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Drawing the Park – The Answer

Yesterday I posed the question of how to draw a map of a playing field.  The first thing to do is decide on the scale. I like to see people use as much of the paper as possible, but there is another consideration too, as I shall show.

With some students, getting them to make good use of the paper is challenging for me. They draw in just the bottom corner. Though this experience is more to do with bar charts and similar graphs, its equally the case with maps and scale drawings.  But make mistake, choosing a good scale is NOT a trivial matter – I’ll make another post on this at the moment, directed at other tutors.

In this case we can use all the paper by using a scale of 1cm = 6m – Note that if we think centimetres for metres, 150 = 6 x 25 and 120 = 6 x 20.   I’d advise against this. In fact I’d advise against any scale that uses factor not based on 5, 2 or 1 (That is 50, 500, 0.5, 20, 200, 0.2, 10, 100 and so on).  These scales make the Maths much easier to understand, both in making the drawing and interpreting it.

I say this from experience without wanting to justify it much further now. Just think of times you have been abroad and the exchange rate is £1 = 60 of the local currency. The mental arithmetic working out how much you are spending becomes tricky.

I’ve waffled a lot today, lets get down to business.   I recommend a scale for this map with  1cm = 10m – 1:1000.

We would draw the full park as rectangle 15cm by 12cm. This only uses part of the paper but we do need space to a title, key and scale – and its better than the 1/4 of the space available I sometimes see.

We draw on the football pitch now. This will be 10cm by 6cm. Where we place it can be ‘trial an error. In fact, if you have scissors to hand (which won’t be likely in an exam!) we can cut out a rectangle with those dimensions and move it round the larger rectangle. We can also cut out a rectangle that is 2cm by 2cm for the play area.   The trees need to be 1cm from the edge and each other and at least 2cm from the football pitch. There is more than one solution but here is mine, with the distances shown in cm in my drawing, as your browser size won’t show the same distances
When you think you’ve finished its worth checking each of you positions and measurements again, to check they comply with the rules

Drawing our own map

In my last post I describe how we could get information about distances from a map using a scale. In this post we will look at how we can use the idea of a scale to draw a map of our own

Think of the following question : Source, my own imagination but I have seen similar questions in Exam papers

My local park is 150m long and 120m wide. We need to plan the park to have one football pitch (100m by 60m), a play area for younger children (20m by 20m) and places to plant 5 trees. Each tree should be at least 10m from the edge of the park and 20m from the football pitch. There should be 10m at least between each item.

We are given a sheet of paper measuring 25cm by 20cm.

See if you can have a go. The answer will be posted tomorrow.

So how do Maps work?

Since I’ve spent a weekend studying maps, I thought a quick post on the maths involved would be in order. Although it might not be immediately obvious, what we are looking at are Ratios – and this can be a pleasant change from mixing paint which is what many questions in Exams about ratio seem to be about!

Now here is a map of one of my favourite places in the world – and comments from anyone who agrees are welcome!  The point is, a map represents the place you want to visit so obviously has to be a lot smaller than the place itself!  (Ok so this reminds me of a favourite Blackadder joke but lets not get off the point)

The map shown here is to the scale 1:25000 if you have it before you. (I can’t make a similar claim from the picture you can see, as that will depend on the size of your browser!)

If you measure on the map that you are 2cm from the car park and pub*, how far do have left to walk?

2 * 25000 = 50000cm.  Which is great but we don’t usually measure walking distances in centimetres. 50000cm = 500m or 0.5km

Activity  : Find a map of your town and measure the distance to a place your often visit.  Don’t assume your map has the scale 1:25000.  The scale for your map will be written somewhere, perhaps even the front of the map.

*Actually the first pub I ever had a pint of beer!

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!)

What exactly is a number machine?

This is a question I asked myself when I returned to tutoring Maths a few years ago. Number machines often turn up in questions of Foundation papers. They provide a useful introduction to quite a few things

  • Algebra
  • Functions
  • Computer Programming – Which is what I was doing before I turned to Tutoring.

 

Here is an example Number Machine. The idea is we ‘feed in’ a number on the left, and see what comes out on the right.

A simple starter question would be : If a 4 in entered into this number machine, what would the result be. Number machines work left to right, just like reading. In this case, if the input is 4, we follow the boxes left to right and get  4 x 4 = 12 then + 2 = 14 .

A more advanced question would be – If the output is 8, what is the input.  This asks us to move right to left but also do the operations if reverse. Remember reverse of adding is taking away. The opposite of multiplying is dividing.

8 – reverse of + 2 is -2   so the number between the boxes is 6.  6 Divide by 3 is 2.  So the in number must be 2.

Try putting 2 into the machine and see how this is a reverse of what we did above.

The more complicated questions miss out the instructions. For example.

If we put 6 into this number machine we get 6 out. What is missing in the second box?

Clue – Its an subtraction sum.

I’ll show how to answer that in my next post.

 

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!

 

 

A short trip to Scotland

There is a story about mathematicians that I think I first read when I studying for A-level.

 

An astronomer, a physicist and a mathematician are on a train in Scotland.  The astronomer looks out of the window, sees a black sheep standing in a field. “All the sheep in Scotland are black!”, comments

“Oh no” says the physicist. “Only some Scottish sheep are black.”

This is the cue for the mathematician to get involved. “No,” he tells his friends. “In Scotland, there is at least one sheep, at least one side of which appears to be black, some of the time.”

