How would your children tackle these questions?
1997 + 998 96 ÷ 4
All too often, we see children automatically turning to a formal written method rather than taking a moment to consider whether this would be the best approach.
There is, of course, a place for written algorithms – they are often the most efficient way to calculate with large numbers. However, if we want children to become true mathematicians, if we want them to reason mathematically, they need to be asking themselves these questions:
- Can I do this in my head?
- Do I need to use a formal written method?
- Is there a more efficient strategy I could use?
Consider the first calculation:
Here, two alternative methods are shown.
Using the column addition method here involves three lots of exchanging. This could provide lots of scope for error. If instead, we consider the number 1,997 and realise that it is only 3 away from 2,000, we could partition the 998 into 3 and 995. We can then combine the 3 with the 1,997 to regroup the numbers into 2,000 + 995 which can be calculated mentally.
Similarly, 96 ÷ 4 could be solved using short division. However if instead, we partition the 96 into 80 and 16, or 40 + 40 + 16 each part can be divided by 4 using our knowledge of times table facts.
This idea of flexible partitioning is not just for children in KS2. Children can use concrete manipulatives to explore how numbers are made up of smaller numbers almost as soon as they begin school. For example, using counters in the part-whole model below, they see that 6 can be made up of 5 and 1, 4 and 2 or 3 and 3.
As they begin to add and subtract across the ten boundary, they can use this idea of flexible partitioning or regrouping to make a whole ten. Consider the calculation 8 + 7. Using the ten frames below, the children see that if they partition the 7 into 2 and 5 they can regroup the numbers to make 10 + 5. They can now see that there is fifteen in total.
Similarly to develop efficient mental strategies for subtraction, it is useful to be able to partition the number which is being taken away in order to cross the tens boundary more easily. For example 32 – 5 can be seen as 32 takeaway 2 to reach 30 then takeaway 3.
This idea of flexible partitioning does not come naturally to most children. It needs to be modelled and practised. Comparison of different methods, through discussions, is essential in order to develop the children’s reasoning and enable them to consider the best approach to use as they meet each calculation.
As teachers, we all want to arm children with the tools to become efficient and confident mathematicians who have a range of strategies at their fingertips and the reasoning ability to help them select which to use when. In order to do this, developing these flexible partitioning strategies right across school is key.
Jane Brown, Maths Specialist at White Rose Maths