Helping a Struggling Math Student: Utilising a foreign number system with a base 4

Please excuse the terrible photos.  The children are rehearsing a play they have written and are intending to act out for my birthday.  The living room has been turned into a theatre complete with scenery, stage and moveable curtains!  It was therefore very dark when I took the photos!!

The children discussed first what base ten meant and what it’s columns would look like and what base 60 (Mesopotamians) looked like and then base 20 (Mayans).  These were bases they were familiar with.  I wanted to see if they were able to handle any old base I threw at them.  I chose base four and wrote it down as shown below:

Simple no?

The children understood from last week’s lesson that numbers represented values and didn’t have to take on the form of 1, 2, 3, 4 etc.  In this case the shapes represented the amounts from 0 to 3.

We discussed why we didn’t need any other ‘numbers’ and the children then went about showing me how numbers greater than 3 were easily made using base 4:

This was the simple part.  I then had them attempt some addition and subtraction.  Only T11 was able to do these without help.  The girls needed some guidance but got there in the end.  Here is their addition in base four:

And subtraction:

Then I had them attempt some multiplication.  Here is where they all came unstuck.  So, breathing deeply I ‘patiently’ showed them how to do it.  Only I couldn’t.  It took me at least half an hour to figure out how to do it, by which point the children had lost interest!

I did take them back to it at a later occasion and walked them through it myself.  It was very interesting to me to see how absolutely ingrained into us the base 10 decimal system is.  In many ways it was weirdly gratifying to see the children naturally want to do the multiplication in base 10 rather than base 4.  Here was our group attempt at multiplication in base four:

I then set them some simple multiplication to do themselves.  One of the main reasons I am including T11 is to improve his ability to work in a team.  He is a very independent, self motivated child, for which I am very thankful.  However, with this comes a slight impatience towards others if they are not doing it exactly his way.  Ahem, much like his mother.  I want to gently encourage him to lead his sisters gently, working as part of a sibling team to complete the odd living maths questions I set them all.  Every member of our family is a strong character, with strong opinions and a will of steel.  Whilst I am blessed that in general they all get on like a house on fire, there are times they are required to work together and there are too many leaders and not enough followers.

I shared with them all my goals of T11 learning to lead gently and by example (his eventual biblical role) and for the girls to submit to his leadership, without losing their voice, so to speak (their ultimate biblical roles).  They all agreed it was something that needed work.  I left them to it, instructing them not to kill each other!

You know, I was very proud.  There were a few raised voices and I could hear T11 audibly take deep breaths to stop himself from saying something impatiently to his sisters.  But I could also hear the effort put in by all three children to work as a team, according to the roles I had set them.  And there were no deaths.  Something I was most grateful for.

I was also pleased by how well they seemed to understand base systems by the end, and how well they could do sums using my base four system.

Over the next few weeks I will be giving the children five sums a day, linked with the work we have already covered this summer.  This is an idea I got from the very excellent Blog She Wrote.  This will show me whether we need to do some more work on these concepts or whether we can move on to new things.

Next week I will outline everything I hope to cover during this five-week term.