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.
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:
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.
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:
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
This is a nice development of the classic “Think of a Number” problems that are a good way of introducing algebraic equations. Nrich covers this introductory activity with Your Number Is… and other linked activities.
This one takes it a stage further. Proving algebraically why this works would be a nice extension activity. Note that with the numbers below, this will only work in 2015. I’ll leave you to work out what you need to change for subsequent years!
Work this out as you read.
Be sure you don’t read to the bottom until you’ve worked it out.
First, pick a number 1 to 7
Multiply this number by 2.
Multiply it by 50. (Come on, you can do that without a calculator).
If you have already had your birthday this year, add 1765
If you haven’t, add 1764
Now, subtract the four digit year that you were born. e.g. 1984
You should have a 3 digit number:
The first digit of this was your original number.
The next two numbers are your age.
Having been inspired by some ideas from Resourceaholic and also from Robert Wilne @NCETMsecondary here, I thought I’d create these True or False cards for use with Year 7. We have covered fraction of an amount, equivalent fractions, adding fractions, and top heavy / mixed numbers so far, but you could easily adapt these if you want to include, for example, multiplying and dividing fractions. I designed the false ones first which really got me thinking about misconceptions that I think could be occurring. The true ones were relatively straightforward after that. Enjoy!
Here they are as a Word doc: Fractions True False cards And if the formatting looks a bit weird, as a pdf doc: Fractions True False cards