Praxis I & II Mathematics Flashcards
This file contains 400 mathematics flashcards for the Praxis tests. The flashcards were created by Jolene
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PRAXIS FLASHCARD #1
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
First 10 Square Numbers
PRAXIS FLASHCARD #2
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
First 10 Cubed Numbers
PRAXIS FLASHCARD #3
Divisibility Rules
Knowing the divisibility rules is not critical to your knowledge
of teaching elementary school; however, the divisibility rules
will make it easier for you to find factors, GCF, LCM, and LCD.
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
PRAXIS FLASHCARD #4
Classify Numbers in Real
Number System
PRAXIS FLASHCARD #5
Complex Numbers
Complex Numbers are numbers made up of a real
number plus an imaginary number; usually written in
the form
where “a” is the real number and “bi”
is the imaginary number. The letter-i is used to denote
the imaginary number:
√
PRAXIS FLASHCARD #6
Real Numbers are numbers that can be located on the
number line. The opposite of real numbers are
imaginary numbers. The symbol used for the set of real
numbers is ℝ.
Real Numbers
PRAXIS FLASHCARD #7
Imaginary Numbers are numbers that contain the
imaginary number “i”, which is the square root of
negative one:
√
Imaginary Numbers
Note that if the discriminant portion of the quadratic
equation is negative, the function or quadratic equation
has no real solutions.
The symbol used for the set of complex numbers is ℂ.
PRAXIS FLASHCARD #8
Rational Numbers = integers, fractional numbers, and
those decimal numbers which terminate or repeat;
examples are 12.5 and 23.666666… The symbol used
for the set of rational numbers is ℚ.
Rational Numbers
The opposite of rational numbers are the irrational
numbers. All real numbers are either rational or
irrational.
PRAXIS FLASHCARD #9
Irrational Numbers
Irrational numbers = all real numbers except the
rational numbers; those numbers that are square roots
of non-square numbers or are non-terminating, nonrepeating decimal. Perhaps the most well-known
irrational number is pi (π) which is approximately equal
to 22/7 or 3.14 The opposite of Irrational numbers are
the rational numbers – all real numbers are either
rational or irrational.
PRAXIS FLASHCARD #10
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.
Integers
PRAXIS FLASHCARD #11
Whole numbers = the counting numbers and zero
(0, 1, 2, 3, 4, …). . The symbol used for the set of whole
numbers is W.
Whole Numbers
PRAXIS FLASHCARD #12
Natural numbers = same as the counting numbers
(1, 2, 3, 4 …). . The symbol used for the set of natural
numbers is ℕ.
Natural Numbers
PRAXIS FLASHCARD #13
Counting numbers = same as the natural numbers
(1, 2, 3, 4, …). A bold capital letter-N is often used to
represent the set of natural numbers.
Counting Numbers
PRAXIS FLASHCARD #14
Prime Numbers = Integers greater than 1 with exactly 2
factors or divisors; numbers that are evenly divisible by
only 1 and themselves.
Prime Numbers
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
PRAXIS FLASHCARD #15
Composite Numbers are numbers that have more than
two factors or divisors; numbers that are not prime.
All whole numbers except for 1 and 0 are either prime
or composite.
Composite Numbers
PRAXIS FLASHCARD #16
Factors are numbers that divide evenly into other numbers –
without a remainder. For example, 5 divides into 40 evenly so
5 is a factor of 40.
Factor
We often create a factor tree or a prime factorization of
numbers to help us recognize the factors of a number:
Thus, the prime factorization of 120 = 2 x 2 x 2 x 3 x 5
PRAXIS FLASHCARD #17
Factor Tree
A factor tree is a method used to find a number’s prime
factorization. Start by splitting the number into any two
factors that multiply to make that number. Then split each of
those two factors into two factors each. Keep splitting each
“branch” of the factor tree until you reach a prime number,
which cannot be split into factors. The final prime numbers at
the end of each “branch” of the factor tree are the prime
factorization of the number.
Thus, the prime factorization of 120 = 2 x 2 x 2 x 3 x 5
PRAXIS FLASHCARD #18
Prime Factorization
A prime factorization of a number is a list of all prime factors
that multiply together to make that number. A factor tree is
often used to find the prime factorization of a number. The
prime factorization can be written as the product of individual
factors or exponents can be used to write the product of
repeated prime factors:
Thus, the prime factorization of 120 = 2 x 2 x 2 x 3 x 5
3
This can also be written as 120 = 2 x 3 x 5
PRAXIS FLASHCARD #19
GCF
The Greatest Common Factor (GCF) of two or more numbers
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
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.
PRAXIS FLASHCARD #20
The Lowest Common Multiple (LCM) of two or more numbers is the
smallest number that is a multiple of all the numbers. One way to find
the LCM is to count by each of the
numbers and find the first number
that is a multiple of all. For example,
find the LCM of 9, 12, and 18 
LCM
Another way to find the LCM is to
write the prime factorization of each of the numbers above each other
with the factors all lined up. Then “bring down” one of each factor and
multiply them:
PRAXIS FLASHCARD #21
Properties of Number System Operations
PRAXIS FLASHCARD #22
When you are asked to simplify or evaluate an
expression, you must follow the Order of Operations:
Order of Operations
1. Simplify inside parentheses or grouping symbols
2. Simplify any expressions with exponents
3. Perform multiplication & division from left to right
4. Perform addition & subtraction from left to right
Several algebra textbooks teach one or both of the
following mnemonics to remember the Order of
Operations:
PEMDAS
Please Excuse My Dear Aunt Sally
PRAXIS FLASHCARD #23
Parentheses
Parentheses are a way to group numbers. Other
grouping symbols are braces { }, square brackets [ ], and
the vinculum or fraction bar. To remove parentheses,
we distribute the number immediately outside the
parenthesis (with its sign). We distribute by multiplying
by the number. If the outside sign is “hidden,” it is
understood to be positive (see Example 1 below). If the
outside number is “hidden,” it is understood to be a 1
(see Example 2 below). For example:
3(3x +1) = 9x + 3
-(2x – 5) = -2x + 5
PRAXIS FLASHCARD #24
There are four ways to indicate multiplication:
4 Ways to Indicate Multiplication
1. Using a small “×”, such as 3 × 5. Note that the “×” is
not used to indicate multiplication in algebra
because it might be confused with the variable “×”.
2. Using a small, raised dot, such as 3 • 5
3. Using parenthesis, such as (3)(5) or 3(5) or (3)5
4. Using no symbol, such as 3y (which means 3 times
y). Note that the “no symbol” is not used between
two numbers because it might be confused – for
example, 35 represents the number thirty-five, not
3 times 5.
PRAXIS FLASHCARD #25
Average
The common meaning of an average is to find the
arithmetic mean. To average a group of numbers, add all
the numbers together and divide by how many numbers
there are. For example, the average of 5, 7, 12, and 8 is (5
+ 7 + 12 + 8) / 4 = 8.
Some related statistical measures are median, mode,
range, maximum, and minimum.
PRAXIS FLASHCARD #26
Rounding Numbers
Rounding a number requires that you understand place
value. Rounding a number is a type of estimation.
Rounding is also called “rounding off.” To round a
number, 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.
PRAXIS FLASHCARD #27
A number in standard form is marked into groups of three
digits using commas. Each of these groups is called a period.
Whole Number Place Value
Within each group, the place values are always the 100’s
place, the 10’s place, and the 1’s place (from left to right).
Understanding place value is key to understanding our number
system. Decimal numbers simply extend the place values to
the right and use “ths” to identify the places (e.g. 100
millionths place).
PRAXIS FLASHCARD #28
When you are asked to order large numbers, write them above one
another with the place values lined up. Then, starting from the left, look
for the largest value. For example, if you are asked to order:
5,139
Ordering Whole Numbers
986,733
3,950
77,922
Write them above each other with the place values lined up as you
would if you were going to add the numbers. Looking at the place
values from left to right, the largest number is 986,733. The next largest
number is 77,922. Both the first and third numbers start in the same
place value but 5 is larger than 3 so 5,139 is larger than 3,950:
PRAXIS FLASHCARD #29
A rectangle is any quadrilateral with four right angles.
Rectangle
(Definition, area, and perimeter)
PRAXIS FLASHCARD #30
A triangle is a polygon with three angles/vertices and
three sides made up of line segments. A triangle can be
named by its three vertices:
.
Triangle
(Definition, area, and perimeter)
PRAXIS FLASHCARD #31
A square is a regular polygon made up of four equal
sides and four equal angles of 90 degrees each.
Square
(Definition, area, and perimeter)
PRAXIS FLASHCARD #32
A parallelogram is a quadrilateral with two pairs of
parallel sides.
(
Parallelogram
(Definition, area, and perimeter)
)
PRAXIS FLASHCARD #33
An even number is an integer that is evenly divisible by
2 (without a remainder). Note that the number zero is
an even number.
Odd vs. Even Numbers
An odd number is an integer that is NOT evenly
divisible by 2.
PRAXIS FLASHCARD #34
Trapezoid
A trapezoid is a convex quadrilateral with at least one
pair of parallel sides. (Outside of the United States, a
trapezoid is called a trapezium.) The parallel sides are
called bases and the other two sides are called legs.
(
)
(Definition, area, and perimeter)
PRAXIS FLASHCARD #35
To find the area of irregular shapes, divide the shape
into regular two-dimensional shapes or picture a
regular shape that has been removed:
Area of Irregular Shapes
Rectangle with a
right triangle
removed
(subtract)
Square and a half
circle combined (add)
PRAXIS FLASHCARD #36
A circle is a closed figure made up of all points that are
equidistant from another point (the center). The
distance from the center point to the edge of the circle
is called the radius.
Circle
(Definition, area, and circumference)
or
PRAXIS FLASHCARD #37
Rectangular Solid
A rectangular solid (also known as a cuboid) is a threedimensional solid where all angles are right angles and
opposite faces are equal. A rectangular solid is also
informally called a rectangular box.
(Definition, volume, and surface area)
PRAXIS FLASHCARD #38
A cube is a three-dimensional solid where all angles are
right angles and all faces are squares. A cube is also
informally called a square box.
Cube
(Definition, volume, and surface area)
PRAXIS FLASHCARD #39
A cylinder is a three-dimensional solid where the top
and bottom are circles
Cylinder
(Definition, volume, and surface area)
PRAXIS FLASHCARD #40
The two-dimensional measure of how many square
units can fit inside the interior of an object
Area
(Definition)
PRAXIS FLASHCARD #41
Decimal System Place Value
A number in standard form is marked into groups of three
digits. Each of these groups is called a period. Within each
group, the place values are always the 100’s place, the 10’s
place, and the 1’s place. Decimal numbers simply extend the
place values to the right and use “ths” to identify the places
(e.g. 100 millionths place). There is not a oneths place.
PRAXIS FLASHCARD #42
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.
Decimal Number Addition
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.
PRAXIS FLASHCARD #43
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.
Decimal Number Subtraction
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.
PRAXIS FLASHCARD #44
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.
Decimal Number Multiplication
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)
PRAXIS FLASHCARD #45
Decimal Number 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).
PRAXIS FLASHCARD #46
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.
Comparing Decimal Numbers
PRAXIS FLASHCARD #47
Fractions
(Definition, Meaning, and Parts)
A fraction represents equal-sized parts of a whole. The top
number of a fraction is called the numerator because it is the
number of parts. The denominator is a denominate number
(measurement of size) telling how many of the equal-sized
parts are in the whole. The line between the numerator and
denominator indicates division and is called a vinculum.
Note that the parts MUST be equal-sized. The parts are usually
indicated by coloring them or shading them:
PRAXIS FLASHCARD #48
A proper fraction has a numerator smaller than the
denominator and indicates a fraction less than one
whole:
Proper Fraction
PRAXIS FLASHCARD #49
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:
Improper Fraction & Mixed Numbers
(Definition)
An improper fraction can be changed into a whole
number or a mixed number by dividing the
denominator into the numerator. A mixed number is
the sum of a non-zero integer and a proper fraction.
PRAXIS FLASHCARD #50
Multiplying Fractions &
Mixed Numbers
Change any mixed numbers to improper fractions.
Change any whole numbers to improper fractions with
a “1” on the denominator. Write the two fractions to be
multiplied horizontally beside each other. 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:
PRAXIS FLASHCARD #51
Division of Fractions &
Mixed Numbers
Change any mixed numbers to improper fractions. Change 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.
* A reciprocal is the inverse of a fraction. Multiplication is
the inverse of division. The inverse of an inverse is the
same as the original problem. As such, multiplication of a
reciprocal is the same as dividing by the original divisor.
