Dividing unit fractions and whole numbers

Activities to try

1. 1

   Build the model, then write the equation

   Why it works: The story tells you which division it is: sharing an amount out among more than one group gives each less than the whole, while asking how many sub-one-unit portions fit inside an amount gives more than you started with — and the picture either matches the words or it visibly does not.

   1. Read a word problem, e.g. 'How many 1/3-cup servings of raisins are in 2 cups?'
   2. Decide whether the story is sharing an amount out among groups (partition) or finding how many portions fit inside an amount (measurement).
   3. Choose and build the matching model on a bar or number line: either share one bar among N people, or fill a 2-cup bar with 1/3-cup portions and count them.
   4. Write the matching division equation (2 ÷ 1/3 = 6) once the picture matches the words.
   5. Lay the story, the picture, and the equation side by side so the three represent the same situation.
2. 2

   Sort the two structures, name the dividend, then compute

   Why it works: Naming the structure and the dividend first helps prevent reversing dividend and divisor and reaching for multiplication out of habit; classifying is a start, but the setup is not complete until the model and equation match the words.

   1. Take several word problems and classify each as 'sharing an amount out' (partition: shared among more than one group, each share is smaller than the amount) or 'how many portions fit' (measurement: when the portion is smaller than one unit, as in these problems, the count is larger than the amount).
   2. Name, for each, which quantity is being split and which is the dividend.
   3. For the sharing problem '3 friends share 1/2 lb of chocolate equally', identify that the half pound is what is cut up, so it is 1/2 ÷ 3, not 3 ÷ 1/2.
   4. After classifying and naming, build the matching bar model and only then write and solve the equation.
3. 3

   Confront 'the answer must get smaller'

   Why it works: A quotient can be larger than the dividend: when you count how many portions smaller than one unit fit inside an amount, the count exceeds the amount.

   1. Predict, before computing, whether 'how many 1/3-cup servings in 2 cups' should be more or fewer than 2.
   2. Fit 1/3-cup portions into 2 cups on a bar or number line (or with real measuring cups) and count them: there are 6.
   3. State in words why counting portions smaller than one unit inside a larger amount gives a count bigger than the amount.
   4. Contrast: if servings were 3-cup servings instead of 1/3-cup, would there be more or fewer in 2 cups, and why?