Geometry Snacks

I’m very pleased to announce that my second book, Geometry Snacks is now shipping from Tarquin Publications. The book is coauthored by Vincent Pantaloni, a fantastic French mathematician who helped design over 50 geometric puzzles as well as some of the creative solution approaches to each one.

Geometry Snacks is a mathematical puzzle book filled with geometrical figures and questions designed to challenge, confuse and ultimately enlighten enthusiasts of all ages.

Each puzzle is carefully designed to draw out interesting phenomena and relationships between the areas and dimensions of various shapes. Furthermore, unlike most puzzle books, we offer multiple approaches to solutions  so that once a puzzle is solved, there are further surprises, insights and challenges to be had.

As a teaching tool, Geometry Snacks enables teachers to promote deep thinking and debate over how to solve geometry puzzles. Each figure is simple, but often deceptively tricky to solve – allowing for great classroom discussions about ways in which to approach them. By offering numerous solution approaches, the book also acts as a tool to help encourage creativity and develop a variety of strategies to chip away at problems that often seem to have no obvious way in.

The inspiration for the book came from the responses to puzzles I have created here, and the different ways in which people solved them. Take the figure below for example:

(not included in the book!)

This puzzle requires you to find what fraction of the whole regular shape the shaded section represents (constructed using midpoints).

Here is one possible solution:

You can see that the pink area covers half of each vertical pair of rectangles, and that is equal to 4 of the congruent triangles. So the answer must be 4/12 = 1/3

But there’s more life in this puzzle yet! Can you see an alternative approach? Can your students?

Shearing the parallelogram into a rectangle, then reconstructing the entire shape into a quadrilateral! It’s a lovely approach, and one I wouldn’t necessarily think of myself.

By sharing multiple approaches both to the reader and potentially with a class, everyone learns from new insights and styles of problem solving in geometry – and each puzzle teaches us something new even if we can solve it!

As such each puzzle in the book includes at least 2 solution approaches.

If you have enjoyed my puzzles over the last five years, consider buying this book. It ships internationally, and I think you’ll like it a lot.

Unit Conversion

I have often found that students really struggle to understand what is going on when we convert units of area and volume. It’s pretty straight forward to demonstrate conversion of units of distance: 1 metre = 100 cm etc just by using simple instruments like a ruler or metre stick, or measuring tape. Whilst some may initially struggle with this, really it’s just a question of scale / ratio etc.

When dealing with area however, it gets a lot more confusing. 1 m2 is frequently misunderstood as being equivalent to 100cm2. It’s understandable why: it feels intuitive, and maps nicely against existing knowledge that 1m = 100cm. Unpicking this misconception can be tricky. In what I’d consider to be the least appealing scenario, students are left just memorising that they multiply/divide by the conversion ratio ‘squared’ so 1m2 would be equivalent to (1 x (100)2) cm2 and so forth. This, as stand-alone information, feels a little empty – and of course I’m keen to expand on why this works. The most common explanation I’ve seen is using a diagram of, say, a square (below is taken from BBC Bitesize):

I have no issue with this explanation. It makes sense and it helps students overcome the somewhat counter-intuitive result upon first inspection, that 50000cm2 = 5m2

However, what I’ve often found is that even with this explanation, students resort back to making the same misconception/error regardless of demonstrations to try and make it feel more intuitive. Perhaps then, the issue needs to be tackled further by adjusting the way in which we approach these problems.

Take the following example:

“Convert 250cm2 to mm2

Students who get it right will (again, in my experience) typically mull it over, recall something about areas being a bit different to distances, and then decide upon a multiplier. They would literally write down 250 x 102, or 250 x 100, then arrive at the answer and move on.

I prefer a pictorial approach, which I’ve found tends to bring out more right answers from those other students, and maintains the intuitiveness of the answer being right.

Forget the conversion element of the question for a moment. We are dealing not just with a number (250), but an area of 250cm2. We can represent that as follows:

This gives us our required area.

Now convert the units for each side into our new units, millimetres:

Our rectangle is the same size, it has the same area, and the lengths are unchanged. We’re just using equivalent amounts in different units of measure.