PRAXIS FLASHCARD #52
Simplifying Fractions
Simplifying fractions is sometimes (incorrectly) called
reducing a fraction. Expand the numerator and
denominator into prime factorizations. “Cancel” any
ones such as 3/3 or 5/5. Then multiply straight across
those numbers that are left. The resulting (simplified)
fraction is equivalent to the original fraction but is in
simplified form.
PRAXIS FLASHCARD #53
To add fractions & mixed numbers:
1.
Addition of Fractions &
Mixed Numbers
2.
3.
4.
5.
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.
PRAXIS FLASHCARD #54
To subtract fractions & mixed numbers:
1.
Subtraction of Fractions &
Mixed Numbers
2.
3.
4.
5.
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.
PRAXIS FLASHCARD #55
A ratio is a comparison of two numbers using division.
Write a ratio using a fraction bar, a colon, or the word
“to”:
Ratios
(Definition & three ways to write)
3:2
3/2
3 to 2
PRAXIS FLASHCARD #56
A proportion is when two ratios are equivalent:
Proportions
When two ratios are equivalent, the cross products are
equal :
(Definition)
3 x 6 = 18
2 x 9 = 18
PRAXIS FLASHCARD #57
Equivalent fractions are fractions which simplify to the
same simple fraction.
Equivalent Fractions
When two fractions are equivalent, their cross products
are equal :
3 x 6 = 18
2 x 9 = 18
PRAXIS FLASHCARD #58
Rates
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 (second
measurement).
13 miles per gallon
$4.59 per pound
12 inches per foot
PRAXIS FLASHCARD #59
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)
Percentages
(Definition)
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.
PRAXIS FLASHCARD #60
Change a percent to a decimal by moving the decimal
point two places to the left and removing the percent
sign: 14% = 0.14
Converting Percentages
to Decimals & Fractions
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
PRAXIS FLASHCARD #61
Change a decimal to a percent by moving the decimal
point two places to the right and appending a percent
sign: 0.14 = 14%
Converting Decimals
to Percentages & Fractions
Change a decimal to a fraction by writing the decimal
over a power of 10 representing the right-most place
value in the decimal, and then simplifying:
0.146 = 146/1000 = 73/500
(An easy way to find the power of 10 is to put one zero
in the denominator for each decimal place in the
number.)
PRAXIS FLASHCARD #62
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.
Converting Fractions
to Decimals & Percentages
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.
PRAXIS FLASHCARD #63
To solve percentages using the percent proportion, use
the means-extreme property of proportions (cross
multiply).
Solving Percentages
The percent proportion can be written as:
(using the Percent Proportion)
PRAXIS FLASHCARD #64
To solve percentages using algebra, write the problem
as an algebraic statement where
Solving Percentages
(using the algebraic equations)
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
PRAXIS FLASHCARD #65
Probability
Probability is the ratio of how likely a specific event is
to happen when compared to all possibilities of events
that might happen. Probability is most often written as
a fraction, but it may also be written as a decimal or a
percentage. The numerator of the fraction tells how
many possibilities of a specific event, the denominator
tells how many total possibilities. For example, in a
deck of face cards, the probability of drawing a heart is
13 out of 52 (13 hearts in a deck of 52 face cards). That
is written as 13/52 but simplified to 1/4.
The three types of probabilities are
classical, empirical, and subjective.
PRAXIS FLASHCARD #66
Probability of Multiple Events
The probability of multiple events has different
calculations depending on whether the events
are independent (OR) or dependent (AND) and
whether the events are mutually exclusive (have
possibilities in common)
 Independent, mutually exclusive  add
 Independent, non-exclusive  add then subtract
the events they have in common
 Dependent  multiply (first event doesn’t affect
probability of the second event)
 Dependent  multiply first probability and the
conditional second probability
PRAXIS FLASHCARD #67
A combination is a way of selecting several things out
of a larger group, where order does not matter.
Combinations
n is the number of items selected
k is the number of items in the larger group
(Note that this question may not be on Praxis 1)
PRAXIS FLASHCARD #68
A permutation is a way of selecting several things out
of a larger group, where order does matter. (permute =
changing order)
Permutations
n is the number of items selected
k is the number of items in the larger group
(Note that this question may not be on Praxis 1)
PRAXIS FLASHCARD #69
Coordinate Plane
(Definition to include five terms)
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)
PRAXIS FLASHCARD #70
Plotting a Point on
a Coordinate Grid
To plot a point with the coordinates of (x, y), we follow
along the x-axis until we get to the first coordinate, and
then we follow along the y-axis until we reach the
second coordinate. We mark a small dot at the location
where these two coordinates intersect.
PRAXIS FLASHCARD #71
Writing Algebraic Expressions
(from English statements)
To write an algebraic expression from an English
statement, we convert each word or phrase to the
equivalent algebraic symbol. For example, “of” means
to multiply, “per” means to divide, “total” means
addition, and “is” means equals. Other flashcards
contain ALL common words that can be converted to
algebra. Here’s an example: The sum of two numbers is
16. One of the numbers is twice the other number.
x + y = 16
x = 2y
PRAXIS FLASHCARD #72
To simplify algebraic expressions, use the distributive
property and combine like terms. This can also be
stated in step-by-step fashion:
1.
Simplify Algebraic Expressions
2.
3.
Clear the parenthesis (by following the
distributive property or the rules of
exponents)
Add the coefficients of like terms
Add the constant terms
PRAXIS FLASHCARD #73
Evaluate Algebraic Expressions
To evaluate an algebraic expression, substitute the
given values for each variable into the expression, and
then follow the order of operations (PEMDAS) to
simplify the expression.
1. Perform the operations inside a parenthesis first
2. Then follow rules for exponents
3. Then multiplication and division, from left to right
4. Then addition and subtraction, from left to right
PRAXIS FLASHCARD #74
Solving Linear Equations
To solve a linear equation, isolate the variable. Apply
the addition principle of equality (add the same number
to both sides of the equation), and then apply the
multiplication principle of equality (multiply the same
number to both sides of the equation). It is often
helpful to clear parentheses and clear fractions first.
PRAXIS FLASHCARD #75
Translate verbal statements into English words using an
equal symbol (=) for the word "is" or using comparison
symbols for statements of inequality:
Solving Algebraic Word Problems
(involving number relationships)
Then, solve the equation or inequality by isolating the
variable.
PRAXIS FLASHCARD #76
Solving Algebraic Word Problems
(involving consecutive numbers)
Numbers following each other in counting order are
called consecutive. They can be denoted as x, x + 1, x +
2, x + 3, etc. Consecutive odd numbers are x, x + 2, x +
4, etc. (assuming x is odd), and consecutive even
numbers are also x, x + 2, x + 4, etc. (assuming x is
even). Example: Find three consecutive, even numbers
whose sum is 90.
x + (x + 2) + (x + 4) = 90
3x + 6 = 90
3x = 84
x = 28 so the numbers are 28, 30, 32
PRAXIS FLASHCARD #77
Problems involving area & perimeter require use of
formulas: P = 2L + 2W, where P = Perimeter, L = length,
W = width. A = LW, where A = area
Solving Algebraic Word Problems
(involving rectangular area & perimeter)
Example: The length of a rectangle is twice its width. If
the perimeter of the rectangle is 60 in, find its area.
L = 2W so P = 2(2W) + 2W
60 = 4W + 2W
60 = 6W
W = 10 and since L = 2W
L = 20
2
Knowing this, A = LW = 20(10) = 200 in
PRAXIS FLASHCARD #78
Problems involving triangles require you to know facts
about triangles, such as: Two angles that form a straight
line (180⁰) are supplementary. The sum of the
measures of the interior angles of a triangle is 180⁰.
Solving Algebraic Word Problems
Example: Find the unknown angle (x):
(involving triangles)
PRAXIS FLASHCARD #79
Set up unit conversion problems as a proportion. Then
cross multiply (Means- Extremes Property) and simplify.
Example: Convert 16 yards to feet.
Because there are 3 feet in 1 yard:
Solving Algebraic Word Problems
(involving unit conversion)
Cross multiply: 3 x 16 = 48 feet
(Unit analysis may also be used – see the flashcard on
unit analysis.)
PRAXIS FLASHCARD #80
Problems involving mixtures use M1V1 + M2V2 = MtVt,
where M is the percentage of each mixture, and V is the
volume or amount of each mixture.
Solving Algebraic Word Problems
(involving mixtures)
Example: How much of a 16% solution is needed to
combine with 34 ml of a 12% solution to make 50 ml of
a 15% solution?
0.16x + 0.12(34)
16x + 12(34)
16x + 408
16x
x
=
=
=
=
=
0.15(50)
15(50)
750
342
21.375 ml
PRAXIS FLASHCARD #81
Solving Algebraic Word Problems
(involving investments)
Problems involving investments require use of the
interest formula: I=Prt, where I = interest earned,
P = principal (original amount), r = annual rate of
interest, and t = time in years.
Example: An investment is made at 5% simple interest
for 12 years. It earned $420 interest. How much was
originally invested?
420
420
4200
P
P
=
=
=
=
=
P(.05)(12)
0.6P
6P
4200/6
$700
PRAXIS FLASHCARD #82
Problems involving discounts require use of the
discount formula: S = r - rd, where S is the sale price, r
is the retail price, and d is the rate of discount.
Solving Algebraic Word Problems
(involving discounts)
Example: A coat is on sale for $125. The coat was
discounted 20%. What was the original retail price?
125 = r – (0.2)r
125 = (0.8)r
(multiply both sides by 10 to clear decimal)
1250 = 8r
r = 1250/8
r = $156.25
PRAXIS FLASHCARD #83
When we think of sales commissions, we often think of
car sales. Thus, it is appropriate that the formula for
commission is C = ar, where C = commission earned,
a = amount of sale, and r = commission rate.
Solving Algebraic Word Problems
(involving commissions)
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
PRAXIS FLASHCARD #84
The distance formula is D = rt, where D is the distance
traveled, r is the rate of speed, and t is the time.
to calculate the rate of speed
Solving Algebraic Word Problems
(involving distance-speed-time)
to calculate the time
PRAXIS FLASHCARD #85
In problems where you are given data about an object traveling with
and then against a moving object, use a table, and then set the SAME
quantities equal to each other & solve. The distance formula is D = rt
Solving Algebraic Word Problems
(involving uniform motion)
Example: A boat can travel 12 mi/hr in still water. If the boat can travel
5 mi downstream in the same time it takes to travel 3 mi upstream,
what is the rate of the river's current?
PRAXIS FLASHCARD #86
To solve for a variable in a formula means to find an
equivalent equation in which the desired variable is
isolated. Follow the same general strategies as solving
any equation.
Algebraic Symbol Manipulation
Example: P = 2L + 2W, solve for W
P = 2L + 2W
-2L -2L
Addition Property
P - 2L = 2W
P - 2L = 2W
2
2
Multiplication Property
PRAXIS FLASHCARD #87
Multiples
A multiple is the product of any quantity and
an integer. Multiples may be found by counting by the
number. A calculator may be used to find multiples –
enter the quantity, hit the plus sign, enter the quantity
again, and then hit the equal symbol. Then keep hitting
the equal symbol to see successive multiples.
23 + 23 = 46, = 69, = 92, = 115, etc.
23 x 1, 23 x 2, 23 x 3, 23 x 4, 23 x 5, etc.
PRAXIS FLASHCARD #88
Geometry Symbols
(13 of them)
PRAXIS FLASHCARD #89
Volume is the measure of the number of cubic units
that can fit inside an object. Volume is also known as
capacity.
Volume
(Definition)
PRAXIS FLASHCARD #90
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.
PRAXIS FLASHCARD #91
pi
The number π is a mathematical constant that is
the ratio of a circle's circumference to its diameter. The
constant, sometimes written pi, is an irrational number
approximately equal to 3.14159 or 22/7. (i.e., a circle’s
diameter can be wrapped around its circumference pi
times – 3 times and a little bit more.)
PRAXIS FLASHCARD #92
A vertex (plural: vertices) is a point that describes
the corners or intersections of geometric shapes.
Vertex
PRAXIS FLASHCARD #93
A straight angle is an angle that measures 180°
Straight Angle
PRAXIS FLASHCARD #94
A right angle is an angle that measures 90°
Right Angle
PRAXIS FLASHCARD #95
An obtuse angle is an angle that measures between
90°-180°
Obtuse Angle
PRAXIS FLASHCARD #96
Complementary angles are two angles whose measure
adds to 90°.
