Tag: geometry

Jenga Views

This blog post is about a sequence of visual perception and geometry puzzles I have created called Jenga Views. You can download a file here with 39 puzzles, roughly in order of difficulty.

In my previous blog post, I showed two variations on traditional Jenga that I think are more interesting and more fun. But long before those were thought of, I had already been doing something non-traditional with Jenga blocks.

Back in May 2018, I saw a tweet that shared a maths/art activity called Fun With Orthoprojections by JD Hamkins. It was a list of challenges where 1 by 2 by 4 wooden blocks had to be arranged to match views from the top and two sides. I thought it was really cool, but I didn’t have any blocks with those proportions. I did have a quadruple set of Jenga blocks, but Jenga blocks have very different proportions to 1:2:4, so in order to use them I’d have to make my own.

After a furious effort in Inkscape and Word over one day and night, I had made the first version of Jenga Views, where Jenga blocks had to be arranged to match views of the shape from the top and two sides. There were 20 puzzles in total: the first eleven were based on JD Hamkins’s Fun with Orthoprojections, but the later ones were all my own. (The SVG Inkscape file with the carefully measured block faces is here, if you’re ever interested in making any yourself.)

I put them out at One Hundred Factorial the next day, where they were an immediate success. Check out these two students working on one of the challenges on that very first day:

In general, over the years, I have found that the Jenga Views challenges are strangely compelling to anyone for whom they spark interest. Some people aren’t intrigued by them, but those who are intrigued tend to just stick around and do all of them, one after the other.

One thing I particularly like about them is that you don’t need an answer key, because you can just look at your construction and tell if it looks right or not. There’s something empowering about being able to check it yourself. (I’m sure there’s a lesson there for maths problems we give to students for practice, but I’m not going to explore that thought more right now.) I love watching students crouching down to see their construction from the correct angle and holding up the pictures to compare.

One of those people who found it compelling was my younger daughter Charlotte. In April 2020, she was 11 years old and only a couple of months into Year 6 (which here in South Australia is the final year of Primary School) and suddenly we all went into pandemic lockdown. My wife, who is a trained teacher, cobbled together her own school away from school curriculum for Charlotte before Charlotte’s school sorted anything out, and I provided some maths activities I had brought home with me from university. I had brought a selection of things from One Hundred Factorial home so I didn’t go insane and so I could try to run One Hundred Factorial from lockdown. One of those things was Jenga Views.

And Charlotte loved it. Because of the quadruple set of Jenga blocks, she actually had enough to do all 20 challenges separately and lay out all the answers at once.

(Note that the file has been updated since this image was taken, so don’t use it as an answer key, because almost all the numbers refer to different challenges now! Also you can tell if your solution really is one without an answer key, as I said earlier.)

Charlotte was so very proud of herself and I was proud of her. I shared the photo and the Jenga Views document on Twitter, and it got a lot of attention. There were a lot of people who were also in lockdown who were extremely grateful for something fun and mathematical to do using resources they had in their house already.

I was all inspired too, and I created more puzzles, adding five more the next day and then five more the day after that, bringing the total to 30. I also created a print-and-cut net that would allow people to make their own blocks in the Jenga proportions, since someone complained they didn’t have any in their house, as well as a video of how to fold it up into a Jenga block. (Two nets are on the last page of the Jenga Views document, if you want something better quality than this picture.)

Eventually, the interest settled down on Twitter. But over the years I’ve pulled out Jenga Views at One Hundred Factorial regularly, and every time, there’s always people who do what Charlotte did and just work through all of them.

Now we’re here five and a half years later (and seven and a half years after its original creation), and I’ve finally gotten around to writing all this up as a blog post.

To reward everyone who waited – including me – I have created nine more puzzles. They are mixed in among the 30 that were already there, so be careful if you’ve downloaded it before because almost all the numbers have changed! Also at the end of the document there are four empty challenges for you to draw your own, and the net of the make-your-own blocks too.

As an aside, the reason I did nine more and not ten is because then you can print them two-to-a-page or three-to-a-page including the title page and they will line up neatly without annoying blanks. It’s been something that’s bothered me every time I’ve printed them over the last seven years.

So there’s the whole story of Jenga Views. I hope you enjoyed reading about it, and I hope you enjoy doing the challenges yourself.

11 December, 2025
Quarter the Cross: Colouring
Quarter the Cross is one of my favourite activities of all time, whether in maths or just life. I learned about it way back in 2015 and have been mildly or very obsessed with it ever since. This blog post is about one particular version of the Quarter the Cross problem you might like: the colouring version!