So now the area, which is unchanged, is represented as 250 x 100 = 25000mm2

Students would literally draw both rectangles, one below the other like so:

Now, all is well and good so far, however, what if we get a question like this:

“Convert 17m2 to mm2

17 is prime, and less easy to put into a nice rectangle. Or is it? In fact, this gives us the perfect adaptation to the rectangle representations. Rather than continuously adapting the heights depending on the numbers we’re given in the question, just make them all of height ‘1’, forever, for every question:

“Convert 25mm2 to cm2

The idea can be further adapted for volume conversions:

“Convert 3m3 to cm3

Helping trainee teachers

I often get asked by new mentors how best to help trainees on placement in their departments. It can be daunting for many reasons – you want them to excel, you want them to have a good and supported experience, but you may be worried about how much work you have to put in and how much time you actually (don’t) have. I thought I’d share some key ideas here.

1. Time
The most fundamental thing you need to give to a trainee is time. Time for them to develop, but also your own time to work with them. They are unlikely to get better if they are not being guided or being given advice and support with planning. There is a balance to be had here. Pointing them towards the shared resources folder on a computer isn’t nearly enough, but sitting and going over every single lesson plan in fine detail with them is far too much. They are adults and should be treated as though they can do a lot on their own once they’re enabled (although some trainees struggle with this initially!) – but they’re also new to this so they’ll need more support and reassurance that what they’re doing is right (or wrong) in the early stages. Allocating time on their timetable as ‘mentor meeting’ or similar is essential. Trainees typically struggle when their mentor meetings are ad hoc and therefore often on different days / times. This inevitably leads to weeks where there is no meeting at all. If you can, have lunch with them too as this can serve as an informal opportunity to raise or discuss any issues or queries.
2. Give them a balance of classes
Don’t give a trainee all the best behaved classes, or all the worst ones. Tougher classes are good to develop behaviour strategies that are less essential to survive in a class where the children rarely stray off task (although bad teaching can cause any class to do that!). A typical strategy is to give them one tougher group, and, if possible, a pair of the same year group to allow for easier planning / more obvious requirements for differentiation strategies (eg if you give a high ability group and a middle/ low ability group in the same year – the topics will likely be the same but the pitch will often be different). That of course assumes classes are setted in some way, which is still pretty common in maths.
3. Hold them to account
Some trainees can initially struggle with the difference between being a student, and being a trainee – they are to be treated as if they were an employee at the school, so if they are ever late / miss deadlines / slow to respond to emails etc, hold them to account for that. Similarly, whilst it’s expected that they will make mistakes in the classroom, if they do not respond to feedback – eg, you tell them they need to do something different, and they ignore it, then again, hold them to account for that.
4. Give actionable feedback
It’s easy to give feedback that makes sense to you, but no sense to a trainee. “Improve your behaviour management” for example, or “be more consistent with behaviour systems”, or “make differentiation more explicit” . All of these can feel clear, but all a trainee will be thinking is ‘how?’. So tell them how. Give them one or two specific actions that they can follow. For example “give students a choice of difficulty for the main task” or “ensure there are two extension tasks available – one exam style question and one algebraic generalisation question”. These are far easier to follow, especially if you provide examples too. Behaviour targets are particularly difficult to implement. How do I be more consistent? What is inconsistent at the moment? A good way to overcome this, is to ensure you are having regular conversations about behaviour. Scenarios and post-lesson discussions such as ‘when student x did this, I would have done this instead’ or ‘when student x did this, that warrants a C1’ etc. Trainees understandably struggle with knowing what actions in a class constitute suitable escalation of behaviour systems, especially when those systems are themselves sometimes nuanced and flexible (rightly or wrongly).