Complementary Angles
PRAXIS FLASHCARD #97
Supplementary angles are two angles whose measure
adds to 180°.
Supplementary Angles
PRAXIS FLASHCARD #98
The hypotenuse is the longest side of a right triangle,
often labeled “c”.
Hypotenuse
PRAXIS FLASHCARD #99
The legs of a triangle are the sides of the triangle. In a
right triangle, the legs are usually labeled “a” and “b”. The
legs are the two shorter sides of a right triangle. The legs
of a right triangle are perpendicular to each other.
Legs of a Triangle
PRAXIS FLASHCARD #100
Parallel lines are lines in a plane that do not intersect or
touch at any point. The lines are equidistant from each
other.
Parallel Lines
PRAXIS FLASHCARD #101
Perimeter is the measure of the distance around a
polygon
(peri = around, meter = measure)
Perimeter (Definition)
PRAXIS FLASHCARD #102
Perpendicular lines are lines that meet or intersect at
90° angles. The slope of one is the negative reciprocal
of the other.
Perpendicular Lines
PRAXIS FLASHCARD #103
Plane Transformations
Plane transformations:
1. Translation (move)
2. Rotation (turn)
3. Dilation (scale)
(enlarge/reduce)
4. Reflection (flip)
(four types)
PRAXIS FLASHCARD #104
The Pythagorean Theorem is used to determine the
measure of an unknown leg or hypotenuse of a right
triangle.
Pythagorean Theorem
PRAXIS FLASHCARD #105
A rhombus is a two-dimensional quadrilateral where all
four sides are the same length. Thus, a square is a
specialized rhombus.
Rhombus
(Definition, perimeter, & area)
PRAXIS FLASHCARD #106
A right triangle is a triangle in which one angle is a right
angle. The right angle is usually marked with a small
square:
Right Triangle
PRAXIS FLASHCARD #107
Scalene Triangle
A scalene triangle is a triangle in which all three sides
are of different length. In diagrams representing
triangles (and other geometric figures), "tick" marks
along the sides are used to denote sides of equal
lengths:
PRAXIS FLASHCARD #108
Similar Shapes
Similar Shapes are shapes that have the same angles
but the size of the sides is different; they are the same
shape but not the same size. The similar shape may be
flipped or rotated, but it is still similar if the two shapes
are merely dilations of each other.
PRAXIS FLASHCARD #109
Surface area is the total area of the faces and curved
surfaces of a solid figure.
Surface Area
PRAXIS FLASHCARD #110
Symmetry (also known as reflection symmetry) is when
both halves of an object are exact copies of each other.
The line down the middle between the two halves is
the line of symmetry.
Symmetry
PRAXIS FLASHCARD #111
The area of an object is the number of square units that
can fit inside the object. For example, if the object is
measured in feet, the square unit of measurement for
the area of that object is a one-foot by one-foot square.
Square Units of Measurement
Be careful when finding area of objects when the
answer is wanted in a different unit of measure:
PRAXIS FLASHCARD #112
Units of Capacity
(Define & List Units of Measure)
Capacity (also known as volume) is the amount of
space inside an object. The measurement of
capacity/volume in the metric system is the liter. In the
customary or U. S. English system, refer to the table
below:
1 pint (pt) = 2 cups (c)
1 quart (qt) = 2 pints (pt)
1 quart (qt) = 4 cups (c)
1 gallon (gal) = 4 quarts (qt)
1 gallon (gal) = 8 pints (pt)
1 gallon (gal) = 16 cups (c)
PRAXIS FLASHCARD #113
Mass
(Define & List Units of Measure)
Mass (also known as weight on Earth) is the amount of
matter in an object. Technically, weight is a measure of
the force of gravity against an object, but on Earth,
mass and weight can be thought of as the same thing.
The metric measure of mass is the gram. In the
customary or U. S. English system, refer to the table
below:
PRAXIS FLASHCARD #114
Length is a measure of distance. The metric measure of
length is the meter. In the customary or U.S. English
system, refer to the table below:
Length
(Define & List Units of Measure)
PRAXIS FLASHCARD #115
Acceleration is the measure of how speed (velocity)
changes over time. It can be expressed as the change
in velocity divided by the change in time:
Acceleration
PRAXIS FLASHCARD #116
Angle
An angle is the figure formed by two rays, called
the sides of the angle, which share a common endpoint,
called the vertex of the angle. An angle can be named
by its vertex or by naming a point on each leg with the
vertex point in the middle. The angle below is Angle B
or Angle ABC, also written as ∠ABC.
PRAXIS FLASHCARD #117
A 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.
Denominate Numbers
PRAXIS FLASHCARD #118
Circumference is the distance around a circle.
(circum = around, fer = carry)
Circumference (definition)
PRAXIS FLASHCARD #119
Congruent Shapes
Congruent shapes are two shapes of exactly the same
size and shape. The two shapes may be rotated or
flipped. The common way to mark the matching sides
and angles of congruent shapes is with hash marks as
shown below:
PRAXIS FLASHCARD #120
To convert from larger units of measurement to smaller
units, multiply. To convert from smaller units of
measurement to larger units, divide. To convert
denominate numbers, use unit analysis.
Converting Units of Measurement
Video explaining unit analysis for English units
Video explaining unit analysis for Metric units
(both videos are at JoleneMorris.com, Math 115, Week 6)
PRAXIS FLASHCARD #121
Density is the measure of how compact the material is
inside an object. Density is measured in terms of the
mass per unit volume.
Density
We have all heard the story of
Archimedes who discovered the
concept of density when he saw
how his body displaced the water in
his bathtub. He cried, “Eureka!
Eureka!”
PRAXIS FLASHCARD #122
Diameter of a Circle
The diameter of a circle is the longest chord in a circle.
It is a line segment that passes through the center of
the circle and whose endpoints are on the circle. The
symbol for the diameter is ⌀, which is made on a
Windows computer by ALT+8960.
PRAXIS FLASHCARD #123
A chord of a circle is a line segment whose endpoints
are on the circle. The diameter is the largest possible
chord in a circle.
Chord of a Circle
PRAXIS FLASHCARD #124
The English System of Measurement is also called the
Customary System of Measurement (or the Common
System of Measurement). This is the primary
measurement system used in the United States.
English System of Measurement
Length = inch, foot, yard, rod, mile, etc.
Weight = ounce, pound, Ton, etc.
Volume = liquid ounces, cup, pint, quart, gallon, etc.
PRAXIS FLASHCARD #125
Customary System of Measurement
The Customary System of Measurement is also called
the English System of Measurement (or the Common
System of Measurement). This is the primary
measurement system used in the United States.
Length = inch, foot, yard, rod, mile, etc.
Weight = ounce, pound, Ton, etc.
Volume = liquid ounces, cup, pint, quart, gallon, etc.
PRAXIS FLASHCARD #126
Equilateral Triangle
An equilateral triangle is one where all three legs
(sides) are of equal measure. An equilateral triangle is
also equiangular, which all three angles are of equal
measure (60°). In diagrams representing triangles (and
other geometric figures), "tick" marks along the sides
are used to denote sides of equivalent lengths:
PRAXIS FLASHCARD #127
Isosceles Triangle
An isosceles triangle is one where two of the three legs
(sides) are of equal measure, which means two of the
angles are of equal measure. In diagrams representing
triangles (and other geometric figures), "tick" marks
along the sides are used to denote sides of equal
lengths:
PRAXIS FLASHCARD #128
An exterior angle is an angle on the outside of a
polygon that is formed by extending the side of the
polygon. In the diagram below, ∠d is an exterior angle:
Exterior Angle
PRAXIS FLASHCARD #129
A line is an infinite collection of collinear points. A line
has no width or depth—it has only one dimension:
length.
Line (geometry)
How is a line named?
A line is named with a lowercase letter or by two points
on the line. The symbol for a line is a double headed
arrow. Thus, the line below is line l, or
l
l
A
B
PRAXIS FLASHCARD #130
A line segment is a portion of a line with two
endpoints. The line segment is named by the two
endpoints. The symbol for a line segment is a line
segment:
Line Segment
PRAXIS FLASHCARD #131
Unit Analysis is the process of multiplying by successive
conversion units (written in fraction form).
unit analysis video: JoleneMorris.com, Math 115, Wk 2
Unit Analysis
Video explaining unit analysis for English units
Video explaining unit analysis for Metric units
(both videos are at JoleneMorris.com, Math 115, Week 6)
PRAXIS FLASHCARD #132
Angles are measured in degrees using a protractor.
Place one ray of the angle along the zero edge of the
protractor. The other ray of the angle points to the
number of degrees that are in the angle.
Measuring Angles
PRAXIS FLASHCARD #133
Protractor
A protractor is a geometry tool used to measure angles.
Place one ray of the angle along the zero edge of the
protractor. The other ray of the angle points to the
number of degrees that are in the angle.
PRAXIS FLASHCARD #134
Metric System of Measurement
The Metric System is an
international system of
measurement based on the
decimal system. The prefixes
of the Metric System are
shown in the table to the right:
length  meter
capacity (volume)  liter
mass (weight)  gram
King Henry Doesn’t Usually Drink Chocolate Milk
PRAXIS FLASHCARD #135
Ordered Pair
An ordered pair is a pair of numbers indicating the
location of a point . The first number, called the first
coordinate, tells how far the point is right or left on the
horizontal x-axis. The second number, called the second
coordinate, tells how far the point is up or down on the
vertical y-axis. The actual point is the intersection of
those two coordinates.
PRAXIS FLASHCARD #136
Ray (geometry)
A ray is often mistakenly called a half line. A ray is part
of a line that has an endpoint but goes infinitely in a
straight line from that endpoint. Two rays that have the
same endpoint form an angle. A ray is named by its
endpoint and any other point on the ray. The symbol
for a ray is a small ray (single headed arrow).
PRAXIS FLASHCARD #137
The following five symbols are called comparison
symbols:
< less than
Comparison Symbols
> 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: ≠
PRAXIS FLASHCARD #138
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.
Reciprocal
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.
PRAXIS FLASHCARD #139
Fibonacci Sequence
The Fibonacci Sequence is formed by starting with 0
and 1, adding those two terms to obtain the third term,
adding the second and third terms to obtain the fourth
term, and continuing by adding the last two terms to
find the next term. The Fibonacci sequence is named
after Leonardo of Pisa, who was known as Fibonacci.
The Fibonacci numbers appear in biological settings and
are used in computer programming.
PRAXIS FLASHCARD #140
The percent of change is also known as the percent of
increase or the percent of decrease. To calculate the
percent of change,
Percent Change
1. Find the difference between the new amount and the
original amount
2. Divide that difference by the original amount
3. Multiply by 100
PRAXIS FLASHCARD #141
The domain of a function is the set of all possible
x-values. The range of a function is the set of all
possible y-values.
Domain & Range of a Function
PRAXIS FLASHCARD #142
An equation is a statement where an algebraic
expression is equal to another algebraic expression or
constant.
Equation
Equations and expressions are often confused. The
equation has an equal symbol; whereas, the expression
does not have a comparison symbol (merely a
collection of terms). We solve an equation but we
evaluate or simplify an expression.
PRAXIS FLASHCARD #143
An expression is a collection of terms that have been
added or subtracted.
Expression
Equations and expressions are often confused. The
equation has an equal symbol; whereas, the expression
does not have a comparison symbol (merely a
collection of terms). We solve an equation but we
evaluate or simplify an expression.
PRAXIS FLASHCARD #144
A function is a relation between a set of inputs and a
set of potential outputs with the property that each
input is related to exactly one output.
Function
PRAXIS FLASHCARD #145
An inequality is similar to an equation, but the two
sides are NOT equal.
Inequality
PRAXIS FLASHCARD #146
A pattern is a type of theme of recurring events or
objects, sometimes referred to as elements of a set of
objects. It is said that algebra is a study of patterns.
Patterns
PRAXIS FLASHCARD #147
Sequence
A sequence is a set of members where order is
important. For example, the sequence of letters ABC is
entirely difference from the sequence of letters ACB –
although we are using the same three letters, they are
in different order.
PRAXIS FLASHCARD #148
A pictograph is a graph or chart where pictures are
used to indicate a specific number of objects.
Pictograph
PRAXIS FLASHCARD #149
Pie Chart
A pie chart (also known as a circle graph) is a circular
graph where sections of the circle represent parts of
the whole. A pie chart ALWAYS gives parts of a whole.
The pie sections are usually labeled with percentages.
