# Standard Form using a calculator

My year 10 class have their Physics GSCE tomorrow. They won’t have much of an appetite for learning new maths so I thought we’d practise using our calculators in Standard Form by getting them to recreate and interpret these. Hope it comes up!

# The beauty of the very very big and the very very small – Standard form

Scale of the Universe is one of the nicest websites I think I have ever seen. It’s fascinating in its own right, but is also a nice resource for looking at very small and very big numbers.

I put together this sheet to use with students to give them some tasks to do whilst playing with the site. The idea is that it gets them used to using standard form numbers and demonstrates why it is actually useful and better than writing all those zeros!

After browsing the ever excellent Don Steward’s Median site for sequences, I found this on Linear and Quadratic growths.

It got me thinking about using shapes to represent sequences and in particular using different colours to represent different sequences laid on top of each other.  For example, the following patterns produce a linear sequence.

However, there are a couple of ways you could look at this which combine sequences which are arguably simpler.

Or:

We can then extend this on to quadratic sequences.  This is nice: one way we can see the sequence of square numbers, the other way we can see how multiplying one dimension by the other leading to some brackets which can be multiplied out.

There are a bunch of these in this file.

There are also over a hundred sequences at visualpatterns.org a website entirely dedicated to, well, visual patterns.

Get your students to create their own once they have got the idea.  Could make for some great wall displays!

On quadratic sequences specifically, a nice worksheet here from @solvemymaths.

Finally, back to Don Steward with this which is actually just a series of terms stacked on top of each other.

# Are you sure you can prove that?

A nice couple of demonstrations of what makes a proof, i.e. just because you’ve got lots of examples doesn’t mean you’ve proven it.

Mathematicians of the 18th century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are all primes.  This was no mean feat without a calculator.  It was a big tempation to think that all numbers of such kind are primes.  But, the next number is not a prime:

333333331 = 17 x 19607843

Another classic example is the question of how many areas you get when you cut a circle with chords formed by joining points on the circumference.

You might think you’ve spotted a pattern of doubling each time (or 2^n).  And indeed the next one is 16.  But the one after that is 31.

The formula is not quite so straightforward and involves combinations:

In its expanded form it looks even more crazy!

Wolfram has more details on this problem here.

And some nice discussion of the problem here.

# A practical M2 lesson on finding centres of mass

Here is a simple, but effective idea that I used with my M2 class. Using bits of card to create shapes and trying to balance them on a ruler to find the centre of mass

They don’t get to do much of this stuff at A level, but it was a really effective way of getting them to appreciate what a centre of mass is before developing a method of calculating it for different shapes.  Full details in these files:

OnaKnifeEdge

OnaKnifeEdge