You can read the rest of this blog post, and four other related posts, in PDF form here
The titles of the five posts in the series are:
- Quarter the Cross (2016)
- A Day of Maths: Quarter the Cross (2016)
- David Butler and the Prisoner of Alhazen (2016)
- Quarter the Cross: Colouring (2020)
- Quarter the Cross: Connect the Dots (2020)
Some resources linked from this post:
25 August, 2020
Book Reading: Which One Doesn’t Belong – Teacher Guide

This is another post about a teaching book I’ve read recently. This one is about the Which One Doesn’t Belong Teacher Guide by Christopher Danielson.

(You can read this blog post and all other Book Reading posts in PDF form here. )

It goes with a beautiful little picture book called “Which One Doesn’t Belong?”, which is a shapes book different from any you’ve ever seen before. In this book, each page has four pictures, and asks the readers to say which one doesn’t belong, and why. The fabulous thing about the book is that there is at least one reason why each of the four pictures doesn’t belong, and talking through these with children (or indeed anyone) is a rich conversation about the properties that shapes have and don’t have.

The Teacher Guide is all about these rich conversations: why it’s important to have them, what you and your students/children can learn through them, and how to facilitate them. Chris has a friendly and welcoming style which draws you easily into a new appreciation of the sophisticated thoughts of children as they make sense of geometry and the world.

There are a few key things Chris talks about that really impacted my thoughts about teaching and learning maths. I’ve organised them by quotes from the book:

Commonly in maths class, student responses are compared to a standard answer key – the measure of what’s right is what’s in the back of the book, or what the teacher has in mind. In a conversation about a well-designed Which one doesn’t belong? task, the measure of what’s right is what’s true. – page 3

I read this quote first when someone else tweeted it out of the book and it struck me as awesome then. In my job at the Maths Learning Centre, students are always asking me if things are right, as if the measure of rightness is if I say it is. But in most places in maths, correctness is measured by truth. Your vectors will either be an orthonormal basis for the subspace or not. A number is either prime or it’s not. You can tell if you’re right by thinking about whether it’s true. I very much want to see opportunities to talk about the truth of things with students, to put the measure of rightness outside an authority figure.

The van Heiles haven’t argued that it is difficult to go from level 1 thinking directly to secondary school geometry; they have argued that it is impossible. If students don’t have experience and instruction building informal geometry arguments, they will not learn to write proofs. – page 8.

Chris is referring to the van Hiele model of “how childrens’ geometric thinking develops over time”. In this model, there is a build-up from noticing that shapes look like things they’ve seen (level 0), to noticing properties that shapes have and don’t have (level 1), to relating properties between properties of shapes (level 2), to logically supporting claims about these relationships (level 3).

The thing in the above quote that really struck me is the idea that it’s impossible to learn to write proofs without experiencing informal arguments first. I see so many students at university every day who struggle with proofs, and it makes me wonder that they maybe need more experience with informal arguments. Indeed, it makes me wonder if they need more experience simply noticing properties, since that’s an even earlier level. This is essentially applying the van Hiele models to other types of maths, but certain aspects of the progression still feel right to me, especially for things vaguely geometrical like vectors or matrices or graphs of functions.

I wonder if a student struggling with proofs might benefit from talking through a progression like this, and then helping them have experiences at the earlier levels before helping them with proofs.

Of course being able to state new facts is an aspect of learning, but much more important to me is being able to ask new questions. – page 21

I had never thought of this idea explicitly before, but immediately I saw that new questions were important to me as well. I was reminded of the time someone asked me if my students were understanding my statstics lectures. I said that I wasn’t completely sure, but certainly the students were asking very deep and complex questions. Instinctively I knew that a new type of question indicated learning.

Also, in the Drop-In Centre, there’s a certain joy when a student asks new questions you’ve never thought of before. They are wondering about the connections between things, which means they are learning, because learning is all about connections.

I am excited to listen out for new questions as a sign of learning, and to tell the students that it’s a sign of learning to have new questons!

… I hope you will begin to see geometry through children’s eyes as well as through the eyes of a mathematician. Mostly, I hope you will come to understand that these two views of geometry are not nearly so distant as the school curriculum might lead us to believe. – page 37

Now, I already believe that children’s investigations and ideas are actually very close to the way mathematicians work. You can’t be married to a very excellent early childhood educator without coming to some appreciation of this! It’s so nice to have someone publish a book telling teachers and parents the same.

Even more, this whole section is all about noticing and naming things and their properties. It’s about whether properties need names at all, or whether the objects that share those properties need names. It’s about what properties are important to make a thing a special thing and what aren’t, and in what context. It’s about the relationships between things. All of these are the work of professional mathematicians both pure and applied. And they are the work of children sorting out how the world works.

