# Colin Foster's Mathematics Education Blog

## 24 November 2022

### Fractions as factors

Can a factor be a fraction?

People sometimes agonise over whether a fraction such as $\frac{2}{3}$ can be called ‘a factor’ of another number, such as 6 (Foster, 2022). If factors are defined as “numbers that divide exactly into another number” (BBC Bitesize, n.d.), then, since $6÷\frac{2}{3}=9$, and $9$ is an integer, shouldn’t $\frac{2}{3}$ be regarded as a factor of $6$?

Perhaps we decide that we want to be able to say that the number $6$ has exactly $4$ factors ($1, 2, 3$ and $6$), and so we don’t allow numbers like $–2$ or $\frac{2}{3}$ to be factors. If so, we could use a tighter definition of factor, such as: A factor is a positive integer that divides into another number a positive integer number of times (or, equivalently, we could say ‘without any remainder’). However, in many instances we might want to think of factors more broadly than this. For example, when factorising $x^2±5x+6$, it might be helpful to think of the $6$ as having four possible factor pairs ($\{1, 6\}, \{2, 3\}, \{–1, –6\}$ and $\{–2, –3\}$). Similarly, when using the factor theorem, we typically treat negative numbers as ‘factors’. In other contexts, it might seem natural to regard a number like $\frac{2}{3}$ as being a factor of $\frac{4}{3}$, since it ‘goes into it’ twice, or even $x$ as a factor of $x^2$, since it goes into it $x$ times, regardless of whether or not $x$ might be an integer. We might even pull out an irrational number, such as $\pi$, from an expression like $2\pi r-\pi l$ to give $\pi(2r-l)$, and call this 'factorising', and refer to $\pi$ here as a ‘common factor’, although it is certainly not an integer, and is not even rational (Foster, Francome, Hewitt, & Shore, 2022).

‘Factor’ seems to be one of those words that is used differently in different contexts, even within school mathematics, and I think it isn’t really possible to settle on a fixed definition which will always apply (see Foster, Francome, Hewitt, & Shore, 2022, for a similar discussion about the word ‘fraction’). Perhaps the best approach to awkward issues like these is to acknowledge them and explore them. Turn the issue into a task: What would happen if we allowed fractions to be factors? Perhaps we call them ‘fraction factors’.

#### Exploring 'fraction factors'

Students often think of fractions as ‘numbers less than 1’, and they may initially think that any fraction would be a fraction factor of any integer, but of course this isn’t right. Although $\frac{2}{3}$ would be a fraction factor of $6$, it wouldn’t be a fraction factor of $5$, since $5÷\frac{2}{3}=\frac{15}{2}$, or $7.5$, which is not an integer. All unit fractions ($1/n$, where $n$ is an integer $\neq 0$) would be fraction factors of every integer, since they are by definition fraction factors of $1$, and $1$ is a factor of every integer. But when would a non-unit fraction be a fraction factor of an integer? Could we ask for all the fraction factors of $6$? Clearly not, because this list would include all of the unit fractions, and there are infinitely many of them.

There are many opportunities here for students to form conjectures and to find counterexamples – and, in each case, to try to find the simplest counterexample they can. It can also be helpful to look at the question the other way round, and ask what integers a fraction like $\frac{5}{12}$, say, would be a fraction factor of.

The conclusion is quite simple, but perhaps not that easy for students to arrive at without quite a bit of useful exploration. A fraction $\frac{p}{q}$, with $p,q \neq 0$, in its lowest terms, will divide an integer $m$ if and only if $m÷\frac{p}{q}$ is an integer. This is equivalent to saying that $\frac{mq}{p}$ must be an integer. Since $p$ and $q$ are co-prime, $\frac{mq}{p}$ will be an integer if and only if $p$ is a factor of $m$. So, the fraction factors of $6$ are fractions that, when simplified, have the (positive integer) factors of $6$ as their numerators:

$$\frac{1}{n}, \frac{2}{n}, \frac{3}{n}, \frac{6}{n}$$ for all integer $n \neq 0$.

Perhaps this seems obvious, but I think it can be quite unintuitive that, say, $\frac{2}{17}$ is a fraction factor of $6$, but $\frac{4}{3}$ isn’t.

A less algebraic – and perhaps clearer – way to appreciate why the numerator matters, but the denominator doesn’t, is to realise that $\frac{1}{q}$, as a unit fraction, will always be a fraction factor of any integer, regardless of what $q$ is, because $q$ of them will always fit into every $1$. For the same reason, $\frac{p}{q}$ will necessarily be a fraction factor of $p$, because $q$ of them will fit exactly into the integer $p$. So, $\frac{p}{q}$ will be a fraction factor of any integer of which $p$ is a factor.

### Things to reflect on

1. Do you agree that it isn't possible to have a single definition of 'factor' that applies across all of school mathematics? Why / why not?

