Ratio

This is a contribution to the series of writings called 'Dose of Don', begun by Anne Watson, which delve into the collection of tasks on Don Steward’s blog. This piece is written by Sam Blatherwick (@blatherwick_sam). Many thanks to Sam for giving me permission to share his writing here. For the background on this series, please see the post Lines and Angles on Square Grids. 

Don was very generous with his tasks and it is hoped that you will return this generosity in the way he requested before he died, namely to donate to justgiving.com/fundraising/jessesteward.

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Don Steward’s median blog was, and remains, a great resource for tasks and there are certain sets of questions that resonate when you come to teaching a topic. You’ll have your own soft spots, but exercises that come to mind for me are the recurring decimals calculations task and the small data set averages problems. Above all though one that stands out when I come to teaching ratio is the ‘Asterix and Cleopatra’ problem sheet. When I first saw it, it felt so modern, simply because we were suddenly encountering questions like this all over the GCSE papers but it didn’t feel like there were any questions in textbooks that were replicating them.

My initial reaction to teaching this was, how? I had my own elaborate algebraic procedures, but I had been chastised in the past by the students for reaching for algebra whenever the going got tough, it wasn’t something that was coming to naturally to students. And then, when you work this out by hammering it with algebra, you were kind of disappointed – like there must be a faster way.

My initial reaction to breaking this down then was to list equivalent ratios, and this is still something I recommend to students when they don’t know what to do. If you do this the answer drops out pretty fast:

What if they had 3:5 – you’d get 1 and 7 – this doesn’t work.

What if they had 6:10 – you’d get 4 and 12 – this does work.

Don crushes the idea that this will be a cinch with the next one:

This takes a lot longer. Fine – we need another method. This is where we hit upon the “what part of the ratio stays the same” approach. The number of marbles Kim has stays the same so we can compare the ratios like this:

Initially: 5:6 = 40:48

Now: 7:8 = 42:48

So we can now see those two marbles in action – Jan had 40 marbles and Kim has 42.

Later we come to a problem again – and if we’d looked back at Anthony and Cleopatra we’d have noted that a method of things being fixed wouldn’t really have worked for us.

What’s fixed here? Not the red socks, and not the white socks. For a while here I was under the impression we were in a pickle when it came to questions like this and just had to go back to listing. Alas, if we go back to listing here, we have to go a very long way.

Until I was introduced to what was actually fixed – and in this case our fixed part was our number of socks – the total of the ratio.

So initially we have: 9:5:14

And subsequently we have 3:2:5

Before: 9:5:14 = 45:25:70

After: 3:2:5 = 42:28:70

So each part of the original ratio is 3 pairs of socks, so we have 135 white pairs of socks and 75 pairs of red socks.

And this is where I thought the story ended, with a lovely straightforward universal method for these problems – decide what you want to be fixed and fix everything in line with it. Then I was in a training session and someone stuck up these questions and asked us to have a go at them, before sharing the algebraic methods that his students did.

And I scoffed and went “oh ho ho, how enlightened I am knowing a better way..!”

There was a discussion around the Sine Rule on Twitter recently where the opinion was aired that as maths teachers we have a tendency to “algebratise” everything too quickly. I’m not sure I saw it at the start of my career, but I think I tend to agree now. Isn’t there something to be said that maths teachers do “algebratise” though? And how do we develop that as a skill?

What about that ‘better way’ I was so smug about? It leads us to the answer, but is it just an endpoint? Where does this lead to? In fact I considered, what if in fact these questions are a lovely vehicle for also introducing some algebraic work? Even after the ‘universal’ method has been introduced there’s running to be had on playing around with algebra in these questions and playing around with how we introduce unknowns.

So going back to the original question:

There are so many ways we can introduce an unknown here. What if we said the number of shares that Asterix has is x?

Originally:

Afterwards:

So: 

What about if Cleopatra has x?

What about if the total was x?

A potential stepping stone from ratio to algebra depends on the models you use for ratio. I tend to build from a “boxes” method of ratio to demonstrate 3:5 as

Where the open box is a vessel that has to contain the same as all the other boxes.

If we use that open box as x then we are solving a question where x is one part of the original ratio, and we can apply that to the Asterix and Cleopatra problem.

Originally: 

Then: 

So:

It’s interesting to note that Don himself notes in his introduction to the tasks that the algebraic way is “relatively unthinking”!