The geometry of children and the geometry of mathematicians are definitely not so far removed.

I have come to understand that talking about this difference is more important than defining it away. – page 54

Along with the rest of this chapter, this quote got me thinking about a whole new way to approach definitions in mathematics. As a pure mathematician, definitions are very important to me, and I always used to start with the definition. But I know those very definitions took years and even centuries to come to their current forms, and I also know that humans don’t learn through definition but through comparison of things that do and do not fit an idea. I think this is precisely what Chris is getting at here.

By skipping straight to the definition, we’re robbing people of a key part of mathematical thought, and we’re skipping them through the van Hiele levels before they’re ready. You don’t need a definition until you have a need to distinguish a thing from the other things around it. You don’t need a definition until you’ve noticed the properties you can use to define something.

The classic example in my own teaching is subspaces in linear algebra. The properties used to define a subspace aren’t even discussed until the definition is given. Little wonder, then, that the definition is meaningless to students!

It’s not just definitions either. I help a lot of students learn statistics, and one of the things that is never explicitly taught in your traditional statistics course is how to choose what is the most appropriate statistical procedure for the situation. I have been teaching this by focussing on some specific aspects of these procedures that statisticians use to distinguish things. Reading this chapter and this quote in particular helped me realise what I was doing was exactly “talking about this difference”. To distinguish between things you need to notice the properties that make them different, and to notice them, you need to compare things. I now have a much clearer idea of what I’m doing when teaching in the way I do.

I want to spend more time putting students in situations where they notice the differences between things and have to talk about them, so that they can distinguish between things they need to, and so that the properties I use to define things make more sense.

Thanks Chris for a most thought-provoking book.

19 January, 2017
Quadrilateral family tree

I have always loved the naming of quadrilaterals, right from when I first heard about it in high school. I’m not entirely sure why, but some of it has to do with the nested nature of the definitions – I like that a square is a kind of rectangle and a rectangle is a kind of parallelogram.

Most of the time you’ll see this classification organised in a sort of “family tree”. Like this:

(Warning: if you search for quadrilateral family tree in Google Images, you’ll get all sorts of things, many of which use American terminology, which is different to that here in Australia, and many others are just plain wrong.)

Of course, almost all people I meet are not aware of these nested definitions. Most students I work with are bamboozled when the answer to the problem of the largest area rectangular garden of fixed perimeter gives them a square. To them, a square is not a rectangle. Then again, there are others who aren’t aware that orientation is not one of the factors mathematicians use to name shapes. Just the other day someone asked me why a shape was called a diamond if it’s this way and a parallelogram if it’s this way. (That was a most interesting conversation about language and maths and maths language.)

Partly I think this is because the only examples we give of rectangles are non-squares — we never include squares in a collection of rectangles. Also, almost all examples we give of these things have their edges parallel to the top and bottom of the page. The casual viewer comes to believe that the orientation is part of the defining properties. A serious case of Beware of the Toast.

For a long time, I’ve wanted to make some posters showing collections of quadrilaterals including ones not parallel to the page edges and including the more special ones further down the tree. And yesterday I just decided that I would finally do it.

Here’s a picture of the majority of them. (You can find them in pdf form here .)

And finally, an upgraded family tree, showing heaps of examples of each type of quadrilateral. (You can find this in pdf form here .)

PS: If you want the original svg files in case you want to edit them (for example, to make them into local terminology), just ask.

This comment was left on the original blog post.

Anne Kelly 30 August 2016:

Hi David
Many thanks for putting together and sharing the Quadrilateral Family Tree. I plan to share it with the classes I teach.
Anne Kelly
Geographe Primary
WA

30 August, 2016
David Butler and the Prisoner of Alhazen

Once upon a time, I did a PhD in projective geometry. It was all about objects called quadrals (a word I made up) – ovals, ovoids, conics, quadrics and their cones – and the lines associated with them – tangents, secants, external lines, generator lines. During the first two years, I did talks about my PhD research, which I could not resist calling “David Butler and the Philosopher’s Cone” and “David Butler and the Chamber of Secants”.

At that time, my use of JK Rowling’s titles had to stop because there was no suitable mathematical thing to insert into the third title. It’s been ten long years since “David Butler and the Chamber of Secants”, and finally I have found something to use. Hence, welcome to…

You can read the rest of this blog post in PDF form here.

19 July, 2016

The line at infinity

I foolishly said this on Twitter about a month ago:

David Butler:	My PhD supervisor used to say 2 conics share 4 points – maybe coincident, maybe complex, maybe on the line at infinity.
Tina Cardone:	on the line at infinity? Tell me about those!
David Butler:	Can of worms! Will need to write something and post later.

