Friday, 28 August 2015

Some 'Whys' Behind our Base 10 Number System

Lead In:

To begin our new unit exploring the central idea:

       Mathematicians saw advantages to creating a base 10 number
       system which extends infinitely in both directions.

we began with a think - pair - share of the following:

We brainstormed ways our lives would be different if we didn't have a number system.

Some fascinating ideas were shared and discussed (see below).

This helped us to take a look at our number system from perhaps a slightly different perspective we might normally see it.  It also helped us to gain a sense of the reasoning behind why people needed to create a number system which they will hopefully draw upon when we begin exploring ancient number systems later in our unit.

One of the interesting discussions we had was when someone suggested people wouldn't exist anymore.  We wondered about that and eventually decided that there would be less people alive if we didn't have a number system. This theory we created was supported by ideas such as people wouldn't be able to grow as much food efficiently as we can today and therefore the human population wouldn't have risen as fast as it has. 

Isn't it amazing that 10 year olds can develop such profound thoughts when we give them opportunities?

Another suggestion that we wouldn't be able to build factories because we couldn't measure mass or length etc led to a further thought that therefore there would be less pollution in the world.  Is our number system partly responsible for pollution?  We thought that was a really interesting theory and felt that despite all the positives of having a number system, we had also discovered a negative impact!

Next, we sat in a circle and discussed this question:

Some really fascinating ideas were discussed.  

We wondered if numbers already existed before people started to think of them or not. Is this like the chicken or the egg?

Two camps emerged in our discussion: 'Yes, they do and would' and the 'No, they don't and wouldn't' and they both used supporting evidence when sharing their theories.

One student theorised that numbers would still exist as they would still remain carved in buildings and printed in books etc.  Another wondered though if there was no one to read those numbers in books and on buildings would they really still be numbers or just random symbols.

Woah!  That's some pretty heavy existential philosophical thinking happening right now.....

We then had half a minute to think about this question before we could share our thoughts:

Varied ideas were shared - zero, one, eight, five and eventually we thought of 10.

I explained that there is a theory as to why our number system is based upon ten and the clue is found on our bodies.........

No suggestions were being shared, so to help out, I had them think how people thousands of years ago might communicate numbers with each other.  That raised the idea of our using our fingers!  We have ten fingers!

Could this be why people created a base 10 number system?

I asked them to then show with their fingers the number 14.

They flashed ten fingers at me and then showed four fingers.

"Exactly. Did you notice what you just did?  You just showed the number 14 is based on 10.  14 is a ten with an added 4."

"What happens when we count with our fingers?"

- When we get to ten we have to start again with new hands.

"So let's look at our hundred grid.  How does it work?"

- The numbers go from 1 to 10 and then we have to start a new line for 11 - 20.

"Why do we think it was structured that way?"

- It's the same as when we use our fingers to show numbers larger than ten. When we start again with two new hands it's like starting a new line on the hundred grid.

To help deepen this new connection we were formulating together, I asked to 
show with your fingers 38.

They flashed all 10 fingers 3 times and then 8 fingers.

"See how the number 38 is based on 10 three times?"

It started to make sense to more of us that this could be the reason our number system was created as base 10.

I showed this image and asked them to think to themselves about what it explains for a minute.  (One of my personal goals this year as a teacher is to remember to give all children enough thinking time and not just respond to kids as soon as they start raising their hands.  Tough habit to break, but consciously did so at this moment)

We shared our different thoughts of how it shows our number system is based on 10.

To liven things up, I pointed to the 10 squared and asked what does this mean? 

= 10 x 10

Which equals?


And so there is our next place value after ten - hundreds

I pointed to the 10 cubed.

10 cubed means?

10 x 10 x 10

10 x 10 equals?


x 10 

1 000

And so there is our next place value - thousands

I pointed to the 10 to the power of 4

10 to the power of 4 means?

10 x 10 x 10 x 10

10 x 10 equals


x 10 

1 000 

x 10 

10 000

And so there is our next place value - ten thousands

I pointed to the 10 to the power of 5.

10 to the power of 5 means?

10 x 10 x 10 x 10 x 10

10 x 10 equals


x 10 

1 000 

x 10 

10 000

x 10

100 000

And so there is our next place value - hundred thousands

More laughter continued as they could see how much further we were going to go on with this pattern, which we did all the way to billions.

Knowing that our number system is based upon 10 is a key understanding to using it.  Later in our unit we will explore not only how but WHY we can multiply and divide whole and decimals numbers easily by 10, 100, 1000.  On Monday, when we revisit this key understanding they will be asked to create their own number system imagining an alien planet where the aliens had a different number of fingers such as 3 or 7 etc.  They will need to think a lot about how and why our number system works in order to create their own.

From this a student remarked whether zero was also a base number for our system as the number ten is made up of a digit 1 and a zero.  Someone replied zero is just nothing.

Is it?

We then shared our ideas of what zero was.

I shared how till this day no mathematician nor scientist in history has been able to prove the existence of zero.  If you could prove it existed, you would probably win a Nobel Peace Prize and become one of the most famous mathematicians in history.  This sparked a lot of interest and excitement and they asked if they could have some time in groups trying to prove it.

Such a great idea. So that's what we did and we shared some of our thoughts:

I was completely blown away by the thoughts being generated!!!

I particularly liked someone sharing that the symbol zero is circular and shows there is nothing inside it!

And though it sounds rather simplistic, the thought shared that 'It is the whole of everything' gets more and more profound the more you think about it.

Some thought how Roman numerals didn't have a zero. Oh! So it must be a NEW number!   Wow!  But, is it a number?

We discussed that every number CAN be proven.  We can prove 2 exists because 2 is 2 ones OR 1 split in half to create 2.  Every number therefore CAN be proven, but zero?  How can we prove nothing exists?  And where did 1 come from? How can 1 come from nothing?!?!?

Someone then connected that's like the Big Bang theory - everything in the universe came from nothing suddenly! Wow!  Others then sparked from this connection and suggested zero could be like a black hole. Another thought zero is like God who must have come from nothing because He created everything. about deep thinking (and bare in mind these are 10 year olds coming up with these amazing thoughts!)

Someone then mentioned zero can be proven because I am a 1. I am a 1 person and when I die, I don't exist anymore so that is like zero.    But does our body really stop existing we wondered?  When it decomposes it just changes into different smaller materials so the body hasn't realély ever stopped existing, it just changes it's existence!  Again - 10 year olds thinking like this!!! Amazing!

As the bell rang for recess it was suggested we try to keep thinking about this next week and the week after.  Wow! Love this so much!!  Totally engaged, really passionate and directing / taking ownership of what we should be doing in our classroom- brilliant!

To keep this fresh in our minds to keep thinking about, here is how our classroom door looks now as we enter:

Later in the day, after lunch, a student came in with her note book sharing that she might have cracked the existence of zero.  She figured that 1 + 1 = 2. Therefore 1 - 1 = 0.
She thought about positive 1 and negative 1 and that if the zero place wasn't there the positive 1 and the negative 1 would be arguing for that prime position of zero.   Does that beginning thinking help us to prove zero exists?  I'm not sure, but what I do know is that she is really thinking deeply about place value and the thoughts the students generated as a whole class, individually and in pairs about our place value system could not occur had this been a worksheet place value activity.

Who says 10 year olds can't have deep philosophical discussions about maths?


What do you think? ...........