In Part 1 of this article I described my thoughts and actions before and during a recent #beingmathematical session provided by @ATMMathematics on Twitter. In this, Part 2, I shall share some thoughts I had afterwards in response to contributions from other participants in the event. I continue to explore the links between how children and more experienced learners might approach mathematical tasks and thinking. In particular, I will discuss conjecture and proof using a range of concrete manipulatives and diagrams, alongside linguistic and symbolic representations.
Here’s a reminder of the task that prompted the mathematical thinking and discussions. It involves isosceles right-angled triangles (or “IRATs”)
Here is one of two threads that raised two related questions I would like to consider here, and in Part 3 of this series. For those who haven’t experienced a #beingmathematical event, it gives a nice flavour of the kind of interchanges that arise.
Elli’s conjecture that “You can make a triangle with a square number of triangles” is illustrated by the five triangles in the bottom right-hand corner of Mary’s lovely image below.
Here is my initial response to the conjecture and to @ATMMathematics’ Socratic prompt about becoming convinced. (I did this a few days after the Twitter event and have not shared it until now.)
This is the kind of approach I would consider using with some of my secondary-school learners from around S3 onwards. As you can see, it begins by systematically tabulating observed results. Below is some work by two primary 3 & 4 children my wife, Fiona, teaches; the pupils listed their mathematical observations in a similar manner.
Daniel’s & Ewan’s record of their observations on a task from ATM’s “Learning and Teaching Mathematics Without a Textbook” by Mike Ollerton, which they enacted using Lego characters.
However, while I have tried to keep my proof simple and informal, the introduction of n to represent a generalised side-length, thus allowing an algebraic treatment, results in a level of abstraction I have found even much older learners can struggle with, at least when they first encounter it. It also assumes familiarity with the formula for the area of a triangle. In fact, by invoking this formula to explain the observations, I have changed this from a counting problem to a problem involving areas. While this was unproblematic for me (in fact, I only became aware of the change when I began to compose this reflection, some three weeks after I first wrote down the proof), I now wonder if less experienced learners would have been able to make this connection. As a final concern, my argument relies upon reasoning involving division; I often find that many learners, across the secondary-school age range, are not secure with such reasoning (especially if, as in this case, it involves division by a fraction).
In the Twitter thread above Becky seems also to have been thinking about the number sequence 4, 9, 16, 25, 36, … in terms of area. She refers to areas in general (not just to areas of isosceles right-angled triangles) and, like me, makes use of the letter n to refer to the generalised case. This made me wonder whether I could extend my own proof to shapes other than IRATs. After jotting down some thoughts, I believe a similar treatment would allow extension to squares, rectangles, parallelograms, rhombuses and to triangles of all types: in short, to any 2-D shapes that could, intact, tile an enlarged scale-copy of themselves.
However, I shall, for now, avoid venturing further into the realms of algebraic abstraction. I want to start by exploring those approaches a younger learner might take, whether independently or with guidance from a teacher. What do I mean by younger? I suppose I’m referring to learners who haven’t been exposed to a great deal of formal mathematical content and are more likely to take an intuitive approach to mathematical tasks.
Trying to take a more childlike perspective myself, I came up with this much simpler “proof”. Does this convince you that you can make a triangle with a square number of triangles?
If you are convinced (I think I am) then we have “proved” the conjecture without even needing to know explicitly what a square number is!
Still, I wanted to explore square numbers and how, as a sequence, they grow, as suggested by Mary’s image and Becky’s link to area scale factors. I remembered this pictorial representation I have both seen and used before.
I think some younger learners (as a secondary school teacher, I’m not sure how young) could make sense of this representation, perhaps being able to extend the sequence both diagramatically and numerically. The algebraic representation is for you, the reader; I would not share this with young learners. However, at the right stage, it could form a meaningful bridge from the iconic to the symbolic. I shall return to this point later.
I wanted to try and find a triangular analogue to this sequence of squares. (At this point, I’m thinking as an adult but still using pictures, rather than abstract symbols, to make sense of what’s going on.) Here’s what I came up with.
The rationale for the orange strip of triangles along the sloping side felt rather contrived, but that’s where I got to in terms of sense-making by analogy to what happens with the squares. I tried it again with equilateral triangles.
This approach, then, has resulted in another attempt to convince myself and my readers that we can build triangles using a square number of unit triangles, be they IRATs or equilateral.
In the equilateral version there are two “sloping” sides, not one (assuming, for both types of triangle, an orientation involving a horizontal base). The fact that the orange strip involves two interlocking sets (one for each dimension of the plane?) was dominating my awareness.
