- Concepts
- - Digit
- - Whole
number place values
- - Scientific Notation
- - Exponential Notations
- - Decimal
place values
- - Factors
- - Reciprocal
- - GCF
- LCM
- - Square
- - Square root
-
- Whole Numbers
- Place Values
- - Expanded
Form of Whole Numbers
- - Rounding
- Comparison
- Decimals
- Place Values
- Comparison
- Terminating
- - Non-Terminating
- Conversion Decimal to
Fraction
-
- Fractions
- Comparison
- Denominator
- - Numerator
- - Conversion Fraction
to Decimal
- - Conversion of Mixed Number to
-
Improper Fraction.
- - Converting Improper Fraction to a Mixed Number.
- - Adding
Fractions with Like
-
Denominators
- - Adding Fractions with Unlike Denominators
- - Subtracting
Fractions with Like
-
Denominators
- - Subtracting Fractions with unlike Denominators
- - Multiplying
Fractions
- - Multiplying
Fractions and Whole
-
Numbers
- - Multiplying
Mixed Numbers.
- - Dividing Fractions
- - Dividing Fractions by Whole Numbers
- - Dividing Mixed Numbers
- Integer
- Addition
- Subtraction
- - Multiplication
- - Comparison
- - Graphing
- - Double Negatives
-
-
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Digit
A number (0 - 9) used to make up larger
numbers.
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Example: The digits 4, 6, and 8
are used to write the numbers:
- 468
- 684
- 846
- ......
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Comparison Symbols:
- <
is less than
- >
is greater than
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3 < 5
7 > 3
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Place values of a whole number
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Expanded Form of Whole Numbers
 |
Determine the multiples
of 10 that can be added together to get the number you started with. |
|
Write them as an addition
problem. |
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1463 = 1000 + 400 +
60 + 3
64201 = 60000 + 4000 +200 + 1
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Rounding Whole Numbers:
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Identify what
increments are to be used (count by 10s, 100s, 1000s, ....) |
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Surround the
original number by counting using the increment (10, 100, 1000, etc.) |
|
Figure out which
of the incremental numbers is closest to your original number and
choose that number as the answer. |
|
If the original
number is half way between two numbers, chose the larger number as the
number that is closest. |
|
-
Rounding 749 to the nearest 10s.
- - Think of 740, 750, 780
- - 749 is between
- 740 and 750.
- - 749 is closest
- to 750.
- - The answer is 750.
-
Rounding 45 to the nearest 10s.
- - Think of 30, 40, 50.
- - 45 is half way
- between 40 and 50.
- - Assume 45 is closest
- to 50.
- - The answer is 50.
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Comparing
Whole Numbers
Using a comparison symbol with
whole numbers:
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When writing the symbol, always
point to the smaller number, wherever it is ....
|
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When reading the symbol, always
say whether the first number is greater or less than the second.
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975 <
983
2478 >
195
8 <
19
34 >
19 |
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Scientific Notation
A x 10N
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7 x 103
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Place values of decimal
digits
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Factors
Things that can be multiplied together to give a specified number.
|
3
x 6 = 18
so,
3 and 6 are FACTORS of 18.
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Greatest Common Factor
| Given two or more whole numbers, the largest factor that can be
found in all whole numbers. |
|
8:
2, 4, 8
36: 2, 3, 4, 6,
9, 12
GCF: 4 |
| |
Least Common Multiple
| Given two or more whole numbers, the smallest number that is a
multiple of the numbers. |
|
6:
6, 12, 18, 24
8: 8,
16, 24
LCM: 24 |
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Base
| The thing that is being repeated in the multiplication. |
|
Given:
73
7
is the BASE
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Exponent
| The number of times that a thing is multiplied times itself. |
|
Given:
y5
5 is the
exponent.
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Exponential Notation
| Short hand notation for the repeated multiplication of a number(or
variable)
times itself. |
|
12
x 5 y4 |
| |
The square of a number "n"
| The number that you get if you multiply the number
"n" x "n". |
|
square
of n = n2
square of 7 = 49
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The square root of a number "n"
| The number "a" such that |
a x a = "n"
|
square
root of n = 
square root of 49 = 7
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Fact:
| a0 = 1 |
|
70
= 1 |
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Denominator
Number on the bottom of a fraction. |
 |
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Numerator
Number on the top of a fraction.
