21 July 2022

Making rounding interesting

Are there any 'boring' topics in mathematics? Understandably, mathematics teachers tend to be kind of professionally committed to the idea that all mathematics topics are interesting. If even the teacher doesn’t find the topic interesting, then what hope is there for the students? And yet, perhaps, if we are completely honest about it, we find some topics a bit harder to be enthusiastic about. For me, ‘rounding’ is that kind of a topic. But can it be made interesting?

I wonder if rounding is any mathematics teacher’s favourite topic? Somehow I doubt it, although perhaps, following this post, lots of people will write in the comments that it is theirs, which would be interesting! Even if it perhaps isn’t the most exciting topic, it’s certainly one that contributes to success in high-stakes assessments. Students will be repeatedly penalised throughout their examination paper if they don’t correctly round their answers to the specified degree of accuracy, so it’s certainly something that needs teaching. Boring but important?

When I suggest that rounding is not a very intrinsically interesting topic, I am not talking about ‘estimation’. That is certainly something that can be extremely interesting and engaging. I really like beginning with some scenario, such as a jar of sweets, and asking students for their off-the-top-of-their-head guesstimates of how many there are, and then coming up with a variety of different ways to improve on this (Note 1). Ideally this leads to the notion that quicker, rougher estimates are not necessarily ‘worse’ than more accurate ones, and choosing the optimal level of accuracy depends on the purpose for which you need the estimate and how much time you have available and how much effort seems worthwhile. Level of accuracy needs to be fit for purpose. A good way to promote this is to ask questions like:

  • Which weighs more - a cat or 10,000 paperclips?
  • Which mathematics teacher in our school do you think is closest to being a billion seconds old?
    [Apologies, but I don't know where I first came across these examples - please say in the comments if there is someone I should acknowledge for these.]

In these, it is clear that your estimates only need to be accurate enough to answer the question. There is no point obtaining more accuracy than you need to do that.

But here I’m not thinking about contextual estimates like that but the more abstract kind of questions that you see in textbooks and on examinations, like:

Estimate the answer to $0.278 \times 73.4-\sqrt{48.3}$.

These questions that ask for ‘an estimate’ but don’t specify how accurate it should be are a bit nonsensical, it seems to me. You could always answer any question like this with ‘zero’, and the only hard part would be working out what degree of accuracy this was to (which the question never asks for). For example, the answer to this calculation turns out to be 13, to the nearest integer, so this would be 0 to the nearest 100, 0 to the nearest 1000, or (to be on the safe side) 0 to the nearest billion! Any point on the number line is always ‘close’ to zero if you zoom out far enough. So, you will technically never go wrong with a question like this by answering ‘0 to the nearest trillion’ – although of course mark schemes would not reward you for that!

More seriously, the usual approach that is taught within school mathematics is to round each individual number to 1 significant figure, with the possible exception of when you are about to find a root, where you might fiddle it to the nearest convenient number instead. So, in this example, although 48.3 would round to 50 to 1 significant figure, or 48 to 2 significant figures, we might instead choose to round it to 49, because $\sqrt{49}=7$. Doing that, we would get something we should be able to do easily in our head: $0.3×70-\sqrt{49}=21-7=14$.

The issue of how good our estimate might be (and therefore what it might be good for) is not really addressed at this level, and students would be expected simply to leave their answer as 14, without any idea how close this is likely to be to the true answer, or even whether it is an underestimate or an overestimate. But is this $14±1$ or $14±1000$ or what? This is really a bit strange, as, in any real situation, a lot of the value in estimating is in getting bounds. We may not care exactly what the answer is, but it is usually important to know that it is definitely between some number and some other number. Simply throwing back an answer like ‘14’, which we know is almost certainly not exactly right, without having any idea how wrong it is, doesn’t seem very useful. Usually, we are estimating a number in order to enable us to make some real-life decision – how much paint to buy, or how many coaches to order – and those all require us to commit to some actual quantity. So, really, I don’t want to know ‘roughly 14’ – I want to know ‘definitely between 10 and 20’ or definitely between 10 and 15. So perhaps we should teach it this way? (Note 2)

