Rebalancing sums – and the ripple effect

    Published: 29 May 2017

    In the tried and tested articles, advisers will share some of their interesting tried and tested approaches to teaching mathematics.   In this sequence, the Rachel Rayner focuses on developing the skill of rebalancing as a mental strategy.

    Rachel Rayner is a Primary Mathematics Adviser for Herts for Learning

    Have you ever noticed how introducing a new drop in the mathematics ocean causes pupils to think differently about already learned concepts?  What follows is a sequence of learning, much loved by me, as it has caused all kinds of new links with number to be made for pupils as well as deeper understanding of core concepts such as conservation and sum from Year 2 to Year 6.  It also challenges pupils preference for working left to right in their calculations (I certainly found that my children’s favourite option) but instead attend to the numbers involved, allowing a far better informed decision about the strategy selected.   The strategy also focuses on the nearness of landmark numbers and the skill of rounding.  This supports estimation and a sense of what is reasonable for further development of number sense.

    To begin with introduce the concept of sum and then check pupils understand conservation of number i.e. that we can partition the number in many ways and it will stay the same number.  Note we are using numbers that can easily be manipulated on our chosen resource i.e. the beadstring.  The attention should be on what is happening and why.


    Next ask the pupils to attend to the sum being conserved while the addends (numbers being added) are redistributed. We will also ask pupils to reason
    about which resulting calculation was easier.


    The next stage is to draw pupils attention to the numbers that are close to helpful landmark numbers (here the I refer to them as ‘treat’ numbers as in trick or treat) and then to rehearse with. We can send the pupils away to work together and bring them back to discuss.

    Throughout the teacher encourages discussion about how many beads could be moved and in which direction. We need the pupils to understand that we
    can ‘move beads’ in either direction. Finally we can ask pupils to apply this to larger or smaller numbers depending on the number range they are working in.
    I demonstrated using the following examples, asking pupils to look at the addends in each case, what do they notice about their distance from multiples of ten?

    16 + 39 = 55     78 + 18 = 96    98 + 36 = 134

    ***NB the challenge for some will be to try this using two different rebalance sums.

    Rehearsal: Provide pupils with three or four practical calculations to rebalance in pairs. They could use beadstrings to physically rebalance each sum (this may need more than one beadstring) or Base 10 in two sets.

    ***NB It is not the size of number here that is important but the strategy and for some cases the pupils to see more than one option

    39 + 17 = 56        15 + 19 = 34         98 + 44 = 142       65 + 57 = 122

    At this point ensure the sum is provided and focus on the rebalancing aspect. Make sure pupils can explain why rebalanced calculations are easier than the original.

    The work below is from a year 6 class from Hemingford Grey Primary School, Cambridgeshire, who were taught this method. Note how pupils draw the ‘beads’ and arrows even with numbers they can’t use on the beadstring and are very happy in rehearsal to consider rebalancing the addends in either direction and that they are well able to identify the ‘treat’ number.



    Even with larger numbers laid out in the column, pupils were able to see how many fewer error points there would be by rebalancing and adding in this way.

    Now onto rebalancing with equal difference…!

    Contact Details

    Latest blogs

    Receive our latest posts direct to your inbox...