PRAXIS FLASHCARD #150
In statistics, 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
Quartiles
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
PRAXIS FLASHCARD #151
The term random variable is used in statistics and
probability to indicate a variable where the value of the
variable may change according to the probability.
Random Variable
This term will probably not appear on the Praxis test
but is included here for completeness of mathematical
topics.
PRAXIS FLASHCARD #152
The term regression is used in statistics to identify a
technique or algorithm for estimating relationships
between variables.
Regression
This term will probably not appear on the Praxis test
but is included here for completeness of mathematical
topics.
PRAXIS FLASHCARD #153
Scale on Bar or Line Graph
Bar graphs and line graphs will often have a
symbol on the vertical axis to indicate that
some numbers have been left out. Be aware
of these “broken scales” when analyzing
graphs. The break in the scale is used when
one or more of the bars/lines are
significantly out of range of the other
bars/lines. These broken scales can visually
give a wrong impression. In the bar graph to the right, the red
bar looks about twice the size of the green bar; whereas, the
red bar is actually six or seven times the size of the green bar!
PRAXIS FLASHCARD #154
The mean is used in statistics to describe one of three
measures of central tendency. Find the mean by adding the
data values and dividing by the number of data values. (Also
known as average.)
Mean
PRAXIS FLASHCARD #155
Median
The median is used in statistics to describe one of three
measures of central tendency. To find the median, list all the
data values in numerical order. The median is the middle
data value. If there is an even number of values, find the
mean (or average) of the middle two values. Medians are
used instead of means when there are outliers in the list of
values that would distort the average.
A mnemonic used to remember this term is to think of the
median in the center of a freeway. A statistical median is in
the center of the data values.
PRAXIS FLASHCARD #156
Mode
The mode is used in statistics to describe one of three
measures of central tendency. To find the mode, list all the
data values in numerical order. The data value that appears
the most in the list is the mode. A data set of values can have
no modes, one mode, or more than one mode. Mode is used
to determine popularity or commonly recurring events.
A mnemonic used to remember this term is to think of the
word "most," which is similar to “mode.” Mode is/are the
values that appear the most.
PRAXIS FLASHCARD #157
Range (of Data)
The range is used in statistics to describe the difference
between the smallest value and the largest value in a data
set. To find the range, find the smallest value (the minimum
or min) and find the largest value (the maximum or max).
The range is the difference when you subtract the min from
the max.
PRAXIS FLASHCARD #158
Sample (statistics)
In statistics, often the population is too large to test
everything in the population. Instead, a subset of the
population is tested. This subset is called a sample. An entire
college class is offered at most large universities to explain
the best way to sample a representative subset of the
population without introducing bias.
PRAXIS FLASHCARD #159
Algorithm
An algorithm is a step-by-step process for solving a problem.
An example of an addition algorithm is:
1. Line up the numbers
2. Add each column starting on the right
3. Carry any tens-place digits to the next column
4. Place commas between periods in the answer
An algorithm is often written as a flowchart showing steps,
branches, and decisions.
PRAXIS FLASHCARD #160
Scatter Plot
A scatter plot is a graph showing a collection of twocoordinate points. The points are not connected with line
segments, but the points may demonstrate a trend. A
technique called the “line of best fit” determines a line
through the points where about half of the points are above
the line and about half the points are below the line. This line
of best fit visually demonstrates a trend.
PRAXIS FLASHCARD #161
Simulation
A simulation is a game, computer program, or device that
approximates a real-life event. Simulations are used when
actual events are dangerous or impractical to duplicate.
Examples of simulations are driving teaching machines,
dissection of frogs, and chemical experiments. Simulations
are used in statistics to approximate results from a real-life
event.
PRAXIS FLASHCARD #162
Standard Deviation
The term standard deviation is used in statistics to describe
how much the results deviate (differ) from the mean
(average). A small standard deviation indicates that most of
the data values are close to the mean. A large standard
deviation indicates that the data have a large range of values.
This term will probably not appear on the Praxis test but is
included here for completeness of mathematical topics.
PRAXIS FLASHCARD #163
Stem & Leaf Plot
The stem & leaf plot is a table used to show all data values,
but in an organized manner. The table has two columns. The
right column gives the ones place and the left column
displays the tens place values:
tens
ones
0
1588
1
36667
2
3
125
The stem & leaf plot above displays the numbers 1, 5, 8, 8,
13, 16, 16, 16, 17, 31, 32, and 35.
PRAXIS FLASHCARD #164
A counterexample is one example used to prove a
hypothesis false.
Counterexample
PRAXIS FLASHCARD #165
Bar Graph
A bar graph is used to demonstrate relative values in a data
set. The primary purpose of a bar graph is to compare values:
The height (or length) of the bars shows how the values
compare to other values in the data set.
PRAXIS FLASHCARD #166
Bell Curve
A bell curve is a line graph where the data values peak in the
middle and fall off drastically in both the positive and
negative direction. It is named a bell curve because the line
resembles the shape of a bell. Another name for a bell curve
is a Gaussian function. A bell curve is often used in grading—
employing the belief that the majority of the students are Cstudents and a small minority are A-grade or Failing students.
This term will probably not appear on the Praxis test but is
included here for completeness of mathematical topics.
PRAXIS FLASHCARD #167
Binomial Distribution
A binomial distribution is a number indicating the number of
results in a two-way experiment. The experiment tests only
two outcomes (YES/NO, TRUE/FALSE, etc.). In the simplest
explanation, the binomial distribution indicates the
probability of the outcomes.
This term will probably not appear on the Praxis test but is
included here for completeness of mathematical topics.
PRAXIS FLASHCARD #168
A circle graph (also known as a pie chart) is a circular graph
where sections of the circle represent parts of the whole. A
pie chart ALWAYS gives parts of a whole. The pie sections are
usually labeled with percentages.
Circle Graph
PRAXIS FLASHCARD #169
A table is used to display the information in an organized
manner. A table usually has column headings -- each record
is in a row of the data table.
Table (of Data)
wwwneic.cr.usgs.gov/neis/general/handouts/mag_vs_int.html
PRAXIS FLASHCARD #170
Trend
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.)
PRAXIS FLASHCARD #171
Variance is a statistical measure of how far the data are
spread out from the mean (average).
Variance
This term will probably not appear on the Praxis test but is
included here for completeness of mathematical topics.
PRAXIS FLASHCARD #172
Correlation is a statistical relationship between dependent
variables.
Correlation
This term will probably not appear on the Praxis test but is
included here for completeness of mathematical topics.
PRAXIS FLASHCARD #173
A correlation coefficient is a statistical measure of the
relationship between dependent variables.
Correlation Coefficient
This term will probably not appear on the Praxis test but is
included here for completeness of mathematical topics.
PRAXIS FLASHCARD #174
Data collection is the process of collecting and preparing
data for statistical purposes.
Data Collection
PRAXIS FLASHCARD #175
Dependent data is the opposite of independent data. Data
that is dependent means that it changes depending on how
other data changes. Independent data is independent of
changes in other data.
Dependent (Data)
PRAXIS FLASHCARD #176
An experiment is an attempt to prove or disprove an
hypothesis. For the experiment to be valid, it must be
repeatable and verifiable.
Experiments
PRAXIS FLASHCARD #177
Frequency Distribution
A frequency distribution is an organized way to display the
results from an experiment. Frequency distribution tables are
often used in probability for situations that do not use
sample spaces (a set listing ALL possible outcomes in an
experiment). The three types of probabilities are classical,
empirical, and subjective. The probability for situations that
use a frequency distribution is called Empirical Probability or
Relative Frequency Probability. Classical probability uses a
list of ALL possible outcomes. Subjective probability is based
on a person’s knowledge of the situation (an educated
guess).
PRAXIS FLASHCARD #178
Histogram
A histogram is a bar graph where the bars are vertical; the
bars represent continuous groups of numerical data; and the
bars touch. The term comes from the late 1890’s meaning an
historical diagram. Example: children’s ages at the movie
A bar graph would have a bar showing how many children of
each age attended the movie, i.e. 6-year olds, 7-year olds, 8year olds, etc.
A histogram would have a bar showing how many of each
age GROUP attended the movie, i.e., 6-9-year-olds, 10-13year-olds, 14-17-year-olds.
PRAXIS FLASHCARD #179
Dependent data is the opposite of independent data. Data
that is dependent means that it changes depending on how
other data changes. Independent data is independent of
changes in other data.
Independent (Data)
PRAXIS FLASHCARD #180
A line graph is a statistical graph showing the data points
connected by line segments. A line graph is used to visualize
trends in the data. The graphic below is a line graph showing
two trend lines.
Line Graph
PRAXIS FLASHCARD #181
Normal distribution is used in statistics to describe the
normal patterns of data. When graphed, data that is a
normal distribution is shaped like a bell curve.
Normal Distribution
PRAXIS FLASHCARD #182
Venn Diagram
A Venn diagram is a diagram using circles to show logic
statements. Most Venn diagrams display overlapping circles.
Venn diagrams are named after John Venn who developed
the concept of Venn diagrams about 1880. By shading
portions of the overlapping circles, set theory concepts such
as UNION and INTERSECTION can be shown.
PRAXIS FLASHCARD #183
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.
PRAXIS FLASHCARD #184
Variable
A variable is used in algebra to represent a value that
changes within the parameters of the problem. The opposite
of a variable is a constant. Lowercase letters of the alphabet
are generally used to denote a variable. There are two types
of variables: dependent and independent.
PRAXIS FLASHCARD #185
Closure
Closure is a property of operations. Number sets are closed
under addition and multiplication, but not under subtraction
or division. This means that you can add any two whole
numbers and the result will be a whole number. But you
can’t subtract any two whole numbers and be guaranteed
that the result will be a whole number (it might be a negative
number, which is not a whole number).
PRAXIS FLASHCARD #186
Ordering Integers
To order or compare 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 is (remember negatives act
in an opposite way from positive numbers).
PRAXIS FLASHCARD #187
The RULE for adding integers is:
If the signs are the same, add the absolute values of the
numbers and give the result their same sign.
Addition of Integers
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.
PRAXIS FLASHCARD #188
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:
Subtraction of Integers
Change the sign of the second number to its opposite and
change the operation to addition. Then, follow the rules for
adding integers.
PRAXIS FLASHCARD #189
The RULE for multiplying integers is:
If the signs are the same, multiply the absolute values of the
numbers and give the result a positive sign.
Multiplication of Integers
If the signs are different, multiply the absolute values of the
numbers and give the result a negative sign.
PRAXIS FLASHCARD #190
The RULE for dividing integers is:
If the signs are the same, divide the absolute values of the
numbers and give the result a positive sign.
Division of Integers
If the signs are different, divide the absolute values of the
numbers and give the result a negative sign.
PRAXIS FLASHCARD #191
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).
See the flashcard on the “Properties of Absolute Value.”
PRAXIS FLASHCARD #192
Multiples of integers are formed when the integer, n, is
successively multiplied by the whole numbers. As such, the
multiples of the integer 3 are 0, 3, 6, 9, 12, 15, etc.
Multiple of an Integer n
The way the “multiple of an integer n” is written in algebra is
kn where “k” is a whole number.
PRAXIS FLASHCARD #193
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.
PRAXIS FLASHCARD #194
Consecutive integers are integers that differ by 1.
Example: -3 and -2 are consecutive integers
Consecutive Integers
PRAXIS FLASHCARD #195
Positive Integers
Positive integers are those integers greater than zero.
Positive integers appear to the right of zero on a number
line. The positive sign is understood if it isn’t written.
Negative integers are the opposite of the positive integers.
The number zero is neither positive nor negative.
PRAXIS FLASHCARD #196
Negative Integers
Negative integers are those integers less than zero. Negative
integers appear to the left of zero on a number line. The
negative sign is always written and never understood.
Negative integers are the opposite of the positive integers.
The number zero is neither positive nor negative.
PRAXIS FLASHCARD #197
Zero is an integer and divides a number line into -|+
Zero (as an Integer)
The number zero has some interesting properties:
 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
PRAXIS FLASHCARD #198
An exponent is a symbol to indicate a shortcut of
multiplication by the same number. The exponent
signifies the base is multiplied by itself (not by the
exponent) the exponent number of times.
Exponent
Example:
Careful:
PRAXIS FLASHCARD #199
A number squared is the same as the number times
itself. The exponent of 2 is often called squared because
the area of a square is the side times itself.
Number Squared
PRAXIS FLASHCARD #200
A number cubed is the same as the number times itself
times itself again. The exponent of 3 is often called
cubed because the volume of a cube is the side times
itself times itself.
