PEMDA
Also Known as BODMAS, BEDMAS, or BIDMAS. It is the order you need to evaluate a math problem starting with the parentheses.
Meaning
- Parentheses - ( ), or [ ], or { } come first
- Exponents - x2
- Multiplication and Division occur at the same time (work left to right)
- Multiplication
- Division
- Addition and Subtraction occur at the same time (work left to right)
- Addition
- Subtraction
It is useful to come up with a trick to remember the order. For instance: "Paul Eats Many Desserts And Snacks" or "Please Excuse My Dear Aunt Sally" or even "Pandas Eeat Many Dead Adder Snakes." There are countless phrases that will work to help remember the order. The best idea is to make up your own.
Example
Let's take a complicated problem and work it step-by-step through the order of operations...
- 2 x 3 (4 + 3(2 - 1)2) + 4(2(3)2)
- 2 x 3 (4 + 3(1)2) + 4(2(3)2) - First, work out the inner-most parentheses
- 2 x 3 (4 + 3(1)) + 4(2(9)) - Next work out the exponents
- 2 x 3 (4 + 3) + 4(18) - Parentheses right next to a number means multiply (times)
- 2 x 3 (7) + 4(18) - Back to the parentheses
- 2 x 21 + 72 - There are no more parentheses or exponents, so now we multiply again
- 42 + 72 - Multiplication comes before addition
- 114 - Last is the addition
Resources and Extra Help
Books
- Transition Math Textbook on Pages 176-181
Videos
- [BODMAS] - Video stepping through using the BODMAS version of the same idea. The letters stand for the same thing, just using different words.
- [U2BMath PEMDAS Video] - A series of instructional videos on Youtube
- [PEMDAS Rap] - Youtube has lots of strange things on PEMDAS...
- [MATHMAN explains Order of Operations] - It has explosions...what more could you ask for?
Games
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Variable
A variable is something that changes or is unknown. A variable is usually shown as a letter.
For instance. 3 + w = 10
w is something we don't know, and is called a variable. We can easily find out what w is equal to. It is whatever we have to add to three to get to ten. W = 7
Videos: A clip from the TV series [Good Neighbors] about variables.
Patterns
A pattern is a rule that determines a series of numbers. For instance, the pattern "Start at 2, and then add 2" would look like: 2, 4, 6, 8, 10, 12, 14, . . .
There are many ways to represent patterns, and there are a lot of problems that you will encounter that use patterns.
Patterns and Finding Missing Values
nth Term
Patterns and Algebra
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Changing Words Into Algebra
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Quadrilateral
Quadrilaterals are polygons with four sides 
Types of Quadrilaterals:
Because there are s many ways to arrange four connecting lines, there are many names used to help group quadrilaterals. The image at the right (click on it to make it bigger) offers some examples of what is discussed below, or you can watch an animation I made [here]
There are essentially three conditions to keep track of:
- Parallel Lines
- Both pairs of opposite sides of the shape are parallel (square, rectangle, parallelogram, rhombus)
- One pair of opposite sides of the shape are parallel (trapezoid)
- No pairs of opposite sides of the shape are parallel (kite, arrowhead)
- Equal Sides
- All sides of the shape are equal (square, rhombus)
- Opposite sides of the shape are equal (rectangle, parallelogram)
- Adjacent sides of the shape are equal (kite, arrowhead)
- No sides of the shape are equal (trapezium)
- Angles
- All angles are 90º (square, rectangle)
- Both pairs of opposite angles (diagonally) are equal (parallelogram, rhombus)
- One pair of opposite angles (diagonally) are equal (kite, arrowhead)
- None of the opposite angles are equal (trapezoid, trapezium)
Area
The way to find the area of a quadrilateral changes depending on the shape. There are two ways to find the areas of a shape. The first way is to count squares. This isn't a very exact way. The best way to fin the area of a shape is by using a formula. Below are the area formulas for each shape.
When finding the area it is very important to remember to write the unit of measurement. If a shape is measured in inches (in), then its area will be measured in inches squared (in2). It is very important to write this in your answer and not just the number part of it.
Square
The area of a square is just its side length squared. In other words, its side length times its side length. For example, a square with a side length of 3cm would have an area of  . You could also use the method for rectangles (below).