In conclusion, not everything needs to be ‘algebratised’ and often other numerical ways of solving problems can give you solutions to problems in elegant ways. However, the answer need not be the end and exploring different algebraic ways of exploring these problems can help your students fluency and competency with algebra.

Symmetry

This is the seventh article written in the 'Dose of Don' series - this one is written by Sudeep Gokarakonda (@boss_maths). Anne Watson 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 Sudeep for giving me permission to share his writing here.

Dose of Don 7: Symmetry

This is a contribution to series of writings, begun by Anne Watson, which delve into the collection of tasks on Don Steward’s blog and pull out threads about key ideas in mathematics that run through several of his tasks. Direct links to all tasks mentioned are included below.

Don was very generous with his tasks and it is hoped that you will return this generosity in the way he requested before he died, namely to donate to justgiving.com/fundraising/jessesteward.

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This post is about symmetry. Mention symmetry, and many people’s first thoughts might involve symmetry in a firmly geometric context. At school, almost all students explicitly encounter reflection and rotational symmetry. For many, their appreciation of symmetry might not extend beyond these ideas.

Symmetry is something that permeates mathematics, and it is something mathematicians can recognise in situations that aren’t presented in a typical geometric context. Here, for example, is such a situation:

You have the following coins totalling 70 pence. In these questions, “amount” refers to a whole number of pence.

a) Using only these coins, what is the smallest amount you cannot make?

b) Using only these coins, what is the largest amount under 70 pence that you cannot make?

c) What do you notice about your two answers?

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I remember a lesson involving Pascal’s triangle during which I causally mentioned its symmetry. One student appeared convinced that I was wrong to suggest any symmetry here. On exploring, I realised they were considering the numerals rather than the numbers, so for them:

This slightly constrained conception of symmetry was not totally surprising, knowing that the student’s exposure to the idea had been limited to the geometric contexts in the GCSE specification.

Here are some slides from Don’s tasks. Not one of the chosen tasks is primarily about symmetry. Nevertheless, each gives us opportunities to notice and explore symmetry.

Mean of a frequency distribution

These are the first two slides from the task. Let’s consider question 16. The frequencies y, w, y, w, y read the same forwards and backwards. It is possible to prove that the mean is always 6 without spotting this, by setting up and simplifying the following:

But spotting the symmetry leads to a chance to generalise. In question 16, if I replaced y, w, y, w, y with any symmetric sequence (e.g.) a, b, c, b, a, would the mean still be 6? If the frequencies were instead a, b, c, d, a but the mean was still 6, what could we conclude?

I also like tying question 16 back to questions 1 and 2 using an alternative approach - appreciating how scores below the mean must be perfectly “balanced out” by scores above the mean.

For example, in question 1, given that the mean is 5,

• the single score of 3 gives a “deficit” of 2;
• the scores of 5 have no impact on the mean;
• the two scores of 6 give a “surplus” of 2; and
• the four scores of 7 give a “surplus” of 8.

Overall, I therefore still need a further “deficit” of 8. Since a single score of 4 results in a deficit of 1, I need eight such scores, so a = 8.

This approach is, to me, is mentally less taxing than setting up and solving the following in my head:

The “balancing out” approach of course works in all questions, but it’s perhaps best illustrated using those questions where the mean is an integer (i.e. questions 1, 2 and 16), because the arithmetic is kept relatively simple. I love how this section of the task can be bookended in this way.

Geometry of the reciprocal function

This task includes five slides. Shown are the first four, which all feature the curve y = 12/x.

The line y = x is a line of symmetry on all four slides - even where we only have the first quadrant. This line of symmetry may not initially be obvious to all students. On the very first slide, however, is the opportunity to spot that e.g. (1, 12) and (12, 1); (2, 6) and (6, 2) etc. are all points on the hyperbola. Is it necessarily the case that if (a, b) is on the curve, then so is (b, a)? What about (–a, –b)?

These questions involve symmetry in a geometric context, but an opportunity to consider symmetry in a more subtle context pops up on slide 4. I see that 3y + x = 12 is just y + 3x = 12 with the x and y swapped around. Without even seeing the graphs of these, I sense symmetry in these coefficients. Here is an opportunity to ask what happens, in general, to a graph if I swap x and y around in its equation. Students could make predictions, and then check by trying several functions using graphing software. Can they come up with an intuitive explanation for what they observe?

This moment may be a natural one to take a detour to visit (or revisit) self-inverse functions—or even sample another of Don’s tasks, such as self-inverse and periodic functions.