Number Cubed
PRAXIS FLASHCARD #201
Square Root of a Number
The square root of a number is the value when
multiplied by itself makes the number. A number’s
square root is always smaller than half of the number.
The symbol for square root is called a radical sign. There
is an index of 2 on the radical sign, but an index of 2 is
rarely written -- it is understood to be 2 if not written.
Finding a square root is the inverse of squaring a
number. Although -6 times -6 also makes 36, we always
use the positive square root (also called the principal
square root).
√
√
PRAXIS FLASHCARD #202
Rules for Exponents
(
(
)
)
( )
√
PRAXIS FLASHCARD #203
√
√
√ √
√
Rules for Square Roots
√
√ √
√
√
| |
(√ )
PRAXIS FLASHCARD #204
There are three primary properties of operations:



Operation Properties
(Three of them)
commutative
associative
distributive
Other operation properties are closure, zero, inverse,
and identity.
PRAXIS FLASHCARD #205
Associative Property
The associative property says that three numbers may
be added/multiplied using the first two numbers first
and then the third -or- they may be added/multiplied
using the last two numbers first and then the first
number. The sum/product will be the same regardless:
(
(
)
)
(
(
)
)
PRAXIS FLASHCARD #206
The commutative property says that two numbers can
be added/multiplied in any order and the sum/product
will be the same regardless:
Commutative Property
PRAXIS FLASHCARD #207
Distributive Property
The distributive property involves both addition and
multiplication. If two numbers are added inside
parenthesis but multiplied by a third number outside
the parenthesis, that third number may be distributed
to each of the numbers inside parenthesis and then
added. The distributive property is one case where
multiplication comes before parenthesis in the order of
operations:
(
)
PRAXIS FLASHCARD #208
Inverse Property
The inverse property defines what happens when you
add or multiply inverse numbers.
The additive inverse is the negative or opposite of a
number. When you add a number and its opposite, the
result is zero.
The multiplicative inverse is the reciprocal of a number.
When you multiply a number and its opposite, the
result is one.
(
)
PRAXIS FLASHCARD #209
The identity property defines what happens when you
add or multiply by the identity numbers.
Identity Property
The additive identity is zero. When you add zero and a
number, the result is that number.
The multiplicative identity is one. When you multiply
one and a number, the result is that number.
PRAXIS FLASHCARD #210
Properties of Zero
The number zero has some interesting properties:
 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
PRAXIS FLASHCARD #211
Properties of One
The number one has some interesting properties:
 One is the multiplicative identity
 Any number multiplied by one equals the number
 One is neither prime nor composite
 A number with an exponent of 1 equals the
number
 The number one raised to any power equals one
 Any nonzero number divided by itself equals one
PRAXIS FLASHCARD #212
Steps to Solve Praxis Problems
1. Read the question carefully, circling, underlining, and/or
writing down what you are looking for.
2. Pull out important information.
3. Draw, sketch, or mark in diagrams or on scratch paper.
4. If you know a simple method or formula, work the problem
out as simply and quickly as possible.
5. If you don’t know a simple method or formula…
a. Try eliminating some unreasonable choices
b. Work backwards from the answers
c. Substitute in numbers—work a simpler problem
d. Try approximating to clarify your thinking
6. Check to be sure your answer is reasonable.
PRAXIS FLASHCARD #213
Questions to ask yourself about your
Praxis answer
Always ask yourself if your answer is reasonable. If you
have time left over at the end of the test, go back
through each answer to be sure it is reasonable – but
do not change your answer unless it is clearly wrong.
PRAXIS FLASHCARD #214
1.
2.
3.
Determine if it’s customary or metric units
Determine if it’s length, volume, mass, time, or
temperature
Change all measurements to the same unit
Tip for working with Praxis
measurement problems
PRAXIS FLASHCARD #215
Change all numbers to fractions or change all numbers
to decimals before beginning your calculations.
Tip for Praxis problems with both
fractions and decimals
PRAXIS FLASHCARD #216
Change all fractions to common denominators before
beginning your calculations.
Tip for Praxis problem with
fractions of unlike denominators
PRAXIS FLASHCARD #217
Remember that negative numbers are always smaller
than positive numbers. Also remember that the further
right on the number line, the larger the number.
Tip for comparing positive and
negative numbers in Praxis
PRAXIS FLASHCARD #218
Make sure that all numbers are in the same format (all
fractions or all decimals or all percentages).
Tip for solving comparison problems
in Praxis
PRAXIS FLASHCARD #219
Study the chart or graph carefully before reading the
question. Note whether the vertical axis is broken (see
Flashcard #153).
Tip for Praxis problems containing
a chart or graph
PRAXIS FLASHCARD #220
1.
2.
3.
4.
Tip for solving Praxis problems of
area, perimeter, and volume
Determine if it’s customary or metric units
Determine if it’s length, volume, or mass
Change all measurements to the same unit
Use the appropriate formula
PRAXIS FLASHCARD #221
x + (x + 2) + (x + 4) + (x + 6)
Using algebra, how can you
express the sum of 4
consecutive odd numbers?
PRAXIS FLASHCARD #222
After taking a practice test, do an item analysis of each
problem you missed:
Item Analysis (Praxis study tip)
1.
2.
3.
4.
5.
What competency/domain is it assessing?
What specific concept or skills it is assessing?
What is the reason you missed the problem?
What were the possible stem distractors that
caused the error?
What vocabulary should you learn?
Add this concept to your study plan as an area of focus.
PRAXIS FLASHCARD #223
Tip about Praxis problem words
in ALL CAPITAL LETTERS
One of the four question types on the Praxis test is an
exception question where you need to find the answer
that does NOT fit the pattern or does NOT answer the
question. The signal words that this is an exception
question will be written in ALL CAPITAL LETTERS, such
as EXCEPT, NOT, and LEAST.
PRAXIS FLASHCARD #224
For making educated guesses,
what is good to know
about the choices?
Except in cases where the problem asks you to compare
or order numbers, all answer choices are listed in
numerical order. If you must guess, start with the
middle answer choice and adjust up or down from there
until you find the correct answer.
PRAXIS FLASHCARD #225
Arithmetic Sequence
An arithmetic sequence is an ordered list of numbers
where each number is formed by adding a constant
number to the previous number. An example of an
arithmetic sequence is: 3, 6, 9, 12, 15, 18, etc. where
each number is formed by adding 3 to the previous
number. To find the nth term of an arithmetic
sequence:
Do not confuse this with an arithmetic series where
numbers are not is a list but form an addition problem.
An example of an arithmetic series is 3 + 6 + 9 + 12 + …
PRAXIS FLASHCARD #226
Geometric Sequence
A geometric sequence is an ordered list of numbers
where each number is formed by multiplying a constant
number to the previous number. An example of a
geometric sequence is: 3, 9, 27, 81, 243, etc. where
each number is formed by multiplying by 3. To find the
nth term of a geometric sequence:
Do not confuse this with a geometric series where
numbers are not is a list but form an addition problem.
An example of a geometric series is 3 + 9 + 27 + 81 + …
PRAXIS FLASHCARD #227
An harmonic series is the sum of progressive unit
fractions:
Harmonic Series
This term will probably not appear on the Praxis test but is
included here for completeness of mathematical topics.
PRAXIS FLASHCARD #228
The additive identity is zero.
Any number added to zero results in a sum of that
number. The additive identity does not change the
number when it is added to it.
Additive Identity
PRAXIS FLASHCARD #229
Approximate Conversions
(English/Customary units to/from Metric)
A meter is a little more than a yard.
A gram is about the weight of a paper clip.
A liter is a little more than a quart.
PRAXIS FLASHCARD #230
1/2 = 0.5 = 50%
(1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5,
1/6, 5/6, 1/8, 3/8, 5/8, 7/8, 1, 2, 3 1/2)
2
1
1
1
1
5/6 = 0.83 /3 = 83 /3%
2
2
2/3 = 0.66 /3 = 66 /3%
1/8 = 0.125 = 12.5%
1/4 = 0.25 = 25%
3/8 = 0.375 = 37.5%
3/4 = 0.75 = 75%
5/8 = 0.625 = 62.5%
1/5 = 0.2 = 20%
7/8 = 0.875 = 87.5%
2/5 = 0.4 = 40%
1 = 1.0 = 100%
3/5 = 0.6 = 60%
2 = 2.0 = 200%
4/5 = 0.8 = 80%
3 1/2 = 3.5 = 350%
1/3 = 0.33 /3 = 33 /3%
Common Equivalents
2
1/6 = 0.16 /3 = 16 /3%
PRAXIS FLASHCARD #231
Addition of Whole Numbers
The algorithm for whole number addition is:
1. Line up the numbers vertically so place values are
in the same column
2. Add beginning in the ones place
3. If the sum is greater than 9, write the tens place
digit above the next column to the left
4. Put commas in the answer to separate the digits
into periods
PRAXIS FLASHCARD #232
An acute angle is an angle that measures less than 90°
Acute Angle
PRAXIS FLASHCARD #233
An acute triangle is any triangle where all three angles
are less than 90°:
Acute Triangle
PRAXIS FLASHCARD #234
Corresponding Angles
When a transversal line crosses two other lines, it forms
eight angles that are often used in geometrical
problems. Corresponding angles are angles that are in
the same position on each of the lines. In the figure
below, Angle 2 corresponds to Angle 6.
PRAXIS FLASHCARD #235
Additive Inverse
The additive inverse is a number such that when a
number and its additive inverse are added, the resulting
sum is zero (the additive identity). The additive inverse
is the negative of a number and the opposite of the
value of a variable:
( )
(
)
( )
(
)
PRAXIS FLASHCARD #236
Algebra
Algebra is the study of numbers, number patterns, and
relationships among numbers. Algebra generalizes
these numbers, number patterns, and relationships. It is
often said that algebraic thinking is the study of number
patterns.
PRAXIS FLASHCARD #237
Adjacent Angles
Two angles are adjacent angles if they share a common
vertex, they share a common side, AND they do not
share any interior points. In other words two angles
that are side-by-side are adjacent.
PRAXIS FLASHCARD #238
www.ets.org/praxis/
Official website where you can
find more information about
the Praxis tests
PRAXIS FLASHCARD #239
An angle bisector is a ray (or line or line segment) that
divides one angle into two angles of equal measure. In
other words, an angle bisector cuts an angle in half.
Angle Bisector
PRAXIS FLASHCARD #240
An arc is a section of a circle. It is a set of points all
equidistant from a center point. Arcs of the same size
cut equal measured central angles in the circle.
Arc
PRAXIS FLASHCARD #241
Base 10
The Base 10 number system is the system we use –
each place value is ten times the value of the place to
the right of it. There are 10 digits in the Base 10 number
system: 0-9. These 10 digits are all that is needed to
make any number. Another name for the Base 10
number system is the decimal number system.
PRAXIS FLASHCARD #242
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
PRAXIS FLASHCARD #243
Cardinal Numbers are numbers used to indicate
quantity. The Cardinal Numbers are the same as the
Natural Numbers (for the purposes of elementary
school students’ understanding).
Cardinal Numbers
PRAXIS FLASHCARD #244
When a list of numbers has TWO numbers that appear
the most, this distribution of numbers in the list is
called a bimodal distribution.
Bimodal Distribution
For example, this list is a bimodal distribution with 3
and 7 as the two modes:
1, 2, 3, 3, 3, 4, 5, 6, 7, 7, 7, 8, 9
PRAXIS FLASHCARD #245
The four basic binary operations are:
Binary Operations
(name the four basic operations and any
special relationships between them)
1.
2.
3.
4.
Addition
Subtraction (the inverse of addition)
Multiplication (repeated addition)
Division (repeated subtraction and the inverse of
multiplication)
PRAXIS FLASHCARD #246
A box and whiskers plot is a visual way to show the five
statistical number summaries:
Minimum, Q1, Q2, Q3, and Maximum
Box and Whiskers Plot
PRAXIS FLASHCARD #247
Central Tendency Measures
Measures of Central Tendency are statistical measures
such as mean, median, and mode. Data has a tendency
to cluster or center on certain values. The term
“average” is also used to indicate measures of central
tendency.
PRAXIS FLASHCARD #248
Algebraic Thinking
Algebraic thinking is the mathematics we teach and
learn to prepare us to understand algebra. In
elementary schools, algebraic thinking is the study of
our number system, patterns, representations, and
mathematical reasoning.
PRAXIS FLASHCARD #249
A coefficient is the number part of a term in an
algebraic expression. For example, negative two is the
coefficient of the following expression:
Coefficient (algebra)
-2x3
The coefficient is a factor of the term.
