Mixed numbers and improper fractions
Two whole loaves and three quarters of another, or eleven quarters. Which way should the amount be written, and how do you add and subtract amounts like it?
Where you are 4 inside
What this area is about
An amount bigger than one can be written two ways. A mixed number such as \displaystyle 2\frac{3}{4} is a whole number and a fraction less than 1 side by side, and it means their sum, \displaystyle 2 + \frac{3}{4}. An improper fraction such as \displaystyle \frac{11}{4}, whose numerator is at least its denominator, counts only pieces. Both name the same number, and each is easier for some jobs.
How these fit together
Rewriting. Mixed numbers and improper fractions turns the wholes into pieces and back again, and compares amounts written either way.
Adding and subtracting. Because a mixed number is a sum, wholes can be combined with wholes and fraction parts with fraction parts. Like denominators is the case where the pieces already match, trading pieces for a whole when the fraction parts make more than one, and a whole for pieces when the fraction part taken is the bigger one. Adding mixed numbers and subtracting mixed numbers handle denominators that differ, renaming the fraction parts first.
A path through it
The one fixed step: rewriting comes before the like denominators case. Adding and subtracting with different denominators have no fixed place among the others.
Next come multiplying and dividing mixed numbers, which lean on rewriting a mixed number as an improper fraction.