Perimeter
The perimeter is the distance around an object.
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Basic Operations
Adding
Adding is the most basic operation where two items are combined.
Methods of Addition
Column Addition
Column addition is the traditional way to add two numbers together. First, you need to place two numbers into columns such that the numbers on top of each other have the same place value.
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Chunking
Chunking is a way of using easier numbers to add. For instance, 23 + 39 could be seen as 20 + 30 = 50, and 3 + 9 = 12. Add these two answers together and you get 62, which is the answer to the original problem.
Commutative Property of Addition
Addition is commutative. That is to say that you can switch numbers around and get the same answer. So for any numbers a and b, a + b = b + a
Example:
4 + 3 = 7
3 + 4 = 7
You get the same answer!
Videos
Associative Property of Addition
Addition is associative. This means that numbers can be grouped in different ways, and still you will get the same answer. For any numbers a, b, and c, (a + b) + c = a + (b + c). Remember with PEMDAS, brackets must be done first. Even so, with addition it doesn't make a difference.
Example:
(1 + 2) + 3 = (3) + 3 = 6
1 + (2 + 3) = 1 + (5) = 6
Videos
Subtracting
In subtraction order is very important. 3 - 2 is different from 2 - 3. Make sure that when you are subtracting a value from another one that you have the correct order. The number you start with is always first, and the number you are subtracting comes second. This may seem obvious, but it is surprising how often this gets confusing especially where PEMDAS is involved.
The Add-Opp Property
The algebraic definition of subtraction is simply adding the negative version of the subtracted value. x - y = x + -y
Example: 3 - 2 = 3 + -2
Of course adding a negative number is harder than subtracting itself, but it is important to know that you can switch from one to the other if required.
Multiplying
Videos
Dividing
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Property of Opposites
The property of opposites states that for any number n, n + -n = 0. In other words, when a number is added to its opposite (i.e. the negative version of itself), then the result is zero.
Examples
1 + -1 = 0
 + - = 0
0.213 + -0.213 = 0
x + -x = 0
Opposite of Opposites Property
The Opposite of Opposites Property states that for any number, n, -(-n) = n. In other words, this is just a fancy way of saying that the opposite of a negative number is a positive one.
Examples
-(-2) = 2 (The opposite of a negative 2 is 2)
-(-x) = x (The opposite of -x is x)
Zero
Zero has many properties. Wikipedia has a great article about the history and properties of [zero]. Zero is a number that was invented a few hundred years BC. It is debatable whether the Olmecs of Mexico or the people of India invented it first, but since its invention zero has created many useful abilities in mathematics and some very unexpected problems.
Additive Identity
An identity in mathematics is an operation that gives the original number back. In addition, the identity belongs to zero.
n + 0 = n.
If you take any number, and add zero to it, you get the original number. This may sound obvious, but the idea behind it is very important.
Division
Dividing by zero is a huge no-no. If there is anything that can cause mathematics to explode, an operation like 5 ÷ 0 will give an error in any calculator because it just can't be done!
Absolute Value
An absolute value is a number's distance away from zero. The absolute value of something is written with two vertical lines. For instance, the absolute value of 3 is written |3|. Another way to think of absolute value is that the absolute value of a positive number is the same number. An absolute value of a negative number is the positive version of it.
Examples
Resources
Books
- Transition Mathematics p248
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Angles
An angle is a way of measuring the distance between two connected lines. Angles are usually measured in degrees (º). The size of an angle is usually determined by using a protractor. Protractors are usually a half-circle like the picture to the right, but they can also be found in full-circle (360º) versions.
Another way to determine the measurement of an angle, although less accurate than a protractor, is to use names. An acute angle is one that is less than 90º. A right angle is one that is exactly 90º. An obtuse angle is one that is bigger than 90º, but less than 180º. An angle that is exactly 180º is known as a straight line. Finally, a lesser known angle type is one that is bigger than 180º. These are called reflex angles.
Angle Facts
- A circle is 360º
- A triangle has a total of 180º when you add together all three angles
- A straight line is 180º
- A half-circle is 180º
- All quadrilaterals have 360º when you add all four angles
- What's known as a full turn is 360º
- What's known as a half-turn is 180º
- What's known as a quarter-turn is 90º
- A right angle is 90º
Special Types of Angles
Supplementary Angles
Supplementary angles are angle pairs that add to 180º.
Complimentary Angles
Complimentary angles are angle pairs that add to 90º.
