Equal groups and sharing
Solving one-step multiplication and division problems with objects, pictures and arrays
About three lessons of 30 to 40 minutes
Can everybody get the same?
Put 12 counters on the table and ask two children to share them fairly. Giving one counter to each child in turn makes two equal groups of 6. The fair-sharing action is division, even before pupils use the division sign.
Now place 3 counters on each of 4 plates. There are equal groups again, but this time the question asks how many altogether. Counting 3, 6, 9, 12 builds the meaning of multiplication without asking pupils to memorise a rule.
- 4 plates with 3 counters on eachfour equal groups of 3 make 12 altogether
- 12 toys shared between 2 boxesdeal one at a time until both boxes hold 6
- 3 rows of 4 countersan array keeps each group equal and easy to count
What students will be able to do
Pupils will represent equal-group and fair-sharing stories with concrete objects, pictures and arrays, then calculate the total or the number in each equal group.
- I can make equal groups with objects.
- I can share objects fairly so every group has the same number.
- I can use an array to show equal groups.
- I can explain whether I am finding the total or the number in each group.
Standards this unit teaches
- uk-ks1-y1-ma-nmd-01UK National Curriculum (England)One-step multiplication and division
Solve one-step multiplication and division problems by calculating with concrete objects, pictorial representations and arrays, with teacher support.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Equal groups
- groups that contain the same number of objects
- Share
- deal a total fairly into equal groups
- Array
- objects arranged in equal rows and columns
- Altogether
- the total number in every group combined
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Make and count equal groups
ConcreteGive pairs of pupils counters and small hoops or plates. Ask them to make 4 equal groups of 3, then count the total by touching one whole group at a time: 3, 6, 9, 12.
Keep the language precise: 4 groups, 3 in each group, 12 altogether. Change one counter so a group is unequal and ask pupils to repair it.
- How do you know the groups are equal?
- If one more counter is added to only one row, is it still an array?
3. Move from objects to an array picture
PictorialReplace loose counters with a drawn dot array. Circle or point to each row as one equal group. Pupils should say the structure before calculating: 3 rows of 4, then count 4, 8, 12.
- How many rows are there?
- How many counters are in each row?
- What is the total?
Common misconceptions and how to address them
MisconceptionAny collection of groups represents multiplication.
Why it happens: Pupils notice separate groups but not that every group must be equal.
How to address it: Compare an array with rows of 3, 3 and 4. Ask pupils to move one counter so the rows become equal before counting in equal steps.
MisconceptionSharing means splitting a total into any two parts.
Why it happens: A pupil may make 7 and 3 from 10 and stop because every counter has been used.
How to address it: Deal one counter to each recipient in repeated rounds, checking that all recipients stay level.
MisconceptionThe number of groups and the number in each group are interchangeable answers.
Why it happens: Both numbers are visible in an array, so pupils may report the wrong one.
How to address it: Underline what the question asks, then finish with a labelled sentence such as '4 counters in each row'.
Guided practice (with answers)
1. How many counters are in 3 equal rows of 2?
Answer: 6 counters, because 2 + 2 + 2 = 6.
2. Share 8 counters equally between 2 boxes. How many are in each?
Answer: 4 counters in each box.
3. There are 5 bags with 2 apples in each. How many apples altogether?
Answer: 10 apples, because 2 + 2 + 2 + 2 + 2 = 10.
4. 12 counters are arranged in 4 equal rows. How many are in each row?
Answer: 3 counters in each row.
Independent practice worksheets
Use the computed dot-array worksheet after pupils have physically made and shared the same structures.
Differentiation
- Keep totals within 10 and use only 2 equal groups at first.
- Give pupils physical counters to place directly over the printed dots.
- Say the full sentence frame: __ groups, __ in each, __ altogether.
- Ask pupils to draw two different arrays with the same total.
- Include a total that cannot be shared equally and discuss the leftover counter.
- Ask pupils to write their own equal-groups story for a given array.
Assessment: exit ticket
Use one grouping question, one sharing question and one array-reading question.
1. There are 3 plates with 4 cakes on each. How many cakes altogether?
Answer: 12 cakes.
2. Share 10 counters equally between 2 children. How many does each child get?
Answer: 5 counters each.
3. Draw an array for 4 equal groups of 2 and state the total.
Answer: Four rows of 2, with 8 objects altogether.
Teacher notes and timings
- Year 1 statutory wording expects teacher support and concrete or pictorial calculation. Formal multiplication and division notation is not required for success in this unit.
- Use both interpretations: finding the total from the group size and finding the group size from a shared total.
- The dot diagrams and answer keys are generated from the same row and column values, so the printed model cannot disagree with its answer.