5. Observations
Try to give a mix of general full lesson observations and targeted foci. For example, you might watch a lesson and give feedback on all elements of it, but often that can be overwhelming for a trainee when you inevitably give and discuss ten points of improvement. Often it can be more effective to focus on only one or two particular things, eg behaviour, or quality of tasks. If you can film lessons, then seriously consider it. Watching a lesson back and being able to stop it and discuss is probably the most effective tool you can use to develop a trainee. Don’t be afraid to give live feedback during a lesson either. It’s much better to signal to a trainee that a student is off task, or that a student needs a target to keep them focused, or to praise a particular student for something they did during the lesson, than to tell them they should have done it afterwards. Obviously this requires forethought. Yelling at the teacher “John is off task, sort it out” is unhelpful! You could develop a simple signal system, or discreetly call the trainee over when students are on task etc. I’ve never understood why observers (myself included at one time) watch a lesson go to shit in those instances where they could have subtly intervened and helped avoid it with just a few simple signals. Obviously this isn’t always possible, but moreoften than not, a little live feedback can steer a lesson into a better direction. Remember too that if they are observing, they’ll need training on what to look at / for, and a post lesson discussion explaining the decisions the teacher made. Without those, the trainee will often become a simple spectator or even a pupil in the room.
6. Planning and Marking
Give trainees all the tips and shortcuts that you use yourself. We all know that marking policies in some schools are completely unwieldy – how do you manage it? Simple tips such as marking a little during a lesson, carrying around a multipen for those ridiculous marking in green / red / purple things you may be expected to adhere to, collecting books in with the page open that students were working on, in ‘confidence’ piles to help prioritise who needs the most marking etc. Those tips may seem trivial, but if they’re the difference between marking for an hour and marking for three, then they really are significant. One of the hardest lessons for a trainee is learning when a lesson is planned ‘enough’. Typically trainees will overplan a lesson, and refine it over and over again – then they despair that it took them 3 or 4 hours to plan. Learning that the lesson would have been almost, if not just as good, much earlier and that all that extra time is effectively time wasted, is hard. Lesson templates can be useful, as can giving trainees two or three websites to source material from rather than the seventeen billion that currently exist online. If they can’t find what they want in those two or three sites, then start creating. Early on, I frequently find a trainee who has spent upwards of two or three hours looking for a perfect resource that doesn’t exist. Trainees will need guidance and support on this – which is best done face to face if time allows for it. Email feedback can often end up being a constant back and forth of minor refinement in the wrong directions – costing an enormous amount of time for both trainee and mentor / host teacher. And don’t hold back with sharing resources. I find it less and less now, but a few years ago I would still come up against mentors who would refuse to hand over their resources because ‘I never had them when I trained – I had to do it all by myself’. Honestly, we need people to join teaching and stay, not leave because staff are trying to make it more difficult!
7. Keep in touch with the trainee’s provider based mentor.
No-one has a more vested interest in the success of a trainee than their university / school direct / scitt mentor. If you have any worries / concerns, or even better, want to celebrate the success of a trainee, let them know! It’s much harder to bring a trainee back from the brink than it is to intervene early on, even if it’s just something really minor. Similarly, it’s always great to hear that a trainee is getting on really well, particularly in those early weeks when they start teaching new groups.

