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How to construct a ….

When I was teaching at the local FE college, there was one part of the GCSE where I sort of cheated,,,  I let YouTube do the work for me!

This was the set of skills called ‘Constructions’ – which can be hard to recreate on a whiteboard, especially when was one my ‘Motor Skills’…

The premise of these skills is that the student needs to

  • Construct a perpendicular bisector of a line
  • Construct a perpendicular to a line from any given point
  • Construct a line that bisects an angle

The extra catch is these have to be done using a pair of compasses and a straight edge only

There is something a little ‘old fashioned’ about these skills, but they remain on the GCSE Maths syllabus. I’ll also add that I haven’t seen them as much on recent exams papers – Maybe once at most across all three appears for any given season…

For all that they are rather fun to do and understand.

They are hard to do in front of a class; and on a blog too!   So let me give you some links to show how to do them.

 

Angling for some Fun – Part two

In the last post I started to describe how patterns in angles can help us find the size of missing angles without doing any measuring. In this post I am continuing with that theme. First I will show what we mean by Corresponding and Alternate angles.

Then I will show how we can put all this together to answer a question.

For both of these situation, we need one line crossing two others which are parallel.  If we don’t know the two lines crossed are parallel, we can’t use these rules, got it?

And you know how to tell if two lines are parallel? The two lines are never going to meet, however far we extend them.

So, the two angles marked in red – they are going to be equal.  So if we are told the lower one is 50˚  then the higher one is also 50˚.

These is what are called ‘Corresponding angles’

This is like the rule in the last post where we know two angles must be the same if they are ‘opposite’. Angles are also the same if the are ‘corresponding’

Now look at this diagram.

The two red angles are ‘corresponding’ and the green angle is ‘opposite’ one of the red angles.

So all three angles will be the same.

We say the bottom Red angle and the green angle are ‘Alternate’.  ‘Alternate’ is like a combination of the ‘Corresponding’ rule and the ‘Opposite’ rule.

Some people think of this as the Z rule – because the angles in a Z are the same.

I’ve been looking for an exam question that uses all of these but they are hard to find, and I only want one for this post, which would be too long otherwise…  so…  let’s have a look at this

 

A few points about the wording and notation here
1. I across two lines shows those lines are of equal length : AB = AC
2. > on two lines shows they are parallel. AP is parallel to CB
3. ‘Diagram Not Drawn to scale’ means take the information given as true. Don’t check them with your ruler or protractor.

ABC is an isosceles triangle, so <ACB = <CAB = 70˚.  < ABC = 180 – 2 x 70 = 40˚ – because of angles in Triangle ABC add to 180˚

<BAP = <ABC because they are Alternate – (to see the Z shape turn it round a bit)

So x = 40.

Angling for some fun? Part One.

Not a post about fish!  But about Angles….

There are a lot of questions you will see that ask you to find the size of an angle, or to show that two angles are the same.

Answering these questions, you also have to give reasons, and in giving reasons you need the language for describing different patterns with angles, and different ways of how you can find the size of angles from other angles.

So over this weekend I’m going to post some of these rules, and the names you need to give while describing those rules…..

And I’ll be honest, I have to look some of them up!  Not the rules themselves – I know all of them – but some of the names for them.

After these posts, I’ll give a couple of exam questions and show how to use the names of the rules in your explanations>

The first thing is to remember that there are 90˚ in a right angle and 180˚ in a straight line. How can we use that information to find missing angles?

  In all these three examples, we are looking to find the number of degrees in angle x.

In this case x and 55˚  make up a right angle, or 90˚ – so x = 35˚ due the ‘angles in a right angle’.  And that is what you should write, to show that the reason for the answer is understood.

 

In this example, x and 140˚ make up a straight line, which is 180˚.

So x = 40˚ due to ‘angles along a straight line’. Again, that is what you should write.

In this case, angles x and the one marked 47 are the same size. This is the law of opposite angles. You might be able to see that this is the angles on a straight line, applied twice, but remember the term ‘opposite angles’ and there no need to work out the other angles.  We write

x = 47˚ , opposite angles.

In my next post I will add alternate angles and corresponding angles to the mix. Then we will be ready to tackle a lot of angle questions.

Factors and Multiples – and a clever disguise

 

 

 

For today’s post I am going to look at the  exam question here.

You can see from the question that we are in the area of common factors and multiples, but the question does come with a bit of a disguise!

Look at that last line – We have been asked to find the highest value of a/b.   So how does a fraction have the highest value?

By making the top as high as possible, and the bottom as low as possible.

So a is the HIGHEST common factor of 72 and 96
and b is the LOWEST common Multiple of 6 and 9.

Its a question about our old friends LCM and HCM – we have see through the disguise!

Let’s find the answer now though. We start by giving our four naumbers in Prime Factor form.

9 = 3 x 3  = 32
6 = 2 x 3

72 = 2 x 2 x 2 x 3 x 3  = 23 x 32
120 = 2 x 2 x 2 x 3 x 5 = 23 x 3 x 5

For the LCM we take the largest power of each prime number included

LCM = 2 x 32 = 18 (=b)

For the HCF we take the largest power of each prime number included

HCF = 23 x 3 = 24 (=a)

So a/b  = 24/18 = 4/3 or 1 1/3

To Infinity and Beyond

I’ve been posting a lot recently on answers to GCSE questions – mainly because that is what I have been doing with students –  and I do have ten more days of that.