The idea of interlocking sets of triangles (also evident in Mary’s image) caused me to think of the “upright” and “inverted” triangles (Mary’s reds and blues) separately. I started thinking about triangular and square numbers, compiling a quick table to see if anything new emerged. (I include two versions of the table later.)
Then I noticed this!
At first I thought I’d discovered something new (to me, at least). Then I recognised it as:
every square number is the sum of two consecutive triangular numbers.
This well-known mathematical truth is usually represented pictorially by joining two different-sized “triangles” (staircases?) to make a square.
Here is my alternative representation of this square version using Cuisenaire rods.
Note that the “triangle” on the lower left contains one more rod (an orange one) than the one in the upper right. I rearranged the right-hand triangle to create the version below, which I think better captures a progression of growing squares as the component pairs of triangular numbers get bigger.
The important feature to note in each of these representations is that two “consecutive” triangular arrays combine to make a square whose side-length corresponds to the larger triangle.
My initial objective for this article was to prove the conjecture that “You can make a triangle with a square number of triangles“. Before the #beingmathematical event, I was unaware of this fact. In trying to prove it, I have discovered an alternative representation of a fact I did know, namely that two consecutive triangular numbers sum to a square number. Since all mathematical proofs use established facts (or, alternatively, axiomatic truths) to demonstrate a “new” fact, it seems to me that this discovery constitutes a further proof of Elli’s conjecture:
if you can build a triangle by combining two smaller triangles, each representing consecutive triangular numbers, then the composite triangle must represent a square number.
The fact that this connection was unexpected reminds me that an important part of the appeal of mathematics is it’s potential to surprise. That I discovered this surprise using non-algebraic approaches to address the original conjecture is evidence that people across a wide range of age and/or experience can access mathematical ideas and investigations in rewarding and enjoyable ways that are not limited by a lack of formal training.
Of course, as a maths educator, I want learners ultimately to be able to develop the skills that allow manipulation of abstract symbols as well as objects and pictures. For my part, I can deepen my mathematical understanding by developing my ability to express relationships not only symbolically, but using concrete manipulatives and pictorial representations too. To complete Part 2, I shall provide a proof, in both pictorial and symbolic forms, of the fact that two consecutive triangular numbers sum to a square whose side-length corresponds to the larger triangular number. This will lead to a derivation of the formula for calculating the nth triangular number (i.e. the sum of the first n natural numbers) and, finally, to a full algebraic treatment.
Let’s start with some definitions, notation and exemplars.
Now consider the first ten triangular and square numbers. Can you spot any patterns?
Here are some triangles to help.
Will this pattern continue? Let’s investigate using Cuisenaire rods. Look what happens when we combine two copies of the sixth triangular number.
We get a rectangle with one side that is longer than the other by one unit.
Here are two rectangles, one made from two copies of a the fifth triangular number and the other from two copies of the sixth.
Let’s combine them to make a bigger rectangle.
We can now split the rectangle to make two equal squares.
Here’s a logical conclusion for our observation.
What if we added a different consecutive pair – would we still get a square? If so, can we tell from the triangular numbers which square number we would obtain? Remember, we can make a rectangle from two copies of any triangular number. If we make another using two copies of the next triangular number and then combine both rectangles, we can separate the resulting rectangle into two equal squares.
Are you convinced that the pattern we spotted in the table of the first ten triangular and square numbers will hold for any pair of consecutive triangular numbers?
While it’s straightforward to calculate the nth square number, it’s more difficult to obtain the triangular numbers using the definition given above, especially when n becomes large. For example, to find the 100th square number just multiply 100 x 100; for the 100th triangular number you would have to add together all the natural numbers between 1 and 100. We can use the idea of combining two triangles, like we did above, to find a simpler way of calculating triangular numbers.
The following three pictures illustrate how.
So the 100th triangular number is 1 + 2 + 3 + … + 99 + 100 = 1/2 of 100 x 101 = 5050.
We can use the formula for evaluating a triangular number to give an algebraic proof of the conjecture that “You can make a triangle from a square number of triangles” which, as we have seen, is equivalent to the observation that if you add two consecutive triangular numbers you obtain the square number corresponding to the larger triangle.
This is consistent with all our other approaches.
The late Nobel-prize-winning physicist Richard Feynman (1918 – 1988) is reputed to have said that, “If you cannot explain something in simple terms you don’t understand it.” I can’t speak for anyone else, but making the effort to explain my thinking in simpler ways than I normally would has definitely deepened my understanding of the mathematical ideas discussed here. Once again, taking on the perspective of a younger or less experienced enquirer has helped me move forward in relation to learning, teaching and doing maths.
Part 2 has proved significantly longer than I had first anticipated. Thank you for staying with me. I hope you’ve found it worthwhile and that you can join me for Part 3.
