Category Archives: spatial reasoning

US Flags

I’ve been on my travels recently in the United States. I was staying with some friends in Arizona and they had this flag on the wall.  It’s an old flag with 48 stars neatly arranged in 6 x 8 matrix. It was from the days before Hawaii and Alaska joined the Union, sometime before 1959.


Now, I’m sure that all American school children know most of this stuff already, but it got me thinking about the US flag and specifically the arrangement of the stars and it gave me an idea for a lesson.

So, I know that there are now 50 stars for the 50 states of the USA.  But how are they arranged?  And how could we describe this mathematically?  I might start by showing students this zoomed-in portion to give them a hint.

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It is, of course a 5×4 matrix inside a 6×5 matrix.

Was there a 49? Yes, briefly when Alaska joined in 1959. But what did 49 look like? Clearly it’s a square number, but they decided that to keep the aspect ratio looking more like a rectangle they offset the rows like this.


There were actually lots of different versions of the stars and stripes detailed on this wikipedia page:

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It might be better to start with a simpler example from the early days when there were only 13 states (2×2 + 3×3). What other numbers can be represented as a sum of two square numbers in this way?


Is there a better way of doing this one? (It’s 36 stars, maybe 6×4 & 4×3)


You might want to give this table as a handout (or here as a pdf).  Get students to work in pairs to come up with their own designs and compare those to the ones actually used.

And then, what if we go beyond 50? I don’t want to think too much about the political implications of such of move and who might end up as the 51st state, so let’s stick to this as a mathematical exercise!






A broken clock

I walked into my classroom this morning and noticed my clock was broken. Not just stopped but really broken, can you see why?


It reminded me of a nice problem solving task which is sort of to do with angles but actually much more to do with ratio and proportion.

I wrote the following on the board:

For a normal clock, what is the angle between the hour hand and the minute hand at the following times:
1) 15:00
2) 13:30
3) 10:15
4) 17:45
5) 9:26

There is a significant range of challenge in these questions.  15:00 – straightforward, right.  As soon as you start moving the minute hand away from 12, you need to consider the fraction of (360/12) degrees that the hour hand moves.  12.30 might be the best option for a question 2 if you really want to scaffold it.  You also might want to squeeze a few more in between Qu 4 and 5.  e.g. 14.40, 15.20.

Next time I do it, I won’t write them up all in one go, but will keep adding to them as I can see learners making progress. Or ask students to challenge themselves by creating their times which might work better in a mixed attainment classroom.

These can all be done without a calculator. It demonstrates how useful it is to have 360 degrees in a circle and 60 minutes in an hour because they have so many factors.

A nice build on this is this question from an OCR Booklet of problem-solving questions.


Reflections from Teaching: Fraction Talks

Some great low threshold, high ceiling ideas here. I could see a whole lesson talking about fractions and getting students to create their own. Would work well for mixed attainment groups.

Also, this is a great digital manipulative for lots of things fractions, decimals and percentages.



For the Love of Maths

In this post, I want to describe my experience with the Fraction Talks Activity I originally borrowed from Nat Banting’s site. He has since become a curator to, a website devoted to various templates teachers may use – if you haven’t seen it yet, it is very worthwhile to check out!

After pondering about the original blog post for a few minutes, I was struck with just how versatile these templates are at connecting certain mathematical aspects together, and providing a visual representation of certain operations/ideas. The fraction talk template that I used when I worked with two classes of Grade7/8 was the following:

UntitledConcept of Fraction

Perhaps one of the most basic uses of the template is to connect it to the general concept of a fraction: a shaded portion of a whole, in which the whole has been divided into equal parts. To do this, we…

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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.

Screen Shot 2015-03-22 at 14.06.32I 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