Visualising
The idea of symmetry crops up when considering arrangements, probabilities, and many topics in statistics. Don’s tasks often include beautiful yet simple visualisations illustrating symmetry:

From Galton's quincunx

From Probability with coins

From Probability arrangements

What I like in this final diagram is that, as drawn, it does not have rotational or reflective symmetry. This opens the possibility to discuss what mathematicians might mean when talking about symmetry here. Such discussions should hopefully help broaden the kind of conception of symmetry exhibited by my aforementioned student of Pascal’s triangle.

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As stated earlier, none of the above tasks are primarily about symmetry. Don created other tasks that I’ve not detailed here, where it is quite possible to work through them - and gain richly - without spotting the symmetries in them. But spotting them gives classes the opportunity to perhaps take a “scenic route” through the tasks - one that helps build up students’ sense of symmetry in mathematics.

Angle bisection, incircles and reasoning with ratios

This is the sixth 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 6: Angle bisection, incircles and reasoning with ratios

This is the sixth of a very irregular series of writings in which I (and, I hope, others) delve deeply into the collection of tasks on Don Steward’s blog 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.

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This ‘Dose of Don’ is different in flavour to my previous posts. Instead of following what has been stimulated for me in Don's work I am following a thread of his own inquiry.  For a workshop on angles he presented some tasks that depended on defining an angle by its tangent ratio. I talked about this in my first ‘Dose of Don’ blog as he had a theory that if you approached angle by limiting it to those that could be expressed on a square grid, then many angle and trigonometric facts, and other geometrical insights, could be deduced in special cases and that possible generalisation to all angles could be explored.  The angles that can be expressed on a square grid and those whose trig ratios can be constructed on the lattice points, e.g. in triangles for which some of the side lengths can be expressed as rational multiples of each other. To simplify this he limited exploration at first to those angles whose tangent ratio was ‘on the grid’, so angles are an inverse of ratios, not yet expressed in degrees or radians.

OK so far?

Then he began to explore angle bisectors.  I had forgotten this direction but found it again while tidying my desk and finding my scribblings. Most of what I found is at : https://donsteward.blogspot.com/search/label/angle%20bisector

He starts by suggesting you use a compass/straight edge approach to bisecting an angle and observe whether and where your bisector passes through lattice points.  There is then the following summary slide about what you might have found (a typo for tan B/tan 2B is easy to spot).

Now the reason I had put this to one side for over a year is because my knowledge of double angle formulae is robust even 60 years after I first met them and this seemed to be getting in the way of imagining how learners might answer the question about a relationship between K and 2K. Could I honestly reconstruct a relationship I knew without using it as a ‘goal’? In other words, could I treat the K question as a goal-free problem? I could pick out features of the diagram and confirm them by using existing knowledge. I don’t recall, however, ever using the tangent double angle formula to find the tangent of a half-angle – the information I need to bisect angles on the grid. After some manipulations I found out how to do this and realised that this is how Don must have developed the particular examples he offered and why (you may have noticed this) they seemed to need Pythagorean triples to ‘work’. But I have not answered for myself the question of how anyone who was not familiar with double-angle formulae might approach the bisection question.

Another way to approach the bisection question is to use knowledge of incircles to do some reverse reasoning: if angle bisection gives me the incentre, then the incentre will give me clues about angle bisectors. Triangles on grids, particularly right-angled triangles with their legs on the grid, offer several reasoning routes and – hey presto! – the tangent ratio for the half-angle plops out before your very eyes!

This line of reasoning depends on some lines of thought that might be more familiar than inverse tan and Don’s slide number 30 can take you there.  It is a free-standing exploration that depends on knowing about areas of triangles and Pythagoras.

A diagrammatic approach I particularly like follows:

I found it is possible to reason the relationship between the half angle and the full angle without thinking about incentres but instead by reasoning with ratios. To show that the two half angles are equal (and therefore must be halves of the full angle) I only have to show that their tangents are equal, i.e. that their defining ratios are equal. The unit lengths in which the ratios are constructed don’t matter. I love this. For me, it is about seeking similar triangles using lattice points.

During our time with Don, John Mason and I raised an issue that has been hanging around for years: reasoning with ratios can be very powerful but we rarely see diagrams from which ratio ‘jumps out’ obviously as an important underlying relationship. We cannot ‘see’ ratio; we have to reason it out. I suppose the same can be said for multiplicative relations more generally; we cannot ‘see’ them in the same way as we can ‘see’ addition or difference. We think Don worked on this in his grid tasks but never had a conversation with him about it. However, I have found four diagrams that suggest he had found something. I cannot find them on his website but maybe have not been looking in the right place. I have reconstructed them here, because my copy is covered with my scribbled workings-out which could be a distraction (e.g. ‘can’t see ratios’, ‘try 3:2 along hyp’ etc.)

