Transcript Averages (means, medians, and modes)

An understanding of these 8 skill sets will help you
ace the COMPASS:
Integers
Decimals
Exponents
Square roots
Scientific notation
Fractions
Percentages
Averages (means, medians, and modes)
An understanding of these 10 skill sets will help you
ace the COMPASS:
Substituting Values
Setting up equations
Factoring polynomials
Exponents and radicals
Basic operations/polynomials
Linear equations/one variable
Linear equations/ two variables
Rational expressions
Quadratic formulas
Functions
What are Integers?
Integers are the set of whole numbers
and their opposites
 What
are decimals?
• The zero and the counting numbers (1,2,3,...) make up the
set of whole numbers. But not every number is a whole
number. Our decimal system lets us write numbers of all
types and sizes, using a clever symbol called the decimal
point.
• As you move right from the decimal point, each place
value is divided by 10.
 Examples:
• Three hundred twenty-one and seven tenths
 321.7
• (6 x 10) + (3 x 1) + (1 x 1/10) + (5 x 1/100)
 63.15
• Five hundred forty-eight thousandths
 0.548
• Five hundred and forty-eight thousandths
 500.048

What are exponents?
 Exponents are a shorthand way to show how many times a number,
called the base, is multiplied times itself. A number with an exponent
is said to be "raised to the power" of that exponent.



