Lessons · Teaching · division: sharing, grouping, and the remainder
Two different questions both called division
Twelve divided by three asks one of two questions: share twelve into three equal groups, how many each? Or take twelve and make groups of three, how many groups? The answer is four either way, and the question matters because only one of the two survives dividing by a fraction.
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What it is for
'How many halves are in three?' is a question every adult can answer with a picture, and it is the same question as three divided by one half. A teacher who only has the sharing picture will say 'share three into a half of a group' and hear themselves make no sense. The grouping picture is the one to keep.
How to think about it
Before dividing, say which question is being asked in words. Sharing gives a size per group; grouping gives a number of groups. Then decide what the remainder means in the situation, because the arithmetic never decides that: it depends on whether the leftover can be split, rounded up, or thrown away.
Worked example
Sharing: 12 sweets between 3 children is 4 eachThe answer is a size. Grouping would ask a different question of the same numbers.
Grouping: 12 sweets in bags of 3 is 4 bagsThe answer is a count. This is the picture that survives fractions.
3 divided by 1/2 asks how many halves are in three, which is 6Six halves. Dividing by a number below one makes the answer bigger.
62 children, 8 to a minibus, is 7 remainder 6, so 8 minibusesThe arithmetic says 7 remainder 6. The situation says you cannot leave six children behind.
Your turn
Name the division question that still makes sense when the divisor is a fraction.
The question: how many of these fit into that
Solve one, graded on the server
The trap
Reading the remainder off the calculator. 62 divided by 8 shows 7.75 on a screen, and 0.75 of a minibus is not an answer to anything. Decide what the leftover means before touching a calculator.