Dividing fractions (unit fractions)

Activities to try

1. 1

   Share a third four ways on a bar

   Why it works: Sharing one-third among 4 cuts the whole into twelfths because the second cut (into 4) multiplies the first partition (into 3): 3 × 4 = 12. The answer is smaller than 1/3, and the denominator grows because the whole is re-partitioned more finely — this is the picture, not a memorised rule.

   1. Start with a whole bar that has one-third shaded.
   2. Apply a '÷4' cut to the shaded third so it splits into 4 equal parts.
   3. Re-rule the WHOLE bar the same way: if your cuts run along the bar (crossing the third-lines), extend them across the whole bar; if they run across the bar like the third-lines, cut each of the other two thirds into 4 equal parts too. Count: the whole is now ruled into 12 equal parts.
   4. Read off one little shaded piece and name it: it is one-twelfth, not a smaller third.
   5. Say out loud where 12 comes from: 3 original parts, each cut into 4, gives 3 × 4 = 12.
2. 2

   Share vs. how-many-fit: the sibling contrast

   Why it works: Dividing a unit fraction by a whole number (sharing) and dividing a whole number by a unit fraction (measuring how many fit) are different meanings with opposite size effects. Naming which is happening HERE prevents over-generalising 'dividing by fractions makes it bigger'.

   1. Work two questions side by side on bars: 'share 1/3 among 4' (1/3 ÷ 4) and 'how many fifths fit in 4?' (4 ÷ 1/5).
   2. For each, say aloud which action is happening: splitting one piece into more pieces, or counting how many pieces fit.
   3. Predict before computing whether each answer is bigger or smaller than the starting number, and why.
   4. Match short stories ('share this third between four of you' / 'how many 1/5 cups fill 4 cups?') to the correct ÷ expression.
   5. State the rule-of-thumb in meaning terms for these cases: sharing among 2 or more makes each part smaller than the amount shared; counting how many unit fractions (smaller than 1) fit into a whole number gives an answer larger than that whole number.
3. 3

   Build and solve your own sharing story

   Why it works: The partition method generalises: for any unit fraction 1/n ÷ m, splitting each of the n parts into m re-rules the whole into n × m parts, so each piece is 1/(n×m). Whenever the fraction is shared among 2 or more (m ≥ 2) the piece is smaller than the original; sharing among 1 would leave it unchanged.

   1. Take a new case such as 1/6 ÷ 3 and write a short sharing story for it (e.g. 'share one-sixth of a pizza between 3').
   2. Draw a bar with one-sixth shaded and split that sixth into 3 equal parts, then re-rule the whole the same way (extend cuts that cross the sixth-lines across the bar, or cut every other sixth into 3 too).
   3. Count the total equal parts the whole is now cut into (6 × 3 = 18) and name one piece (1/18).
   4. Verify with the inverse: 1/18 × 3 = 1/6.
   5. Explain how you found the bottom number, referring to the two cuts, not an added-on rule.