For today something different a look at something quite unusual…  As the title suggests – INFINITY.

Today’s post is based on some of the work of German mathematician Georg Cantor, a man not often understand by his contemporaries – In fact one said he was ‘aha ed of his time by one hundred years.

What Cantor was trying to do was explore what infinity means, and his revolutionary idea was that there is more than one infinity!  In fact…  there are an infinite number of infinities!

I’m not planning to show where they all come from in this post – I’ll just show that there is more than one.

 

 

If you go out with six friends this week, how will you know there are six of them? Well, OK, that seems like a simple question – because you can count, right?. Yes, but understanding how we can count is the starting point of understanding where all the extra infinities come in. You know you have six friends with you because you have given each a number, starting with 1..  and you got up to six! OK so may not have done that explicitly, but that is what you did.

And that is what we are going to use in a moment to demonstrate infinities.

Two quantities are the same if you can match up one item from one ‘list’ with one ‘item’ of the other… and have none left over. So the numbers 1 to 6 can be matched with your 6 friends with no numbers left over.

Now, this is where we come to some surprising ‘facts’…..  Ask,  are there more whole numbers than ‘even’ whole numbers.  You think, yes? Well look at this…..

 

 

 

 

Even numbers can be paired up with whole numbers with no numbers from each group left out.  This is one of the surprising things we learn about infinity. The number of even numbers is the number all numbers.

Now imagine you are on a ‘chessboard’ floor that stretches for infinity in all directions.

How many squares are there. Infinity – or more than infinity. This, after all ‘infinity squared 

 

 

 

Let’s try counting the squares in the same way that we counted your friends.  Start with one square – it could be the one you are standing on. Thats 1.  Now count the 8 squares around you – 2,3,4… 8

Then the squares around that.

 You can keep going like that for ever  – but that’s OK, we can keep counting for ever.

So even on this infinite board, the number of squares is the same as the number of numbers – Infinity.

 

Now, at this point, if I didn’t know what was coming next, I might be starting to doubt my earlier remark – that there is more than one sort of infinity.

Lets consider now, decimal numbers, by which I mean numbers of the form

1.5434677654433345656

I have 20 digits after the pint there. Lets say we mean numbers that DO end, but we will not specify how many digits they have before they do.

So lets start with the number 1.5

Where do we go to next to count; 1.6  or 1.55? Or 1.5000000001.

There are actually an infinite number of next steps, and thats before we get off our first ‘square’.

And THAT is where the next order of Infinity comes in. Its not possible to ‘count’ the decimals  with the numbers 1, 2, 3 …….  even they go on for ‘infinity’.

There are more than infinity decimals!

Crazy Curves

One last post before I move on from ‘continuous’ curves….

I was going to include this in the last post, but that was already too long.

We have looked at curves which are continuous everywhere, and some which are not – but are continuous for most of the way.

Is it possible for a curve to be discontinuous everywhere?  In theory yes, though we need to consider rational and irrational numbers.

A rational number is any that can be written down, accurately, with numbers.  This includes numbers there are ‘recurring’ like 0.3333333 because this can be written as 1/3, and be accurate

Pi is an example of an irrational number

So if we say that y = f(x) where f(x) = 1 when x is rational and f(x) = 1/x where x is irrational…  then that would define a curve, but one that is so chopped up is would be continuous in only very small sections between rational numbers

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.

Let’s have a look at curves

I’ve been looking at exam questions a lot recently in this diary – well it is coming up to exam time!

In June, the focus is going to be on famous Mathematicians, Over the summer I’m looking to write some entries on mathematical puzzles – If you know some good ones please send them in!

Today though, I’m looking at curves – the lines we can draw to represent certain mathematical equations.

In particular, today and tomorrow I am writing about curves that are ‘continuous’ and curves which are not, which we call ‘discontinuous’. Also, what that means for how we can use these curves to answer questions – That bit is for tomorrow!

What do I mean by continuous? Basically this means that an ant can follow the curve and get to all parts of the curve, without flying off the page or leaving the line. There is a Mathematically precise way of defining ‘continuous’ but I think my ant gives you the idea.

Lets look at some examples. (For this I am going to borrow screenprints from my favourite websites, the link on the Links page)

y = x2

You can imagine the any being able to walk around this line. In fact any line of the form
Axa + Bxb + Cxc+ ….   will be continuous so long as a, b, c etc are positive integers.

 

 

 

 

y = sin (x)

 

And again the ant can walk up and down these curves.

 

Since the line for y = cos x looks similar, we can say that is continuous too.

Lets have a look at some Non-Continuous curves now, The easiest to show is  y = 1/x

This curve is in two parts. Our ant isn’t going to get from one part of the curve to the other without a jump.

 

 

 

 

 

Going back to our trigonometry, Tan (x)  – unlike Sine and cosine – is discontinuous. There are many and regular breaks in the line.

 

 

In tomorrow’s post I will look at some more eccentric examples and show why its important to understand if a curve is continuous or not.

 

 

 

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.