## Tuesday, 17 May 2016

### Enquiring into Prime Numbers

Prime numbers.

How is it best to 'sell' learning about prime numbers to a child?

I've been wondering and researching a lot about prime numbers in the past few days before we start.

Though, of course, I've introduced prime and composite numbers in the past, I don't think I've found a way to 'sell' them properly.

I've been trying to find out why they are important to real life situations as a 'selling' point. When we see real life applications, our engagement increases.

It seems, so far, that real life applications are a bit of a stretch for a child to connect with, but some of these might help arouse curiosity and wonder:

° If you have a pizza split into a prime number, it can't be evenly shared so you could claim it all for yourself (well, sort of)

° The U.S Defence pays millions for new prime numbers discovered to use in the encryption systems.

° Prime numbers are used on their laptops to secure information. Public & Private keys are based on prime numbers. Eg when we use our credit cards online, our bank gives our card number a prime number to lock/unlock access with our credit card.

° We can find prime numbers being used in nature. Eg, the Cicada Prime was discovered when scientists found that cicadas hibernate underground and emerge to mate and die every 13 or 17 years (prime numbers). The theory is that their biological clock does this for evolutionary pressures.  If cicadas hibernate for X years and had a predator that underwent a similar hibernation of Y years, then the cicadas would get eaten if Y divided by X. So by 'choosing' to awaken in a prime number of years, they have made it less likely for their predators to awaken at the same time.

So, I'm thinking the best way for us to approach prime and composite numbers is to help children further discover the wonder of numbers and patterns we can discover when exploring them.

To make an analogy, we will make a connection that prime numbers are the 'atoms' or building blocks of every whole number.

This might be the best 'selling' point.

For children to discover how every positive, whole number is made up of the factors of prime numbers, that will show why mathematicians get so excited about them.
To add some spice and perk up interest, we will discover how we can world fame and earn millions of francs (by selling to the U.S Defence) if you discover a prime number.

This YouTube showing the current largest known prime number will hopefully arouse a keen interest in the wonder of prime numbers: Largest Known Prime Number

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For our enquiry, I thought it would be more effective if we slowly discover for ourselves what defines a prime and composite number rather than simply being told.

To tune in, we began with this number pattern sequence and with our partners we had 10 minutes to continue making it as far as time allowed.

We had the help of calculators so we could find patterns as they emerged:

Before starting though, we looked at 4 ÷ 3 = 1.333

What does that mean?

- The number continues to repeat itself.

- It goes on infinitely.

Do we know the symbol we use to indicate that?

- We place a dot above the last number.

Straight away, an interesting wondering was shared:

Can we have a number like 1 . 33333333333 with a dot on the last digit and then a 4?

Such a great question.

We thought about it and we shared our thoughts.

Eventually we decided we couldn't because if the 3 goes on infinitely, we would have to wait forever will we could add the 4.

We then started making our sequence patterns and soon after AMAZING discoveries started to be found and excitedly shared:

There was a lot of excitement and genuine interest in the patterns we could find.

I was excited too!

To help us find out about prime numbers, we looked at the numbers and identified how many numbers each was divisible by.

As we shared, I placed the numbers in either column:

Predictably, when we got to 9 some questioned whether it should actually be in the first column.  It makes sense.  They could see a pattern.  Odd numbers in the left and even in the right.

But then we remembered how 9 is divisible by 1, 3 and 9.

After this, I asked if we knew what the numbers in the left column are called.

One student shared how they are prime numbers.

Have we heard of prime numbers before.

Three or four hands.  (quick assessment)

And the other numbers are called......composite.

So, share with your partner what you know now about prime and composite numbers.

We did a quick share.

It's important that children are given opportunities to formulate new understandings in their own words.

I know there are more and perhaps even better ways to define a prime and composite number, but I want the children to discover these themselves.

For now, defining them by how many numbers they are divisible by is a good, solid start.

We then looked at our next learning experience:

There was some amazement remarks that even back in Ancient Greece people were thinking about prime numbers!

With our partners and a hundred grid, we were to find the prime and composite numbers.  This is a pretty standard primary school activity.

To make it a bit more interesting, we would record both successful and unsuccessful strategies we test out.

Why should we record and share unsuccessful strategies?

- They help us to make more effective strategies.

- All successful strategies come from unsuccessful ones.

- I don't think we should call them unsuccessful strategies because belongs we learn from them then they become successful to our learning.

We liked that last idea a lot so we put quotation marks around the 'UN' in unsuccessful.

Lots of different strategies and great number talk happened during this as the children tested theories:

Half way through our time, we paused and shared strategies we created and were testing out:

Theories we're formulating and wanting to test out:

And wonderings:

Some were beginning to wonder if 1 was prime, composite or neither.

This is great thinking.

I know it's neither, but I want the children to come to their own conclusions.

We looked at our initial 'definitions' of what makes a prime or composite number.

Prime- divisible evenly only by itself and 1.

Composite - divisible evenly by more than 2 numbers

Based on our initial definitions which we will expand upon later, where does 1 fit?

- As a prime number.

We have this as a wondering, so let's keep coming back to this the more we find out about primes and composites:

Wondering about the largest prime number, we realised this is unknown since numbers go on infinitely.

I did share though that a few years ago a new large prime number was discovered.

- How big is it?

It's REALLY big. So big that when printed it is this thick of double-sided paper.

- Wow!

(I'll show them tomorrow)

Some people get paid millions for discovering new prime numbers by the U.S military and other organisations.

- Wow!

Using these shared thoughts, we then continued finding the prime and composite numbers up to 100.

There were quite a lot of 'errors' being made, but I wanted the children to have the opportunity to find these out themselves.

After our quick share, some children were looking at the numbers differently and finding 'errors' they had made and many began trialling different strategies and testing out different theories being formed.

By the buzz level in the room, it seems we have are to a great start enquiring into prime and composite numbers.

To increase this further, for home learning we have a choice of 4 YouTubes to watch and record.

I'm hopeful that by watching these, they will also gain amazement about prime numbers and how they are used.

The link is:  Some Wonders of Prime Numbers