Finding the square root of a number is the inverse operation of
squaring that number. Remember, the square of a number is that
number times itself.
The perfect squares are the squares of the whole
numbers.
The square root of a number, n, written below is the
number that gives n when multiplied by itself.
By using exponents, we can reformat numbers. For very large or very small
numbers, it is sometimes simpler to use "scientific notation" (so called, because
scientists often deal with very large and very small numbers).
The format for writing a number in scientific notation is fairly simple: (first digit
of the number) followed by (the decimal point) and then (all the rest of the digits
of the number), times (10 to an appropriate power). The conversion is fairly
simple.
•
Write 124 in scientific notation.
•
This is not a very large number, but it will work nicely for an example. To
convert this to scientific notation, I first write "1.24". This is not the same
number, but (1.24)(100) = 124 is, and 100 = 102. Then, in scientific notation,
124 is written as 1.24 × 102.
Actually, converting between "regular" notation and scientific
notation is even simpler than I just showed, because all you really
need to do is count decimal places.
•
Write in decimal notation: 3.6 × 1012
Since the exponent on 10 is positive, I know they are looking for a
LARGE number, so I'll need to move the decimal point to the right, in
order to make the number LARGER. Since the exponent on 10 is
"12", I'll need to move the decimal point twelve places over.
• In other words, the number is 3,600,000,000,000, or 3.6 trillion
•
Adding and Subtracting Fractions:
•
Like fractions are fractions with the same denominator. You can add
and subtract like fractions easily - simply add or subtract the numerators
and write the sum over the common denominator.
•
Before you can add or subtract fractions with different denominators,
you must first find equivalent fractions with the same denominator, like
this:
•
Find the smallest multiple (LCM) of both numbers.
•
•
When working with fractions, the LCM is called the least common denominator
(LCD).
Rewrite the fractions as equivalent fractions with the LCM as the
denominator.
Method 1:
Write the multiples of both denominators until you find a common multiple.
The first method is to simply start writing all the multiples of both
denominators, beginning with the numbers themselves. Here's an example
of this method. Multiples of 4 are 4, 8, 12, 16, and so forth (because 1 × 4=4,
2 × 4=8, 3 × 4=12, 4 × 4=16, etc.). The multiples of 6 are 6, 12,…--that's the
number we're looking for, 12, because it's the first one that appears in both
lists of multiples. It's the least common multiple, which we'll use as our least
common denominator.
Method 1:
Write the multiples of both denominators until you find a common multiple.
The first method is to simply start writing all the multiples of both
denominators, beginning with the numbers themselves. Here's an example
of this method. Multiples of 4 are 4, 8, 12, 16, and so forth (because 1 × 4=4,
2 × 4=8, 3 × 4=12, 4 × 4=16, etc.). The multiples of 6 are 6, 12,…--that's the
number we're looking for, 12, because it's the first one that appears in both
lists of multiples. It's the least common multiple, which we'll use as our least
common denominator.
Example:
To multiply fractions:
Simplify the fractions if not in lowest terms.
Multiply the numerators of the fractions to get the new numerator.
Multiply the denominators of the fractions to get the new denominator.
Simplify the resulting fraction if possible.
Using Percent
Because "Percent" means "per 100" you should think
"this should always be divided by 100"
So 75% really means 75/100
And 100% is 100/100, or exactly 1 (100% of any
number is just the number, unchanged)
And 200% is 200/100, or exactly 2 (200% of any
number is twice the number)
•
Mean
• There are three main types of average:
mean, mode and median. The mean is what
most people mean when they say 'average'.
It is found by adding up all of the numbers
you have to find the mean of, and dividing
by the number of numbers.
• So the mean of 3, 5, 7, 3 and 5 is 23/5 = 4.6 .
Moving Averages
A moving average is used to compare a set of figures over time. For
example, suppose you have measured the weight of a child over an eight
year period and have the following figures (in kg):
32, 33 ,35, 38, 43, 53, 63 ,65
Taking the mean doesn't give us much useful information. However, we
could take the average of each 3 year period. These are the 3-year moving
averages.
The first is: (32 + 33 + 35)/3 = 33.3
The second is: (33 + 35 + 38)/3 = 35.3
The third is: (35 + 38 + 43)/3 = 38.7, and so on (there are 3 more!).
To calculate the 4 year moving averages, you'd do 4 years at a time instead,
and so on...
Mode
The mode is the number in a set of
numbers which occurs the most. So the
modal value of 5, 6, 3, 4, 5, 2, 5 and 3 is 5,
because there are more 5s than any other
number.
Range
The range is the largest number in a set
minus the smallest number. So the range
of 5, 7, 9 and 14 is (14 - 5) = 9. The range
gives you an idea of how spread out the
data is.
The Median Value
The median of a group of numbers is the number in the middle,
when the numbers are in order of magnitude. For example, if the set
of numbers is 4, 1, 6, 2, 6, 7, 8, the median is 6:
1, 2, 4, 6, 6, 7, 8 (6 is the middle value when the numbers are in
order)
If you have n numbers in a group, the median is the (n + 1)/2 th
value. For example, there are 7 numbers in the example above, so
replace n by 7 and the median is the (7 + 1)/2 th value = 4th value.
The 4th value is 6.
• suppose x = 9
and you have an equation like
8x - 15 = 6x + 3
•
you put the number 9 in
place of the x's in the
equation
8(9) - 15 = 6(9) + 3
57 = 57
Here is an example of a word problem
A child's piggy bank has 3 times as many dimes as nickels. Altogether she has $3.85. How many
dimes and nickels does she have?
•
To solve this problem first use this format
Let D = number of dimes (worth 10¢ each)
The total value of the dimes is: 10D cents
•
Let N = number of nickels (worth 5¢ each)
The total value of the nickels is: 5N cents
The total value is: 10D + 5N = 385
We are told that "dimes = 3 times nickels": D = 3N
Substitute into our equation: 10(3N) + 5N = 385
Then we have: 35N = 385 → N = 11
Therefore, there are 11 nickels and 33 dimes
An example would be
Factor 3x-12
First identify what the two terms have in common
Both terms are divisible by 3.
When I divided the "3" out of the "3x", I was left with only the "x"
3x – 12 = 3(x
)
When I divided the "3" out of the "–12", I left a "–4" behind, so I'll put
that in the parentheses, too:
3x – 12 = 3(x – 4)
•
This is my final answer: 3(x – 4)
Exponents are shorthand for repeated multiplication of the same
thing by itself. For instance, the shorthand for multiplying three
copies of the number 5 is shown on the right-hand side of the
"equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this
example, stands for however many times the value is being
multiplied. The thing that's being multiplied, being 5 in this
example, is called the "base
A "radical" equation is an equation in which at least one variable expression is stuck
inside a radical, usually a square root
Example
For instance, this is a radical equation:
There are a couple of issues that frequently arise when solving radical equations. The
first is that you must square sides, not terms. Here is a classic example of why this is so:
I start with a true equation and then square both sides:
3+4=7
(3 + 4)2 = 72
49 = 49
...but if I square the terms on the left-hand side:
3+4=7
32 + 42 "=" 72
9 + 16 "=" 49
25 "=" 49 ...............Oops!
Linear Equation
An equation that can be written in the form
ax + b = 0
where a and b are constants
example
7x-1=10
Solving a Linear Equation
in General
Get the variable you are solving for alone on one side and
everything else on the other side using INVERSE operations
7x-1=10
+1
+1
+1 on both sides
7x=11
Then divide by seven to get the variable by its self
7x=11
/7 /7
X=11/7
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everything:
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
polynomials:
http://www.coolmath.com/algebra/algebra-practice-polynomials.html

algebra:
http://www.math.com/practice/Algebra.html

pre-algebra:
http://www.math.com/practice/PreAlgebra.html

COMPASS specific:
http://www.act.org/compass/sample/math.html