I have been a runner for more than fifty years. Looking to attempt to fire an enthusiasm for maths or at the least for arithmetic, here are some examples of things I have wanted to know. A little trigonometry is required, so this will suit a good Y9 class and upwards.

1. As I get older, I notice that I need to stretch a lot more. Despite that, my legs no longer straighten in a typical running stride and the straightest the leg will reach might easily be 15º short of straight. Try to work out what you think the shortening of stride is. Since there are several ways of doing this, aim to find a range of answers, so you can say “More than this and less than that”. And, preferably, someone else reading your work can see what you have done. Don’t think of this as ‘finding an answer’ quite as much as ‘persuading someone my answer is about right’.

2. Suppose that, or whatever reason, you’re able to change your stride length by a single centimetre for no other change except that you go faster by that 1cm per stride. How much difference will this make to your time per kilometre? By al means use your own figures. I’m going to suggest that 5mins/km is ‘usual’ running speed, that 5km/hour is walking speed and that there are usually 70 walking strides in a 100m, with more like 40 when running at ‘usual’ pace. You could go find out what you do (and so could a whole class).

3. Having done Qs1&2, how do you think a loss of 5º of leg extension affects a 5km parkrun time?

I note that if you are trying to adjust 300 seconds by a change of 5cm in 250cm, it doesn’t matter if you use 245/250 or 250/255, because they’re very much the same. Just decide if you expect the 300 seconds to be bigger or smaller and chose numbers accordingly. You’re going to round the result heavily anyway, so it really doesn’t matter. Your objective is to discover if the difference is big enough to notice. A second look at the problem might decide to try to make the numbers less vague, but until you’re doing that, it really doesn’t matter.

4. A favourite problem included on an earlier page asked about the use of sheeptracks on the moor. Here it is again in different words.

I think I walk faster on a sheep track than on the long grass or heather that is between sheep tracks, so I’ve been following the sheep track that most nearly goes in the direction I want. After walking on Dartmoor on the same general route as a group of students, I think we all walk about a minute faster every kilometre on a sheep track than on the nicest of the rougher ground. If that is correct, how far away from the chosen direction can we go on a track so as still to be faster than going the direct way on the rougher ground? Use figures around the 5kph mark. Look what happens if you go 10º away from your preferred direction.

1. The critical part is the knee joint and, whatever the thigh angle, we assume that ‘before’ has a straight leg and ‘after’ has the lower leg (of length L) at 15º less straight. At the point when the bent leg hits the floor, the heel of the ‘straight’ leg is x ahead of the bent one, so the smallest change is that x/L = sin 15º, so x=L sin 15º. On me L is around 50cm, so x is around 13cm.

For the other extreme, imagine the straight leg continuing forwards as in my diagram. The leg didn’t get longer, but this is where it landed. I think the worst case would be if the lower leg was vertical, as in the smaller diagram, which makes the horizontal x=L tan 15º, or still close to 13cm.

So I’m going to say that, on this evidence, a *change* of 15º in the flexibility of that joint is around 13cm. Similarly a change of 5º makes a change of about 44mm.

2. Walking 70 strides/100m =>143cm/stride, changing to 144 cm, making 0.7% difference, or about 5 seconds per kilometre.

Running, 40 strides/100m => 2.5 m per stride changing to 2.51, 1.2 seconds per kilometre. Would you notice? - That’s half a minute across a half-marathon.

3. So a change of 44 mm, 4.4 cm suggests a change of 5 seconds per kilometre when running. This is a change you would definitely notice, most of a half-minute straightaway in a parkrun. A minute is about the variability I show across a year (I don’t run when injured, so it’s pretty consistent); so half a minute is the difference between an average run and a good day or a bad day. Adding or losing half a minute because failing to stretch has reduced the stride length (or at my age, doing enough stretching has improved stride length) is a factor one might decide is worth some of that boring stretching so as to gain it (or gain it back, or not lose it). Similarly, you might see advantage in stretching the stride on a downhill stretch, or with the wind on your back. Is that altogether too technical, or is it the result of a little bit of arithmetic applied sensibly?

4. Let’s first assume 5km/hour is improved to 5.5 kph. Equivalence gives a right angle triangle two sides 5 and 5.5 (and something shorter). So 5/5.5= cos 25º => anything up to 25º away from the preferred direction is still faster than the direct route. Going at only 10º off is still 5.4kph in the desired direction.

Second, use 4.5 and 5kph, meaning base speed is 4.5 on rough ground, but 5 on the better surface, then the figures move a little (1.2º). I don’t think we can tell a single degree apart when sheep tracks change direction as much as they do. 10º off at 5kph is 4.9 kph - that’s still a 9% improvement, an hour quicker on an 11-hour walk.

Of course, what you do is take the next sheep track going a bit more in ‘your’ direction, so if you’ve been off to the right, you take the next to the left as long as it’s not more than 25º off the target, which means that you might easily have a 45º change because the more you push the extreme, the more you want a new path closer to the ‘right’ direction. You don’t do this in a mist, obviously. The technique assumes decent visibility. Ten Tors teams assumed I was simply being weird; some persisted in this attitude even after being shown the maths. The argument that ‘made sense’ was that the irregular route was far too much like ‘thinking’ or ‘work’. Funnily enough, the people with that attitude didn’t often make it all the way to selection, mostly because the walking day is so very long that it is actually worthwhile doing the work involved in thinking. Some teams left the thinking to a couple who could do this easily, but that dos require significant trust that all these ‘silly changes of direction’ add up to something ‘good’. This is what I see as a psychological problem far more than a maths one.

Going all day, say 50km and successfully staying within 10º of the target direction for a gain of 5.5 instead of 5kph adds up to 770m to 2 sf. (the added distance may be that big, if the course is always off bey 10º). But is is still 45 minutes faster than doing 5kph all day. I think that one would be less tired too, which, experience tells me, says that there would be less slowing down, so the difference between the two strategies can only grow.