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Addition of Fractions with like
denominators
| Add the top numbers. |
| Copy the bottom number to answer. |
| Reduce/Simplify the answer. |
|
 |
| |
Subtraction of Fractions with like
denominators
| Subtract the top numbers. |
| Copy the bottom number to the answer. |
| Reduce/Simplify the answer. |
|
 |
| |
Converting a Fraction to decimal
| Divide bottom number into top number.
 | Add a decimal point. |
| Copy the decimal point straight up. |
| Add a zero after the decimal point. |
| Divide. |
|
| Reduce/Simplify the answer. |
|
 |
| |
Converting a Decimal to a Fraction
| Determine the place value of the last decimal
digit. |
| Use the place value of the last decimal position to rewrite the
decimal as a fraction |
| Reduce/Simplify the fraction. |
|
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Terminating Decimals
| Decimal that has a definitive end. |
|
0.23
0.532
|
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Non-Terminating Decimal
| Decimal that does not have a definitive end. |
|
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Comparing Fractions
| Use a number line. |
| Fractions to the left are smaller than
fractions to the
right. |
|
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| |
Comparing Fractions
| Cross multiply from the bottom to the top numbers.
(Place the result over the top of the fraction). |
| Point the comparison symbol to the
fraction under the smaller number. |
|
 |
| |
Comparing Fractions
| Adjust fractions to have common
denominators |
| Compare numerators. |
| Point the comparison symbol to the
fraction with the smaller numerator on the fractions with common
denominator. |
|
 |
| |
Multiplying
a Fraction with a Fraction
| Multiply top numbers together. |
| Multiply bottom numbers together. |
| Reduce/Simplify the answer. |
|
 |
| |
Multiplying
a Fraction with a Whole Number
| Place a ONE under the whole number. |
| Multiply the top numbers together. |
| Multiply the bottom numbers together. |
| Reduce/Simplify the answer. |
|
 |
| |
Multiplying
a Fraction with a Mixed Number
| Convert the mixed number to an improper fraction. |
| Multiply the top numbers together. |
| Multiply the bottom numbers together. |
| Reduce/Simplify the answer. |
|
 |
| |
Multiplying
a Mixed Number with a Mixed Number
| Convert the mixed numbers to improper fractions. |
| Multiply the top numbers together. |
| Multiply the bottom numbers together. |
| Reduce/Simplify the answer. |
|
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Reciprocal
| A fraction turned upside down. |
|
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Dividing Fractions
| Rewrite the problem as a multiplication problem
| Change the division sign to multiply sign |
| Change the fraction on the RIGHT to it's reciprocal |
| Bring down the fraction on the LEFT. |
|
| Multiply the fractions. |
| Reduce/Simplify the answer. |
|
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Convert Mixed Number to an Improper Fraction
| Multiply bottom number time the integer,
add the top number. |
| Copy the bottom number to the answer. |
|
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| |
Convert
Improper Fraction to Mixed Number
| Divide bottom number into the top number. |
| Turn remainder into a fraction. |
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- Integer Addition
Case 1: Both terms have the same
sign ...
| Add the absolute values of the terms. |
| Place the sign of the added terms in front of
the answer. |
Case 2: Both terms have different
signs ...
| Subtract the numbers. |
| Place the sign of the larger number in
front |
- of the answer.
|
- -5 + (-6) =
-11
- 7 + 6 = 13
- -5 + 6 =
1
- 9 + (-11) = -2
|
| |
Integer Subtraction
| Convert the problem to an addition
problem |
| Do Integer Addition. |
|
- -6 - 11
- -6 + (-11)
-
- 5 - 8
- 5 + (-8)
-
- 8 - (-7)
- 8 + 15
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Integer
Multiplication
Case 1: Both terms have the same sign
| Multiply the numbers. |
| Make answer positive. |
Case 2: Both have have different signs
| Multiply the numbers. |
| Make the answer negative. |
-
|
- (-5)(-6) = 30
- 8(5) = 40
- -9(4) = -36
- 8(-7) = -56
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Double Negatives
-(-a) = a
|
-(-5)
= 5
-(-(-9))
= -9
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Integer
Comparison
| Graph numbers on the number line. |
| Numbers to the left are smaller. |
|
-5 < 8
-8 < -3
-12 > - 30
8 > -2
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Ratio
| Comparison of two numbers. |
|
3
to 1
3 : 1
3 / 1 |
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Proportions
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Ratio &
Proportion |
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Percent
| Ratio of a number to 100. |
|
7%
7 : 100
15%
15 : 100
125%