2. If you agree, which other technical mathematical terms do you think may be problematic in this kind of fashion? (See Foster et al., (2022) for a discussion of 'fraction' in this regard.)

### References

BBC Bitesize (n.d.). What are factors? https://www.bbc.co.uk/bitesize/topics/zfq7hyc/articles/zp6wfcw

Foster, C. (2022, October 13). How open should a question be? [Blog post]. https://blog.foster77.co.uk/2022/10/how-open-should-question-be.html

Foster, C., Francome, T., Hewitt, D., & Shore, C. (2022). What is a fraction? Mathematics in School, 51(5), 25–27. https://www.foster77.co.uk/Foster%20et%20al.,%20Mathematics%20in%20School,%20What%20is%20a%20fraction.pdf

## 10 November 2022

### Is area more difficult than volume?

I have a tendency to assume that concepts get more difficult as the number of dimensions increases. Length is pretty straightforward, surely (how long is a piece of string?). Area is a bit harder, because we are in 2 dimensions now, and volume is even harder, because that's 3 dimensions. However, I'm not sure that this is right or that it makes much sense to think of building up in this way. After all, a point is zero-dimensional, and that is certainly not a simple thing to get your head around at all!

I think it is much easier to get an intuitive sense of volume than it is of area. From a very young age, children build with blocks and pour sand and water into containers, so they are engaging with 3-dimensional concepts such as volume right from babyhood. We are 3-dimensional beings and live in a 3-dimensional world, so we really ought to feel at home working with a concept like volume. By contrast, I think that area and length may be more inherently difficult concepts conceptually, as we never see these things in their true reality. A 1-dimensional horizontal line segment is represented on paper by a very thin rectangle, because it has to have a little bit of vertical height, otherwise it would be invisible. Indeed, this 'rectangle' is really a very shallow cuboid of ink, sitting on top of the sheet of paper, so it's an approximate cuboid rather than a 'line'. Everything we see that is intended as an approximation to or representation of something 1-d or 2-d is really forced to be actually 3-d just because that's the kind of world we live in.

I was thinking about this recently when considering how to design some lessons on area. What does it mean for two shapes to have the same area? By contrast, equal volumes is fairly easy to grasp. Two hollow shapes have the same capacity (or volume of space/air inside them) if you could fill them up using the same quantity of liquid (Note 1). If they are solid objects, then they have the same volume as each other if they displace the same quantity of a liquid that they are submerged within. For children who have played with floating and sinking objects in the bath, this is familiar territory. You can easily be sure whether two objects have the same volume or whether one has greater volume than the other, and which way round it is. It might not be easy to estimate volumes at a glance in practice - I am always surprised that an ordinary drinks can contains 330 cm$^3$ (it never looks big enough to contain that many centimetre cubes) - but it's clear what 'greater volume' means and how we could, in principle, find out (Note 2).

This is all much harder with area. Two shapes have the same area if you can cut up one of them and fit the pieces exactly inside the other one, with no gaps and no overlaps. This is not easy at all. What if the shapes have awkward edges? Can a disc of radius $1$ be 'cut up and fitted exactly inside' a square with side length $\sqrt{\pi}$? You would have to make infinitely many cuts to do it: can you be sure that there would definitely be no tiny gaps or overlaps? To a young child, there is nothing exceptional about a circle - they see circles all the time - so this should surely not be a 'hard' example to think about. Similarly, we expect children to accept things like the fact that shearing a rectangle parallel to one of its sides doesn't change the area. Maybe we push over a stack of paper to illustrate this. But is it really so clear that the area is definitely not changing?

I was thinking about lessons I've seen in which the teacher aimed for more intuitive understandings of area. It is normal to choose a small square 'unit' and try to cover a given shape with a finite number of these units. So children might place small plastic squares over a drawing of a shape to see how many are needed to cover it completely without any gaps. Sometimes the answer is an integer (e.g., for a carefully-chosen rectangle), or it might be a non-integer, but still exact, if half-squares, say, can be carefully totalled. Sometimes, it may involve estimation (e.g., tracing a hand or foot onto squared paper and approximating the total surface area). Occasionally units other than a square are used, and so long as the units tessellate we can often still tile many shapes with them. But I think all of this is much harder than pouring liquids into and out of various vessels.

