Equivalent fractions
Two thirds of a pan, cut again, becomes four sixths. Is that a different amount, or only a different name for the same one?
Where you are 4 inside
What this area is about
Two fractions are equivalent when they name the same amount of the same whole, the same point on a number line. Cutting every piece the same way changes the name and not the amount, so one number has many fraction names. Choosing the right name is what makes fractions comparable and addable.
How these fit together
The idea. Seeing equivalent fractions shows it in pictures and on the number line: cut every piece in two and both numbers double, while the shaded amount stays put.
Two moves. Multiplying renames a fraction in smaller pieces: \displaystyle \frac{a}{b} = \frac{a \times k}{b \times k} for any whole number k \neq 0, with b \neq 0. Simplifying runs it backward, dividing the top and the bottom by a common factor; when the top and bottom share no factor but 1, the fraction is in simplest form.
The working tool. Common denominators renames two fractions in one size of piece, the smallest such being the least common denominator.
A path through it
Pictures first, then multiplying. After multiplying come simplifying and common denominators, in either order.
With these you can compare and add fractions with unlike denominators, find a fraction between two fractions, and later scale ratios and convert fractions to percents.