Two meanings of division
Twelve divided by three has two different pictures, and children need both. Sharing: twelve sweets shared between three children, how many each? Grouping: twelve sweets put into bags of three, how many bags? The answer is four either way, but the actions are different, and word problems use both.
Start with sharing, because young children have done it in real life. Deal counters one at a time into three piles and count a pile. Then introduce grouping: take three counters at a time until none are left and count the groups. Ask children which picture matches each word problem you give them.
Link division to multiplication early
Division is the inverse of multiplication. If three times four is twelve, then twelve divided by three is four and twelve divided by four is three. Those four facts (two multiplications, two divisions) are a fact family, and children who know their times tables already know the division facts; they just have not been told.
Teach children to read a division question as a missing-number multiplication: twenty-four divided by six is really asking six times what makes twenty-four. That single habit turns division facts from a new thing to memorise into something they can already do.
Arrays: the picture that shows both
An array of dots, three rows of four, shows three times four, four times three, twelve divided by three and twelve divided by four all at once. Draw arrays for every division fact while the idea is new. Ask children to write the whole fact family from one array.
Remainders
Remainders arrive naturally with counters: fourteen shared between three gives four each with two left over. Write it as fourteen divided by three is four remainder two, and check it by multiplying back: three times four is twelve, plus the two left over makes fourteen. That check catches nearly every remainder error.
The remainder must always be smaller than the divisor. If a child writes fourteen divided by three as three remainder five, the check fails: five is bigger than three, so another group of three fits.
Word problems then ask what to do with the remainder. Fourteen children in cars of three need five cars (round up). Fourteen dollars shared by three people is four dollars each with two dollars left (leave the remainder). Fourteen cookies shared by three is four and two thirds each (make a fraction). Discussing which is right for each story is a lesson in itself.
A teaching sequence
- Sharing with counters, then grouping with counters. Say the division sentence out loud each time.
- Arrays and fact families: from one array, write two multiplications and two divisions.
- Division facts by table: start with dividing by 2, 5 and 10, then 3 and 4, then the rest, matching the order the times tables were learned.
- Missing-number multiplication: read every division as a times question.
- Remainders with counters, then with facts, always checking by multiplying back.
- Dividing bigger numbers by splitting: eighty-four divided by four is eighty divided by four plus four divided by four, which is twenty plus one, so twenty-one. This is the idea long division formalises later.
- Only then, the written method for long division.
Worked examples
Thirty-five divided by five. Five times what is thirty-five? Five times seven. Answer: seven.
Twenty-seven divided by four. Four times six is twenty-four, leaving three. Answer: six remainder three. Check: 4 x 6 = 24, 24 + 3 = 27.
Ninety-six divided by eight. Split ninety-six into eighty and sixteen: eighty divided by eight is ten, sixteen divided by eight is two. Answer: twelve.
A baker has 50 buns and packs them in boxes of 6. How many full boxes? Six times eight is forty-eight, leaving two. Eight full boxes, two buns left over.
Common mistakes
- Writing the numbers the wrong way round: twelve divided by three is not the same as three divided by twelve. Reading it as twelve shared by three helps.
- Remainder bigger than the divisor. Always multiply back to check.
- Thinking division always makes numbers smaller. Dividing by a fraction, later on, makes them bigger, so avoid teaching the shortcut as a rule.
- Guessing division facts because the times table underneath is not secure. Fix the table first.