Then, we can consider that how much effort it is worth going to in order to get a more accurate estimate depends on how narrow I want my bounds to be. It’s foolish to act as though more accuracy is simply an absolute good. (Sitting down and calculating more and more digits of $\pi$ forever would not serve any useful purpose.) Sometimes, when peer marking, students will be told to give themselves more marks if their answer is closer to the ‘true’ answer, but I think this reinforces an unhelpful view of estimation. It also encourages students to 'cheat', by first calculating the exact answer, rounding this answer, and then constructing some fake argument for how they legitimately got it. If more accuracy were always better, we would always use a calculator or computer and get the answer to as high a degree of accuracy as we possibly could. But, with estimation, the sensible thing to do is to spend your effort according to how useful any extra accuracy would be in the particular context that applies. These seem to me the important issues in estimation, and they go largely unaddressed in the lessons on estimation that I see.

Exploring rounding

One way to make the topic of ‘rounding’ a bit more interesting is to begin to explore some of these issues. For example, in the calculation above, since (almost) all of the numbers were rounded to 1 significant figure, it might seem sensible to give the answer to 1 significant figure, which would be 10, suggesting that this means $10±5$. In this case, the exact answer to the original calculation (13.45537…) is also 10, to 1 significant figure, which is good, and there seems to be an assumption within school mathematics that almost has the status of a theorem:

Claim 1: If you round each number in a calculation to 1 significant figure, then the answer will also be correct to 1 significant figure.

However, there is no reason at all why this should be true, and you might like to consider what the simplest counterexample is that you can find. When can you be more confident using this heuristic, and when should you be more cautious?

***

A simple counterexample would be $3.5+3.5$, which comes to 8 if you round each of the 3.5s to 4 (to 1 significant figure) before you add them, but 8 is not the correct answer to 1 significant figure, because of course it should be 7.

A slightly more complicated scenario that might be worth exploring with students involves rounding two numbers in a subtraction, so we could begin with a question like this:

Estimate the answer to $14.2-12.9$.

(You might ask, "Why estimate something so simple, and not just calculate?", and the point of this is not to be a realistic rounding scenario, but a simplified situation to help us see what is actually going on with rounding.)

So, here's another claim:

Claim 2: If we round each number to the nearest integer, then the answer will be correct to the nearest integer.

Is this claim always, sometimes or never true?

Students will need a bit of time to figure out what the claim even means. Using the notation $[x]$ to mean “round $x$ to the nearest integer”, we could write:

$[14.2]-[12.9] = 14-13 = 1$

And $[14.2-12.9] = [1.3] = 1$.

So that checks out in this case.

So, in this notation, the question is:

When is $[a-b] = [a]-[b]$?

I think this is a potentially interesting task, where there is plenty to think about, but it also generates some repetitive but somehow acceptable routine practice. (I call such tasks mathematical etudes – see Foster, 2018). You might like to try it yourself before reading on.

***

Running through this for all possibilities of $14.x-12.y$, where $x$ and $y$ are single digits between 1 and 9, we find this situation: 
Table 1. $[14.x-12.y]$ and $[14.x]-[12.y]$ compared, where $x$ and $y$ are single digits between 1 and 9.
Ticks indicate where equality holds.

So, a counterexample to Claim 2 would be any of the empty cells in this table; for example, $[14.7-12.3] = [2.4] = 2$ but $[14.7]-[12.3] = 15-12 = 3$.

We might prefer to write this on one line as: 

$$3 = 15-12 = [14.7]-[12.3] ≠ [14.7-2.3] = [2.4] = 2$$

Applying some deduction, we might say:

1. If both numbers in the subtraction are rounded up, or both are rounded down, we should get a tick. These are the ticks in the green squares in the table above.