What about more informal approaches to area? I saw a lesson in which area was informally understood as 'the amount of ink you need to colour in the shape', whereas perimeter was 'the amount of ink you need to draw around the edge of the shape'. Students had felt-tip pens and were illustrating area and perimeter of some shapes by colouring them in (Note 3). Of course, there is a lot of imprecision here, but perhaps the act of colouring in helps to reinforce the nature of the concept we are focused on. Time can be brought in as a proxy for area, so the shape with larger area is the one that takes longer to colour in (colouring races to determine which shape has greater area). There are some assumptions here about things like the same pen being used, students with equal 'colouring speed', and so on. And, of course, features of the shape that might mean that some shapes of area 10 cm$^2$, say, are quicker to colour than others - compare these two, for example:

Other possibilities for building helpful intuitions include things like cutting out shapes in thick card (with uniform thickness) and weighing them against each other to determine which has larger area. Or cutting them out of pastry and baking them and seeing which weighs more or takes longer to eat. All of this kind of work is a long way from the more common emphasis on quickly getting to the calculations. If you ask a child, "What is area?" they may be quite likely to say, "Base times height". Admittedly, "What is area?" is a hard question for anyone to answer. But stating a formula for calculating the area of one specific type of shape (i.e., a parallelogram) is not the same as having a good fundamental sense of what area is all about.

### Questions to reflect on

1. Do you agree that area is more difficult than volume? If not, why not?

2. How important do you think it is to develop an intuitive understanding of concepts like area?

3. Would you use any of the approaches mentioned here? Why / why not?

### Notes

1. We'll assume throughout that the shapes are made of something like plastic, which doesn't absorb the liquid or get deformed by it! There are all kinds of other physical assumptions in play, such as the assumed incompressibility of liquids, and so on.

2. For reflections on some of the ambiguities around volume and surface area, see Foster (2011).

3. Colouring in is not always the most educational of activities in school, but in this case it seemed that it might be.

### Reference

Foster, C. (2011). Productive ambiguity in the learning of mathematics. For the Learning of Mathematics, 31(2), 3–7. https://www.foster77.co.uk/Foster,%20For%20The%20Learning%20of%20Mathematics,%20Productive%20Ambiguity%20in%20the%20Learning%20of%20Mathematics.pdf

## 27 October 2022

Sometimes a superficially small tweak to a task - changing just one little thing - can dramatically alter it, and mean that a lot more thinking - or very different thinking - is needed. In many cases, students don't need to be taught any additional facts or methods beyond what they have already learned - they know everything necessary to be able to figure out what difference the change makes.

The details matter in mathematics task design and adaptation. Here are just a few examples of small ways in which tasks can be adapted to make them more challenging and so that they may provoke deeper thought. For more on ways to adapt mathematics tasks, see Prestage and Perks (2013). The tasks below all take the form "You know this, but what about this?"

#### 1. Change the base

It's sometimes said that You don't understand long division until you can do it in any base.

You know how to work out $252 \div 6$ in base $10$, but can you do it in base $16$?

In which other bases does $252 \div 6$ give an integer answer? Why?

Doing arithmetic in different bases used to be much more common in UK mathematics teaching, and it is still quite prevalent in schools in many countries. I find that people often assume that it's much more complicated than it is. I once heard a staffroom conversation that went something like this:

A: I think we're just too obsessed with base 10 and we don't take opportunities to work in other bases.
B: Yes, I guess it's just a convenience thing - it's so handy, for example, to be able to multiply by 10 by just placing a zero on the end.
A: Do you know how to multiply by 7 in base 7?
B: Er, no, not off the top of my head, but I should think it's very complicated. That's my point about why we always resort to base 10.

Exploring factors of numbers in different bases can be a good way to see why person B is wrong about this!

You know the factors of $12$ in base $10$, but what are the factors of $12$ in base $16$?

In which bases is $12$ prime? Why?

By changing something, away from the familiarity of base 10, we can 'make the familiar strange' and see things in new ways. It is such an apparently small change, but there are many tasks in which a change of base can be insightful (see Foster, 2007). For example,

You know that, in base 10, $\frac{1}{2}$ is a terminating decimal and $\frac{1}{3}$ is a recurring decimal. Which fractions terminate and recur in other bases? Is there a base in which $\frac{1}{2}$ is a recurring decimal and $\frac{1}{3}$ is a terminating decimal? Or in which they are both recurring or both terminating? What is $\frac{1}{2}$ in base 16? What about in base 15? What about in other bases?

#### 2. Introduce modulus signs

The definition of the modulus function is very simple. You could use the ideas below when teaching students about the modulus function for the first time. But you could also use them with any students - even quite young ones - who don’t ‘need’ to learn about the modulus function. You would be using the modulus function as a way to get them to think more deeply about things they do 'need' to think about.

The modulus function $\lvert x \rvert$ is defined as:

$$\lvert x \rvert=\left\{ \begin{array}{@{}ll@{}} x, & \text{if}\ x \ge 0 \\ -x, & \text{if}\ x \lt 0 \\ \end{array}\right.$$

But the idea is much simpler and more accessible than this formal notation might suggest. The value of $\lvert x \rvert$ is the absolute 'size' of $x$, regardless of its sign. So $\lvert 18 \rvert = 18$, but $\lvert -18 \rvert = 18$ also. This means that $\lvert x \rvert$ is never negative (Note 1).

