There has been much written about many adults having been taught at school to complete written calculations by ‘carrying’ and ‘borrowing’ mystery numbers, with little understanding and hoping that somehow the right answer will appear. Some of you may remember being told to put milk bottles on the doorstep! At Herts for Learning we have been working with teachers to banish these little mysteries. Increasingly our schools encourage a range of representations to show the partitioning and exchanging, enabling pupils to articulate and understand what is happening and why. They have also, with our advice, moved away from the jumping number line towards application of known facts and an attention to learning complements of key mathematical units. In recent weeks, we have been asked why our training, ESSENTIALmaths planning and other published resources have regular references to ‘regrouping’:
What is regrouping?
What happened to partitioning and exchanging?
Why is being able to regroup so crucial?
As a maths team we are now committed to the use umbrella term ‘regrouping’ to capture any time where numbers are flexibly manipulated to support calculations. Often the experience children were provided with partitioning was limited to seeing numbers as just separate hundreds, tens and ones, for instance: 400 + 20 + 5 = 425. Below I share five ways where flexible regrouping (or partitioning) can support children to confidently tackle a range of calculations by making the numbers, including those in the form of fractions and decimals, more ‘friendly’.
1.Regrouping for addition within 20
To be able to regroup increasingly more complex whole numbers, fractions and decimals, children need to experience and understand all numbers can be regrouped flexibly to help support calculations. This example is taken from Year 1 of our Progression for Mental Fluency document (available here), where counters and tens frames are used to allow children to regroup either the 8 or the 6 to ‘think 10 for addition’.
A full understanding that we can see smaller numbers that live inside bigger numbers is necessary. Knowing that a 2 and a 4 live inside 6 is fundamental knowledge that is applied here. For pupils who have not secured conservation of number i.e. knowing that we can arrange 6 differently and it still remains 6, gain a heightened understanding using the counters and tens frames as the 6 yellow counters remain visible at all times. Teachers should take the opportunity to ask, “Where is the 6 now?” and ensure that frequent opportunities to clarify that it hasn’t disappeared are taken.
2. Regrouping to support subtraction
A key feature of our ESSENTIALmaths plans is that we provide opportunities for children to experiment with, compare and evaluate different strategies, through discussion with others. This allows pupils to make informed choices about the methods they choose in order to develop strong number sense. The below images shows two different regrouping strategies to solve 74-27. Which one do you prefer?
Regrouping the minuend (the number being subtracted from) is not a very well-known strategy; we tend to be more familiar with counting back or regrouping the subtrahend (the number being subtracted). However, we have found regrouping the minuend is very well understood by our pupils and is more likely to be accurate. This could be because working backwards is quite a lot more complex than this approach and pupils we have spoken to prefer this method when they feel confident with number bonds and complements to multiples of ten and hundred, and because they feel they can track the arithmetic better.
3. Regrouping to support mental division
96 ÷ 4 regrouping the 96 into 90 and 6 isn’t much help for this division calculation as a mental strategy, but flexibly regrouping the 96 into 80 + 16 or 40 + 40 + 16 can support children to use known multiplication and division facts to tackle the calculation without a formal method. It also allows pupils to see that just as with multiplication we can deploy distributive law to the dividend (number being divided). Pupils can relate this to what actually happens in formal written division, which is of course that the 96 would be regrouped into 80 and 16.
4. Regrouping fractions to support addition in LKS2
Regrouping is certainly not limited to whole numbers and this example shows two ways of adding 4 ÷ 7 and 5 ÷ 7 . The images shows two ways of regrouping either fraction to then find complements to , 7 ÷ 7 one whole, plus the extra part. This supports the child to find the total as both mixed fraction, 1 2 ÷ 7 , and improper fraction, 9 ÷ 7
5. Regrouping in UKS2
In UKS2, regrouping can be applied to decimal and fraction concepts, as well as measures such as time and money:
Jenny’s flight was 4 hours 35minutes, she watched a film for the first 85 minutes. Once the film had finished, how much of the flight was left?
Explain how regrouping can be used to solve this problem.
4 hours 35 minutes can be regrouped as 3 hours + 95 minutes. This makes it much easier to subtract the duration of the film, which results in 3 hours and 10 minutes of the flight remaining. Regrouping meant that the calculation could be made without converting all of the timings to just minutes.
Whether called regrouping, exchanging, decomposition or any term, these examples demonstrate the key is for children to have the opportunity to be playful and flexible with numbers to be able to apply their understanding to increasingly varied and complex contexts.