*This is the third 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 3: Squares**

Squares as geometric objects have the properties of equal
sides, right angled vertices, several kinds of symmetry and equal diagonals
that bisect each other at right angles. Any of these properties can be
expressed algebraically. Observation and measurement of components of squares can
support deductive reasoning about other properties. For example, any straight
line segment going through the point at which the diagonals intersect cuts the
square in half by area. Area is a very useful feature of squares because it is
the square of the side length (hence the similarity in the names) and that
provides a bridge into meaningful algebra. But before I go over that bridge I
will stay for a while with the equality of sides. If learners have internalised
the reasoning power that depends on these equalities – the legitimate journeys
from ‘these sides are equal’ to some less obvious statements – they are on the
road towards mathematical reasoning. I propose that reasoning with squares
based on their property of equal sides is good preparation for later geometric
reasoning and a suitable arena for engaging in mathematics that is not
primarily about calculation.

There are classic types of problem that can be solved using
that property, see https://donsteward.blogspot.com/search/label/squares%20inside%20rectangles.
The numbering of the squares indicates the length of the side.

Another version of the second diagram appears in the
‘equations’ task sequence on the same blog. This starts with the 7 and 28 being
given. One feature of these puzzles is that a choice has to be made about which
square to label as the unknown. In this diagram it is not the smallest that he
has chosen to label as d. There is also some potential confusion about whether
a length or a square has been labelled so you have to work through it to see
what he was thinking. I have always thought it is important for learners to see
someone else’s ‘work in progress’ so they understand that – yes - maths can be
done messily and then it has to be tidied up to communicate it to others.

One interesting thing about this task sequence is that he called it ‘equations’ but there are no actual equations written out, only expressions. Several ‘opposite side’ equations have been in his head in order to generate the expressions, and then the expressions can be gathered together and organised into equations from which the unknown will be found. Here, height can be expressed as 5d + (d + 7) and also as (42- d) + 28. Simpler examples are available.

The algebraic manipulations that arise in these puzzles are purposeful and meaningful. In my view these problems do not need prior experience of gathering like terms or dealing with brackets – they can themselves generate a need for these tools so could be a starting point rather than an end point of a ‘gathering terms’ teaching sequence.

Don offers a few simpler examples, and making them up takes
a great deal of patience to ensure that solutions are always integers (there is
no point in complicating matters with fractions when introducing deductive
reasoning and algebraic formulation).

Let’s suppose learners have done several of these and become
adept at ‘reasoning from the equality of sides’. Where can we go from
here? For me an obvious direction is
towards the use of area models – those two-dimensional representations of
algebraic relationships that relate the word ‘square’ as a shape to the word
‘square’ as the second power. This model depends on the internalisation of
equality of sides of a square, of the equality of opposite sides of a
rectangle, and of expressing area of rectangles as the product of side lengths,
which in squares gives x^{2}. I
am not being patronising to list these, rather I am drawing attention to the
need for learners to have internalised them not as verbally memorised facts but
as components of the network of concepts that go with squares and rectangles.
If they have to think hard to recall these they may not be in a suitable state
to use such diagrammatic representations – they are the necessary tools for
thinking.

So when and how does Don use area models? There is an example of his creative insight at https://donsteward.blogspot.com/2020/03/two-2-digit-multiplications.html which caught my eye. In this task sequence Don explores what happens when 35, 45, 55 … are squared. He asks the learner to work these out without a calculator and then hopes they will notice the appearance of 25 as the rightmost digits. The diagram he offers to explain this is:

There is a sense in which the leftmost digits of the answer are out of the way of the ‘25’ because of … well why?

He then generalises this particular family of numbers to
‘10n + 5 squared’ showing that what is being represented by the diagram can
also be represented, and calculated, by multiplying each term of the second
bracket by each term of the first bracket and hence getting 100 as the coefficient for both n and n^{2}.
Why might we be interested in these rather special cases? One possible answer
is the availability of suitable grid paper to facilitate the transformation between
head and pencil-&-paper. Another possible reason is that Don the offers a
further sequence about 31^{2} and 31 x 29 which are represented as
extensions of the use of area diagrams,
and from which can be learnt something about the formation of the middle term
of the polynomial format of a quadratic (other formats are available) and difference
between two squares, and more, under the general heading of ‘a add b squared’: https://donsteward.blogspot.com/search/label/a%20add%20b%20squared.

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