We might also be tempted to say:

2. If one number is rounded up and the other number is rounded down, we should not get a tick.

As you can see from Table 1, this is false, as can be seen by the ticks in some of the white squares. Why does this happen? For example, in $14.3-12.8$, the minuend rounds down and the subtrahend rounds up, and we get no tick, since the difference, 1.5, rounds up to 2, whereas $14-13 = 1$. However, this doesn’t always happen. For example, in $14.2-12.8$, as before, the minuend rounds down and the subtrahend rounds up, giving $14-13 = 1$, but this time the difference is only 1.4, which rounds down to 1, so we do get a tick.

There is lots to explore here, and the idea of comparing the result from applying some function before and after some composition - i.e., $f(x \pm y) \stackrel{?}{=} f(x) \pm f(y)$ - is a highly mathematical question.

When I look back at mathematics tasks that I have designed over the years, I now notice that they often cluster around certain ‘favourite’ topics. Without meaning to, I have unintentionally avoided certain topics – perhaps those that, like ‘rounding’, seem intrinsically less interesting – and cherrypicked other topics to design tasks for. At Loughborough, we are currently working on designing a complete set of teaching materials for Year 7-9, so we are now working in the same kind of situation as teachers – we can’t miss anything out! And this has led me to thinking about how to address some of those potentially ‘less interesting’ topics, which is proving fun.

Questions to reflect on

1. Are there mathematics topics that you personally find less interesting to teach? Which ones? Why?

2. What tasks can make these topics more interesting for you and for your learners?

3. For rounding, what other tasks can you devise? Is $[a+b] \stackrel{?}{=} [a]+[b]$ an easier or harder problem? What about $[ab] \stackrel{?}{=} [a][b]$?

Notes

1. Dan Meyer is the expert at designing tasks like this; e.g., https://blog.mrmeyer.com/2009/what-i-would-do-with-this-pocket-change/; https://blog.mrmeyer.com/2008/linear-fun-2-stacking-cups/; https://www.101qs.com/70-water-tank-filling 

2. I think the various versions of the “approximately-equal sign” $≈$ are not really our friends here, because a statement like $13≈10$ doesn’t really have a precise meaning.

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.


07 July 2022

A football on the roof

I am always on the lookout for 'real-life' mathematics that is of potential relevance and interest to students but where the mathematics isn't trivial and the context isn't contrived. Too often the scenario is of potential interest but the mathematics is spurious, and doesn't really offer anything in the actual situation that couldn't be done more easily without mathematics. It is not easy to find good examples, but I think this is one that might provide some opportunities to work on topics such as similar triangles and ratio.

Some students lost a football on a flat roof and wanted to know whether the ball had rolled off and fallen down behind the back of the building (i.e., gone forever) or whether it was worth climbing up to retrieve it (Note 1). There weren't any tall buildings nearby that they could access to get a good view of the roof. What they needed to know was how far back from the building they needed to stand so as to be sure that if the ball was there they would be able to see it.

"Can you see it?"
"No, but I just need to go back a bit further."
"If it was there, you'd be able to see it by now."
"I'm not sure."

In particular, going as far back from the building as possible, given the constraints of the surrounding buildings, did the fact that they couldn't see the football mean that it was definitely not there, or could it be that it was just not visible over the edge of the building (Figure 1)?

Figure 1. A football on the roof

This problem is reminiscent of lessons in which students determine the height of a tree near the school using a clinometer, but it feels somehow different. There is not usually any good reason for wanting to know the height of a tree, and it is usually hard to find any way to decide afterwards whether the students' estimates are reasonably accurate or not. In this case, there is a clear 'need to know' and, ultimately, when the site manager brings a ladder, the students would discover if they were right or wrong, so it feels as though something is at stake.