At the time I declared this was a bit of a can of worms and I promised to write something and post it later. Well, here it is. But it might take a few blog posts to untangle it.

You can read the three blog posts in this series in PDF form here.

11 July, 2016

The crossed trapezium

Recently I started thinking about the properties of the following shape, which I like to call the “Crossed Trapezium”. It has two parallel edges, which are joined by two crossing lines.

You can read the rest of this blog post in PDF form here.

These comments were left on the original blog post:

Five Triangles 6 April 2016:

Related factoid: in the figure you dub a “crossed trapezium”, the two triangles that were removed from the original trapezium have equal area.

Also, this useful figure pops up in a lot of ratio problems we tweet about, such as “method 4” solution in this octagon problem:
https://casmusings.wordpress.com/2014/11/13/squares-and-octagons-a-compilation/

David Butler 6 April 2016:

Interestingly, not only are they equal area, but their areas together are hab/(a+b). Most interesting.

Tom A 13 September 2017:

The relationship between the formulas for the ordinary and crossed trapezia is rather elegant. Unfortunately, the formula for the ordinary trapezium isn’t numerically well-behaved: if the two bases are the same length, then the a-b term is zero, and the result is undefined; if the two bases are nearly the same length, a-b is very small, and calculations on a computer may be inaccurate.

An alternative formula without that problem is:

a = 1/2 * ((b^2 + a^2 + 2ab) / (b + a)) * h

Which is still similar to the formula for the crossed trapezium, but doesn’t have an obvious (to me) geometric interpretation! Something about the triangles cut out of the sides?

Peter 19 September 2017:

Thanks for sharing this interesting work. I first started thinking of areas for crossed trapezia whilst preparing lessons on motion in a straight line for my students. I have been looking at the vel-time graph for a period of constant acceleration where the sign of the velocity changes. The area trapped between the graph and the t-axis is a crossed trapezium. Interestingly, if you consider the area in the first triangle to be postive and the second to be negative (consistent with calculating displacements), the formula for the area of the trapezium is unchanged from the canonical one.

6 April, 2016
The frustrated cone

If someone asked you what your favourite 3D shape was, what would you say? A cube? A sphere? A dodecahedron?

Well, my favourite 3D shape is the frustrated cone. A frustrated cone is a cone with the pointy bit cut off (some people call it a “conical frustum” or a “truncated cone”, but “frustrated cone” sounds so much cuter).

And why is it my favourite?

Well firstly it’s the name. It makes it sound like it has a life of its own – like it wants to be a cone, but has been prevented from making it all the way to the vertex. This is one more example of my general personality-based approach to understanding all sorts of things. Moreover, most people I talk to have never heard the name, and so if I bring it up then I have to explain it, thus extending my opportunity to talk maths.

Secondly, the formulas associated with it are pleasantly simple: the frustrated cone in the picture has a curved surface area of π (r+R)s, and a volume of ⅓ π h (R² + Rr + r²). Aren’t they nice? Moreover, the formulas connect neatly to the formulas for cylinders and full cones: If r=0, then you have a cone and the formulas both give the right answer, and if r=R you have a cylinder and the formulas still give the right answer. That appeals to my mathematical sense of connectedness.

Thirdly, the method for coming up with these formulas is an excellent work of problem-solving. What you do is imagine that your frustrated cone is a bigger cone that has been cut off, and you give a letter to represent the unknown height of the vertex above the top face. Using the formulas associated with a full cone and the properties of similar triangles, you can get the formulas I had before. (I’m not putting the working here because that would rob you of the fun of figuring it out yourself!)

And finally, it just looks cute, sitting there all squat and round like a creme caramel!

18 August, 2013
Four triangles and three squares

The picture here holds something really cool:

To make a picture like this, you begin with a triangle, then you construct a square on each side (like with Pythagoras’ Theorem). After that, you join the corners of the squares together to produce three more triangles. So in the end you have four triangles – which I’ve coloured in purple in the picture – and three squares – which I’ve coloured red.

When you construct the picture in this way then the following is true: the four triangles each have the same area!

Isn’t that cool?!

The thing that makes it even cooler is why the four triangles all have the same area. If you rotate one of the side triangles until its edge meets the middle triangle, you’ll make one bigger triangle. The two triangles have the same base length and their top points are in the same place, so they also have the same height. So they must have the same area, since the area of a triangle is determined by the length of its base and the height of the point above the base.

Again I say: isn’t that cool?

8 February, 2011