The coefficient is multiplied and is, therefore, a
multiplicative factor.
PRAXIS FLASHCARD #250
Convex describes an object such as a polygon that is not
concave. All vertices of a convex polygon are less than
180-degrees in measure:
Convex
PRAXIS FLASHCARD #251
Concave describes an object with a hollowed out or cut
out portion—a part of the object has been “caved” in.
The opposite of a concave polygon is a convex polygon.
Concave
PRAXIS FLASHCARD #252
Coplanar describes two-dimensional figures that are on
the same plane.
Coplanar
PRAXIS FLASHCARD #253
Collinear describes two or more points that are on the
same line (they are in a straight row or lined up).
Collinear
PRAXIS FLASHCARD #254
A compass is a tool used in geometry to draw arcs and
circles. These arcs may be used to bisect lines and
angles.
Compass (geometry)
PRAXIS FLASHCARD #255
A common fraction is also known as a simple fraction. It
represents parts of a whole and is written as a division
problem:
or 1/4
Common Fraction
The 1 in the example above is the numerator.
The 4 in the example above is the denominator.
PRAXIS FLASHCARD #256
Comparing Integers
To order or compare 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 is (remember negatives act in an opposite way
from positive numbers—the larger the absolute value,
the smaller the negative number).
PRAXIS FLASHCARD #257
As opposed to a variable, an algebraic constant is a
known number that does not vary.
In the trinomial
Algebraic Constant
, the 5 is a constant.
A constant term is a term in a polynomial without a
variable in it.
PRAXIS FLASHCARD #258
Coordinate Grid
A coordinate grid 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).
PRAXIS FLASHCARD #259
A cone is a three-dimensional shape with a circular
base. A cone can be formed by spinning a triangle in
three-dimensional space.
Cone
(definition and volume)
PRAXIS FLASHCARD #260
All counting numbers (except 1) have at least two
factors. Common factors are those factors that are in
common with two or more numbers.
For example,
Common Factors
The factors of 6 are 1, 2, 3, and 6
The factors of 9 are 1, 3, and 9
The common factors of 6 and 9 are 1 and 3
PRAXIS FLASHCARD #261
All counting numbers have an infinite number of
multiples. Common multiples are those multiples that
are in common with two or more numbers.
Common Multiples
For example,
The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, etc.
The multiples of 3 are 3, 6, 9, 12, 15, 18, etc.
The common multiples of 2 and 3 are 6, 12, 18, etc.
PRAXIS FLASHCARD #262
A mixed number indicates the whole amounts and the
parts (a fraction). For each whole number, there are a
complete number of parts (for example, if the parts are
measured in thirds, each whole has three thirds).
Convert Mixed Number
to Improper 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.
2
For example, 6 /3 means there are 6 whole amounts of
3
18
/3 (6 x 3 = 18) so there are /3. Adding the remaining
2
20
2
/3 results in a total of /3 in 6 /3.
PRAXIS FLASHCARD #263
An improper fraction shows a total number of parts, but
in those parts is at least one whole. If the parts are
measured in thirds, each third in the numerator of the
improper fraction makes one whole.
Convert Improper Fraction
to Mixed Number
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
2
/3 equals 23 divided by 3, which is 7 r 2 or 7 /3
PRAXIS FLASHCARD #264
To convert measures of length in the Metric System,
merely move the decimal point to the “place value” of
the new unit of measure:
kilo- hecto- deka- m deci- centi- milli-
Converting Measures of Length
To convert measures of length in the U. S. Customary
System, use Unit Analysis (multiply by the conversion
factor in such a way that all units cancel out except the
unit you want). For example, to change 16 feet to
inches:
PRAXIS FLASHCARD #265
To convert measures of length in the Metric System,
merely move the decimal point to the “place value” of
the new unit of measure:
Converting Measures of Area
kilo- hecto- deka- m deci- centi- milliTo convert measures of length in the U. S. Customary
System, use Unit Analysis (multiply by the conversion
factor in such a way that all units cancel out except the
unit you want). For example, to change 16 feet to
inches:
PRAXIS FLASHCARD #266
To convert measures of mass in the Metric System,
merely move the decimal point to the “place value” of
the new unit of measure:
kilo- hecto- deka- g deci- centi- milli-
Converting Measures of
Mass/Weight
To convert measures of weight in the U. S. Customary
System, use Unit Analysis (multiply by the conversion
factor in such a way that all units cancel out except the
unit you want). For example, to change 5 pounds to
tons:
PRAXIS FLASHCARD #267
To convert measures of capacity in the Metric System,
merely move the decimal point to the “place value” of
the new unit of measure:
kilo- hecto- deka- l deci- centi- milli-
Converting Measures
of Capacity/Volume
To convert measures of length in the U. S. Customary
System, use Unit Analysis (multiply by the conversion
factor in such a way that all units cancel out except the
unit you want). For example, to change 5 quarts to
ounces:
PRAXIS FLASHCARD #268
To convert measures of time, use Unit Analysis
(multiply by the conversion factor in such a way that all
units cancel out except the unit you want). For example,
to change 16 hours to seconds:
Converting Measures of Time
PRAXIS FLASHCARD #269
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.
PRAXIS FLASHCARD #270
Deductive Reasoning
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.
PRAXIS FLASHCARD #271
Dependent Variable
A dependent variable is one that depends on another
variable (independent variable) for its value. In an
equation with two variables, you can choose any value
you want for one of them (the independent variable),
but once that value is chosen, the dependent variable
has limited values in order for the equation to be true.
Dependent and independent variables are often
interchangeable – in an equation with an x-variable and
a y-variable; it doesn’t matter which variable you
choose a value for, but that sets the value of the other
variable.
PRAXIS FLASHCARD #272
Discounts
Discounts are usually written as percentages. A
discount is an amount by which the purchase price is
reduced. Be careful when working with discount
problems because the answer may be the discount
percentage, an amount of money discounted, the sales
price after the discount, or the original sales price.
Discount amount is the original sales price multiplied by
the percentage of discount. The 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.
PRAXIS FLASHCARD #273
Dividing by Powers of 10
Because we use a Base 10 number system, each place
value is 10 times the place value to its right. As such,
dividing by powers of 10 is simply a matter of moving
the decimal point to the left (one place for each power
of 10). Example:
In this example, we were dividing by 10 to the power of
2, which is a 1 followed by 2 zeroes, so we merely move
the decimal point 2 places to the left.
PRAXIS FLASHCARD #274
Division of Whole Numbers
Division of whole numbers is the inverse
of multiply and is repeated subtraction.
We divide to find how many groups of
a number (the divisor) can be created
from another number (the dividend). The
symbol is used to signify division; it means
“divided by.” The algorithm for division Is:
Divide the divisor into the digit in the largest
place value of the dividend, multiply, subtract,
bring down the next digit, and divide again.
PRAXIS FLASHCARD #275
An equiangular polygon is one where all angles of the
polygon are the same measure.
Equiangular Polygon
If a polygon is both equiangular and equilateral, it is
called a regular polygon.
PRAXIS FLASHCARD #276
An equilateral polygon is one where all sides of the
polygon are the same measure.
Equilateral Polygon
If a polygon is both equiangular and equilateral, it is
called a regular polygon.
PRAXIS FLASHCARD #277
Even numbers are integers that are evenly divisible by
two (2). Zero is an even number.
Even Numbers
PRAXIS FLASHCARD #278
Estimation
Estimation is a mathematical process of finding an
approximate value of a variable, expression, or
operation. The most common way to find an estimate is
to round off the numbers in the problem to numbers
that are easy to calculate and then find the answer to
the rounded problem. The symbol for an estimated or
approximate answer is .
For example:
287 + 321 + 878
300 + 300 + 900
1,500
PRAXIS FLASHCARD #279
Euler’s Polyhedron Formula is V – E + F = 2
which means the number of vertices subtract the
number of edges add the number of faces in any
polyhedron always equals 2.
Euler’s Polyhedron Formula
PRAXIS FLASHCARD #280
Expanded Notation is writing a number to show each
digit’s place value.
Expanded Notation
Example: Write 123,456 in expanded notation
(
) (
) (
(
) (
) (
)
or this number can be written using exponents:
(
)
(
(
)
)
(
(
)
)
(
)
)
PRAXIS FLASHCARD #281
Sieve of Eratosthenes
The Sieve of Eratosthenes is a technique to teach young
children about prime numbers. A paper is numbered
from 2-100. Circle 2 because it is prime. Then count
every 2 numbers and cross them off (cross off the
multiples of 2). The next number after 2 that is not
crossed off is 3, circle it. Count every 3 numbers
(multiples of 3) and cross them off. The next number
after 3 that is not crossed off is 5, circle it. Count the
multiples of 5 and cross them off…. This process
continues for the entire chart. All crossed-off numbers
are composite; the prime numbers are circled.
PRAXIS FLASHCARD #282
1. Factor out any common factors in all terms.
2. If the polynomial has four terms, factor it by
grouping.
3. If it is a binomial, look for a difference of squares,
Factoring Polynomials
4.
5.
a sum of cubes, or a difference of cubes. (Note
that a sum of squares cannot be factored.)
2
If it is a trinomial and the coefficient of the x
term = 1, un-FOIL to factor.
2
If it is a trinomial and the coefficient of the x
term is not 1, use the AC method to factor.
PRAXIS FLASHCARD #283
Factorial is a unary operation. The exclamation point is
the symbol used to denote factorial. Factorials are most
commonly used in permutations and combinations.
Factorial
To find the factorial of a number n, multiply all the
numbers from 1 to the number n. Example: Find 6!
By convention, 0! = 1
PRAXIS FLASHCARD #284
A flow chart is a diagram used to visually describe an
algorithm. A flow chart shows a step-by-step path for
the algorithm.
Flow Charts
Small circles are used to show the start and end points.
Diamonds are used for decisions -- to ask questions and
branch the flow chart depending on the answer to the
question. A rectangle is used to show a process or
action step. Input and output are represented by
parallelograms. Other symbols are used in more
complex flow charts.
PRAXIS FLASHCARD #285
Fundamental Counting Principle
The Fundamental Counting Principle: If there are m
ways for one event to occur and n ways for another
event to occur, there are m x n ways for both to occur.
These events (in a sample space) are listed using a tree
diagram or a table.
PRAXIS FLASHCARD #286
Function Machine
A function machine is a visual device to help young
students understand the concept of a function. Each
function machine has a rule it applies to numbers put
into the machine (the inputs). After the machine applies
the rule, it outputs the result.
PRAXIS FLASHCARD #287
Golden Ratio
The Golden Ratio is a common ratio found in nature,
arts, and mathematics. It is also known as the Golden
Section or Divine Proportion. The Greek letter phi (Ф) is
used for the Golden Ratio. A Golden Ratio exists
between two numbers a and b (with a being the larger
value) if
Leonardo Da Vinci used the Golden
Ratio in creating his statues and
paintings. The Golden Ratio drawing of
Da Vinci (shown to the right) is famous.
PRAXIS FLASHCARD #288
Inductive reasoning is making sufficient observations
that conclusions can be formed.
Inductive Reasoning
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 conclusion is true. It often helps to draw logic
circles when working with deductive reasoning.
PRAXIS FLASHCARD #289
To lines on the same plane that share a single point are
said to be intersecting lines.
Intersecting Lines
PRAXIS FLASHCARD #290
On the Praxis exam, it is important to interpret answers
that have remainders correctly. Do you round up? Do
you round down? Do you use the remainder as part of
the answer (or as the entire answer)?
Interpreting Remainder Problems
1. How many boxes can be filled? (use only the
quotient; ignore the remainder)
2. How many cans are needed to paint the wall?
(round the quotient to the next greater whole #)
3. How many in the last box that isn’t completely full?
(use only the remainder)
PRAXIS FLASHCARD #291
There are basically two kinds of interest: simple and
compound. Simple interest is paid on the principal
amount only. Compound interest is paid on the
principal amount plus accrued interest.
Interest
The formula to find simple interest is
where p
is the principal, r is the interest rate, and t is the time
period.
(
) where
The formula to find compound is
p is the principal, r is the interest rate, and n is the
number of interest periods.
PRAXIS FLASHCARD #292
LCD
The Lowest Common Denominator (LCD) of two or
more fractions 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, lined up by
factors. Then “bring down” one
of each factor and multiply.
PRAXIS FLASHCARD #293
Logic Diagram
A logic diagram is a visual way to determine the truth or
logic of statements. A truth table may
also be used. With a logic diagram,
use circles to show relationships.
