Helping a Struggling Math Student: Relationships between the number bonds

I began this lesson asking the girls to tell me what number bonds were in their own words.  C10, whilst it was obvious in our lesson that she understood them, was unable to articulate clearly her understanding.  I was looking for a definition which included the words: picture, relationship, whole, parts.  L10 was able to tell me very clearly.  I understand this is a simple topic for a ten year old to grasp, but for L10 who really struggles with maths, I felt this was somewhat of a breakthrough.  We were both very happy.

In this lesson I wanted to extend the idea of the whole making up the parts and the relationship between all three numbers.  I reminded them that the picture below showed the relationship between any number and the parts that together made it:

Playing with Words

I thought for fun we would explore the truth that everything in life was a sum of it’s parts.  I used L10 as an example, stating that she was a sum of all of her body parts, her two legs, arms, head, body etc.  I asked her what she would be without her legs?  She replied legless!  I asked her…’a legless what?’  She replied giggling, ‘a legless L10!’  She was having fun.  She gave me lots of concrete examples of something whole (table) being a sum of it’s parts (table top and four legs).  These were simple examples but both girls were able to pictorially see the diagram above.

Transferring the Knowledge to Blocks

I gave the girls some blocks, first number bonds of ten again.  I had them play about, asking them to remove one of the pieces and work out what they had to do with the other pieces to find the value of the missing piece.  They did this with lots of numbers.  I then set out a hundred block (100) which I labelled c, three tens and a four (34) which I labelled a and left a space blank, which I labelled b:

I asked them to work out b.  They were able to do it quickly and accurately without using fingers!  A big improvement for L10.  This exercise was to introduce the idea of an unknown.  These had always baffled L10 and was one of the areas of maths she just could not get, no matter how it was explained.  I think the reason was that she didn’t understand the relational aspects of number bonds.  We practiced lots with the blocks until I was confident she understood.

Transferring Knowledge to Numbers

This has always been a problem for L10.  She is able to work out sums when they are to do with something concrete, that she can see: cooking for example.  But ask the same question just with numbers and she struggles.  I set out simple two figure problems to begin with.  She did them really well until I asked the following:

x-96=57

She immediately put the 96 in the whole part of the diagram, not realising that x was the whole and 96 was one of the parts.  I tried to guide her, but it was clear she could not get it.  So I replaced the 96 with an 8 and the 57 with a 2:

x-8=2

She immediately told me x=10 and then saw that ten was greater than 8.  I asked her to apply her understanding to the previous sum.  This was easy now she understood.  We did a few more.  I then set her a harder sum, using larger figures.

x-3214=1111

Tears immediately sprung into her eyes and she was convinced she could not do it.  My goal in setting her a sum with such large numbers was to show her that no matter the number, if she understood the sum and therefore the calculation to do the sum, she would always be able to solve it.  I thought it was an important point to realise – that maths follows the same rules regardless of the size of the numbers.  Know the rules and you can solve the maths.  I told her not to think of the whole sum as this would likely panic her.  Instead use the tool she knew (the diagram) and fill in what she knew.  Reluctantly she did so.  And then she was away.  Can I tell you how chuffed both she and I were?

It may have been that she could have done it in her head.  My other two certainly could.  I didn’t want L10 to even attempt it.  I wanted her to use a tool I had given her to solve a problem that had automatically made her nervous.  This was so that the next time she was faced with this type of problem and felt she could not do it, she had something in her tool box to call upon, hopefully reducing her anxiety.

So far I had given the girls just one tool, and that was the whole and parts diagram.  This was a tool I would have mentally used, even at university.  My next step was to have the girls tackle harder 2 step problems.  I took these from a maths curriculum for their age group.  I wanted them to see how, when the foundations were firmly in place, they could apply their knowledge to progressively harder problems and be confident of the outcome.  But before I did that, I needed to give them a second, very small but nevertheless important, tool to use.  The tool of ‘doing what you can before you start to panic over the whole sum’.  This tool would be key to keeping L10’s anxiety in check.