What is the difference between sharing and grouping division?

In sharing you know how many groups there are and you are finding the size of each, as with twelve cookies between three children. In grouping you know the size of each group and are finding how many groups, as with twelve cookies into bags of three. The arithmetic is identical; the meaning of the answer is not. Ask which kind a problem is before solving it, because most word-problem errors start there.

How do I explain remainders?

With real objects and a real story, so the leftover has a meaning attached to it. Thirteen crackers shared between four children gives three each with one left over, and what happens to that cracker depends on the story: kept, split in half, or given away. Remainders written as r1 with no context are the reason children later cannot decide whether an answer should be rounded up or down.

Does my child need to know his multiplication facts before dividing?

Multiplication comes first in the sequence, and for good reason: written division without the facts becomes slow guesswork, and long division becomes impossible to hold in his head. He can still share and group objects earlier than that, because the model does not need fluent facts. A practical rule is to teach the model whenever he is ready, and to hold off on written division inside a fact family until that family's multiplication facts are quick.

When should long division start?

Usually fourth grade, and only when three things are in place: multiplication facts that arrive without counting, subtraction with regrouping, and a solid grip on place value. If any one of them is shaky the algorithm collapses, and the child is generally blamed for the wrong thing. Spending a few weeks on the missing piece is faster than teaching the steps a third time.

How to Teach Division — Readiness, Practice, Mastery

Division splits a total into equal parts, and it asks two different questions. Sharing asks how many each person gets; grouping asks how many groups can be made. Both are written 24 divided by 4, which is exactly why word problems catch children out. Teach the two aloud with objects, and keep the matching multiplication fact beside every division.

Two questions, one symbol

Deal 24 cards to four players and each player gets six. That is sharing, and you knew the number of groups before you began. Put the same 24 cards into piles of four and you get six piles. That is grouping, and you knew the size of each group instead. Both are 24 divided by 4, but the six means something different: six cards each, or six piles. A child who has only ever shared will freeze on the second one.

Naming the two out loud is most of the teaching. Ask which kind the problem is before any calculating: are we finding how many in each group, or how many groups. A child who can answer that can usually do the arithmetic, and a child who cannot will sometimes reach the right number for entirely the wrong reason and be unable to repeat it.

Half the time on the model, half on the facts

Division facts are not really separate facts at all; they are multiplication facts read backward, so every division sentence should be said aloud with its multiplication partner until the pairing is automatic. That single habit is what makes long division bearable when it arrives in fourth grade. Split the ten minutes evenly, and do the objects before the recitation.

Three minutes, dealing: "Share these 20 counters between four people. How many each?"
Two minutes, grouping the same 20: "Make piles of five. How many piles?"
Two minutes, fact families aloud: "Six fours are 24, so 24 divided by four is six."
Two minutes on a word problem: he names the kind before he touches the numbers.
One minute on paper: three problems, and one of them leaves something over.

A deck of cards does most of it

A deck of cards teaches more division than any worksheet, because dealing already is division and every child knows the rule is fair shares. Crackers at snack time cover the sharing question; a jar of pennies and four paper cups covers it with a number you choose. For grouping, use a scoop and a bowl of rice, and ask how many quarter-cup scoops come out of the bowl. Leave the leftovers on the table rather than sweeping them away: two counters that cannot be shared are the remainder, and they are far easier to discuss while they are still sitting there.

What secure division sounds like

It is working when he names the problem type before calculating, reaches for a multiplication fact instead of counting up in ones, and can say what the remainder means in that particular story, as two crackers left over rather than only a written r2. He should also notice that an answer bigger than the total he started with must be wrong.

He solves 24 divided by 4 on a page but cannot start the same problem in words.
He writes 4 divided by 24, taking the numbers in the order the sentence says them.
He counts up in ones rather than using a multiplication fact he already knows.
He drops the remainder, or writes one on problems that divide exactly.
He can share but has never once been asked to make groups of a given size.

What to do when division stalls

Return to dealing objects, with totals under twenty that divide exactly, and do only the sharing question for several days. Then introduce grouping with the same totals so the contrast is unmistakable. If the arithmetic rather than the model is the problem, the fault is upstream: a child who does not know that six fours are 24 cannot divide 24 by four, and those ten minutes belong to the multiplication facts for that family instead.

The pace to expect

The sharing idea often arrives before it is taught, from splitting candy between children, and can be formalized in a few weeks in late second or third grade. Division facts follow multiplication by a few months and usually settle in fourth grade. Remainders take longer than families expect, and long division commonly runs across fourth and fifth grade with more than one restart. That is the normal pace rather than a delay.

First, your child should already…

This skill builds directly on:

Multiplication — master this first.

What your child is learning

Division is multiplication in reverse. It can mean sharing equally or measuring out equal groups from a total.

Signs your child is ready

Understands equal groups
Knows related multiplication facts
Can model a total with counters or arrays

The common stumbling point

Not identifying whether the problem asks for the number in each group or the number of groups.

Practice this skill

Worksheets and decodable practice for Division are in the library.

Grade 1 — Division as Equal Groups 20 worksheets
Grade 2 — Division as Equal Groups 20 worksheets
Grade 3 — Long Division 20 worksheets
Grade 4 — Long Division 20 worksheets
Grade 5 — Long Division 20 worksheets

Every set opens its first three worksheets to everyone; the rest are included with All Access.

Browse the whole practice library

How to know it’s mastered

The child models and solves division problems and names the related multiplication fact.

Common questions

What is the difference between sharing and grouping division?: In sharing you know how many groups there are and you are finding the size of each, as with twelve cookies between three children. In grouping you know the size of each group and are finding how many groups, as with twelve cookies into bags of three. The arithmetic is identical; the meaning of the answer is not. Ask which kind a problem is before solving it, because most word-problem errors start there.
How do I explain remainders?: With real objects and a real story, so the leftover has a meaning attached to it. Thirteen crackers shared between four children gives three each with one left over, and what happens to that cracker depends on the story: kept, split in half, or given away. Remainders written as r1 with no context are the reason children later cannot decide whether an answer should be rounded up or down.
Does my child need to know his multiplication facts before dividing?: Multiplication comes first in the sequence, and for good reason: written division without the facts becomes slow guesswork, and long division becomes impossible to hold in his head. He can still share and group objects earlier than that, because the model does not need fluent facts. A practical rule is to teach the model whenever he is ready, and to hold off on written division inside a fact family until that family's multiplication facts are quick.
When should long division start?: Usually fourth grade, and only when three things are in place: multiplication facts that arrive without counting, subtraction with regrouping, and a solid grip on place value. If any one of them is shaky the algorithm collapses, and the child is generally blamed for the wrong thing. Spending a few weeks on the missing piece is faster than teaching the steps a third time.

What comes next

Once this is solid, move on to:

Fractions and Decimals →