For example: (1) ALL cats have tails.
(2) SOME cats are black. (3) Goldy is
a cat.
ALL means the circle is completely inside another circle.
SOME means the circle is partially inside another circle.
NONE means the circles are completely separate.
From the logic diagram, we see that Goldy definitely
has a tail but may or may not be black.
PRAXIS FLASHCARD #294
A mixed number is a whole number and a proper
fraction combined. Mixed numbers may also be called
mixed fractions.
Mixed Number
This graphic shows two whole pizzas and a fraction of 3
3
pieces out of 4  2 /4
PRAXIS FLASHCARD #295
National Council of Teachers
of Mathematics
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. States have the option
of accepting the NCTM standards as is, modifying them
for state use, or writing their own standards. Most
states have accepted the NCTM with some limited
modifications. www.nctm.org
PRAXIS FLASHCARD #296
Monomial
A monomial is a polynomial with a single term. A term
contains a coefficient with possibly one or more
variables all multiplied. Neither a term nor a monomial
has any addition or subtraction. Here are samples of
four monomials:
PRAXIS FLASHCARD #297
Multiplicative Identity
The multiplicative identity is the number one (1). We
can multiply any number by one and the result is the
original number. Multiplying by the multiplicative
identity does not change the value of the number.
We use the multiplicative identity when we simplify
fractions. If the numerator has a factor of 2, and the
denominator has a factor of 2, that is the fraction 2/2,
which is the whole number 1. Since multiplying by 1
does not change the value of the original fraction, we
can “cancel” that 1 to simplify the fraction:
PRAXIS FLASHCARD #298
The multiplicative inverse of a number is the reciprocal
of that number. We can multiply any number by its
multiplicative inverse and the result is the number one.
Multiplicative Inverse
PRAXIS FLASHCARD #299
Multiplication of Whole Numbers
The algorithm for whole number multiplication is:
1. Line up the numbers vertically so place values are in
the same column
2. Multiply beginning in the ones place -- multiply
each digit in the multiplicand. If the product is
greater than 9, write the tens place digit above the
next column to the left (regrouping or carrying)
3. Next, multiply by the tens place -- to signify that
you are in the tens place, put one zero in the partial
product. Multiply each digit in the multiplicand.
4. Add the partial products, and put commas in the
answer to separate the digits into periods
PRAXIS FLASHCARD #300
Multiplying by Powers of 10
Because we use a Base 10 number system, each place
value is 10 times the place value to its right. As such,
multiplying by powers of 10 is simply a matter of
moving the decimal point to the right (one place for
each power of 10).
Example:
In this example, we were multiplying by 10 to the
power of 2, which is a 1 followed by 2 zeroes, so we
merely move the decimal point 2 places to the right.
PRAXIS FLASHCARD #301
Net or Network (geometry)
A net is a two-dimensional
representation of a threedimensional object. If a net is
cut out, it can be put together
to form the three-dimensional
object it represents.
PRAXIS FLASHCARD #302
An obtuse triangle is a triangle where one of the angles
is obtuse (greater than 90-degrees).
Obtuse Triangle
PRAXIS FLASHCARD #303
Odds (Statistical)
Odds and probability are related concepts. With
probability, you compare the number of favorable
outcomes to the total possible number of outcomes.
With odds, you compare the number of favorable
outcomes to the number of remaining (unfavorable)
outcomes. 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
PRAXIS FLASHCARD #304
Order of Magnitude
An order of magnitude is a measure of powers of 10.
This concept is often used in working with scientific
notation. A number that has been multiplied by 10 has
increased its order of magnitude by 1. A number that
has been multiplied by 1000 has increased its order of
magnitude by 3.
PRAXIS FLASHCARD #305
Origin (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)
PRAXIS FLASHCARD #306
Ordinal numbers, unlike cardinal numbers that indicate
a quantity, are numbers that indicate order or rank.
st nd rd th th
Ordinal numbers are 1 , 2 , 3 , 4 , 5 , etc.
Ordinal Numbers
PRAXIS FLASHCARD #307
Continuous variables can assume an infinite number of
values between any two specific values. They are
obtained by measuring. They often include fractions
and decimals.
Continuous Data
PRAXIS FLASHCARD #308
Quantitative Data
Quantitative variables are numerical and can be
ordered or ranked. For example, the variable age is
numerical, and people can be ranked in order according
to the value of their ages. Other examples of
quantitative variables are heights, weights, and body
temperatures.
PRAXIS FLASHCARD #309
Qualitative Data
Qualitative variables are variables that can be placed into
distinct categories, according to some characteristic or
attribute. For example, if subjects are classified according
to gender (male or female), the variable gender is
qualitative. Other examples of qualitative variables are
religious preference and geographic locations.
PRAXIS FLASHCARD #310
Discrete Data
Discrete variables can be assigned values such as 0, 1,
2, and 3 are said to be countable. Examples of discrete
variables are the number of children in a family, the
number of students in a classroom, and the number of
calls received by a switchboard operator each day for a
month.
PRAXIS FLASHCARD #311
There are two ways to solve a percentage problem. To
solve percentages using the percent proportion, use the
means-extreme property of proportions (cross
multiply). The percent proportion can be written as:
Percentages, Solving
The second way to solve a percentage problem is with
simple algebra: Write the percentage as an algebraic
equation where “what number”  variable (x),
is  =, and of  multiply. Then solve the equation.
PRAXIS FLASHCARD #312
There are three types of percentage problems
depending on what value is missing in the equation:
What number is 15% of 45?  x = (0.15) ∙ (45)
Percentages
(3 types of problems)
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
PRAXIS FLASHCARD #313
Numbers that have a whole number square root. The
first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64,
81, and 100.
Perfect Squares (numbers)
Is zero a perfect square? There is a debate about this in
the mathematics community – some believe zero is a
perfect square because 0 times 0 = 0; some disagree
because they say the definition of a perfect square is
“numbers that have a POSITIVE integer square root”
and zero is not positive. ….and the debate continues.
PRAXIS FLASHCARD #314
Polynomial
(terms; degrees; types)
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
PRAXIS FLASHCARD #315
A plane is a two-dimensional surface. It is like a sheet of
paper that has no thickness, yet it extends in all
directions for width and height.
Plane (geometry)
PRAXIS FLASHCARD #316
A point is a non-dimensional location on a plane. A
point is usually labeled with a capital letter of the
alphabet.
Point
PRAXIS FLASHCARD #317
Prism
A prism is a three-dimensional object with two bases of
the same figure. Prisms are named according to their
bases. As such, if the two bases are triangles, it is a
triangular prism. If the two bases are hexagons, it is a
hexagonal prism.
PRAXIS FLASHCARD #318
Pyramid (geometry)
A pyramid (in geometry) is a three-dimensional object.
The base of the pyramid is a polygon. Line segments
connect the base of the pyramid to a single point, called
the apex. Each base edge and the apex form a triangle - thus all faces of a pyramid are triangular.
PRAXIS FLASHCARD #319
A polygon is a two-dimensional shape drawn on a
plane. A regular polygon is where all sides and all angles
of the polygon have the same measure.
Polygon
(definition and names)
PRAXIS FLASHCARD #320
Quadrants (Coordinate Grid)
A coordinate grid 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).
PRAXIS FLASHCARD #321
A proportion is when two ratios are equivalent:
Proportions (How to Solve)
When two ratios are equivalent, the cross products are
equal. Thus, to solve a proportion, cross multiply the
two numbers that are diagonal from each other, and
then divide by the number diagonal from the unknown.
PRAXIS FLASHCARD #322
A Pythagorean Triple is a set of any three integers (a, b,
c) such that
Pythagorean Triple
The three numbers of a Pythagorean Triple describe the
length of the three sides of a right triangle. Perhaps the
most well-known Pythagorean Triple is 3-4-5. There are
16 Pythagorean Triples with c < 100:
(3,4,5)
( 9, 40, 41)
(16, 63, 65)
(36, 77, 85)
( 5, 12, 13)
(11, 60, 61)
(20, 21, 29)
(39, 80, 89)
( 7, 24, 25)
(12, 35, 37)
(28, 45, 53)
(48, 55, 73)
( 8, 15, 17)
(13, 84, 85)
(33, 56, 65)
(65, 72, 97)
PRAXIS FLASHCARD #323
A quadrilateral is a polygon with four sides (and four
vertices). Other names for a quadrilateral are a
quadrangle and a tetragon.
Quadrilateral
The interior angles of a quadrilateral add to 360◦.
An excellent graphic showing the Euler diagram of
quadrilateral types can be found on Wikipedia:
http://en.wikipedia.org/wiki/File:Euler_diagram_of_quadrilateral_types.svg
PRAXIS FLASHCARD #324
Radius
A radius is a line segment that goes from the center of a
circle to any point on the circumference of the circle.
The measure of the radius is half the diameter. Often,
the term radius is also used to denote the measure of
the radius line segment.
PRAXIS FLASHCARD #325
Quadratic Equation
(definition & five ways to solve)
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:
1. Factor and set each factor equal to 0
2
2. If there is no x-term, solve for x and apply the
square root method.
3. Graph the equation (as a parabola) and determine
the solutions where the parabola crosses the x-axis
4. Complete the square
5. Use the quadratic formula
PRAXIS FLASHCARD #326
Relationships (data pairs)
A relationship is simply a set of ordered pairs. If the set
of ordered pairs has only one y-value for any x-value,
this relationship is a function. If the set of ordered pairs
has only one y-value for any one x-value, it is a function
with a one-to-one correspondence.
PRAXIS FLASHCARD #327
Reduce a Fraction
Reducing a fraction is the same as simplifying a
fraction. The term “reduce” is seldom used in
mathematics today. To simplify a fraction, expand the
numerator and denominator into prime factorizations.
“Cancel” any ones such as 3/3 or 5/5. Then multiply
straight across those numbers that are left.
True story: When I did my student teaching 40 years ago, I was
teaching how to add fractions and I used the term “reduce” a
fraction. After the lesson, my cooperating teacher took me
aside and told me to use “simplify” instead -- he said, “Jolene,
only women reduce. Fractions simplify.”
PRAXIS FLASHCARD #328
Regrouping
Regrouping is the modern term used instead of carrying
and borrowing. Children are now taught to add and
subtract by keeping place value in mind. When we need
to carry or borrow, we now teach students to re-group
units into 10s or 10s into units.
PRAXIS FLASHCARD #329
An equiangular polygon is one where all angles of the
polygon are the same measure.
Regular Polygon
An equilateral polygon is one where all sides of the
polygon are the same measure.
If a polygon is both equiangular and equilateral, it is
called a regular polygon.
PRAXIS FLASHCARD #330
Roman Numerals
Roman numerals and the Roman number system are
similar to the Arabic number system used in the United
States. The Roman number system is based on 10 so it
is decimal, but it does not have place value. Letters are
used to represent various numbers (Roman number
names). The rule with Roman numbers is to write the
numbers in descending order (from greatest to
smallest). The exception to this rule is if a smaller
number comes before a larger number, we subtract
that smaller number from the larger number. I = 1,
V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000
PRAXIS FLASHCARD #331
Sales Tax
Sales Tax is written in percentages (which are
converted to decimal to computer sales tax). The final
purchase price of an item = marked price + (sales tax
times marked price). Using FP for final price, MP for
marked price, and ST for sales tax, the algebraic
equations is:
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.
PRAXIS FLASHCARD #332
Any triangle that is NOT a right triangle is an oblique
triangle. As such, acute triangles and obtuse triangles
are in the category of oblique triangles.
Oblique Triangle
PRAXIS FLASHCARD #333
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 × 10
3
PRAXIS FLASHCARD #334
Sets
A set is a collection of objects. The objects in a set can
be numbers, expressions, and other mathematical
objects. Georg(e) Cantor developed set theory in the
late 1800’s. Common operations on sets include
intersection, union, complements, and Cartesian
products. Other concepts include the Universal set, Null
set, members or elements, and sub-sets.
Sets in mathematics include the set of integers (Z),
rational numbers (Q), primes (P), real numbers (R),
natural numbers (N), etc.
PRAXIS FLASHCARD #335
Primary data is data obtained from an observation or
experiment. It is raw data that has not been
manipulated in any way.
Primary Data
PRAXIS FLASHCARD #336
Secondary Data
Secondary data is data obtained from someone else
other than the user. Secondary data can be thought of
as second-hand data; nevertheless, secondary data can
be extremely useful depending on how the data was
original obtained and manipulated.
