Mr. Burke's C.C. Algebra Class Page

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This page will contain notes on all the topics which we have covered so far. You will be responsible for all this material.

WolframAlpha Knowledge Engine

In class, I mentioned the WolframAlpha Knowledge Engine, located at http://www.wolframalpha.com.
You can use this site to solve equations and inequalities, and it will even give you a graph of the answer.

It works for more than math questions. Try typing a Chemistry or Geography question and see what you get!


Table of Contents


STATISTICS REVIEW

We are having an important Statistics exam on Friday, May 2, 2014.
Statistics will be an important part of the Common Core Algebra Exam.
You should review this material.

Measures of Central Tendency: The mean, median and mode. They are numbers used to describe a set of data. See the individual definitions below.

Mean: The average. To find the mean, add up all the data and divide by the number of data items there are. Example: The mean of 6, 3, 5, and 12 is (6+3+5+12)/4 = 26/4 = 6.5.

Median: The middle number when the data are in order. If the data are not in order, than the number in the ”middle” is meaningless. If there are two numbers in the middle, the median is the average of the two numbers. Example: The median of 6, 3, 5, and 12 would be found be first ordering the numbers as 3, 5, 6, 12. The numbers 5 & 6 are in the middle, so the median is 5.5.

Mode: The most frequently appearing piece of data. Unlike mean or median, there can be more than one mode. There might be no mode if no data is repeated. Mode doesn’t have to be a number: consider the data set (red, red, yellow, blue, blue, blue); “blue” is the mode. Examples: (1, 2, 3, 3, 3, 4, 4): “3” is the mode; (88, 95, 92, 95, 88): 88 and 95 are the modes; (7, 8, 9, 10): no mode.

Frequency The number of times something occurs. In the data (10, 10, 11, 11, 11, 12, 12, 13), “11” has a frequency of 3, and the total frequency of the set is 8.

FREQUENCY TABLES

A Frequency Table is a summary of the data, organized as a table. The data are on in the left column. The frequency of each piece of data is one the right.
There are three examples of frequency tables below.

In the first example, every number in the data is listed, along with its frequency. The interval of the table is 1. Because of this we can find the mean, median and the mode of the data. We can re-create the data if we wanted to by writing out all the numbers.

In the second example, the data is collected into intervals of 10. We don’t know the actual numbers; we only have approximate information. Because of this we cannot find the exact mean, median or mode. However, we can find which interval contains the median and which interval is the most frequent. (There are methods to find a mean, but we aren’t going to calculate that right now.)

In the third example, the data aren’t numbers, they’re adjectives (qualitative data). Because of this, we can find a mode, but not a mean or median.

Finding the Mode, Median and Mode of a Frequency Table

In the first table, the total frequency (the sum of the Frequency column) is 20. If we write out the data, we can see that it has a total of 36. Divide 36 by 20 and we get a mean of 1.8. We can also see that there are more 2s than any other number, so 2 is the mode. And out of 20 numbers, the 10th and 11th are in the middle: both of those numbers are 2, so the median is 2.

However, we don’t need to rewrite all the data. (And if the table were bigger, we wouldn’t want to write out all the data!) The number with the highest Frequency is 2, with a frequency of 7 (mode). If you keep a running count as you go down the Frequency column, you will see that the 9th through 15th numbers are all 2 – that includes the 10th and 11th number (median). And if we know how many of each number there are, we can take a short-cut to get the sum of the data: (0 x 3) + (1 x 5) + (2 x 7) + (3 x 4) + (4 x 0) + (5 x 1) = 36. Divide 36 by 20 and we get 1.8 (median).

In the second table, the mode and median can be found using the same method.

In the third table, because the data are descriptive and not numeric (quantitative), there is no median. A middle size somewhere between Medium and Large wouldn’t have any meaning.


Important Vocabulary

Variable: An unknown value, usually represented by a letter of the alphabet, such as x.

Term A number, a variable, or a number multiplied by a variable.
Ex. 3, y, 4x (which means "4 times x")

Co-effiecient The number in front of a variable. If there is no number, then the co-efficient of the variable is 1. If there is only a negative sign in front of the variable, then the co-efficient of the variable is -1.
Ex. 5y, the co-efficient is 5; -x, the co-efficient is -1.

Expression: A mathematical phrase that may contain numbers, variables and operators. Expressions do not have equal signs. Expressions can be evaluated, but not solved.

Equation: A mathematical statement contains two algebraic expressions and an equal sign. Some equations are True. Some are False. Some are Open, meaning that there are some values for the variables which will make the equation true.

Inequality: A mathematical statement that compares two algebraic expressions, where one is greater or less than the other.

less than: < less than or equal to: < greater than: > greater than or equal to: >

Set: A collection of objects, such as numbers, which are also called elements or members of a set.
Examples: {3}, {-5, 5}, {1, 2, 3, 4, ...}, {x real | all numbers not equal to 0}

Solution Set: A set of values which are all the solutions to equations or inequalities.
Example: x2 = 16 has the solution set {-4, 4},

Set-builder notation: A special way of describing a set that has too many numbers to be listed. It has the following format:

{variable number-type | description of set or logical condition}
Example: {x real | x < 3}, which is read as "The set of all real numbers x such that x is less than or equal to 3."




Combining Like Terms



An algebraic expression can be made up of many terms separated by addition and subtraction. If all the variables in two terms are the same and have the same exponent (or no written exponent), these terms can be combined by adding or subtracting them.
Examples: 3x + 4x can be combined into 7x.
        5y2 - 3y2 can be combined into 2y2.
        10cd + 5cd can be combined into 15cd.
Non-examples: 3x and 4y are not like terms. Neither are 5d and 5 nor 2m2 and 2m.



Solving Simple Equations


Some statements are True and some are False. An equation like x + 5 = 12 is an Open statement. It doesn't have a true or false value because of the variable. However, we can solve the equation for x, finding the value that makes the statement true.

To solve a simple one-step equation, you have to look at the operation being performed. We need to do the opposite operation, working backward to find what number that when increased by 5 is equal to 12.

x + 5 = 12
- 5 = -5
Subtract 5 from Both sides of the equation
x + 5 - 5 = 12 - 5
The two fives will cancel because 5 - 5 = 0
x = 7
The solution is 7
Try this one on your own:
x - 12 = 30