Essential concepts your need to know extremely well if you don't want to look like an idiot in any math class. Equations are something that you will constantly be using throughout your math life. Learning and understanding the basics is an integral part of succeding in maths.

Changing the order of the addends doesn't change their sum.

• `a + (b + c) = (a + b) + c`

Any number plus zero equals itself.

• `a + 0  = a`

If two sides of an equation are equal, you can add or subtract the same amount to both sides, and they will still be equal.

• ```a = b
a + c = b + c
a - c = b - c
```

### Multiplication & Division

Order of operations:

Multiplication can be written three different ways.

```9 * x
9x
9(x)
```

A fraction bar is also a division symbol.

• 4 / 2

Changing the order of multipliers doesn't change their product.

• `ab = ba`

Zero times any number is zero and 1 times any number is the same number.

• ```x(0) = 0
(0)x = 0
x(1)= x
1 * x = x
```

If two sides of an equation are equal, you can multiply or divide each side by the same quantity (number or equation) and it will still be equal.

```a = b, c != 0
ac = bc
(a / c) = (b / c)
```

### Coordinate Plane & Points

The origin's coordinates are:

• ```(0,0)
```

Points are named by an ordered pair:

• ```(4,2)
```

The first number in an ordered pair is the x-coordinate, and the second number listed is the y-coordinate.

### Graphs of Lines

Linear equations can be graphed when in the form:

• ```y = mx + b
m equals the slope of the line.
b equals the point where the line crosses the y-axis.  This is called the y-intercept.
```

### Lowest Common Multiple

The LCM is seful when multiplying and dividing fractions.

When finding an LCM, use only multipliers that are whole numbers.

• `{4, 8, 43, 104}`

Be sure to be aware of all the numbers you are finding an LCM for.

• ```LCM for 4 and 5

Multiples of 4 Common Multiples Multiples of 5
4		                5
8                               10
12                              15
16                              20
20              20              25
...             ...             ...
```

### Greatest Common Factor

The GCF is useful when dealing with fractions.

When finding a GCF, use only whole numbers.

• `{2, 9, 27, 201}`

Be aware of all the numbers you are finding a GCF for.

• ```GCF of 8 and 12

Factors of 8  Common Factors  Factors of 12
1                             1
2                             2
3
4             4               4
6
8
12
```

### Multiplication of Fractions

Do not cross-multiply fractions.

• ```   4    2     4   don't do this!
- *  - != --
2    3    12
```

You can multiply more than two fractions together in one problem.

• ```   1   3   4   12   3
- * - * - = -- = -
2   2   5   20   5
```

### Division of fractions

Cross multiplication is involved in the division of fractions.

Flip the second fraction of the two being multiplied at the time upside down to do the problem correctly.

As indicated above, there can be more than two fractions in a division problem involving fractions, but you can only divide 1 fraction by 1 fraction, so you have to do a problem like that in more than one part.

• ```   1   3   2   4   2   20   5
- / - / - = - / - = -- = -
2   4   5   6   5   12   3
```

### Common Denominators

To add or subtract a fraction from another number (whole or fractional), the denominator needs to be the same.

• ```  1   3        4   3
- + -   ==   - + -
2   8        8   8
```

When a fraction has a numerator and denominator that are the same number, the fraction is equal to 1.

Multiplying by 1 does not change a number, even though the form might change.

• ```  4   4   2    8
- = - * - = --
5   5   2   10
```

I am still struggling in math. The sequences of long symbols depict to me only a disconnect between numbers and common logic. The long formulas fill up with digits like a basin, and tangle down the blue lines in my notebook: dimensional analysis is not my strong point.

The Pre-Algebra 1B teacher always hands back my test with smoke stains. Karisa, the girl behind me, lives in the slums of Trenton, New Jersey: she is of unappreciated genius. Zack, sitting on the far opposite side of the room, doesn't complete any of his homework but aces tests and exams consistently. He doesn't need to waste time on busywork that won't help him learn.

A quarter of my grade is in ensuring that my binder is in order; in making sure that all the class handouts--and all of my smokey, graded, failed homework--is neat and tidy. Failing a test or missing an assignment is received with belittlement from the teacher, while success is met with professional apathy.

There is no motivation.

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