratios
• A fraction compares PART to WHOLE
• A ratio compares any two numbers
• Convert a fraction into a ratio by changing
denominator to the difference (D – N)
• Example: 2/3 becomes 2:1
What is a Proportion?
How to Solve Proportions
What is a Percentage?
• A percentage (or a percent) is a way of
expressing a number, especially a ratio, as
a fraction of 100.
(per = divided by; cent = 100)
• The percent key on a calculator merely
divides by 100. If your calculator doesn’t
have a percent key, hit the divide key and
then 100.
3 Types of % Problems
• What number is 15% of 45? 
x = (0.15) ∙ (45)
• What percent of 45 is 15? 
45 ∙ x = 15 or 45x = 15
• 15% of what number is 45? 
(0.15) ∙ x = 45 or 0.15x=45
Convert % to Decimals & Fractions
• Change a percent to a decimal by moving
the decimal point two places to the left
and removing the percent sign: 14% = 0.14
• Change a percent to a fraction by writing
the percent as a fraction over 100 and
simplifying:
14% = 14/100, which simplifies to 7/50
• Remember: per = divided by; cent = 100
Convert Decimals to %
• Change a decimal to a percentage by
moving the decimal point two places to the
right and appending the percent sign:
0.14 = 14%
Solve % Using Proportions
Solve % Using Algebra
To solve percentages using algebra, write the
problem as an algebraic statement where
what number  variable (x)
is  =
of  multiply
What number is 15% of 45?  x = (0.15) ∙ (45)
What percent of 45 is 15?  45 ∙ x = 15 or 45x = 15
15% of what number is 45?  (0.15) ∙ x = 45 or 0.15x=45
Unit Analysis
Unit Analysis is the process of multiplying by
successive conversion units (written in
fraction form).
Unit Analysis Video: JoleneMorris.com, Math 115, Wk 2
Video explaining unit analysis for English units
(JoleneMorris.com, Math 115, Wk6)
Video explaining unit analysis for Metric units
(JoleneMorris.com, Math 115, Wk6)
Denominate Number
• Specifies a quantity in terms of a number
and a unit of measurement.
• For example, 7 feet and 16 acres are
denominate numbers.
Rate & Unit Rate
• A rate is a ratio between two
measurements with different units.
• In addition to the three ways to write a
ratio, rates may also use the word “per”.
• Rates are usually simplified to a one in the
denominator (unit rate).
13 miles per gallon
$4.59 per pound
12 inches per foot
Interest
Sales Tax
Sales Tax is written in percentages (which are
converted to decimal to computer sales tax).
FP = MP + (ST × MP)
If you know two of those three amounts, you can use
basic algebra to find the missing number. Remember to
state the sales tax as a percentage in application
problems.
Discounts
• Usually written as percentages.
• Amount by which the purchase price is reduced.
• Discount amount is the original sales price
multiplied by the percentage of discount.
• Discounted price is the difference of the original
price and the discount amount.
OP – (OP * D) = DP
where OP is the original price, D is the
discount percentage written as a
decimal number, and DP is the discounted price.
Percent of Change
• The percent of change is also known as the
percent of increase or the percent of
decrease.
• To calculate the percent of change,
– Find the difference between the new amount
and the original amount
– Divide that difference by the original amount
– Multiply by 100
Sales Commission
• C = ar, where C = commission earned,
a = amount of sale, and r = commission
rate.
• Example: Juana sells cars on a 3%
commission rate. She just sold a car for
$23,500. What was her commission?
C = ar
C = 23500(.03)
C = $705
Ratios, Proportions, & Percentages
ANY QUESTIONS?
Integers
What are Integers?
• The counting numbers, their negatives, and
zero (…, -3, -2, -1, 0, 1, 2, 3 …)
• The symbol used for the set of integers is
ℤ.
Number Line
• A number line is a straight line where each point
of that line corresponds to a real number.
• A line is made up of an infinite number of points
and there are an infinite amount of real numbers.
• Usually the line is marked off to show the
integers, including zero.
• A number line is generally written as a horizontal
line.
Absolute Value
• The value portion of a number without a
sign.
• Also described as the distance on a
number line from 0.
• Zero is the only number that is its own
absolute value (because zero is neither
positive nor negative).
Properties of Absolute Value
Ordering & Comparing Integers
• Remember that negative numbers are
always smaller than positive numbers.
