Helping a Struggling Math Student: Two Step Number Bond Problems

So the girls know and understand their number bonds – what next?

I had grabbed some sums from a year 6 curriculum to see if the girls would be able to apply what they had learnt to work for their own age group.  Why does this matter to me?  I thought long and hard before changing to a non curriculum based maths.  Gary and I talked with the girls themselves.  At the end of the day, if I just want to prepare my children for life maths, timetables would be unimportant.  To be honest it wouldn’t matter to me whether they took their GCSE maths at 14 or 24.  It is one of the obvious advantages of home schooling – to teach where they are at.  That said, I would not like for my children to want to move on with their lives, to (for example) take qualifications in cookery, only to find out that a maths GCSE is required.  I want to be able to give them choice, so a lack of education will never be a stumbling block to that choice.

This means that each topic I teach, though I am very willing to go right back to the start, my ultimate goal is to bring their maths understanding and application skills up to where they should be.  Whilst I won’t be teaching from a curriculum, I will be occasionally referring to one to double-check I am teaching them in a way which will be helpful to them for passing their exams in the future.  I want to emphasise the point that it is to check.  As far as I am concerned, the days are gone when they have to sit down and do 30+ sums each day.  I will give them a couple at most, just to check.  As maths is fairly cumulative, practice will be gained in old areas simply by learning new topics.

Problems set for their age group – year six

I had already skimmed through the papers for their age and set them all the basic one step problems (of which there were only a few), so I looked into slightly trickier problems.  These were either worded a little differently or they were two-step problems.  C10 found these all very simple so went through them on her own.  In addition I set her some extra sums just to keep her busy whilst her sister was calculating.  I set them one per day, alongside some number bond games.

The first one I gave her was a multistep problem which required her to make a new whole and split it into parts:

I asked her to work out how many apples she would have at the end of the second day.  As you can see she did confidently.

The next one read: A 120cm stick is cut into two pieces so that one of the pieces is 30cm longer that the other piece.  How long is each post?

Claire’s bank balance is £2.50.  Claire’s and Mike’s balance add up to £31.  How much does Mike have in his account?

These two posed no problems.

A 240 cm length of wire was cut into three pieces.  The shortest and longest pieces were each 20cm longer or shorter than the third piece.  What was the length of the third piece of wire?

I did this one sitting next to them.  Both girls were unsure where to start.  However once prodded in the right direction they were away.  It always interests me to see the tears roll down L10’s face before she has even started.  She feels immediately defeated by the problem in front of her.  Because I was next to her we were able to go through it using the two tools I have given them so far: Do what you can and then use the number bond diagram for any addition/subtraction problems.  They got there in the end and by themselves, but it seemed a long drawn out process.

The difference between two numbers is 19.  What is the other number if 1) the smaller number is x and 11) If the larger number is y?

Write the sum of the two number using only one letter, x

What are the two numbers if their sum is 40?

This question caused L10’s shoulders to slump, tears spring into her eyes and a look of defeat appear on her face.  All this and she hadn’t attempted anything yet!  I told her to take a deep breath and not concern herself with any part of the question except the first part.  I asked what tool she could use.  She wrote down all she knew:

Big number – small number = 19  No problems so far.

It was the next part she struggled with, crying she had never done anything like this before.  She had, but it was possibly worded differently and that always throws both girls.  Also, I’ve always known she has found unknowns difficult.  Since we started learning together she had become competent at solving for one unknown, but this was two unknowns and it stumped her.  I encouraged her to think of the x and y as labels to describe something they didn’t yet know the value of.  I used the twins as an example.  Often they can not be told apart, so instead of giving them their name people give them a label such as ‘the one wearing a striped shirt’ or ‘the twin with her hair in a ponytail’  These are simply labels to describe something to help minimise confusion.  L10 face cleared.  This was a concrete example and one she could get her head around.  So I asked which label belonged to which number, from the information given.

She was easily able to then write:

y-x=19

I asked her what next?  She burst into tears.  She said she didn’t know the numbers so how could she answer the questions.  I asked her what other tools I had given her?  She said the diagram.

She was easily able to construct her diagram, which allowed her to ‘see’ the sum.  I casually asked her what y was equal to.  She was able to tell me y=x+19 and also for good measure said that x=y-19.  I showed her she had just answered the first part of the question.  She was getting excited now!

The next part of the question was asking to express the sum of x and y just using x.  This introduced a new tool, that of reducing the unknowns to just one.  We reviewed what we knew:

y=x+19

x=y-19

I asked her what the question was asking:

She replied sum =x+y

I again asked her what the question was asking:

She understood she needed to get rid of the y but had no idea how to do it.  I asked what tool she could use to make things a little clearer.  Given we have very few tools right now she guessed at a diagram, which she constructed again with relative ease.

I asked her if we could replace the y with anything, using the information we had written down previously.  She hesitantly suggested replacing the y with what it was equal to in terms of x.  Inside my heart was singing!  I told her to go ahead:

As y=x+19 so sum= (x+19)+x

Now I asked what the sum of x and y were using only x to express it.  She wrote tentatively:

Sum =x+x+19

I asked how many unknowns we now had.  She smiled a wonderful big grin replying ‘one’.  I asked if we had answered the next part of the question.  Yes!

I then asked her to express, using the diagram, a number sentence for what x was equal too.  Tear sprung back.  This time I knew from where they were coming.  I reassured her I didn’t want her to solve it, simply write down what she knew. Which she did using the number bond diagram as before.

From this diagram she was able to then write the number sentence:

40=x+x+19

40-19=x+x

2x=21

x=10.5

And as y=x+19

y=10.5+19=29.5

We had got there, but it wasn’t without a lot of help from me.  Over the next few days I set them similar questions, each of them requiring prompting at certain times in the question.  After a week of practice it was becoming easier.  The test of course, will be whether after having a break away they will be able to answer similar two-step problems.  I decided to leave it for a while, and after testing them with a quick fire verbal quiz on number bonds to 10, I decided we were ready to move on to learning the number bonds to 20.