A good way to start would be to decide on simplifying assumptions that it seems sensible to make; i.e., things that we might sensibly choose to ignore. For example:

  • assume that the ground and all roofs are perfectly horizontal
  • assume that the roof in question is free from any debris
  • assume that the building has height 4 m and goes back 6 m
  • assume that the football is perfectly spherical, with diameter 22 cm
  • assume (worst-case scenario) that the ball is right at the back of the roof, against the brick wall

Students may suggest more outlandish things, such as assuming that light travels in straight lines or that the curvature of the earth is negligible, and I would include these as well if they raised them.

Figure 2. Careful analysis (diagram not drawn to scale)

Let's start with a careful analysis, which uses trigonometry and is 'a sledgehammer to crack a nut' for this simple scenario. This is not the approach that I would envisage students taking.

Lots of the work has been done in the diagram (Figure 2), and our units are metre throughout.

We have

$$\tan \theta = \frac{r}{l}$$

$$\tan 2\theta = \frac{h}{d}$$

Using the identity

$$\tan 2\theta \equiv \frac{2 \tan \theta}{1-\tan^2 \theta}$$

we obtain

$$\frac{h}{d}= \frac{2 \left( \frac{r}{l} \right) }{1- \left( \frac{r}{l} \right)^2},$$

giving

$$d= \frac{h(l^2-r^2)}{2rl}.$$

We can now substitute in some reasonable values:

  • $h=4-1.8=2.2$; the height of the building subtract the maximum eye height of the student when standing on tip toes or jumping,
  • $l=6-0.22=5.78$; the depth of the shed subtract the diameter of the football, and
  • $r=0.11$.

This gives $d=57.8$, so the student would just be able to see the top of the football from about 58 metre back from the shed.

But the trigonometry here is overkill for the nature of this problem and the accuracy required, so it would be much quicker and more reasonable to use the simplified diagram shown in Figure 3.

Figure 3. Rougher analysis (diagram not drawn to scale)

For this rougher analysis, we don't need to use $\tan$ explicitly and can just equate corresponding ratios in similar triangles.

So,

$$\frac{d}{h}=\frac{l}{2r},$$

giving that

$$d=\frac{hl}{2r},$$

so, with the same values as above, this again gives $d=57.8$, correct to 1 decimal place, and the same conclusion that the student would just be able to see the top of the football from about 58 metre back from the shed.

In the situation where $l  \gg r$, we can see that in our first equation

$$d= \frac{h(l^2-r^2)}{2rl}$$

the bracket $(l^2-r^2)$ will, to a good approximation, reduce to $l^2$, giving

$$d \approx \frac{hl^2}{2rl}=\frac{hl}{2r},$$

as before. So, all routes lead to an answer of about 58 metre.

But, what if the playground extends only, say, 40 metre before meeting another building? Would it be worth the students going inside and fetching a chair to stand on? Would that make enough difference to be worth the trouble?

The beauty of having derived a formula is that a question like this can be answered instantly by substitution. All that changes here is that $h$ reduces from 2.2 to, say, 1.7.

So,

$$d=\frac{hl}{2r}=\frac{1.7 \times 5.78}{2 \times 0.11}=44.7,$$

and so this would not quite be enough to work within the available space, since $44.7 > 40$. Rearranging the equation to give

$$h=\frac{2rd}{l}$$

reveals that, unless you can find a stool of height at least $2.2 -1.52 = 0.68$ metre, then there is no point bothering.

The mathematics here is not profound, but the result is not guessable without it. I think we need more tasks like this, where a little bit of mathematics (not pages and pages) tells you something practically useful that you couldn't have guesstimated accurately enough without it.

Questions to reflect on

1. Would your students find a task like this credibly realistic and engaging? How might you improve it?

2. What 'real-life' tasks do you use that are both non-trivial mathematically and non-embarrassing in terms of correspondence with reality?

Note

1. Disclaimer: Nothing in this post should be taken to endorse climbing onto roofs to retrieve footballs!

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