Finally, if you have got this far, a pedagogic question: how do the variation and invariance in these four diagrams help or hinder understanding and generalising the underlying relationships?

Substitution

This is the fifth 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 5: Substitution

This is the fifth of a very irregular series of writings in which I (and, I hope, others) delve deeply into the collection of tasks on Don Steward’s blog 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.

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John and I hosted a day’s online workshop on ‘Substitution’ recently. Don attended our day workshops regularly and always had something extra, interesting and challenging to offer so we co-opted him onto our team. Hence the workshops we have hosted since May 2020 have always reminded us of him.  So I was wondering how Don used substitution in his tasks – whether he made a Big Deal of it or not.  We posed the question at our workshop: ‘Is substitution a Thing?’  Our answer in the day was ‘Yes’ as substitution crops up all over mathematics; the purposeful substitution of one expression for another to simplify, gain insight, clarify, test, modify, make new things possible, etc. etc.

Shortly after this Richard Perring posted a tweet asking the same question but from a different perspective. His question was about exercises in subbing numbers into algebraic expressions in early algebra.  You know the kind of thing, the internet is full of them, e.g. ‘If p = 2; q = -6; r = 10, calculate -pq²r^3’. The exercise is not about algebra, it is about calculating with negative numbers once you have understood the syntax of the symbol system. It changes algebra into numerical answers. By contrast this task: ‘If p = 2; q = -6; r = 10, find at least five different algebraic expressions whose value is 4 using as many of the letters and mathematical signs as you need’ focuses on turning arithmetical understandings into algebraic expressions and launches ‘what if …?’ questions.  

So I began to search Don’s collection of tasks to find places where substitution gives useful mathematical perspectives and handles.

I didn’t have to look very far: https://donsteward.blogspot.com/2020/04/two-types-of-sum.html

Think of substituting expressions for consecutive numbers into a,b, and c (which are generalisations but not very helpful ones) and you have the beginnings of a ‘proof’ for conjectures that arise from doing the calculations. A dance begins between generalisations, structure, examples and relationships that is typical of mathematical exploration and the associated questioning: ‘What can I write instead of ….?’  ‘Can I test that with an example?’ is often about substitutions that are helpful in revealing or expressing structure.

And a later slide of Don’s gives:

This gives a reason for becoming more fluent with such manipulations - ‘doing’ algebra with a purpose. There’s more but you’ll have to go to his website for that while I indicate some other things I found once I had ‘substitution’ in my sights.

I have to force myself to open files with titles like ‘decimal subtraction’ but here goes: https://donsteward.blogspot.com/search/label/decimal%20subtraction.

This does not disappoint; I have had some great fun showing only the first two lines to people and seeing what happens. I got myself hooked on wondering about the cyclic nature of what appears and rewriting numbers as sums of powers of ten, i.e substituting the separate place values into the ‘answers’.  This is  the ‘undoing’ of what is done in primary school to build up multi-digit numbers from the place value components and is the basis for many mental methods in Vedic mathematics and Trachtenberg methods (see Google for these) and an old method that was taught in schools in the 18^th and 19^th century called ‘casting out nines’. Some people immediately substituted ‘nine tenths of…’ for the left hand sides, which explains something about the answers you get but not (for me) the cycles. The immediacy of this response was impressive but they were all people ‘of a certain age’ for whom expressing rational numbers as fractions retains some of the manipulability that can get lost in decimal notation. So there are two kinds of substitution at work here, both about equivalent numerical structures, used to explore and then explain the generic behaviour seen from the specific examples.

This reminds me of a feature of substitution that exposes its purposefulness. Firstly, it is a two-way action in which some things are gained and some are lost: generality/specificity; approximation/accuracy; manipulability/visualisation etc.  This last duo comes from thinking about modelling phenomena, but also from substitutions that change bases, such as are done in order to integrate functions.

Don’s website holds many tasks in which substitution is more explicit than what I have offered so far, see https://donsteward.blogspot.com/search/label/substitution. Martin Wilson of Harrogate is credited with some of  the ideas.  There are several of these that can be explored by trial and adjustment, i.e. purposeful substitution to get a ‘feel’ for what is going on and also raw material for later reflection – why these numbers?  Both the sets below use structures which we hope will become familiar for learners but also have extra features to think about.  I could imagine learners being asked to ‘make up some of your own like these’ and hence writing algebra for themselves, having used substitution to test their inventions – some two-way number/algebra thinking.

