Division with remainders: common errors in grade 3
Which errors grade 3 children make in division and division with remainders, how teachers can recognise them and how to respond in lessons.
In brief: Common errors in division in grade 3 are a remainder larger than the divisor, numbers swapped round in the calculation and a remainder that is not interpreted in a word problem. They often point to a missing basic idea of division, grouping or sharing, or to a missing link with multiplication. The errors can be spotted by using the inverse operation as a check and by asking what the remainder means in the story. Mazio Math practises division in the Basiscamp in the world ‘Multiplying and Dividing’. In the unit ‘Checking methods’, children decide which method fits a question about a real-life situation.
Division describes two different situations and often leaves a remainder. Both make it demanding for many children. This article shows teachers in grade 3 which errors are common, how to ask follow-up questions and how to respond in lessons.
Two basic ideas of division
A sound understanding of division covers two basic ideas. In sharing, the number of groups is known (‘32 cards are shared among 4 children. How many does each child get?’). In grouping, the size of the groups is known (‘32 people travel in cars with 4 seats each. How many cars are needed?’). This is how the didactic commentary on module N4 B of the DZLM support materials on division describes it. Children should be able to use both ideas, but they do not need to assign tasks explicitly to one or the other.
The difference matters when there is a remainder. A card left over when sharing is something different from a person who still needs a car when grouping.
Common errors with examples
| Error pattern | Example | Possible cause |
|---|---|---|
| Remainder larger than the divisor | 23 ÷ 4 = 4 remainder 7 | The child has not looked for the largest multiplication fact that fits. |
| Numbers swapped | 3 ÷ 12 = 4 | The roles of dividend and divisor in the situation are unclear. |
| Different operation with a division sign | 12 ÷ 8 = 4 | The child subtracts and writes a division sign. |
| Remainder not interpreted in context | 30 children, cars with 4 seats: ‘7 remainder 2 cars’ | The calculation is not related back to the situation. 8 cars are needed. |
| Only the result written down | For a sharing situation, only ‘8’ is written. | The link between the situation and the division calculation is missing. |
| No link with multiplication | 45 ÷ 5 is solved by counting, although 9 × 5 is known. | Division is not used as the inverse of multiplication. |
The error patterns with swapped numbers and a different operation are listed in the evaluation table of the diagnostic assessment in the module mentioned above. The page Division with remainder on KIRA covers interpreting the remainder in real-life situations. It notes that children already find it hard to recognise that a remainder has to be interpreted according to the situation.
A note on notation: strictly speaking, 23 ÷ 4 = 5 remainder 3 is not an equation, because 28 ÷ 5 also gives 5 remainder 3. The check in the form 5 × 4 + 3 = 23 shows what the remainder means and guards against a remainder that is too large.
How to ask diagnostic questions
- Ask for a check. ‘Work backwards. What is 4 × 4, and then add 7?’ Children often discover for themselves that 4 fits into the remainder once more.
- Ask the child to explain the remainder. ‘What does the 2 in your answer mean? What happens to the two children?’
- Model the situation. ‘Show me 32 ÷ 4 with counters.’ Whether the child shares into four groups or makes groups of four shows which idea they are using.
- Make up a story. ‘Make up a story for 25 ÷ 6 in which something is left over.’ If the story fits a different operation, the basic idea is missing.
- Ask about the method. ‘Which multiplication fact helps you with 45 ÷ 5?’ A child who cannot name one is calculating without reference to multiplication.
What helps in lessons
- Sharing with counters and figures. The figures stand for the people, and the counters are shared out step by step. What is left over is the remainder.
- Grouping on a dot array. Circle groups of a given size on the hundred-dot array. This also prepares for informal written division.
- Number line. Draw jumps the length of the divisor and count how often they fit and what is left over.
- Discussing the remainder in context. Have the children interpret the same calculation in three stories. Once it is rounded up (cars), once rounded down (full packs), once the remainder is divided further (pizza).
- Fact families. Write down 4 × 6 = 24, 6 × 4 = 24, 24 ÷ 4 = 6 and 24 ÷ 6 = 4 together.
How Mazio Math helps with division
Mazio Math is a learning platform for maths lessons in primary school. The difficulty adapts to each child’s previous work, even when the whole class is working on the same content.
Practice in the Basiscamp. For grades 3 and 4, the Basiscamp includes the worlds ‘Multiplying and Dividing’ and ‘Calculating up to 1 million’. The written methods have their own sections, up to written division. The Basiscamp keeps track of which task types each child has already mastered and brings back content that has not been practised for a while.
Real-life situations in units. Units build up a topic step by step, from first exploration to confident mastery. In ‘Checking methods’, children decide which method fits a question about a real-life situation. In ‘The Mistake Detectives’, they check whether answers and solutions are correct. Both fit work on the remainder and the check.
When a child makes a mistake. The learning assistant Mazio helps directly within the task, with a question, with highlights in the picture and, if needed, with a given partial step. He does not give the solution; the child takes the next step on their own.
For the class. Once a week, Mazio Math checks whether a misconception is showing up in the class. The support card suggests steps for lessons and suitable content for the board. This is how you work with it:
- Open the class under ‘Results’ and tap the flag under ‘Misconceptions in the class’.
- Read the example, the background and the steps under ‘In lessons’.
- Tap ‘Suitable content for the board’, make your selection and put it on the class board.
In the library, sorted by content area, you can look up common misconceptions with examples and see whether there is an entry on your topic.
An example from the classroom. A grade 3 class has been introduced to division with remainders. The teacher starts with three stories about 30 ÷ 4 and has the children work out in groups what happens to the remainder. Then, under ‘Lessons’, the teacher puts the Basiscamp exercise from ‘Multiplying and Dividing’ on the board as compulsory, with ‘Checking methods’ below it as optional content. The teacher had checked beforehand with ‘Open & play’ whether this unit already suits their grade 3 class, because the teacher decides which content is used in which class. As a note, the teacher writes ‘Check every calculation with a remainder in your exercise book.’ Later, under ‘Results’, the teacher sees for each area how many tasks were correct at the first attempt and uses ‘Put on the board again’ to add the sub-worlds that are not yet secure.
There is more about the content in grades 3 and 4 on the page Maths in grades 3 and 4 (in German), and about the foundation in the times tables on the topic page on the times tables (in German). How Mazio Math evaluates a class’s errors in thinking describes how Mazio Math analyses recurring errors in a class. Mazio Math is free for teachers: create a free account. Using it with a class is currently available to schools in Germany; if your school is in another country, registration records your interest.
Frequently asked questions
Why can the remainder not be larger than the divisor?
If the remainder is as large as the divisor or larger, the divisor fits in at least once more. The result is then too small. The check with a multiplication plus the remainder makes this visible.
When is division with remainder introduced in primary school?
The curriculum where you teach sets the timing. Division with remainder builds on the times tables up to 10 × 10 and on division without remainder, and in Germany it is usually part of the content for grade 2 or 3.
How do I explain the difference between grouping and sharing to children?
Children do not need to learn it as a pair of terms. What matters is that they can act out both situations with materials and write a division calculation for each. Ask what the story is looking for: the number of groups or the size of each group.