125 : 100 |
| |
Percent to Decimal
| Shift decimal point two places to the
left. |
| Remove the percent sign. |
(If no decimal point is specified in
the percent, the decimal point is between the last digit and the % sign.) |
7% = .07
23%
= .23
123%
= 1.23
211/3
% = .211/3 |
| |
Percent to Fractions
| Convert percent to a decimal. |
| Convert decimal to a fraction. |
| Reduce the fraction. |
|
7%
= .07 = 7/100
56% = .56 = 56/100
= 14/25
|
| |
Decimal to Percent
| Shift decimal point two places to the
right. (Add extra zeros if necessary.) |
| Add a percent sign. |
|
.2
= 20%
.03
= 3 %
.27
= 27%
.125
= 12.5% |
| |
Fraction
to Percent
| Convert the fraction to a decimal. |
| Convert the decimal to a percent. |
|
1/5
=> 5) 1
0.2
=> 20% |
| |
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Percent 'of'
| Convert 'of' to a times sign. |
| Convert the percent to a decimal. |
| Copy the other number. |
| Multiply the decimal times the other
number. |
|
- 15% of $200
- .15 x $200
- $30
|
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Percent 'off'
| Convert 'off' to a times sign. |
| Convert the percent to a decimal. |
| Copy the other number. |
| Multiply the decimal times the other
number to get the DISCOUNT. |
| Subtract the DISCOUNT from the original
number to get the SALES PRICE. |
|
-
- 20% off $3400
- .20 x $3400
- $680
- $3400 - $680
- $2720
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What Percent?
| Convert 'is' to equal sign(=). |
| Convert the word 'What' to W. |
| Convert the word 'of' to a times sign |
| Divide the solitary number by the number
closest to the W. |
| Convert the result to a percent. |
|
What % of 300
is 15 ?
W x 300 = 15
0.05
300
) 15.00
5% |
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Statistical Display:
Bar Graph |
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Statistical Display:
Pie Graph |
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Statistical Display:
Line Graph |
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Statistical Display:
Histograph |
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Statistical Display:
Venn Diagram |
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Statistical Display:
Box & Whisker Plot
| List the data in order from least to
greatest. |
| Find the median. |
| Find the lower quartile ... the median of
the lower half of the data (lowest number to the median). |
| Find the upper quartile ... the median of
the upper half of the data (median to highest number). |
| Draw a number line ...
| Mark the least and greatest values |
| Mark the lower and upper quartiles |
| Mark the median. |
| Draw a box using the lower and upper
quartile marks as the bounds for the box. |
| Draw a whisker from the least number
to the lower side of the box. |
| Draw a whisker from the greatest
number to the upper side of the box. |
|
|
Given the numbers:
12, 13, 14, 15, 16, 18, 19, 22, 24, 26, 28
| The least is 12, the greatest data is 28 |
| The median is 18, |
| The lower quartile is 14. |
| The Upper quartile is 24. |
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Statistical Display:
Stem & Leaf Plot |
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Statistical Display:
Frequency Table
Table used to organize and display the
frequency of individual numbers in a list of numbers.
|
- Given ages (numbers):
- 12, 13,13,15,15,16,16,16,
- 17,17,17,17,17,17,18,18,18
-
-
| Age |
Count |
- 12
- 13
- 14
- 15
- 16
- 17
- 18
|
- 1
- 2
-
- 2
- 3
- 6
- 3
|
- (In this frequency table
- the mode is 17 ... see the
- explanation of mode.)
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Statistical
Analysis: Data Range
The difference between the greatest number
and the least number in the data.
|
Given student
scores:
79, 80, 91, 68,74
The range is:
91 - 68 = 23
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Statistical
Analysis: Quartiles
| Lower Quartile
| The median between the lowest value
of data and the median of full data set. |
|
| Upper Quartile
| The median between the greatest value
of data and the median of the full data set. |
|
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Interquartile
range
The difference between the
lower quartile and the upper quartile.
|
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Statistical
Analysis: Mean
| Add all numbers in the list of
numbers. |
| Divide by the count of numbers in the
list. |
|
Given class scores:
79, 82, 71,84,80, 77
The mean(average) is:
- 79+82+71+84+80+77
-
6
78.83%
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Statistical
Analysis: Median
| For an odd number of data, the median is
the middle number when the data is arranged in order. |
| For an even number of data, find the mean
of the two middle numbers. |
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Statistical
Analysis: Mode
The number that occurs most frequently in
the data.