• It helps to place the numbers on a number
line to compare them.
– The larger the value of a positive number, the
larger the number is.
– The larger the absolute value of a negative
number, the smaller the number.
Classroom Activity
Adding Integers
• If the signs are the same, add the absolute
values of the numbers and give the result
their same sign.
• If the signs are different, subtract the
absolute values of the numbers and give
them the same sign as the number with
the larger absolute value.
Subtracting Integers
Because a negative number is the inverse of
a positive number, and because subtraction
is the inverse operation of addition, the RULE
for subtracting integers is: Change the sign of
the second number to its opposite and
change the operation to addition. Then,
follow the rules for adding integers.
Multiplying Integers
• If the signs are the same, multiply the
absolute values of the numbers and give
the result a positive sign.
• If the signs are different, multiply the
absolute values of the numbers and give
the result a negative sign.
Dividing Integers
• If the signs are the same, divide the
absolute values of the numbers and give
the result a positive sign.
• If the signs are different, divide the
absolute values of the numbers and give
the result a negative sign.
Review Properties
in Number Theory Packet
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Zero
One
Identity (additive & multiplicative)
Inverse (additive & multiplicative)
Integers
ANY QUESTIONS?
Geometry & Measurement
Geometry Symbols
Definitions
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Point
Line
Plane
Line Segment
Ray
Angle
Parallel
Perpendicular
Vertex
Arc
Coplanar
Collinear
Bisector
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Chord
Diameter
Radius
Circumference
Symmetry
Surface Area
Perimeter
Area
Volume
Transversal
Polygon
Similar
Congruent
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Protractor
Compass
Density
Unit Analysis
Golden Ratio
Angles
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An acute angle measures less than 90°
A straight angle measures 180°
A right angle measures 90°
An obtuse angle measures
between 90°-180°
• A reflex angle is measured in
a clockwise direction as
opposed to the normal
counter-clockwise direction.
More About Angles
• When two lines intersect, they form four angles. The two
angles opposite each other are called vertical angles.
• Complementary angles are two angles whose measure
adds to 90°.
• Supplementary angles are two angles whose measure
adds to 180°.
• Two angles are adjacent angles if they share a common
vertex, they share a common side, AND they do not share
any interior points.
• An exterior angle is an angle on the outside of a polygon
that is formed by extending the side of the polygon
• When a transversal line crosses two other lines, it forms
eight angles -- Corresponding angles are angles that are
in the same position on each of the lines.
Triangles
• Classify triangles by their legs
– Scalene triangle - all legs are of different length.
– Equilateral triangle - all three legs (sides) are of
equal measure.
– Isosceles triangle - two of the three legs are of
equal measure
• Classify triangles by their angles
– Right triangle - one angle is a right angle.
– Acute triangle - all three angles are less than 90°
– Obtuse triangle - one of the angles is obtuse.
Pythagorean Theorem
3-Dimensional Solids
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Sphere
Rectangular Solid (Cuboid)
Cube
Cylinder
Cone
Prism
Pyramid
Euler’s Polyhedron
Formula is V – E + F = 2
Polyhedron
Measurement
• English System or Common System
– Length = inch, foot, yard, rod, mile, etc.
– Weight = ounce, pound, Ton, etc.
– Volume = liquid ounces, cup, pint, quart,
gallon, etc.
• Metric System
– Meter
– Gram
– Liter
Temperature
Converting Units of Measurement
Tessellations
• A tessellation is a two-dimensional plane created
by one or more polygon shapes fitted into each
other so no “open space” remains.
• Equilateral triangles, squares, and hexagons are
the only regular polygons that tessellate.
• Check out the tessellations
of M. C. Escher:
Transformations
A transformation in geometry changes the
position of a shape on the coordinate plane.
There are four forms of transformation:
– translation (slide)
– rotation (turn)
– dilation (scale)
– reflection (flip)
Net (or network)
A net is a two-dimensional representation of
a three-dimensional object. If a net is cut
out, it can be put together to form the threedimensional object it represents.
Geometry & Measurement
ANY QUESTIONS?
Data, Statistics, & Probability
Measures of Central Tendency
Data has a tendency to cluster or center on
certain values. The term “average” is also
used to indicate measures of central
tendency.
– Mean (evenly distributed / “average”)
– Mode (bimodal distribution / most popular)
– Median (outliers)
– Range
Quartiles
• Quartiles are three points that divide a set of
ordered data into four equal groups.
