Fractions
A whole is cut into equal pieces. How does a count of those pieces become a number you can compare, add, multiply and divide?
Where you are 15 inside
What this area is about
A fraction \displaystyle \frac{a}{b}, for whole numbers a and b \neq 0, names a pieces of a whole cut into b equal parts. From that picture, fractions become numbers: placed on a line, renamed, compared, and used in all four operations.
How these fit together
What a fraction is. Part of a whole demands equal parts and a named whole. Unit fractions are single pieces \displaystyle \frac{1}{b}; the numerator counts them and the denominator names their size. Fractions greater than one and whole numbers as fractions count past one. A fraction as division reads the bar as divide: \displaystyle \frac{3}{4} = 3 \div 4.
Fractions as numbers. The number line gives each fraction a point, equivalent fractions give one point many names, and benchmarks like \displaystyle \frac{1}{2} judge size. Comparing uses all three.
Operations. Adding and subtracting counts matching pieces, renaming unlike ones first; mixed numbers write amounts like \displaystyle 2\frac{3}{4} and get added and subtracted too. Multiplying counts copies or takes a fraction of an amount, dividing shares or asks how many fit, and a fraction of a quantity finds a part or works back to the whole.
A path through it
Part of a whole, then unit fractions and the numerator and denominator (either order), fractions greater than one, and whole numbers as fractions. Next the number line with equivalent fractions, benchmarks, comparing; adding and subtracting, mixed numbers, multiplying, dividing, and a fraction of a quantity. A fraction as division fits anywhere after part of a whole.
Multiplying fractions leads on to ratio.