It is possible for more than one number to
be the mode if the count is the same for two or more items that happen
the most.
|
- Given ages (numbers):
- 12, 13,13,15,15,16,16,16,
- 17,17,17,17,18,18,18,18
-
-
| Age |
Count |
- 12
- 13
- 14
- 15
- 16
- 17
- 18
|
- 1
- 2
-
- 2
- 3
- 4
- 4
|
The mode = 17 & 18
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Probability:
Population
The totality of all potential measurements
or observations under consideration in a given problem situation.
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Probability: Sample
A collection of measurements or
observations taken from a given population.
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Probability:
Experiment
Some process or operation that leads to a
well-defined outcome.
|
| Flipping a coin. |
| Counting left handed students in a room. |
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Probability: Outcome
The results of a single trial of an
experiment..
|
| Finding flipping a coin comes up with a
head. |
| Counting 2 left handed students in a
classroom. |
|
| |
Probability: Odds
Ratio of the number of positive outcomes to
negative outcomes.
|
Container
contains 5 red balls and 8 blue balls. The odds of drawing a red ball from
the container are 5 to 8. |
| |
Probability Event:
Simple
The outcome of a single trial in a
particular experiment.
|
The result of
tossing a coin once and having the coin land either heads or tails ... one
simple event. |
| |
Probability Event:
Compound
A set of simple random events or a subset
of the sample space.
|
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Probability Event:
Independent
|
The events
resulting from drawing two cards from a deck, returning the card to the
deck each time. Result is two independent events. |
| |
Probability Event:
Dependent
|
The events
resulting from drawing two cards from a deck without returning the cards
to the deck. The result of the second draw is limited by the
result of the first draw. |
| |
Probability Event:
Mutually Exclusive
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Meaning of
factorial:
Multiplication of a whole number by every
number less than it..
n! = n(n-1)(n-2)
....(1)
with special case
that .... 0! = 1
|
3! = (3)(2)(1)
= 6
6! = (6)(5)(4)(3)(2)(1) = 720
0! = 1 |
| |
Meaning of
Multiplication:
| Shortcut to adding. |
| Shortcut to counting. |
|
-
- Count the cells, you find
...
- 30 cells.
- Count columns and rows
... we find ...
- 3 rows, 10
columns.
- Add the number of cells
found in each column ... we find ...
-
10 + 10 + 10 = 30 cells.
-
- Multiply the number of
rows times the cells in a column ... we find ...
- 3 x 10 = 30 cells.
|
| |
If a number (n) of
acts are carried out, each act can be performed in the same number of ways
(k), then the total number of possible outcomes for n acts is:
(k)(k) ... (k), or kn
|
- How many cookies ?
- 4 cells x 4
cookies = 8 cookies.
-
- How many chocolate chips?
- 4 cells x 4
cookies x 4 chips =
-
64 chips.
-
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| |
If there are n acts
which can be performed in k1, k2, ... kn
ways, respectively, then the total number of different possible outcomes
for the n acts in succession is :
(k1)(k2) ...(kn)
|
- How many cookies?
- 3 cells x 4 cookies
= 12 cookies.
-
- How many chocolate chips
?
- 3 cells x 4
cookies x 5 chips =
-
60 chips.
-
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Statistical: Permutations
The way that objects can be arranged and
grouped. If there is a set of n objects, the number of ordered
arrangements of objects will depend on r, the number of objects selected
and arranged.
Permutations are referenced as ...
Pnr=
n(n-1)(n-2)(n-3) ... (n-r+1)
or
= n!
(n - r)!
|
Given 5 cards, how many ways
can we draw the 3 cards:
Pnr=
5(4)(3)
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Statistical Counting
Combinations
The number of ways r objects can be
selected from n objects without regard to order.
n
(
r )
= n!
r!(n - r)!
|
Given 5 cards,
how many combinations of 3 cards can we draw?
=
5!
3!(5 - 3)!
= 10 combinations
|
| |
|
How many outcomes
can result from throwing 3 coins...
| H |
H |
H |
| T |
H |
H |
| H |
T |
H |
| T |
T |
H |
| H |
H |
T |
| T |
H |
T |
| H |
T |
T |
| T |
T |
T |
8 different outcomes.
2 possibility, 3 touses
|
| |
|
How many distinct
combinations can you find for 3 coins ...
| H |
H |
H |
All
heads |
|
|
|
|
| T |
H |
H |
|
| H |
T |
H |
two
heads |
| H |
H |
T |
|
|
|
|
|
| T |
T |
H |
|
| T |
H |
T |
two
tails |
| H |
T |
T |
|
|
|
|
|
| T |
T |
T |
All
Tails |
4 combinations.
n = number of objects .... 3
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