How many halves are in three?
Tonight's idea
What does \(6 \div 2\) ask? One way to read it: how many 2s fit in 6? Three of them fit, so the answer is 3.
Read \(3 \div \tfrac12\) the same way. How many halves fit in 3?
Each whole holds 2 halves. Three wholes hold \(3 \times 2 = 6\) halves. So \(3 \div \tfrac12 = 6\).
The answer is bigger than 3, and that is fine. Nothing grew. You counted small pieces, and small pieces fit more times. The smaller the piece, the more of them fit.
Dividing by \(\tfrac12\) counts halves. There are 2 in every whole, so you double. Dividing by \(\tfrac14\) counts quarters. There are 4 in every whole, so you multiply by 4.
Watch one: bows from a 4 metre ribbon
A ribbon is 4 metres long. Each bow needs \(\tfrac13\) of a metre. How many bows can you make?
The question is how many thirds fit in 4, so it is \(4 \div \tfrac13\). Each metre holds 3 thirds. Four metres hold \(4 \times 3 = 12\) thirds. So \(4 \div \tfrac13 = 12\) bows.
Check by going back: 12 pieces of \(\tfrac13\) metre make \(\tfrac{12}{3} = 4\) metres.
Next year's version, dividing by any fraction at all: Fractions that fight back (Grade 6).
Your turn
- Work out each one by asking "how many fit?" (a) \(4 \div \tfrac12\) (b) \(2 \div \tfrac13\) (c) \(5 \div \tfrac14\)
- A jug holds 3 litres of juice. A glass holds \(\tfrac15\) of a litre. (a) How many glasses does one litre fill? (b) How many glasses does the whole jug fill? (c) Write part (b) as a division.
- Here are four divisions: \(6 \div 2\), \(6 \div 1\), \(6 \div \tfrac12\), \(6 \div \tfrac13\). (a) Before you calculate, say which answer will be biggest. (b) Work out all four. (c) What happens to the answer as the number you divide by gets smaller?
- A plank is 3 metres long. You cut it into pieces \(\tfrac34\) of a metre long. (a) How many quarter metres are in the plank? (b) Each piece uses 3 of those quarters. How many pieces do you get? (c) Write what you found as a division. Draw the plank to check.
Stretch: when less than one fits
A half is bigger than a quarter, so a whole half does not fit in a quarter. Only part of one fits. What part? Use it to write the answer to \(\tfrac14 \div \tfrac12\). A drawing helps.
Answers
- (a) \(4 \times 2 = 8\). (b) \(2 \times 3 = 6\). (c) \(5 \times 4 = 20\).
- (a) 5 glasses, because 5 fifths make a whole. (b) \(3 \times 5 = 15\) glasses. (c) \(3 \div \tfrac15 = 15\).
- (a) \(6 \div \tfrac13\), because thirds are the smallest pieces. (b) \(6 \div 2 = 3\), \(6 \div 1 = 6\), \(6 \div \tfrac12 = 12\), \(6 \div \tfrac13 = 18\). (c) The answer gets bigger. Smaller pieces fit more times.
- (a) \(3 \times 4 = 12\) quarters. (b) \(12 \div 3 = 4\) pieces. (c) \(3 \div \tfrac34 = 4\). Check: 4 pieces of \(\tfrac34\) m make \(\tfrac{12}{4} = 3\) m.
Stretch. A half is two quarters. A quarter is one of those two, so it has room for half of a half. \(\tfrac14 \div \tfrac12 = \tfrac12\). Check: half of \(\tfrac12\) is \(\tfrac14\).