You might be wondering about a place in the curriculum where the word ‘substitution’ is used explicitly – the solution of simultaneous equations. In our workshop, and also in some of Don’s tasks, a powerful use of substition turns out to be the use of equivalent algebraic expressions, or temporarily equal expressions to simplify the use of variables. A really simple version of this reasoning is: if a = b and a = c then b = c, and a, b and c can be substituted for each other. Here is a development of that, where expressions rather than individual letters can be manipulated to ‘reduce’ the number of variables in a situation (https://donsteward.blogspot.com/2016/03/find-expressions.html ).  The task is to express a, b and c in terms of n. Rather than using the language of moving terms to and fro over the equals sign like deranged chess pieces the language of logical arithmetical reasoning can be used. For example, in the first set: ‘if b – c = n, then I also know that c = b - n; does that help?’ The standard question: ‘if I know …. then what else do I know?’ kicks in big time when transforming algebraic expressions. New expressions for a and b can be substituted into the first equation.

This realisation, that so much mathematics depends on substituting one expression for another when building expressions, equations, mathematical models and so on, seems to get lost in formulaic approaches to simultaneous equations. The idea of substituting expressions into other expressions has recently made more sense to me than the traditional ‘elimination and substitution’ language of methods. For example Don’s ‘where do the lines meet?’ tasks (https://donsteward.blogspot.com/search/label/simultaneous%20equations) cry out ‘substitute for y in the second equation’ rather than the ‘rearrange and match coefficients and subtract the equations’ that appears in some textbooks.

I am not saying that this kind of substitution can be used to solve all such problems, but the awareness of the power of substituting one expression for a variable or another expression pervades mathematics, so a common pedagogic question could be: ‘is there a substitution that can be made from the given information that gives insight/simplifies/gives some traction?’. Being a bit fanciful – this could even be used in angle-chasing situations. I suspect that if John and I wrote ‘Questions and Prompts for Mathematical Thinking’ today we might include a question of this kind that could turn many procedural tasks into something more creative.

In my next Dose of Don I shall return to finding inspiration in his tasks rather than imposing my own perspective on them.

Geometrical Reasoning Within Constraints

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

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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.

 

Squares

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

This is the third 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 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.

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In Dose of Don #2 I quoted a remark Don had made in his blog that ‘legitimately going from one statement to another (is kind of what maths is about)’ [his brackets]. It was said in the context of working with straight lines and linear expressions, so I decided to continue with squares as both geometric and algebraic objects. How does Don ‘legitimately go’ from one statement to another?

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.

These two diagrams appear in his ‘number’ task sequence. Both of these depend on reasoning about equal sides. The first diagram can be completed with verbal reasoning: ‘this must be … because …’. The second diagram is less straightforward: ‘if I knew this length, then I would also know that length’ and therefore needs algebra – the labelling of the unknown. It doesn’t even have to be a letter for younger learners, it could even be a coloured sticky dot to stand for an unknown value. It might be astonishing that many ‘square in rectangle’ problems can be resolved from very few initial measurements and the labelling of only one unknown.

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 x².  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 n². 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 31² 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.

As always, he leaves the progression up to the teacher, but I can’t help thinking that the presence of these tasks on his website means that he is favouring a meaningful, purposeful ‘slow burn’ approach to using and manipulating algebra by using spatial awareness.  That would fit with his commitment to the ideas of Dina van Hiele who classified the components of mathematical understanding as: visualisation, analysis, abstraction, deduction and rigour. These are often regarded as hierarchical and indeed ‘seeing’ is a first response to a new bit of mathematics while imagination and imagery are key experiences of doing maths, not only features of good pedagogy. While the order might be hierarchical in terms of abstraction it is not hierarchical in terms of learner-age. In my next ‘Dose of Don’ I will explore some more of his commitment to spatial reasoning. 

Straight Lines

This is the second 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 my previous post Lines and Angles on Square Grids. My thanks go to Anne for giving me permission to share her writing here.

Dose of Don 2: Straight Lines

This is the second 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 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.

