*This is the fourth article written by Anne Watson in the 'Dose of Don' series. She posted it on her blog here and I have replicated it word for word. For the background on this series, please see the post Lines and Angles on Square Grids. My thanks go to Anne for giving me permission to share her writing here.*

**Dose of Don 4: Geometrical reasoning within constraints**This is the fourth of an irregular series of writings in which I (and, I hope, others) delve deeply into the collection of tasks on Don Steward’s blog

donsteward.blogspot.com and pull out threads about key ideas in mathematics that run through several of his tasks. Where possible I give you a direct link to the tasks; where I have extracted part of a task I direct you to the ‘parent’ from which it came.

Don was very generous with his tasks and I hope that you will return this generosity in the way he requested before he died, namely to donate to

justgiving.com/fundraising/jessesteward

Nichola Clarke’s DPhil research investigated the
mathematical reasoning of students in lower attaining sets in upper secondary
school. She found some students who were perfectly capable of ‘if … then …
because…’ reasoning in geometrical situations but took ages to then calculate
angles; they appeared to be struggling with reasoning when they were actually
struggling with arithmetic. You might think that things would be different now
that arithmetical fluency appears to have a higher profile in primary
mathematics, but because the emphasis is sometimes on column methods instead of
on recognising number bonds and relationships the same obstacles are likely to
apply. Ori Golan had the insight, when first teaching geometrical proofs, that
removing arithmetic by teaching algebraic and logical representation of angle
relationships would fasttrack his students to reasoning rather than a
dependence on calculating, and it did.

I didn’t have this in mind when looking for hidden threads
in Don’s geometry resources but found it in his sequence of work on the regular
dodecagon (confusingly posted under the heading ‘area’). He had told me of the existence of
this sequence, and made it available to participants of our workshop on angles
(see pmtheta.com ‘PMTheta@home’ series) but my mind had been elsewhere and I
did not pick up its potential significance. A superficial look shows pretty
patterns and I thought of it as ‘how you* apply* angle-reasoning’ rather
than as a sustained *introduction* to reasoning with angles.

His sequence is partly inspired by David Wells’ book: ‘Curious
and Interesting Geometry’ but I cannot find that book to check what is in it (after
Covid I will return to downsizing my library to passing visitors – but Wells seems
to have already gone).

In several of his resources Don realised that working with a
limited ‘palette’ of numbers would focus learners’ minds on relationships, properties
and theorems rather than on calculation. In this case the palette involves 360,
180, 90, 45, 60, 30 and various multiples, sums and differences of these
together with the idea that one revolution is 360° and the interior angle sum of
a triangle is 180°. Even this list has some redundancies but you have to start
somewhere. I found it useful to employ the ‘angle subtended at the centre of a
circle by an arc is twice the size of the angle at the circumference subtended
by the same, or a congruent, arc’ but this fact could also be deduced within
the sequence.

Preparation for the sequence could consist of a significant time
finding out ‘what numbers can be made from adding, subtracting, halving,
doubling etc.?’ as a toolkit and sharing it as a public resource in a
classroom. As with many number facts, it is useful to be so familiar with these
that they can be recognised, e.g. ‘how might 120° be made?’ rather than seeking
probable components with a calculator or p&p calculations.

OK, so armed with some familiar angle relationships what can
be said about this?:

The more relationships you know about the more ways there
are of answering this, but both can be deduced from the angles round a point,
the angle sum for triangles, and symmetry. Other reasoning chains are possible
of course, for example a turtle argument for an exterior angle would also be a
good start. Anyone who has played successfully
with the coding for ‘Frozen’ would have ways to think their way into this.

I would follow up with this inquiry:

What are the minimum assumptions necessary to sort this
diagram out? I am imagining a ‘facts board’ collecting the facts that are
necessary and those discovered as they emerge from students’ work. I am afraid
I cannot recall the name of a teacher who had a ‘conjecture’ list and a
‘proved’ list on show in the classroom for work such as this. Is it OK to use
results that are only conjectures? Would a class-owned digital collection give
the same availability for students to browse when they become stuck?

You might think me unadventurous in my choice of a second
slide, given the goodies that Don has offered, but I am thinking that this
second slide gives an opportunity to do some preparatory work on how chains of
reasoning might be talked about and represented. These communication tools need
to be established before moving on.

Here is a flavour of more complex slides:

There are 47 slides in all for this sequence and I have
worked through them, but am stuck on a couple. The central feature of my
working was always the methods of reasoning; usually several pathways are
available. Some of the reasoning uses general facts and
geometry-theorem-development through reasoning about structure; some of the
reasoning uses the specific properties of specific angles.

I presented the benefits of using a limited collection of
numbers also in Dose of Don 1, in which I showed tasks where he had used grids
to constrain the variation of angles, often represented by ratios.

But what I set out to do for this
Dose of Don was look at area, and it was only because the above ideas had been
posted under ‘area’ that I found them again. I had intended to collect some of
Don’s ‘cutty-uppy’ tasks (otherwise known as proofs by dissection or proofs
without words) to identify some of the thinking behind them. Conservation of
area is a concept that most children arrive at when very young, and this sense
is enhanced by playing with sets of 2-D shapes. Indeed, understanding of area
as a concept is difficult since it is not such a lived experience as linear
measure is (by moving things) or volume (by pouring things or trying to hide
under chairs). In my experience, this is why many students grab onto a formula
such as *A = l x w *for area questions*, *often incorrectly,* *instead
of using conceptual understanding. Cutting paper shapes and moving them around
builds on an intuitive idea of conservation and also provides content for ‘if …
then … because …’ reasoning. I am afraid that watching shapes move on a screen
does not give a sense of personal spatial manipulation. Deciding where to cut
and move involves imagination plus reasoning; using dynamic geometry software
limits these decisions to those that are conventionally and geometrically
useful, such as using mid-points, perpendiculars, and so on; origami offers a
palette of useful outcomes that can be achieved by folding. Watching someone
else, or a prepared animation, pre-decides the moves.

I find that with some of Don’s ideas
I have to distinguish between what can be used for initial learning, what for
the development of ideas through sequences of examples that draw attention to
characteristics or complexify those ideas, and what for application and
extension to other ideas. I am offering some that can be used for initial
learning about area of certain shapes alongside using manipulation as an
exploratory tool.

Consider this question, which appears to be about the area of triangles but only needs the concept of conservation of area. Don offers eight different proofs in '

quartering a parallelogram' and I am particularly interested in those that build on a cutty-uppy approach:

Note how the final proof that I have selected labels angles
to transform a cutty-uppy approach into the use of congruence because certain
kinds of movement also conserve angles. To know that congruence is not only
about angles and sides but also area (therefore) seems to be a relevant
fact. Can you find this explicitly in a
school textbook?

Can congruence be deduced from the equality of one side, of one
corresponding angle, and of area? Ditto two pairs of equal angles plus area?
And so on.

Don provides a sequence about the area of a rhombus from an initial idea to one of the possible algebraic representations in '

grid kites and rhombuses'. The sequence uses cutty-uppy with constraints by limiting the exploration to a square-dotted grid.

I shall leave the algebraic development to readers.