PRAXIS FLASHCARD #337
For all real numbers a and b:
| |
|
Properties of Absolute Value
|
| |
| |
|
| |
|
|
| || |
| |
| |
| | |
| |
PRAXIS FLASHCARD #338
An expression is simplified when
How can you tell when
an expression is simplified?




No parentheses appear
No powers are raised to powers
No more than one like term
No negative exponents appear
PRAXIS FLASHCARD #339
Simultaneous Equations
(definition and 3 ways to solve)
Simultaneous Equations are two or more equations
with multiple variables. These are often called systems
of equations. A solution gives values for the variables
that are true for all equations in the system.
There are many ways to solve a system of equations.
Three ways discussed in beginning algebra are:
1. Elimination (sometimes called adding)
2. Substitution
3. Graphing
Another method presented in intermediate/advance
algebra is the use of matrices.
PRAXIS FLASHCARD #340
Simplifying Square Roots
To simplify square roots, take out the square root of
any perfect squares that are factors inside the radicand.
Perhaps the easiest way to do this is the factor the
number inside the radicand so it is obvious which
factors can be taken out.
Example: Simplify √
√
√
√
PRAXIS FLASHCARD #341
A polyhedron is any three-dimensional solid with faces,
edges, and vertices. Euler’s formula describes an
interesting property of convex polyhedron: V - E + F = 2
Polyhedron
(definition only)
PRAXIS FLASHCARD #342
Convex polyhedron are named according to the
number of faces:
4 = tetrahedron
5 = pentahedron
Polyhedron
6 = hexahedron
(names)
7 = heptahedron
8 = octahedron
9 = nonahedron
10 = decahedron
PRAXIS FLASHCARD #343
Solution (algebra)
An algebraic solution is the answer to an equation. The
solution will give a value or multiple values for the
variables in the equation. A solution is also called a root
of the equation. For more than one variable, the
solution will be an ordered pair, an ordered triple, etc.
PRAXIS FLASHCARD #344
A sphere is a three-dimensional, perfectly round shape.
Sphere is from the Greek word for “ball.”
Sphere
(definition, volume & surface area)
Technically, in mathematics, a sphere only includes the
“surface” and not the interior.
PRAXIS FLASHCARD #345
Speed is a measurement that tells how fast an object is
moving. The rate of speed is usually expressed as a
ratio of distance over time.
Speed (measurement)
Distance = rate of speed × time (D = rt)
PRAXIS FLASHCARD #346
In a polygon, the sum of the interior angles is equal to
the number of sides, subtract 2, and then multiply by
180°:
(
)
Sum of Interior Angles
PRAXIS FLASHCARD #347
Subtraction of Whole Numbers
The algorithm for whole number subtraction is:
1. Line up the numbers vertically so place values are
in the same column
2. Subtract beginning in the one’s place
3. Use regrouping (formerly called borrowing) if the
top number is too small to allow subtraction
4. Put commas in the answer to separate the digits
into periods
PRAXIS FLASHCARD #348
Transformations
A transformation in geometry
changes the position of a
shape on the coordinate
plane. There are four forms of
transformation:
1.
2.
3.
4.
translation (slide)
rotation (turn)
dilation (scale)
reflection (flip)
PRAXIS FLASHCARD #349
Tessellation
A tessellation is a two-dimensional plane created by
one or more polygon shapes fitted into each other so
no “open space” remains. Kepler first discussed
tessellations in the early 1600’s. Equilateral triangles,
squares, and hexagons are the only regular polygons
that tessellate. There exists an entire branch of
geometry about tessellations, begun by Russian
scientist Fyodorov in the late 1800’s. Tessellations for
3+ dimensional spaces are also defined.
PRAXIS FLASHCARD #350
Time Measurement
Time can be formatted in using a 12-hour clock with
a.m. and p.m. or using a 24-hour clock (military time). In
most places of the world, time is adjusted twice a year
by one hour for Daylight Saving Time (note that there is
no “S” on Saving -- it is not Daylight Savings Time). The
common units of time are:
60 seconds = 1 minute ; 60 minutes = 1 hour ;
24 hours = 1 day ; 7 days = 1 week ;
28-31 days = 1 month ; 12 months = 1 year
365 days = 1 common year (366 days = 1 leap year)
PRAXIS FLASHCARD #351
There are two scales used to measure temperature.
The majority of the world uses the Celsius scale
(formerly called Centigrade). In the United States, we
commonly use the Fahrenheit scale.
Temperature
Water freezes at 0°C and at 32°F
Water boils at 100°C and at 212°F
To convert temperatures between the two scales:
[ ]
[ ]
[ ]
([ ]
)
PRAXIS FLASHCARD #352
Transversal
A transversal is a line that crosses two or more other
lines. A transversal of two lines forms eight angles that
are often used in geometrical problems (see: vertical
angles, adjacent angles, corresponding angles, interior
angles, and exterior angles).
PRAXIS FLASHCARD #353
Triangles can be classified in two ways:
1.
Triangles
(two ways to classify)
2.
By the angles in the triangle: acute, obtuse, and
right.
By the sides in the triangle: equilateral, isosceles,
and scalene.
PRAXIS FLASHCARD #354
Velocity is a measure of the speed of an object and the
direction in which it is going. For example, the wind is
blowing 22 mph in a NE direction.
Velocity
PRAXIS FLASHCARD #355
Vertical Angles
When two lines intersect, they form four angles. The
two angles opposite each other are called vertical
angles. Vertical angles are always the same measure. In
the drawing below, Angle 1 and Angle 3 are vertical
angles. Angle 2 and Angle 4 are vertical angles.
PRAXIS FLASHCARD #356
The vertex of a parabola is a single point where the
parabola changes direction from upward to downward
(or downward to upward).
The x-coordinate of a parabola’s vertex is found by
Vertex of a Parabola
PRAXIS FLASHCARD #357
The five rules of classical probability theory:
Rules of Probability
1. The probability of any event will always be a
number from zero to one (
( )
).
2. When an event cannot occur, the probability will
be 0.
3. When an event is certain to occur, the probability
will be 1.
4. The sum of the probability of all outcomes in the
sample space is 1.
5. The probability that an event will not occur is
equal to 1 minute the probability that it will occur.
PRAXIS FLASHCARD #358
A Sample Space is the set list all possible outcomes of a
probability experiment. Two ways to list sample spaces
when there are two outcomes done in sequence are
using tree diagrams and tables.
Sample Space
PRAXIS FLASHCARD #359
Words that Signal Addition











add
sum
increase
total
rise
plus
grow
added to
more than
increased by
gain
Except for the phrases
more than,
subtracted from, and
less than, the
translation to algebraic
expressions is virtually
word for word. The
three phrases listed in
red the previous
sentence are translated
in reverse order.
PRAXIS FLASHCARD #360
Words that Signal Subtraction







subtract
subtracted from
minus
difference
take away
less than
decreased by
Except for the phrases
more than,
subtracted from, and
less than, the
translation to algebraic
expressions is virtually
word for word. The
three phrases listed in
red the previous
sentence are translated
in reverse order.
PRAXIS FLASHCARD #361
Words that Signal Multiplication





multiply
multiplied by
product
times
of
 twice
Except for the phrases
more than,
subtracted from, and
less than, the
translation to algebraic
expressions is virtually
word for word. The
three phrases listed in
red the previous
sentence are translated
in reverse order.
PRAXIS FLASHCARD #362
Words that Signal Division





divide
divided by
quotient
per
ratio
 half
Except for the phrases
more than,
subtracted from, and
less than, the
translation to algebraic
expressions is virtually
word for word. The
three phrases listed in
red the previous
sentence are translated
in reverse order.
PRAXIS FLASHCARD #363
A literal equation is an equation made up of only
known, measurable quantities. A literal equation is the
same as a formula.
Literal Equation
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.
PRAXIS FLASHCARD #364
x + (x + 1) + (x + 2)
Using algebra, how can you express
the sum of 3 consecutive numbers?
PRAXIS FLASHCARD #365
face downward (frown)
In the equation y = ax2 + bx + c, a
negative a makes the parabola
_________.
PRAXIS FLASHCARD #366
face upward (smiley-face)
The equation y = ax2 + bx + c,
a positive a makes the parabola
_________.
PRAXIS FLASHCARD #367
y=c
The equation y = ax2 + bx + c
crosses the y-axis at y = _____?
PRAXIS FLASHCARD #368
a parabola:
The equation y = ax2 + bx + c
makes what shape?
PRAXIS FLASHCARD #369
How do you convert a fraction
such as 2/3 into a ratio?
To convert a fraction into a ratio, keep the numerator;
the new denominator becomes the difference of the
denominator and numerator. The denominator of a
fraction is the WHOLE amount; the denominator of a
ratio is the REMAINING part.
Example: The fraction 2/3 is a ratio of 2 parts to 1
remaining part or 2:1
PRAXIS FLASHCARD #370
A fraction compares PART of something to its whole.
A ratio compares two different things – neither thing is
always the whole or sum of the two.
What is the difference between
a fraction and a ratio?
PRAXIS FLASHCARD #371
What is a fraction?
A fraction is a numeral showing a part of a group or a
part of a set expressed as division. The top number is
called the numerator; the numerator indicates the part.
The bottom number is called the denominator; the
denominator indicates the total in the group or set.
PRAXIS FLASHCARD #372
will be
is
Words that Signal Equals
PRAXIS FLASHCARD #373
divided by 100
When translating word problems, the
word “percent” means _________.
PRAXIS FLASHCARD #374
division
When translating word problems, the
word “per” means _________.
PRAXIS FLASHCARD #375
the unknown – use a variable such as x, y, or n
When translating word problems, the
word “what” means __________.
PRAXIS FLASHCARD #376
equals (=)
When translating word problems, the
word “is” means __________.
PRAXIS FLASHCARD #377
multiply
When translating word problems, the
word “of” means __________.
PRAXIS FLASHCARD #378
10 inches
This is the 3-4-5 triangle with a factor of ×2
In a right triangle with legs of 6 and 8
inches, the hypotenuse is _______.
PRAXIS FLASHCARD #379
1:1:√
The ratios of side lengths in a
45-45-90 triangle are ______.
PRAXIS FLASHCARD #380
1:√ :2
The ratios of side lengths in a
30-60-90 triangle are ______.
PRAXIS FLASHCARD #381
What are the three “special
right triangles”?
30°-60°-90°
45°-45°-90°
3n°-4n°-5n°
PRAXIS FLASHCARD #382
(
The total degrees of measure
inside every n-sided shape
)
Subtract 2 from the number of sides and multiply by
180 degrees.
PRAXIS FLASHCARD #383
What is a Tree Diagram?
What is it used for?
A tree diagram is a graphic organizer that lists all
possibilities of a sequence of events in a systematic
way. A tree diagram is used in determining probability –
it is a way to calculate the total possible outcomes and
view each possible scenario.
PRAXIS FLASHCARD #384
mode
What measure of central tendency
is used to track trends or popularity?
PRAXIS FLASHCARD #385
When the data are consistently distributed
and there are no outliers.
When is the mean the best
measure of central tendency?
PRAXIS FLASHCARD #386
When the data contains outliers
When is the median the best
measure of central tendency?
PRAXIS FLASHCARD #387
addends; sum
What are the parts of an
addition problem?
PRAXIS FLASHCARD #388
subtrahend; minuend; difference
What are the parts of a
subtraction problem?
PRAXIS FLASHCARD #389
factors; partial product; product
What are the parts of a
multiplication problem?
PRAXIS FLASHCARD #390
divisor; dividend; quotient; remainder
What are the parts of a
division problem?
PRAXIS FLASHCARD #391
Three dots in a triangle
Symbol for “therefore”?
PRAXIS FLASHCARD #392
∞
Symbol for “infinity”?
PRAXIS FLASHCARD #393
Symbol for “approximately”?
PRAXIS FLASHCARD #394
Symbol for “congruent”?
PRAXIS FLASHCARD #395
Symbol for “summation”?
PRAXIS FLASHCARD #396
||
Symbol for “absolute value”?
PRAXIS FLASHCARD #397
A reflex angle is an angle measured in a clockwise
direction as opposed to the normal counter-clockwise
direction.
Reflex Angle
PRAXIS FLASHCARD #398
A binomial is an algebraic expression with exactly 2
terms
Example: 3x – 2y
Binomial
PRAXIS FLASHCARD #399
A trinomial is an algebraic expression with exactly 3
terms
2
Example: 3x + 2x - 1
Trinomial
PRAXIS FLASHCARD #400
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