In Math 9, I have learned about the following:
- Fractions on number lines
- Comparing fractions
- Adding/subtracting fractions
- Multiplying/dividing fractions
Click on a concept to jump to it.
Introduction to fractions
Before I begin, I will explain fractions. Here is an example of a simple fraction:
In this case, the 2 is what we call the numerator, placed on top of the fraction. 4 would be the denominator.
The 1 next to the fraction is a whole number, representing 
We can convert the whole number back into fraction form and add it to the fraction.
This is an improper fraction. When performing operations with fractions, you should always convert them to improper fractions and converting them back to mixed fractions if needed once you have completed operations.
Fractions on number lines
This is a number line.
On a number line, numbers are arranged from smallest to biggest from left to right. With negative numbers, they move left, starting from 0, as shown here.
With fractions however, the whole numbers are much more spaced out, with the lines between whole numbers representing fractions.
For example, these are the locations of 
Notice how
Another example, this time with 
Notice how the fractions have 2 different denominators. In a situation like this, fractions are placed on an equivalent line. ie: 
Comparing Fractions
Comparing fractions is easiest with common denominators, like so.
Right off the bat, it’s obvious that the first fraction is bigger than the other. To mark that
However, many times, both fractions will not have a common denominator.
In this case, we can multiply both the numerator and denominator by the denominator of the other fraction to acquire a common denominator.
2 
3
3
4
Now, we have these fractions.
These fractions are now easier to compare. Since the bigger fraction is on the right, we use the < symbol. If it were on the left, we would use the > symbol instead.
If we had equivalent fractions, shown below, we would use an = symbol.
Comparing negative fractions is the same case with positive fractions, albeit with larger numbers now being smaller, like so.
In this case, 
Comparing a negative fraction to a positive fraction is no different.
An alternate way to see if 2 fractions are equal is to divide the numerators of each fraction by their denominator, and seeing if the quotient is the same.
3 
6
As the quotients are not the same, the fractions are not equal. Another example:
5
10
These quotients are the same, so the fractions are equal.
When fractions with whole numbers are encountered, you can turn it into an improper fraction by multiplying the denominator by the whole number, then add it to the numerator, like so.
3
9 + 2 = 11
With negative fractions, this is also the case, though the fraction should be treated as a regular fraction during conversion.
5
20 + 1 = 21
Adding and subtracting fractions
When adding and subtracting fractions, the fractions must have common denominators. As shown before, this can be achieved by multiplying the numerators and denominators by the denominator of the other fraction. An example:
4
5
2
3
To add is simple; simply add the numerators like normal numbers.
To simplify the fraction, first convert the fraction into a mixed fraction if needed:
22
22 – 15 = 7
Afterwards, reduce the fraction to it’s simplest form. In this case, the fraction is already in it’s simplest form, so no action is needed. Here’s an example of a fraction that can be reduced.
15
Subtraction is similar.
9 – 3 = 6
When an operation results in a fraction equal to a whole number, the fraction should be converted to a whole number.
6
Negative numbers “invert” operations. Subtracting a negative number from another negative number would actually add to the value while appearing to subtract, and adding a negative number to another negative number would subtract from the value while appearing to add.
Adding or subtracting negative numbers from positive numbers also have “inverted” operations. Adding a negative number would be the same as subtracting from a value for example.
Multiplying and dividing fractions
When multiplying fractions, there is no need to have common denominators. You will have to multiply both the numerator and denominator by the numerator and denominator of the other fraction respectively.
4
6
As with addition and subtraction, you can simplify the resulting fraction.
48
Multiplying negative fractions is quite simple. Each negative number will “invert” the value of the number. An odd amount of negative numbers would make a negative product, while an even amount would mean a positive product. An example:
4
5
As you can see, the resulting fraction is negative. However, if both fractions were negative…
-4
5
Double negatives result in a positive fraction, as you are multiplying a negative number by a negative number (double negative), and the inverse of a negative number is a positive number.
With division, you have two options to effectuate:
- Multiply the numbers for a common denominator before dividing
- Flip the second number to do a multiplication instead (this is called a reciprocal)
Starting with method 1:
Multiply the numbers to a common denominator.
4
5
In this example, simply divide the numerators like integers. Since dividing the denominator always equals 1 with a common denominator, it is simply a matter of dividing the numerator.
8
This is how one would perform the same division with method 2:
Flip
Now, effectuate as if the question were a multiplication.
8
10
Dividing negative fractions functions in a nearly identical fashion to multiplication, where dividing with odd amounts of negative numbers create negative quotients, while even numbers create positive quotients. For this example I will use the reciprocal method of fraction division.
-5
7
And of course, if we do the same question but both numbers are negative…
For the sake of simplicity, the negative sign on the second fraction will remain on the numerator, as the position does not change the value of the fraction.
-5
7
I believe that this covers all of the concepts listed at the top of the page.