Home » Answers

How we can use ‘Continuity’ to answer questions

In the last post I showed the difference between a Continuous curve and a Discontinuous curve, with a few examples from well known curves.

In this post I am going to show that this can be useful to know in answering a certain sort of question.

I’m not going to do this whole exam question.

The first part involves putting the value x = 3 into the given equation. You will get the answer y = -6

Use the value x = 4 and you get the value y = 20.

This is where the fact that this is a continuous line is so important. You know that the line must cross the line y = 0 somewhere between x=3 and x=4 , and that is what we mean by the root.

If you don’t believe that, think of the equation y=1/x.

If x = -2, the y = -1/2
If x = 1 then y = 1.

The same situation; y goes from negative to positive. Does that mean we have a value of x between -2 and 1 for which 1/x = 0?

No it doesn’t, because the curve for y=1/x is not continuous. we can only use that rule for continuous curves.

Note:  Some curves, such as y=1/x in fact, are continuous for much of their length. There is just one place where it is not.  So you can use the rule above if the curve is continuous in the RANGE of numbers you are working in.

Parallel Lines

For today’s question I am looking at one in Linear algebra

I should start by saying what I mean by ‘Linear Algebra’, which is a term I sometimes use with GCSE students and find they haven’t heard of it…  I guess that’s just me using language from ‘further along’ the Maths road, that will become second nature to you later on if you continue with the subject……  anyway it means what it sounds like..  algebra as its used to describe lines.

Anyway, on with the question – there is not a lot of work to do here, but to get to the answer you need to remember how we can get to two lines being parallel through their equations – and the secret lies in their gradients.  Two lines will be parallel if their gradients are the same.  on the graph below I show two lines, one with the equation y = 2/3 x + 2 and one with the equation y = 2/3 x – 1. As you can see on the graph, the lines are parallel.  The equations are in the familiar form y = mx + c so we read off the gradients as the same, 2/3 in each case.

In the question we have from the exam, that is not so. Well, not for both.  We can see the gradient from the first line – that is 3.

Some re-arrangement is needed, and that’s the small amount of work needed.

3y – 9x + 5 = 0  – Add 9x and take 5 from each side gives

3y = 9x – 5 – Divide throughout by 3

y – 3x – 5/3.  We don’t really need the 5/3 part; we can see though from the 3x that the lines have the same gradient so are parallel.

Proved as required.

I’ve been driving in my car

Today’s question is one I did with a student last week about petrol consumption.. and it comes with a confession – I got a bit brain-tied when I first tried to do it. This can happen to anyone.

Its not a spectacularly difficult question – though all question can seem difficult if you can’t see how to do it straight off.

My student started with the right step, by highlighting the important information.  That was good start, I said, commented on what the units for ‘consumption’ were,  and then my brain froze.

I unfroze it later so lets have a look at what to do.

Deal with the first 9 minutes first. What is the speed? Well, whenever you see a situation where it is a ‘mile a minute’ – just think 60mph, since there are 60 minutes in an hour. Makes thinks quicker that way.

At that speed – which is less than 65 mph, we use the first line of data.

1 gallon will take us 50 miles at that speed, but we are going nowhere near that far – just 9 miles.  So the amount of petrol used will be 9/50 of a gallon – which is 0.18 gallons.

The next part we are given the speed, 70mph, so we know it is the second line of the data we need.  First though we need to know the distance.    1 Hour 36 minutes = 1.6 Hours. 36/60 is 0.6 of an hour, and that makes its easier to do the calculation.

70 x 1.6  = 112 miles.

[You could do this on your calculator. I’m always looking for short cuts, and I notice that this is the same as 16 x 7 which I can do with my times tables]

At this speed, we would use more than 1 gallon, because 1 gallon will only get us to 40 miles; 2 gallons 80 miles…. or to put it another way, divide our number of miles by 40 which gives us 2.8 gallons.

Now look at the petrol used in bother parts of the journey. This gives us 2.98 gallons.

This is less tan 3 gallons, which is what the question asked us to do.

Summary

With many questions that say ‘show that’, its best to leave thinking about what you are ‘showing’ until the end.

Then split the journey in two. Don’t try doing a whole question in  one when you can split it into parts.

Answer to the Number machine Question

Here is a confession – If anyone read my last blog post they will have seen a mistake with the question I posed at the end – If you are looking now, this has now been corrected!

The question asked to fill the second box in.  The first box says  x 2 , so if we feed in 6, the number in the middle is 12.   So the second box needs to be an operation that gets from 12 to 6.

Without the clue now added, you could have at least two different answers.   “Take 6” is what I expected, and with the clue that is now the RIGHT answer.  Without the clue, ‘Divide by 2’ would have worked too.

(So would ‘add -6’ if we are getting pedantic, though really thats the same as subtract 6. Another possible answer would be ‘Raise to the power of 0.721, but I wouldn’t expect people studying Number machines to spot that.  There are probably a lot of other operations that get from 16 to 6 in we go that deep!)

All is revealed with regards to dogs – and sheep!

I gave the dog problem from 2 posts again to my Wednesday student yesterday and he solved it in the way it was intended to be solved.

We start by choosing a letter to stand in for the answer we want. Let S be the number of small dogs. Also let L be the number of Large dogs. We have not be asked to find the number of large dogs but this is part of the situation.

So we can say S  = L + 36 – because there are 36 more small dogs than large dogs. Actually L = S – 36 is the same and will lead us to the answer required more quickly.

Also L + S = 49 – the number of dogs. This ‘equation’ I think is intended – but I will return to this point.

We can solve the equations by substituting the first into the second to give  S – 36 + S = 49.

Simplify by adding the S and the S and adding 36 to both sides, we get 2S = 85   and so S = 42.5

“But how can we have half a dog”, asked my Wednesday student, and a very fair question two. This is the more obvious reason why this is a bad question – interestingly bad but bad none the less. If we are going to encourage students to take ‘real life’ problem solving seriously, then the questions we ask should make sense.

But my other reason why I’d want to change this question comes back to the story about the black sheep.  The point of the  story is that as mathematicians we shouldn’t assume anything – or at least we should qualify any answer by stating clearly which further assumptions we have made – I claim the answer to this question is incomplete unless we also say

‘Assuming all dogs are large or small’  – i.e  there are no medium sized dogs!  Without that, we can’t use safely the equation S + L = 49!