## Friday, 20 May 2016

### Prime Factorisation - the DNA of Each Number!

We began by sharing ways we thought prime numbers were amazing.

Lots of great and varied thoughts were shared.

We then looked at some true or false statements. We used our hands to create the letter T or F and then shared our reasonings.

Starting with strategies like this can be a useful way to tune back in and reflect on what we have been discovering in our unit.  It can also help to address some misconceptions being harboured.

We wondered why there wasn't a clear pattern with prime numbers when our orderly number system has so many patterns?

Could this be another reason why prime numbers fascinate people?

That's probably why they are so hard to find, we thought.

Someone suggested there might be a pattern in the units column. Perhaps the next two numbers in that column are prime and the next 3 are composite.

We thought that was plausible and added it to our wonder wall to find out later.

Why do we learn prime factorisation?

I've been wondering this and how to 'sell' it as something interesting to do.

I thought about how we have been making the connection that prime numbers are like the 'atoms' of whole numbers.

So, what connection could we make we prime factorisation?

I thought of how every number is unique and how its prime factorisation could be seen like the DNA of each person.

We discussed what we thought the image (below) was explaining and this led us to sharing what we knew about DNA and genes.

We determined that what makes every person unique is their particular DNA- no one else has the same DNA.

How could prime numbers be like the DNA of a whole number?

We looked at our previous 'atoms' activity to see what we noticed about the numbers.

Sample:

Amongst some of the suggestions, one student explained how the prime factors of 51 are 3 and 17. No other number has those, that's what makes 51 unique.

Could any other number from 51 to infinity have the same prime factors?

We thought not.

So, that is how we could connect prime factorisation with DNA.

We then looked at how we can make factor trees and began together with 12.

We realised how 1 ruins our pattern and thus, it helped us further see why it shouldn't be considered a prime number. (There more evidence we can find about one be neither-the better our understanding of that will be)

We could see how 12 had the same prime factorisation ( 2 x 2 x 3) each time. That's is 12's DNA!

What number should we try next?

24 was suggested because we might find a pattern.

We loved that suggestion so we tried seeing how many factor trees we can make for 24.

Most of us were able to make 3 different factor trees and were quite shocked when one pair had created 4!

So what i sthe DNA / prime factorisation of 24?

= 2 x 2 x 2 x 3
OR
2 cubed x 3

- Wait! There's a pattern!

- We are doubling 12 to get 24 and the number of possible factor trees has also doubled from 2 to 4.

We loved that discovery.

- There's ANOTHER pattern!

- The prime factorisation for 12 is 2 x 2 x 3
and for 24 it is 2 x 2 x 2 x 3

- Maybe prime numbers do make patterns after all!

So what do we predict the prime factorisation of 48 would be? What is its DNA make up?

We predicted it would be 2 x 2 x 2 x 2 x 3.

How many possible factor trees could we make for 48?

- 8 because it doubles.

We all agreed that that was plausible.

Even though time was saying we should move on to reading groups, the children really were eager and excited to see if their pattern prediction would work out, so pairs tried to create as many factor trees for 48 as they could.

You can imagine our shock when one pair had discovered not our predicted 8, but actually 9 possibilities!!!!!!

What?!?!?   Why???   How can that be?!?

Nearly everyone rushed to their table to see the evidence for themselves such was the shock of this discovery.

- Why is it 9?

- Why hasn't our pattern continued?

Then someone shared a theory.  He had thought about how massive prime numbers often end with 'subtract 1'.  He wondered that since 1 is a unit instead of being a prime or composite then the pattern rule might be     x 2 + 1   because there is a connection with 1 making prime numbers.

We liked this theory and really wanted to test it out!

We predicted that there would be 19 possible factor trees for 96.

Pairs and groups eagerly started making them.

We wanted to prove our theory.

SHOCK!  HORROR!

There were 20 possible factor trees!!!!!!!!!!!

- Why?????

- Why isn't our pattern working?!?

Someone then suggested another interesting theory:

- Maybe the pattern is  x2     x2 + 1       x2 + 2     x2 +3

We thought that was really plausible and wanted to test it out, but it was time to go.

A few asked if they could continue trying this after recess!

Unfortunately they couldn't, but we will continue this on Monday.

- Yay!!!

Before leaving, we orally reflected on what we had discovered and any new wonderings were added to our wonder wall to investigate later.

I had hoped that I could have raised some interest in prime factorisation, but really didn't like my chances. So you can imagine how I was really blown away by how excited and engaged they were.

I think the connection that we can see the unique prime factorisation of a number is unique ONLY to that number does help children deepen their appreciation for prime numbers.  I didn't know about this prior to today myself, so I am also blown away and my own appreciation for prime numbers has just skyrocketed.

The DNA for each number is its prime factorisation. I love this analogy :)