Finally, remember that trainees all start from different places. It doesn’t matter how good or bad they are in their first lesson (within reasonable expectations!), what matters is that they respond to feedback and continue to get better and better.

Trig Identities #4 Double Angles

Starting with one of our ‘sum and difference’ identities:

$\sin(x\pm y)=\sin(x)\cos(y) \pm \cos(x)\sin(y)$

If we take $\sin(2x) = \sin(x+x)$

then, using the identity we started with:

$\sin(2x)=\sin(x)\cos(x) + \cos(x)\sin(x)$

$\sin(2x)= 2\sin(x)\cos(x)$

We can do it all again with cos:

$\cos(2x) = \cos(x+x)$

$\cos(2x) = \cos(x)\cos(x) - \sin(x)\sin(x)$

$\cos(2x)=\cos^2(x)-\sin^2(x)$

Now, recall that $\cos^2(x) +\sin^2(x) = 1$

$\cos^2(x) = 1 - \sin^2(x)$

so

$\cos(2x) = 1 - \sin^2(x)-\sin^2(x)$

$\cos(2x) = 1 - 2\sin^2(x)$

That’s one identity, now if we go back to the start and manipulate it all a bit differently:

$\cos(2x)=\cos^2(x)-\sin^2(x)$

$\cos(2x)=\cos^2(x)-(1-\cos^2(x))$

$\cos(2x)=\cos^2(x) - 1 +\cos^2(x)$

$\cos(2x)=2\cos^2(x) - 1$

Finally, tan is pretty straight forward if you use the sum&difference identity, but we haven’t derived that yet, so here we go:

$\tan(x)=\frac{\sin(x)}{\cos(x)}$

$\tan(x + y) = \frac{\sin(x+y)}{\cos(x+y)}$

Using the sin & cos sum/difference identities:

$\tan(x + y) = \frac{\sin(x)\cos(y) + \cos(x)\sin(y)}{\cos(x)\cos(y)-\sin(x)\sin(y)}$

$= \frac{\sin(x)\cos(y) + \cos(x)\sin(y)}{\cos(x)\cos(y)-\sin(x)\sin(y)} * \frac{\frac{1}{\cos(x)\cos(y)}} {\frac{1}{\cos(x)\cos(y)}}$

$= \frac{\frac{\sin(x)\cos(y)}{\cos(x)\cos(y)} + \frac{\cos(x)\sin(y)}{\cos(x)\cos(y)} } {\frac{\cos(x)\cos(y)}{\cos(x)\cos(y)} - \frac{\sin(x)\sin(y)}{\cos(x)\cos(y)}}$

$\tan(x + y) = \frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}$

Phew! Now, if x and y are equal, then just call them both x:

$\tan(2x) = \frac{\tan(x)+\tan(x)}{1-\tan(x)\tan(x)}$

$\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}$

Trig Identities #3 Periodicity

You can hopefully see from the diagram above, that if we add $2\pi$ to our
angle $\theta$, we would come ‘full circle’ (ba-dum tish) and be back where we started from (think of it as adding 360 degrees if it makes you happier).

Hence:

$(\cos(\theta \pm 2\pi), \sin(\theta \pm 2\pi)) = (\cos(\theta), sin(\theta))$

$\cos(\theta) = \cos(\theta \pm 2\pi)$

$\sin(\theta) = \sin(\theta \pm 2\pi)$

And since csc and sec are basically just utilisations of sin and cos, then they have the same property:

$\csc(\theta) = \csc(\theta \pm 2\pi)$

$\sec(\theta) = \sec(\theta \pm 2\pi)$

Now  $\tan \theta$ has a periodic cycle half the size of $\sin \theta$, which you can perhaps visualise easily by studying their respective plots side by side:

and so :

$\tan(\theta) = \tan(\theta \pm \pi)$

$\cot(\theta) = \cot(\theta \pm \pi)$

Trig Identities #2 Pythagorean Identities

Above is a quick refresher on each trig function. You only really need sin, cos, tan – but for convenience:

$tan(\theta ) = \frac{sin(\theta )}{cos(\theta )}$

(think SOHCAHTOA on the unit circle)

$csc(\theta ) = \frac{1}{sin(\theta )}$

$sec(\theta ) = \frac{1}{cos(\theta )}$

$cot(\theta ) = \frac{1}{tan(\theta )} = \frac{cos(\theta )}{sin(\theta )}$

Recall the formula of a circle is

$x^2+y^2=1$

and if our $x$ and $y$ coordinates $(cos(\theta), sin(\theta))$ lie on the circle (which they do) then:

$cos^2{\theta } + sin^2{\theta } = 1$

Take this identity and divide both sides by $sin^2{\theta }$:

$\frac{cos^2(\theta )}{sin^2(\theta)} + 1 = \frac{1}{sin^2(\theta)}$

$cot^2(\theta)+1=csc^2(\theta)$

We could instead have divded both sides by $cos^2{\theta}$:

$1+ \frac{sin^2(\theta )}{cos^2(\theta)} = \frac{1}{cos^2(\theta)}$

$1 + tan^2(\theta) = sec^2(\theta)$