I remembered this story yesterday when posting the problem about the dogs, which I reckon wasn’t written by the same mathematician. I’ll explain why in my next post, where I will also give the answer!

How many small dogs? A maths problem from the Independent

Sometimes the time I spend scouring Facebook pays off, and I find a maths problem posted with the a discussion that follows.

This one comes from the Independent’s Website and the story that goes with it says its for a 7 year old,  I’d be interested to see a seven year old tackle it – I have no doubt that a bright one could!  I’m going to share it will my year 10 (15 year old) student this afternoon, and I’ll report back tomorrow, with an answer!

A look at some new style questions

This is an article I wrote for my profile on Tutor Pages.

Some of this I may have covered in posts here before

As you are likely to know if you are reading this, the grading for GCSE has been changing, with the new method phased in between 2017 and 2019. Maths, my subject, was one of the first to ‘move over’ to the grades from 9 to 1 from A to G.
Unless a tutor has worked in schools during this period and so picked up on training there, he or she needs to do some homework on the changes. I don’t claim to have done all the work I need to on this, but I thought I would share some observations on how exam questions have changed, specifically in Maths.
The change from letters to numbers is not just cosmetic. The distribution of the grades will be different. Under the old system, grades A to C were considered as the old fashioned ‘pass’. When I took my GCSE course in the 1980s, this was shortly after the GCSE qualification had been created from a merger of GCE and CSE. My lecturers on that course were keen to say that all ‘letters’ were a sign of attainment – a student who gained a D had found some level of success – but the idea that one should ‘pass’ – and that A, B and C were the pass grades – persisted. It’s commonly held that you ‘have a GCSE’ if you attained C or higher.  I’m not saying that is good or bad, but I was sure this would happen even while being told otherwise by my lecturers.

A notable feature of the 9 to 1 system is there are more pass grades.  A grade 4 or above will be considered a ‘pass’ – which is not to say that a pass will be easier to achieve.  A 4 will be roughly equivalent to a C, though the grade boundaries are not exactly the same.  It’s at the top end that the change is greatest, with a 7, 8 or 9 all corresponding to an A/A* grade, and differentiation between the brighter students that provides the reasoning behind the change.

For this to be achieved there needs to be an increase in the number of truly challenging questions on exam papers. There will be more questions that students targeting a respectable 4, 5 or 6 will find it difficult to attempt. There is also a different style of question even in the 4/5/6 range of question that will reward not only hard working ‘book workers’ but encourage a depth of understanding and problem solving skills.

Does this make the exams harder? Arguably so.  But if prepared for properly by educators, will raise the level of ability as related to grade.

I’d like to look at a few examples to support my point – though I’ll add here as an aside, for the non school based tutor, the change has the unwanted side effect that we won’t have the bank of old paper to use. This will of course, improve with time. Exams tend to be made available 15 months or so after they are ‘live’, and some boards have published free samples.

This is a favourite question of mine for working with students is this one – taken from AQA’s sample

 

Calculation of surface area and volume isn’t the simplest subject area on the GCSE Maths syllabus, but neither is it the most complex. But this question really tests the exam takers reasoning powers. On the old A-G papers, this might have been ‘Find the volume of a cube with surface area 150cm2’. This question puts the student in a new role, which might be unfamiliar if their tutor has not prepared well.

Actually, role playing aside, I wouldn’t recommend any but the brightest student not to just solve the problem themselves first, and compare their approach with Steph’s. She has made a mistake, and one becomes clear when one has worked out one’s own method. But it’s essential to stress that the answer to the question is ‘Steph is wrong because….’ – not the students own answer to the underlying problem

This is another question designed to strech the student. The three marks for part a can be earnt by understanding how to estimate.  The extra mark requires an understanding of exactly what they did in part a. It’s a deeper level of knowledge, and shows that grades 8 and 9 are there if one is truly at home with numbers.

Its not all reasoning questions that make the new exams more challenging.; I have noticed an extension of the syllabus. I don’t remember any question using the language of functions when I taught GCSE in 2013-14, but this question is on the same sample as Hannah above

I have found my able Year 10 student lapped this up – but for a student aiming for a 4 to 6 grade, I’d probably leave this alone.

Rules for spotting factors

We often need to spot the factors for a higher number. How can we do this without doing the division sums?

For some numbers, its easy to spot. How can you tell if 5 is a factor of your number?

That’s easy. If the last digit of the number is a 0 or 5, then 5 is a factor. If it doesn’t, then 5 is not a factor

5 is a factor of 670 and 1225. It is not a factor of 234 or 1352.

Is 2 a factor? Thats easy in a similar way. The last digit needs to be a 0, 2, 4, 6 or 8.  The even numbers, of course. 2 is a factor of even numbers. Its not a factor of odd numbers.

Is 3 a factor?  Or 9?  For these possible factors we don’t just look at the last digit, but we add up all the digits in the number. If they come to a known multiple of 3, then 3 is a factor. If they come to a known factor of 9 then 9 is a factor.

So 4524 – add up the digits  4 + 5 + 2 + 4 = 15 – 15 is 3 x 5 so 3 is a factor of 4524, but 9 is not.

But 4527  4 + 5 + 2 + 7 = 18  and 9 is a factor of 18, so it is also a factor of 4527.

Note the pattern also holds for 15 and 18 – I used our knowledge of 3 x 5 = 15 and 2 x 9 = 18 above,  but also 1 + 5 = 6 and 1 + 8 = 9.

 

How do I know if 6 is a factor?  use both riles above.  If 2 is a facor and 3 is a factor, then so is 6.