ProfSmudge

And when we represent numbers on a number line, are they represented by points or by segments?

When we introduce fractions as parts of a whole, are we conveying the idea of a number that expresses a ratio, or are we describing a quantity of, say, pizza?

Yes, I've used a diagram like the first with pgce students and about half say the rectangles are similar.

Wow! Actually, the third diagram provides a nice way of analysing the first diagram. If you consider the diagonal of each rectangle (top left to bottom right, say), their slope changes (they don't remain parallel).

Wonderful photo!

I've recently come across this nice task - nice because I imagine many pupils' initial ideas won't work, but they might collectively get to a solution through (guided) discussion.

Like the Dienes blocks. As good as any, I suppose!

How are the total lengths of these yellow+black lines determined? Weird.

Another nice thing about 4 is that it is small compared to 10, so one can get a more coherent sense of the exponential nature of the column headings (shown on their side, here) rather than having to resort to Dienes-type longs and flats and cubes

Nice, but where would you go next? A column of 4 of the big squares, or a row, or a cube...?
A nice thing about 4 is that one could build larger and larger squares in a kind of spiral - but it doesn't really generalise!

'...that's all I'm saying'
Hm, I think that's a bit of a sleight of hand. I feel the table is not as 'dense' as it doesn't show the compound measure and so is (potentially) easier to access; and the within and between relations are simpler to identify and understand.

They need to work through this first....

Might be quite nice to walk it!
Each step back I go down 4, so to go down 28 I take 7 steps, so I land on (18–7=11,-6).

Yes, but I think it is important to be able to go back to a model at any time, as a kind of check or reassurance, rather than forget it completely which seems to happen with some children

Love those red marks!
[And here's the diagram drawn more or less to scale.]

Or, using symbols:

Two subtly different methods for solving an equation with the unknown on both sides

Lawrence Atkinson, a 'minor' British artist (!873 - 1931).
Fab

I've got a rather strange result of just over one half....

Seems like a rationalisation for top-down teaching

Another fun fact:
28 ÷ 7 is equal to exactly 4
[sorry!]

A couple of years ago I wrote an article about the chapter 'Simplifying fractions using common factors' from Book 6A [Mathematics Teaching 290, February 2024, pp 15-17]. The focus seemed to be far more on Instrumental than on Relational understanding.

It does involve working systematically, which certainly helps with combination problems, but I think the key is it is looking at the structure of a single exemplar, not inducing a rule from a set of empirical results, as might happen with a set of different size 'chessboards'.

Yes, complementary addition,
or the shopkeeper's method (which has died out now!)

Or consider the 8x8 chessboard. Where can, say, the top-left tile be placed for different squares?
In just one place for an 8x8 square;
in 2x2 places for a 7x7 square;
in 3x3 places for a 6x6 square, etc.
---
in 8x8 places for a 1x1 square.

Or take 'Handshakes'. If one takes the cases of 1 person, 2 people, 3 people, etc, one might get a rule like 'n people will make 0 + 1 + 2 + ... + (n–1) handshakes.
If one looks just at 10 people say, where each person shakes hands with 9 other people, one can get 10x(10–1)/2, leading to n(n–1)/2

Yes, it's based on my work on the Proof Materials Project (2004 - 2005). Here's an extract from the report, arguing for taking a generic approach to the 'matchstick squares' task.

I'm not sure I can answer your question, but Figure 13, below, is from the LUMEN paper. In the blue region they've switched to that interesting informal method.
I have re-written this at the bottom-right, using their red arrows, with the standard method shown bottom-left.

In #MathsToday we worked on 16÷6.
A good follow up might be to explore '4 chocolate bars shared by 6 people'.

In the January 2026 edition of Maths in School, the LUMENaries Shore, Foster and Francome present an intriguing method for solving simultaneous equations. Here it is applied to a simple linear equation in one unknown: