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How do we combine powers

In today’s post I will be looking at how to Combine Powers. By a ‘Power’ I mean that little number you sometimes see at the top right of a number.

So – What does it mean?

32  is another way of writing 3 x 3.  ‘Squares’ are quite familiar. But we can extend this idea

56  =  5 x 5 x 5 x 5 x 5 x 5 –  count the 5s – there are 6 of them!

And with that, you can probably how to see how any other ‘power’ calculation can be worked out, like 23150 or 31245 – though you’ll understand if I don’t write those out in full!

Answers to ‘power’ sums can get very big!

By the way, sometimes you will see the word ‘index’ and that means the same thing. And sometimes its called ‘order’, which explains why it is an O in the acronym BODMAS.  If you are not sure what I mean by BODMAS, check here

How to Combine Powerspower-pic

Once we understand how something is written in maths, the next step is to see how we can combine powers. How does this idea with things we already know?

For example – what does it mean if we write  63  x 64

The easiest way to see how to make sense of that is the first write this out in full

6 x 6 x 6    x   6 x 6 x 6 x 6  –    and now we have 7 6s  – all times together –  So we can write this as 67

63  x 64 =  67

Now, we don’t want to write things out in full every time, so lets look at what we have really done. Then we can see a short cut.

3 + 4 = 7!  So we can see a rule that might come from this. If you want to multiply power, just add the powers together.

Note that we can only do this if the bigger number is the same. We can’t add the powers and get any sensible answer if we try

54 x  73 – where the 5 and 7 are different. There is a way I would simplify that but I won’t look into that now.

Some special cases

One last thing for today,  I’d like to consider what 21, 20 and 2-1 all mean.

What does it mean to say ‘Multiply 2 and 2 and 2… -1 times! Doesn’t seem to make much sense, does it? But we can see how these things can mean something if we look again at our rule to combine powers.

So that’s not what we do, but we do want our rule of Index Adding to mean multiplying to still work.

So lets look at

22 x 21 = 23

and 22 x 20 = 22

and 21 x 2-1 =20

The answers I’ve shown here have been worked out by the ‘adding powers’ rule – e.g. 2 + 1 = 3

The first one is the easiest to explain –
2 x 2    – how do we get that to 2 x 2 x 2 ?  By multiplying by 2 again!

so 21 = 2. That makes sense if you can get the sentence  2 multiplied together 1 time!

But what does it mean to say ‘2 Multiplied together 0 times’?

22 x 20 = 22 – But what number doesn’t change others in multiply sums?

The only number that does that is 1…  so it can only make sense that 20 -1

In fact, this is the rule for all numbers 50 = 1 430 = 1 – and so on.

This takes me to my special case, which complete the picture on how to use powers.

21 x 2-1 =20

Using the other two cases we can rewrite this as

2 x 2-1 = 1

from that we can see 2-1 = 1/2 – because thats the only number that completes this sum

We can extend that idea to say any ‘-‘ power  – just put the number on the bottom of a fraction with 1 on the top

So 14-12 = 1/1412

Read the Question

My second post on avoiding and correcting errors seems a bit obvious! Its very important to actually read the question. But it doesn’t stop many people, including myself from forgetting this ‘tip’ from time to time!

It’s time to read the question!

I’ll show this by an example from the exam paper I’ve just completed.

BarChart
Bar chart

The first part I answered correctly – see if you can!

Spot the difference!

Its the second part I am writing about here.  Now I know why I made the mistake. I’d tutored students many times before on questions like this – or so I thought – and I wanted to show off!

I read the question as ‘What was the difference in Monthly average expenditure’ between the two years. I have seen students answer this question before – when it was the question asked – by reading off all the data, adding all the numbers together and comparing.

And I usually tipped the more able students that they don’t have to do this. Its quicker to find the difference in each column and add these up. So that is what I did now….

Have you seen this question before? Are you Sure?

Thinking you have seen the question before, like this, is a common reason for misreading – And its an error that is going to effect more able students, or at least ones with a good memory!

But this won’t help when the marks are added up.  Its the questions where you think ‘Oh, I know what to do with this one’ that you need to be most careful with, because they are the ones you are more likely to rush into without proper reading

OH, and have you spotted the important detail I missed in the question?

You will see I did too much work, and would have got no credit for it.

If you’d like to see my other post on very basic errors, then follow this link

 

When the answer just doesn’t seem right : Correcting Maths Mistakes

I’ve completed  a couple of Foundation exam papers this week, as a bit of forward planning for one of my students – and when I came too compare my answers with the official ones,When I looked at my answers I saw I’d made some mistakes, so this is an entry about correcting maths mistakes.

In this post, though, I’d like to share with you a question that I did get wrong initially, but where I spotted my own mistake.

The thing is – My answer just didn’t seem right.  And that’s what I’d like to share with you in this posts – It’s a very important skill with number questions; To be able to feel when your answer feels right.

This is the question

 

 

 

 

 

To make this comparison, you need to work out how much one biscuit cost.

For the 20 Biscuit tin the sum is  £1.50/20. Although this is from a calculator paper, I did this ‘long hand’ and got the answer 7.5p each.

Next I did the calculation for the second tin, again ‘long hand/in my head’  and got the answer 5p per biscuit.

I don’t know how I did this – I made a mistake, and I never pretend I never make mistakes. I just wasn’t taking care.

But what I can do is think ‘Um, that doesn’t seem right. The second box costs a bit more than the first, and has slightly more biscuits.’

The answer just had to be about the same, not as different a 5p and 7.5p. That might seem like a big difference, but 7.5p is 50% more.

So I did the answer again, and found the cost per biscuit was also 7.5p each. The answer was that Nada was wrong, box 2 offers the same value, not better.

This might not work every time – If my wrong answer had been 7.4p per biscuit I may not have spotted my mistake.

But you would be surprised how often just thinking as you write your answer ‘does this make sense’, you can spot some basic errors. Correcting Maths mistakes is essential if you are to get the grade you deserve : Don’t let it effect your grade.

Here is some more useful advise on avoiding errors

A little more on Higher Dimensions

Continuing yesterday’s theme of higher dimensions, in this post I look at a couple more 4 Dimension shapes and how they can be represented on a screen.

Thinking about how to show 4 dimensions its helpful to think of ways to show 3 dimensions on a 2D screen. One such way is to take ‘cross sections’.  For example, one way to show a cylinder is a series of circles.  Stack these together and you get a cylinder.

A cross section of a 4 Dimension shape will be a 3 Dimension shape,

Another way of thinking of this is to consider time as the forth dimension*.  Imagine seeing a sphere appearing as a dot, then growing to ‘full size’, then disappearing again at the same rate.  The diagram here would show stages of the process.

I’ve seen other representations of a ‘hypersphere’ but this is the one clearest to me.

How would a ‘Hypercube’ look like, using the same ‘cartoon technique?

(*Some people think this to be the case, through the 4-Dimension space-time physicists work with isn’t that simple, but it will do for this thought experiment).

Before we move off 4-Dimension shapes, I’d like share one of my favourite shapes. This can only exist if we have 4-Dimensions (The closest 3 dimension idea is a mobius strip)

In the picture, it looks like the bottle goes ‘through itself’. In the ‘4 dimensions’ this would not be the case.  Rather like if we want to get past a wall we step over it, using the third  dimension that a creature who knew only two dimensions could not

 

 

 

 

For more information on 4-dimensional shapes look here

Living on a higher dimension

Living on a higher dimension is something I am sure we’d all like to do; Well, if I meant a level of unbound wisdom; This is a Maths Diary, though, so I probably mean Dimension, as in shapes!

Maths students study shapes that have 2 or 3 dimensions.  The underlying maths regarding shape can be extended to more dimensions.

For example, we all know the area of a square is  L2 where L is the length of one of its sides   The volume of a cube is L3, again where L is the length of one of the sides.

 

A ‘Cube’ in higher dimensions

So what does L4 represent? Its definitely something we can write down, but does it have a meaning.  Its fair to say that, by extension this  would be the ‘amount of space’ occupied by an equal-lengthed shape in 4 Dimensions!

The difficulty lies in trying to relate that to what we know, as we don’t know 4 dimensions. A 4-dimension cube is often called a hypercube. Another name for it is a Tesseract – and that is a word I’ve only just learnt!

Below is representation of one, but there is a problem with trying to show 4 dimensions, using just a 2-dimension screen.  I’ve seen various ways of doing this, and the way I’m showing here is the clearest to me… Think of the ