Today my students were learning about Pi, its origins and its irrationality. We used this handy website to look up things like the date, and our birthdays to see where, in the first 200 million digits of pi, they resided. It helped us visualise what it means to have infinite digits (as much as one can).

Someone came up with the idea of looking at the distribution of each digit 0-9 within the first 200 million digits to see if there were any interesting patterns. Disappointingly, the only pattern is that they are roughly evenly distributed. But that didn’t stop us making a Pi Pie Chart… even though I hate pie charts, I made an exception because punny.

So proving the area formula for a trapezium is turning into something I do when I get bored. Here is my third proof. This one requires two versions annoyingly, as I mapped out all the different types of trapezium I could think of (are there more??)

Type 1 has a base that exceeds the top in width, both to the left and right. Type 2 and type 3 differ only in that type 2 has a top right vertex in line with the bottom left vertex, whereas type 3 does not. Type 4 has a base and top completely unaligned, and type 5 has two right angles.

I needed a slightly different proof for type 1. Or perhaps I’m just missing something so that I can combine these two proofs nicely without writing a completely different one? Seems likely.

Anyway, it’s a simple premise: enclose the shape in the smallest possible rectangle, and show that the area of the trapezium is equal to the area of the rectangle minus the remaining triangle(s). If you play with the algebra a little, you end up with the formula we’re all familiar with. I think with the added arrows it’s pretty clear where everything comes from, albeit a little messy.

Magic Squares are puzzles based around a square array of size n, containing the unique positive numbers 1 – n^{2}.

To solve the puzzle, you must place each number into the grid such that the sum of each column, row and diagonal is the same. This total is known as the magic constant.

Let’s consider then, the 3×3 magic square.

First we must find the target total for each row / column / diagonal. This is often given in the puzzle, but it doesn’t have to be. It can be calculated as follows:

So our target in this case is 15.

By observing the above diagram, you can see that the total across all four straight lines must equal 15 x 4 = 60.

We also know that the total of all 9 numbers is 45 (which is…less than 60). Knowing that the central number is used 4 times, we can now deduce what that number is.

If we call it ‘x’, then after considering our total of 45, we have 3x left over, which sums to (60-45)

60 – 45 = 15

x = 5

So our central number must be 5.

Next we’ll prove that 9 must not go in a corner:

If it is placed in a corner, then we soon find we have 3 numbers that *must* go in only two spaces:

Hence 9 does not go in a corner.

Once it is placed between the corner pieces, the rest solves itself:

Squares of this kind are subdivided into two categories: singularly even and doubly even. The former is divisible by 2, but not 4. The latter by both. There are strategies to solve them here.

Magic constants for these are calculated in much the same way as before. Thus the magic constant for a square of size n=4 is:

136 / 4 = 34

We can in fact generalise for the magic constant.

The sum of consecutive numbers can be written as the formula for triangle numbers, which is:

Our ‘n’ for the formula above is in fact going to be ‘n^{2}‘ , because ‘n’ for us refers to the number of squares in a row, therefore ‘n^{2}‘ refers to the number of squares in the entire grid. So we get:

That will give us the sum of numbers, but we need to divide that by n to get the magic constant. Which gives us:

But what about the number of solutions?

In fact there is only one magic square possible for a 3×3 grid. This square is known as the Lo Shu.

No-one as yet has been able to calculate the number of solutions for an n x n grid, but the number of solutions does grow exponentially.

n = 3 (1 solution)

n = 4 (880 solutions)

n = 5 (275305224 solutions)

n = 6 (unknown).

So between magic squares of n=4 and n=5 you pretty much have enough starters to last a lifetime.

After a successful proffessional development morning back in February, I invited Don Steward back to Huddersfield University this month to run another session for trainees, NQTs and local (and some not so local!) maths teachers as part of our mini Maths Teacher Conference. The event included other speakers and workshops too, which I’ll write a little about later in the week. (Check out http://www.missblilley.co.uk/ to read about one of them, which Don attended and described as ‘incredible’ yesterday).

Anyway, moving on. Don has kindly let me write up his session, and I think the materials will be available on his blog as well, which if you haven’t seen it, is here)

The session was very focused around proofs, how to introduce them, and how to derive them.

He opened with this:

This is intended for Year 7 students. Before students do anything, the question “will we get the same answer if we go in either direction?” is asked. A discussion can be had, then students can place a low number in the top left circle, and work right, then down first, then start again but go down and right. Was the answer the same? Nothing too taxing so far. Students then try again with a second number, and then again with a third. Is there a pattern emerging (spoiler alert – yes).

Students are then encouraged to try a big number (in fact, it was suggested that younger students often choose a big number in a bid to try and show it works for all numbers). This time however, we cannot fit say, a million in that small circle, so we’re going to call it ‘m’ (genius!!).

In this way we’re essentially moving discretely into algebra, whilst keeping it all very approachable for students. Running through creates algebraic expressions now.