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In Dose of Don 1 I focused on a particular feature of Don’s work on grids. I think of this as ‘that little triangle’ – a right angled triangle that, nestled up to a straight line on a grid, can be used to define angle, gradient, tangent, direction, distance between two points, rate of change, ratio, instantaneous change and so on. I am sure I will come back to this triangle again. For now, the multiple roles played by ‘that little triangle’ made me zoom in on expressions of the form ‘mx + c’ or ‘kn + c’ where the common use of x is for continuous variables and n for discrete variables. One thing I realised is that an expression like kn + c can be seen as a member of a family of multiples of k with remainders. For example, any whole number can be written as 4n, 4n + 1, 4n + 2 or 4n + 3. Any whole number is either 0(mod 4), 1(mod 4), 2(mod 4) or 3(mod 4). The parallel lines y = 4n + c are all the same line translated vertically according to remainder c when dividing by 4. The ‘undoing’ of 4n + c can be understood as ‘subtract the remainder to get back to the multiple, then divide by 4 to get back to n’, which is the same as saying  that the equations: y = mx + c and x = (y - c)/m are equivalent, which of course you know but the context or resurrecting a division from its whole number and remainder parts seems to be a good context for thinking about transforming equations (finding equivalent ways to express the same relationships) rather merely going through some manipulations to solve something.

Don poses a raft of questions about properties of any four consecutive numbers, many (but not all) of which can be proved by using expressions 4n + c. For example, if their sum is 130, what would the numbers be? ('4 consecutive numbers mixed questions'). The algebra associated with this question is of the same kind as finding expressions for perimeters.  On the same page he offers an enigmatic slide that combines mod 3 and mod 4 when the ‘n’ in a linear expression is itself a linear expression. I have a bit of a ‘thing’ about substitution when it is given as a pointless exercise which focuses on calculation rather than structure, so I enjoyed this slide because it needs the distributive law and would look good represented by cuisenaire rods or even two connected cogs (If 3n + 2 turns of one cog make a bigger cog turn once, then ….?).

So far, the use and meaning of algebra comes through the questions posed. How does he approach the more procedural necessities of working with linear expressions? The following two slides show a commitment to structure and meaning. My personal approach to algebraic expressions is to avoid doing anything to them unless I know it is necessary and can anticipate its use. So with these slides I did not start by ‘what should I do?’ but ‘what are these telling me?’. 

These are from 'algebra snakes and branches' in which much of the emphasis is on building and transforming expressions so that given expressions can be read with meaning.

I have offered these to various teachers and also young learners and there seem to be two reactions: one is to multiply out all the brackets, simplify and compare whole expressions; the other is to think about their constants and eliminate those that cannot have the right constant, then check the number of ‘n’s’ and eliminate any that cannot have the right number, then check by substitution e.g. 1 for n or d (do you need two values?). This approach uses the meaning of the distributive law and substitution can be used to find out what the effect on ‘-3’ is of subtracting 2 times it on one of the examples on the right hand side. However, asking ‘what are these telling me?’ reveals some care in devising these examples. I am not going to point out everything I observe but, for example, look at how (n + 2) appears in various guises in the left-hand example. Something similar lurks in the right-hand example. ‘Multiplying out’ loses those observations that would significantly reduce the work by recognising structures.

This ‘what is it telling me?’ approach to algebra has echoes throughout his collection of tasks.  Here are a couple of slides that embed the question: ‘if I know this – what else do I know?’

These are from a slides entitled ‘so, linear’. He says on the website that ‘legitimately going from one statement to another (is kind of what maths is about)’ [his brackets]. This is so deep but so understated.

The shift from thinking of linear form as the generalisation of ‘mx + c’ to ‘ax + by’ is one I need to explore more. I recognise that a graph that can be written as ax + by = c is a variation on x + y = 1, with the associated ease of finding intercepts on both axes. It looks as if Don had that in mind with his suggestions of substituting zeroes in the righthand slide. It also looks as if he had the requirement for algebraic solutions to simultaneous equations in mind with some of these transformations.

Here is another example of the need to recognise the algebraic format of linear graphs with a typically playful task: 'muddled rules and graphs'. Why has he chosen to use similar numbers in most of these? How might learners’ approach to this task vary if the axes were not similarly scaled? How many of these can be ‘seen’ as variations on x + y = 1?

Finally I find myself returning to a task I posted in Dose of Don 1 called ‘integer intersection points’. I return to thinking again about straight line graphs as representations of similar triangles, every pair of points on the line being the vertices of a right-angled triangle whose vertical and horizontal lengths are in a fixed ratio. Because my approach to these does not seem to match his, I am left with the intrigue of working out his train of thought and talking to myself about its equivalence to mine.