Rectangle
The area of a rectangle is the base times its height or a = bh. This area is perhaps one of the easiest of all shapes to find and is taught quite early on. You can use this to find the area of a square also since the base and the height of a square are the same.
Parallelogram or Rhombus
For parallelograms and Rhombuses, you use the same formula as rectangles and squares (above). The only problem is that when finding the area, you need to make sure that the base and the height are at right angles to each other. This usually means that you don't use one of the measurements on the shape itself and can cause confusion.
Trapezoid
A trapezoid is one of the more complicated basic areas to find. The formula is ½h(b1 + b2). In other words, the area is the sum of the parallel sides of the trapezoid (called base 1 and base 2) multiplied by half the height.
Kite and Arrowhead/Delta
The area of the kite and the arrowhead/delta can be found by splitting the shape into two triangles and then adding their areas together.
Trapezium
A trapezium, having no parallel sides, can be a problem to find the area. Often times it can be thought of as a compound shape. More often than not, you can divide the shape into a trapezoid and a triangle.
Resources
Videos
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Probability
Probability consists of the following topics:
Frequency:
Frequency is the amount of times something happens. For instance, if we are talking about the weather, if it rains 5 days this week, then the frequency is 5.
Relative Frequency:
The relative frequency is how often something happens in relation to the bigger picture. If we use the example of it raining 5 days this week, then the relative frequency would be:
5 days of rain 5
----------------- = -
7 days of the week 7
Another way to think of this is to find the relative frequency you need to take the part and divide it by the whole.
Part
-----
Whole
Relative Frequency of Many (Independant) Events
When you're talking about more than one thing happening, then probability can get a little more complicated. For instance, most people know that when tossing a coin it will come up heads ½ of the time. But how often will a coin come up always heads when tossed twice?
Here are the possibilities:
| First Toss |
Second Toss |
| H |
H |
| H |
T |
| T |
H |
| T |
T |
So there are four possibilities. The possibility of getting heads twice is only one of these four, so the probability is ¼. The problem with making a table to figure out the probability is that it can take some time. For instance, what is the probability of it coming up heads ten times in a row?
There's an easier way. Take the probability of each toss of the coin and multiply them together:
½ x ½ x ½ x ½ x ½ x ½ x ½ x ½ x ½ x ½ =
1/(2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2)=
1/210 = 1/1024
Since each coin toss has a ½ chance coming up heads, multiplying them together gives us the probability of many coin tosses coming up the way we want.
Videos
Relative Frequency in Algebra
Sometimes you aren't given simple numbers to find the probability of an event. Instead, you are given a formula.
Fair Vs Unfair
An event is fair if the results of an experiment are about what we expect they will be. For instance, if you have a six-sided dice and you toss it 600 times, you would expect that each number would come up about the same number of times. If one number stands out more than the rest, then it would be considered unfair. Here is an example of a fair and unfair dice:
Results of Rolling a 6-sided dice 600 times
|
Fair Dice |
Unfair Dice |
| Roll a 1 |
101 |
70 |
| Roll a 2 |
104 |
68 |
| Roll a 3 |
96 |
45 |
| Roll a 4 |
99 |
72 |
| Roll a 5 |
95 |
75 |
| Roll a 6 |
105 |
270 |
| Total Rolls |
600 |
600 |
tice that with the unfair dice there are far more rolls of a 6 than any of the other values. With this dice we expect the answers to be about the same. This is what makes the dice unfair
Mutually Exclusive
Something is Mutually Exclusive from something else if there is no overlap between them. For instance, Monday and Tuesday are mutually exclusive because there is never a time when it is Monday and Tuesday at the same time in the same place.
Complimentary Events
Resources
Videos
- [Croupier] - In this movie clip they talk about the odds and probabilities in poker.
- [Statistics:Probability] - A silly video that gives a good overview of probability
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Open Sentence
A sentence is a mathematical expression with inequality symbol in it. An open sentence is a mathematical sentence with a variable in it.
Examples of Open Sentences
a + 4 = 7
b + 3 > 2
2m - 3 = 10
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Equation
An equation is a mathematical sentence that uses an equal sign.
Examples
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