• The first quartile, also called the lower quartile,
splits off the lower 25% of the data. It is denoted
by Q1
• The second quartile, also called the median,
splits the data in half. It is denoted by Q2
• The third quartile, also called the upper quartile,
splits off the higher 25% of the data. It is denoted
by Q3
Trends
• A trend is the general direction data tends to move. From
a line graph, a trend can be obvious when the line is going
in an up or down pattern.
• Example: In the stock market, when stocks are trending
down, it is called a bear market. When stocks are trending
up, it is called a bull market. (A mnemonic to remember
which is which: A bear has claws that curve downward
and a bull has horns which curve upward.)
Algorithm
• An algorithm is a step-by-step process for solving
a problem.
• An example of an addition algorithm is:
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Line up the numbers
Add each column starting on the right
Carry any tens-place digits to the next column
Place commas between periods in the answer
• An algorithm is often written as a flowchart
showing steps, branches, and decisions.
Probability
• Ratio of how likely a specific event is to happen when
compared to all possibilities of events that might happen.
• Most often written as a fraction, but it may also be
written as a decimal or a percentage.
• Odds and probability are related concepts. With odds,
you compare the number of favorable outcomes to the
number of remaining (unfavorable) outcomes.
Probability of Multiple Events
• Independent (OR) , mutually exclusive 
add
• Independent, non-exclusive  add then
subtract the events in common
• Dependent (AND)  multiply
(Fundamental Counting Principle)
• If dependent, be sure to use the
conditional second probability
Odds
• Related to probability
• Probability  favorable outcomes /
possible outcomes
• Odds  favorable outcomes / unfavorable
outcomes
EXAMPLE: If you have a box with 2 red balls and 3
blue balls, the probability of randomly picking a red
ball is 2 out of 5 or 2/5. The odds of randomly
picking a red ball are 2 for and 3 against, or 2:3
5 Rules of Probability
Tree Diagram
A tree diagram is a graphic organizer that lists all
possibilities of a sequence of events in a systematic
way.
Factorial
Sequences
• Arithmetic – a common number added
• Geometric – a common number multiplied
• Fibonacci – each term comprised of the
sum of previous two terms
Series
• Harmonic – sum of progressive unit
fractions
Charts & Graphs
(see study packet)
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Bar
(broken scale)
Histogram
Line
Circle or Pie
Pictograph
Table
Scatterplot
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Venn Diagram
Stem & Leaf Plot
Box & Whiskers Plot
Logic Diagram
Flow Chart
Deductive & Inductive
• Deductive reasoning is a form of logic starting
with statements of fact and drawing logical
conclusions. If the laws of logic are followed
from the statements of fact, the conclusions are
true. It often helps to draw logic circles when
working with deductive reasoning.
• Inductive reasoning is making sufficient
observations that conclusions can be formed.
Data, Statistics, & Probability
ANY QUESTIONS?
Algebraic Reasoning
Algebra & Algebraic Thinking
• Algebra is the study of numbers, number patterns, and
relationships among numbers.
• Algebraic thinking is the study of our number system,
patterns, representations, and mathematical reasoning.
• A variable is used in algebra to represent a value that
changes within the parameters of the problem.
Lowercase letters of the alphabet are generally used to
denote a variable.
• The opposite of a variable is a constant.
• The number multiplied to the variable is a coefficient.
-2x3
Real Number System
Properties of Operations
Order of Operations
• Simplify inside parentheses or grouping symbols
• Simplify any expressions with exponents
• Perform multiplication & division from left to
right
• Perform addition & subtraction from left to right
PEMDAS
Please Excuse My Dear Aunt Sally
Expressions & Equations
• An expression is a collection of terms that have
been added or subtracted.
• An equation is a statement where an algebraic
expression is equal to another algebraic
expression or constant.
• A literal equation is an equation made up of only
known, measurable quantities. A literal equation
is the same as a formula.
• An inequality is similar to an equation, but the
two sides are NOT equal.
Words that Signal + - × ÷
• Addition: add, sum, increase, total, rise, plus,
grow, added to, more than, increased by, gain…
• Subtraction: subtract, subtracted from, minus,
difference, take away less than, decreased by…
• Multiplication: multiply, multiplied by, product,
times, of, twice…
• Division: divide, divided by, quotient, per, ratio,
half, …
Function
• Relationship is simply a set of ordered
pairs.
