# 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

# Extending Transformations

All four transformations (Translation, Reflection, Rotation, Enlargement) with Year 10. It feels like one of those topics that we cover in Year 8, they mostly grasp some of the skills but then forget the details (e.g. to specify the centre of rotation). That’s OK, I reckon. But it does imply that the time in Year 8 is better spent mastering the more tricky skills like reflecting in 45 degree lines than answering GCSE style questions, e.g doing translations using vectors.  These sheets are an old favourite which are great for practising those skills.

I was a bit surprised how long my middle set took to get to grips with it. Inevitably there were some who finished quickly because they found it easier to visualise than others. I was a bit stumped for how to extend it. A quick conversation in the maths office gave me the idea of multiple translations. Of course! Do, say, a reflection followed by a rotation. And then what is the single transformation that has the same effect?

I created a simple extension question:

The shape ABCD has coordinates (-1,1) (-3,1)
(-3,4) and (-1,5) respectively. It is reflected in the line y = x to produce shape A’B’C’D’ and then rotated 900 anticlockwise about (2,2) to create A’’B’’C’’D’’.

What are the coordinates of each of the points A’’, B’’, C’’, D’’ after these TWO transformations?

Describe fully the single transformation which would map ABCD onto A’’B’’C’’D’’.

This document can easily be printed and chopped. I used this geogebra file to create it.

It’s easy to create more questions like this by moving the points of the original shape and changing the transformations in the Geogebra file.

# Extension Arithmetic

Here is a mixture of arithmetic questions to ponder with solutions (or here as a pdf in case the Equation editor in Word messes things up).   They are all non-calc although that’s not immediately obvious when you look at them. I used this today with second set Year 9 and they seemed to appreciate that they achieved more than they originally thought they would when I put them all up at once.

Here is an example:

# Make a square with one cut

This is one of those amazing sheets which keep them going for quite a while!  I’ve tried this with a lower set Yr10, an extension class in Yr8 and even a few teachers!  It’s hugely differentiated all on one sheet and the way it’s laid out encourages students to wander to another shape if they get stuck – some of these are really hard.

I quite often offer scissors but most people quickly realise that they prefer just to do it with a pencil and visualising it – which of course is the whole point!

Make a square with one cut