That is all the formal teaching that is needed. The task is then is to try some things that students can already do, but with a modulus function thrown into the mix - a superficially small change, but with a 'butterfly' effect.

You know what the graph of $y=\sin x$ looks like. What about $y=\lvert \sin x\rvert$? Or $y=\sin \lvert x \rvert$?

You know what the graph of $y=x+1$ looks like. What about $y=\lvert x \rvert+1$ or $y=\lvert x+1 \rvert$?

You know what the graph of $y=(x+1)^2$ looks like. What about $y=\lvert x+1 \rvert^2$? Or $y=( \lvert x \rvert + 1)^2$?

You know how to solve $3x-2=10$. What about $\lvert 3x-2 \rvert =10$?

You know how to solve $3x-2=2x+1$. What about $\lvert 3x-2 \rvert = \lvert 2x-1 \rvert$?

You know how to solve $3x-2<2x+1$. What about $\lvert 3x-2 \rvert < \lvert 2x-1 \rvert$?

You know how to find the integral $\int (x^2-1) dx$. What about $\int \lvert x^2-1 \rvert dx$?

Introducing modulus signs is a little like introducing $\pm$ signs (see Foster, 2012) in that you are making a superficially small change that has the effect of requiring students to think quite hard.

Finally, for modulus functions, take a look at this graph:

What combination of modulus functions could produce this graph? (See Note 2 for the answer.)

#### 3. Go non-linear

Older students may know, or have a sense of, what happens if you change a function $f(x)$ into, say, $f(-x)$, or $f(mx)$, where $m$ is a constant. But what happens to well known graphs of $y=f(x)$ if we change:

• $x \mapsto \frac{1}{x}$; e.g., $\sin {(\frac{1}{x})}$, $e^{\frac{1}{x}}$?
• $x \mapsto x^2$; e.g., $\sin {x^2}$, $e^{x^2}$?
This is something that older students may find interesting to explore. It is worth delaying going to graph-drawing software until after you have thought hard about it, and have definite conjectures to check out. Turning to graph-drawing software too soon risks circumventing the thinking, rather than supporting it.

#### 4. Look for in-between values and concepts

You probably know what the graph of $y=x^n$ looks like for positive integer $n$ and for $n=-1$. But what about for other negative integer $n$? And what about for non-integer $n$, both positive and negative (Note 2)?

You probably know about 'highest common factors' (or 'greatest common divisors'). What about 'second-highest common factors'? When do/don't they exist? What can you work out about them?

You probably know about 'common multiples' as the numbers where sequences like $3n$ and $8n$ intersect. But where do sequences like $3n+1$ and $8n-2$ intersect?

You probably know how to construct a perpendicular bisector? How would you construct a perpendicular trisector?

#### Conclusion

Variation theory currently receives a lot of attention, and can be powerful for helping students attend to what is the same and what is different in a sequence of tasks. In this spirit, little changes beyond what students need to know, and into unknown terrain, can be interesting extension tasks and valuable tasks for everyone to deepen their knowledge and understanding.

### Questions to reflect on

1. What other butterfly examples do you have, where 'changing one little thing' has a big effect?

2. When would you use tasks like this?

### Notes

1. Of course, the notation $\lvert x \rvert$ is also used where $x$ (perhaps written as $\textbf{x}$) is a vector, or where $x$ is a complex number, or to indicate the determinant of a matrix (in which case $\lvert x \rvert$ could be negative). But in this post I am only thinking about $x$ as a real number ($x \in \mathbb{R}$).

2. The graph is $y= \lvert x-2 \rvert + \lvert x+2 \rvert - \lvert x-1 \rvert$. Students could invent tasks like this for each other.

3. If you use software to draw $y = x^n$, and set up a slider to vary the value of $n$, you may notice a 'flickering' behaviour for negative values of $x$. It is interesting to think about why this happens. If $n$ is rational and equal to $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0$, and $\frac{p}{q}$ is in its lowest terms, then the domain of $x$ is:

• all of the real numbers, if $q$ is odd, but
• all of the non-negative real numbers, if $q$ is even.

This means that, when $n$ is real, but not rational, the domain is all the non-negative real numbers. So, the graph flickers as you drag the slider because, as $n$ varies, the function keeps switching between being defined and undefined (see also Dobbs, 2017).

### References

Eastaway, R. (2021). Maths on the back of an envelope: Clever ways to (roughly) calculate anything. HarperCollins.

Weinstein, L., & Adam, J. A. (2009). Guesstimation. Princeton University Press.

Weinstein, L. (2012). Guesstimation 2.0. Princeton University Press.

### Teaching specific tactics for problem solving

This is my final blogpost, as my year as President of the Mathematical Association draws to a close, so I've allowed myself to go on at ...