• Function is where each input is related to
exactly one output.
• One-to-one Correspondence is a function
where each output has exactly one input.
• Domain (x)
• Range (y)
Absolute Value
• Absolute value is the value portion of a
number without a sign. Absolute values are
also described as the distance on a number
line from 0. Zero is the only number that is
its own absolute value (because zero is
neither positive nor negative).
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Properties of Absolute Value
Literal Equation
• A literal equation is an equation made up of only
known, measurable quantities. A literal equation
is the same as a formula.
• With a literal equation, you are not solving for an
unknown quantity that varies. Instead, you are
manipulating the letters/variables in the equation
to a different form to substitute values in it.
Solving Algebraic Word Problems
(see the study packet)
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Consecutive numbers
Rectangular area & perimeter
Triangles
Unit conversion
Mixtures
Investments with interest
Discounts & Commissions
Distance-Speed-Time & Uniform motion
Simultaneous Equations
• Simultaneous Equations are two or more
equations with multiple variables. These
are often called systems of equations.
There are many ways to solve a system of
equations. Three ways discussed in
beginning algebra are:
– Elimination (sometimes called adding)
– Substitution
– Graphing
Polynomials
A polynomial is an algebraic expression with one or
more terms. A polynomial cannot have a variable in
the denominator (which is a negative exponent). A
polynomial with one term is called a monomial; two
terms, a binomial; and three terms, a trinomial. The
term with the highest exponent (sum) determines
the degree of the polynomial.
degree
type of
polynomial
1
linear
2
quadratic
3
cubic
Classifying Polynomials
Factoring Polynomials
• Factor out any common factors in all terms.
• If the polynomial has four terms, factor it by grouping.
• If it is a binomial, look for a difference of squares, a sum
of cubes, or a difference of cubes. (Note that a sum of
squares cannot be factored.)
• If it is a trinomial and the coefficient of the x2 term = 1,
un-FOIL to factor.
• If it is a trinomial and the coefficient of the x2 term ≠ 1,
use the AC method to factor.
Quadratic Equation
• A quadratic equation is a second-degree polynomial
equation (the exponent on the leading term is a 2). There
are many ways to solve a quadratic equation, but the five
most common ways are:
– Factor and set each factor equal to 0
– If there is no x-term, solve for x2 and apply the square root
method.
– Graph the equation (as a parabola) and determine the solutions
where the parabola crosses the x-axis
– Complete the square
– Use the quadratic formula
Distance Formula
The distance formula is used to find the distance
between two points. The distance formula can be
obtained by creating a triangle and using the
Pythagorean Theorem to find the length of the
hypotenuse. The hypotenuse of the triangle will be
the distance between the two points.
Coordinate Grid
A coordinate plane is a two-dimensional grid for locating
points. There is an x-axis and a y-axis at 90-degree angles,
which divide the grid into four quadrants that are numbered
counter-clockwise using Roman numerals. The origin is
where the two axes cross (0, 0). A coordinate pair is a pair of
numbers indicating the location of a point (x, y). Sometimes
called a Cartesian grid after the mathematician René
Descartes (1596-1650)
Slope of a Line
The slope of a line is an algebraic concept used to graph
linear equations. In the equation y = mx + b, the variable m
represents the slope of the line. Slope is calculated by
dividing the change in the y-coordinate (the rise) by the
change in the x-coordinate (the run). Parallel lines have
equivalent slopes. Perpendicular lines have slopes that are
negative and reciprocal of each other. To graph a line when
the slope and the y-intercept are known, plot the y-intercept
and then use the slope to count UP and OVER to find
another point on the line.
Graphing a Quadratic Equation
Rules for Exponents
Rules for Square Roots
Simplifying Square Roots
Approximating Square Roots
Approximating square roots means to find the approximate
value of a number’s square root. We find approximate
square roots by comparing the number to perfect square
numbers where the square roots are known. For example, to
find the approximate square root of 51, use the fact that 7 x
7 = 49 and 8 x 8 = 64. Since 51 is between the perfect
squares of 49 and 64 (but closer to 49 than 64), the
approximate square root of 51 is between 7 and 8 (but
closer to 7 than 8). The approximate square root of 51 is 7.1
or 7.2
Algebraic Reasoning
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Praxis Workshop PPT #2
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