Then you can get a student to think of a number, but not say it to anyone. If that student is called Beth, we’ll use ‘b’ for ‘Beth’s number’. We run through again, deriving the same expressions as before, which highlight the pattern, but this time, even though b and m are both algebraic representations, crucially for students, they do not know what ‘b’ is, yet they did for ‘m’. They now have a proof. Beth’s number could be anything, and the pattern still stands. There are variations on this idea too. You could give students the end numbers and work backwards for example.

Here is a more visual proof of the pattern:

Pretty clever.

The second proof exercise looked at the cases where 8n + 1 generates a square number (see below)

Is there a pattern emerging?

You may be able to see that the ‘n’ values that create a square number are in fact triangle numbers. I should point out that this is where a lot of the greatness of the presentation is lost in blog format, as that thinking time and discovery was very much handed over to us, the audience, rather than being given the answers instantly as I’m doing here.

Again, a few different methods of proving this were mooted and considered, and a nice visual proof is shown below (n = 10):

This proof was derived nicely by a trainee in the audience too.

A third proof activity is shown below. This time, looking at the answers generated for a general formula (n+2)^2 – n^2 . No focus is initally given to algebra at all, it’s about playing around with real examples, looking at the answers, and trying to spot the pattern. In this case, the answers are 4(n+1) each time. Which would initially be desribed without algebra before diving into proofs (ie first we spot that each answer is a multiple of 4, then we look at ‘what’ is multiplied by 4 to get that particular answer, then we discover it is in fact the number between 7 and 5, 3 and 1, 10 and 8 etc).

The eventual visual proof for this result is shown below:

It’s a rearrangement of a more intuitive pair of diagrams:

It turns out the diagram is used as a visual proof for several theorems

I recognised it from a Pythagorean proof:

But there are some other nifty proofs too (see the PowerPoint at the end).

Further investigations involved the relationships between area and perimeter:

Using angle bisectors to investigate the (infinitely) more interesting inscribed circles:

and taking a very mathematical method to approach magic squares (which will merit a post of its own soon!)

A very thought provoking, and inspiring presentation once again. Many thanks Don.

I’ve created a series of number bonds ‘workouts’ in the style of (the fantastic) Times Tables Rockstars.

There’s no website I’m afraid(!), these are just excel sheets that randomly generate numbers each time you load them. That means that you have an infinite variety. Hurrah!

There are:

3 Levels of difficulty for number bonds to 10 (add, subtract, mixed)

The same again (3 levels) for bonds to 20:

6 levels of difficulty for bonds to 100 (below is level 6):

and I’ve also added in 2 levels for bonds to 60 (time), 180 (angles) and 360 (more angles).

MAKE SURE YOU READ THIS NEXT BIT:

Each sheet has randomly generated numbers on it. That means that every time you do something (eg press enter), the whole sheet will change. That’s great for making different worksheets each week, but really bad if you want to reload an old one with exactly the same questions on it.

SO!! If you want to make a permanent copy of any single set of questions, then print the page to pdf.

Do that by selecting file / print but selecting PDF instead of a printer.

PS the answers are all in each sheet as well. Just change the font colour of the cells where the answers should go and voila, there are the answers.

Here is the excel file. All the different sheets are listed along the bottom of the file (as…excel sheets).

I have lots of new classes this year, several of which came to me with poor mindsets about their ability and the likely grades they will achieve in maths. I have been fairly aggressive (no, not like that…duh) in my pursuit of a classroom where errors and mistakes are actively encouraged to be openly discussed and not hidden away in shame.

I’ve reacted to certain situations slightly differently this year:

A student laughed at another student’s answer, so I sent them out of the class (with me) to talk to me about their behaviour

A student said ‘oh my god’ when another one answered a question (incorrect answer). I sanctioned them and wrote in their planner.

A student said “well done” sarcastically when another persisted past an incorrect answer and finally, with some questioning, came to a logical answer that was correct. I stopped the class and made a HUGE issue out of what that student was doing, and how every single one of us makes mistakes, ESPECIALLY ME, because guess what, I have to work with all of your calculations in super fast time to keep up with 25-30 students and that leads to silly mistakes QUITE OFTEN. I asked them how they would feel when, inevitably, it’s their turn to talk about their answer, which is incorrect, and they’re anxious, and they’re not sure what I want them to say, when in actual fact all I want them to say is how they got there. How will they feel, when they know there’s another student just waiting to heckle them, to dig a finger in, and make them feel bad about not knowing something straight away. Funnily enough they have been respectful since.

I’ve made a big deal out of the supportive network that the class should be, and how no students are in competition with each other, only with themselves. “What anyone else gets in an exam has no bearing on your life” etc. “If the person next to you doesn’t understand, help them.” etc.

I make a big teaching point out of mistakes, and I make a big point of praising students who talk about their answer, right or wrong. The results so far have been brilliant. Students who at the start of term always answered with “i don’t know, i can’t do it, I can’t do ANY of it” now persist with thinking it through, and usually come up with, at the very least, a considered idea.

