Dividing fractions
Dividing whole numbers means sharing out or counting how many fit. What do those mean when the numbers are fractions, and why does flipping the second one give the answer?
Where you are 6 inside
What this area is about
A division a \div b, with b \neq 0, asks for the number q with q \times b = a. With fractions that question keeps its two everyday meanings, and one short rule answers both.
How these fit together
Two meanings. Sharing into equal parts asks how much one share is when an amount is split into n equal shares. How many fit asks how many copies of one amount fit inside another, and the answer can be larger than the number you started with.
The rule and its reason. Multiplying by the reciprocal turns every division into a multiplication: \displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}, where b, c and d are not 0. Why flipping the second fraction works shows that dividing by a number and multiplying by its reciprocal do the same thing.
Using it. Dividing mixed numbers renames each mixed number as an improper fraction first. Word problems decide which meaning a situation calls for and which amount goes first.
A path through it
Sharing first, then how many fit. After how many fit come the rule and why it works, in either order, and the word problems. Mixed numbers come after the rule.
This area opens multiplying and dividing rational numbers, unit rates, and finding the whole from a fraction.