Vertical Angles 
Vertical angles occur when two lines intersect. The angles opposite of each other are called vertical angles. These angles are equal to each other.
Vertical angles usually come in pairs. In the image on the right, both a and b are equal to each other. Also, the unlabeled angles are equal to each other as well.
Angles in Parallel Lines
These types of angles deserve their own section.
Resources
Books
- Transition Mathematics p145-149 - Measuring Angles and Parts of an angle
- Transition Mathematics p152-154 - Angle Types (Obtuse, Acute, etc.)
- Transition Mathematics p254-259 - Adding angles (including negative angle measurements)
- Transition Mathematics p382 - Interior and Exterior Angles
- Transition Mathematics p376-380 - Supplementary and Complementary
Videos
Games
- [Super Maths World] - Go to shape, and then Angle Rules 1 to test yourself on the fact that angles in a straight line add to 180º, and angles in a circle add to 360º.
See Also
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Angles in Parallel Lines
If two parallel lines are crossed over by a third line, this line is called a transversal. The angles they make carry certain properties that can be used to help find missing angles.
Vertical Angles
Vertical angles are discussed in more detail here. In the diagram to the right, it means that angles a and d are equal to each other since they are vertical. There are a total of 4 pairs of angles that are equal to each other because of this.
Supplementary Angles
Any angles along a straight line add to 180º. Because of this, we can find out what a neighboring angle is equal to since they will both add together to get 180º. More on supplementary angles here.
Corresponding Angles
Also known as F angles.
Alternate Angles
Also known as Z angles.
Solve Equations
Here are some basic strategies that can be used when trying to solve equations of different types
One-step Equations
A One step equation is a problem that requires one step to solve. These are the easiest equations to solve.
x + a = b
These problems can usually be done in your head, but it is important to how to do it step-by-step so that you can later on work on harder problems using the strategies. First, lets give an example of such a problem:
One way to think of this problem is to replace the x with the word something. So this problem becomes:
In other words, what number do you add five to in order to get to 11? 6! So my answer is six, or more accurately:
The best way to solve these types of problem is a bit more tricky. The strategy above won't work with harder problems that we'll cover later, but this strategy will work with almost all problems so it's good to give it a try. The trick is to do the opposite. Let's start with the same problem again:
Now, in order to figure out what x is equal to, you need to do the opposite. Currently, we are adding 5 to x. What is the opposite of adding 5? Subtracting! So if you subtract five from 11 like so:
You get the same answer as before: 6.
x - a = b
To solve problems of this type, you could take the strategy from the problem of solving x + a = b above.
Now, change x with the word something
If you read this out it sounds like: Something take away 4 is equal to 13. So we know that to find out what the something is equal to, you have to figure out what the number was before you subtracted 4. 17 - 4 is 13, so our answer must be 17.
Just as with the x + a = b, a better way is to solve by doing the opposite. Here's an example:
To solve this, do the opposite. What is happening to x to get to 13? We are subtracting 4! What is the opposite of subtracting 4? Adding 4, and so we get:
a - x = b
The equation a - x = b presents a problem. If we use the strategy that we've discussed before, we run into a problem: negative numbers. If you try to solve by doing the opposite you end up with the problem in the example below:
the opposite of adding 8 is subtracting 8 so we get:
minus x equals minus 4? That doesn't make sense! One thing you can do is to take the opposite of both sides. This means that if something is negative, make it positive. You have to do this to both sides. You should by now know the opposite of opposites property that allows you to do this.
- - x = -4
- -(-x) = -(-4)
- x = 4
So our answer is x = 4. You can try it out by putting this number back into the first equation. 8 minus 4 is equal to 4:
- 8 - x = 4
- 8 - 4 = 4
- 4 = 4
ax = b
In algebra, ax means a times x. The way to solve these problems is to do the opposite. What is the opposite of multiplying? Dividing!
In this problem, we need to find out what times five is twenty. You may be able to do this in your head, but the best way is to do the opposite. The opposite of multiplying by five is dividing by five so we get:
x ÷ a = b
Again, to solve this type of problem, we need to do the opposite. The opposite of dividing is multiplying!
In this example we need to figure out what number, when divided by three is nine. You may be able to do this in your head, but the best way is to do the opposite. The opposite of dividing by three is multiplying, so we get:
Resources
Videos
Games
- [Math Kiwi] - Try to solve one-step equations that have negative numbers!