I am totally (and purposely) over-reacting to fairly minor incidents of discouragement, which feels a bit weird. But it seems to have worked, and it worked quickly.

I haven’t blogged anything in a while because BUSY. However, I thought I’d share how I set up my day-to-day teaching to make things as easy and efficient as I possibly can. I’m a big fan of ‘work smart’. A lot of the things I do and use have a big investment of time and effort initially (several hours in most cases, a few weeks in one or two), but the payoff is instant, and usually permanent.

Seating Plans

As soon as I am able to, I create detailed seating plans using PowerPoint. This has evolved over the last three or four years, but essentially I do it like this:

Just to explain what’s going on here: Lots of copy and pasting! I have to get the photos from SiMS, and they only come as a block image of a whole class, so I screen grab that, then crop several copies of it down as shown.

The background colours can be altered to represent different information. I change colours depending on general behaviour / effort. Green for good, blue for less good.

Some names have bold or underline or an asterisk. This is simple codification of more sensitive information.

Because each student photo is grouped to its background and text, they are essentially draggable objects so changing the seating plan is no bother at all. T stands for target grade, C stands for current grade.

Alongside the seating plan, I make a QR Code class sheet. This is a list of all students in the class (with their photo so I know who’s who!) and alongside their photo is a large QR code. It has to be large as it contains a lot of information (simple QR codes can be tiny and still work. Complex ones won’t get read easily by a QR Code reader if they’re too small.

The QR code takes my phone directly to a pre-wrtten email, with the address of the parent already filled in. The email reads something like

“I am trying to improve parental communication and the quality of presentation in student work by trialing a new system I’m calling the #workselfie. Students can volunteer (or be selected) to have their work photographed and sent home. Your child has been chosen this week. Kind regards Mr blah. ”

This took a good few hours to set up. Email addresses are hidden away in the depths of SiMS and we’re not at a point where a class list of emails can be reeled off. So i copied each one into a spreadsheet and did this, using the website www.qrstuff.com

I cannot emphasise enough how handy a screen grabbing tool like… erm, screen grabber, or ‘preview’ on a mac is. I grab the QR code rather than download, insert etc.

Each class took about 45 mins to set up:

That’s obviously a big investment. But here’s the payoff…

I tried the #workselfie idea last year, but it was REALLY hard to maintain. I could send about 6 photos home a week to consider it manageable. But that involved taking a photo on my phone, emailing it to myself, downloading it onto my work computer, going into SiMS to find the child and the email, etc. The whole process took about 10-15 minutes per photo. A pain.

Here is how long it takes me now:

You’ll have to imagine the seating plan on paper rather than lifting from the laptop screen. I sent 6 in ten minutes today! Parental feedback has been great.

I keep all of this stuff on a clipboard when I teach.

Anyway, moving on…

2. Extension Activities

I find that the thing I need at my disposal at all times is resources. The more the better. Finished early? I have resources. Finding it easy? I have resources. And so on.

The extension resources I use are taken mostly from mathschallenge mainly because they’re awesome and the answers are provided.

I printed each difficulty level in sets of 4, with an answer booklet for me:

They sit on my desk, ready for action. You’ll also notice I have my own puzzles as stickers. These are for challenge homeworks (alongside typical ones) if students are finding stuff too easy. I occasionally just stick them in books if students finish early too.

3. Physical Resources

I have very few physical resources. I don’t do the whole sort cardy stuff very much. I try to keep things simple with these few tools:

That’s a whole load of dice, a whole load of blank dry-wipe dice, blank dry wipe cards, decks of cards, and scratch-card stickers. These things are very versitile and most can be appied to any topic on the curriculum. It’s just about imagination. Rounding numbers? Roll a dice 4 times and put all the digits together. Then round it. It stops everyone copying each other, and it’s just a bit more interesting with almost no effort from me. I don’t use any of them a LOT, but I use them more than anything else I suppose.

4. Text Books

I don’t use purchased text books. I collated my favourite resources and ones I had written mysel into my own version of text books. They are fairly crude, but the students like them A LOT more than normal text books. I keep them held together with simple clip binders so that I can add and remove pages as and when I need to.

I have a set of 18 of each book. These took me a long time to collate and produce, but I’ve used them for years now.

I also have equipment boxes on each desk (as seen in the photo above). These have typical stuff in them. The main thing is mini-whiteboards. I use them pretty much all the time. To the extent that I sometimes forget that students should probably write something in their exercise books. I’m not convinced there’s a better way to teach, question, assess, progress all in one, all together.

5. Rewards

I don’t use the school system anywhere near as much as I should. I just use these badboys mostly:

6. Digital Resources

I have *so many* digital resources I cannot begin to describe it. I have collated these, and purge them once or twice a year. They are all on a datastick that sits on my keys. The way to make this very effective is meticulous organisation:

And the most used folder is definitely the ‘generic starters’ folder. Full of powerpoints chock full of puzzles like these and so on. Great for *any* point in the lesson.