- [Equation Basketball] - Try to solve other one-step equations with negative numbers
- [Battleship!] - An interesting take on battleship where you can only hit the enemy ship if you answer a one-step equation correctly.
Two-step Equations
A two-step equation is one that takes two steps to solve. They are often a combination of two one-step equations.
a ÷ x = b
This problem is a bit trickier as a is being divided by x, and so doing the opposite here may not help. Let's try it anyway. The opposite of dividing by x is multiplying by x
So now we get a problem much like the ax = b problem above. Now we can just do the opposite of dividing by 10 and get our answer
ax + b = c
a (x + b) = c
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Harder Equations
ax + b = cx + d
a(x + b) = c(x + d)
(x + a)(x + b) = c
Miscellaneous Resources
Videos
See Also
- Solving Systems of Equations
- Solving by Substitution
- Solving by Elimination
- Solving by Graphing
- Solving Quadratic Equations
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Quadratic Equations
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Polygon
A polygon is a many sided shape, or more literally a shape with many angles. Poly comes from the ancient Greek polus which means many, and gon comes from the ancient Greek gonia which means angle. A good article on polygons can be found at [Wikipedia] which talks about the different aspects of polygons. Here I'll just mention a few characteristics.
Polygon Names
Polygons are named as follows. Most you should be familiar with, but a few will be new. Quadrilaterals are just a name for all four sided shapes including squares, rectangles, trapezoids, etc.
| Number of Sides |
Shape Name |
| 3 |
Triangle |
| 4 |
Quadrilateral |
| 5 |
Pentagon |
| 6 |
Hexagon |
| 7 |
Septagon (also Heptagon) |
| 8 |
Octogon |
| 9 |
Nonagon |
| 10 |
Decagon |
| More than 10 |
n-gon (i.e. a polygon with 12 sides is a 12-gon) |
Concave / Convex
Polygons can either be concave or convex. Concave means that a vertex of the shape goes inside the shape itself. In other words, the shape has a cave in it. This is how I remember the difference between concave and convex. A convex polygon is the other type in which there are no vertices inside of the polygon.
Regular / Irregular
A regular polygon is where all of the sides are the same length, and all of the angles are the same measurement. An irregular polygon is one where this isn't true.
Interior Angles
| Shape Name |
Interior Angle Sum |
Individual angle measurement if all angles are equal |
| Triangle |
180º |
60º each |
| Quadrilateral |
360º |
90º each |
| Pentagon |
540º |
108º each |
| Hexagon |
720º |
120º each |
| Septagon (also Heptagon) |
900º |
128.6º each |
| Octogon |
1080º |
135º each |
| Nonagon |
1260º |
140º each |
| Decagon |
1440º |
144º each |
| n-gon |
180 x (n-2)º |
[180 x (n-2)] ÷ nº |
Exterior Angles
Exterior angles are angles that are outside of the shape itself and they carry certain properties.
Resources
Games
- [Super Maths World] - Go to Kids, and then shapes for some practice. Level 4 is help on naming polygons
Triangle
Triangles are a type of polygon that has three sides. There are several different types of triangles depending on their properties. Triangles are one of the most important shapes used, and are very important in a variety of fields.
Types of Triangles: There are many different types of triangles and their names are taken from what their angles are and what their side lengths are.
Names Based on Angles
- A triangle is a Right Triangle if one of its angles is a right angle (90º)
- A triangle is an Obtuse Triangle if one of its angles is bigger than 90º
- A triangle is an Acute Triangle if all of its angles are smaller than 90º
Names Based on Side Length
- A triangle is Scalene if each side is a different length than the others.
- A triangle is Isosceles if two sides are the same length
- A triangle is Equilateral if all sides are the same length
Resources
Games
Videos
- [Square One Triangle Song] - I'm probably showing my age posting this, but I used to watch Square One as a kid, and this video describes the different types of triangles.
Area
The area of a triangle can be thought of as finding half the area of a parallelogram since a right triangle is essentially half of a parallelogram. A simpler example would be a rectangle, which is a type of parallelogram. When you cut a rectangle in half along the diagonal (see picture below) you get two triangles.
 
Since finding the area of a rectangle is as simple as multiplying the length of the base times the height, the area of a triangle would be half that.
Area of a Rectangle = base x height
Area of a Triangle = ½ base x height
Since this is a lot to write, we typically abbreviate this into algebra: a = ½bh
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Angles in a Triangle
Interior Angles
No matter what, all interior (inside) angles of a triangle add to 180º.
See Also
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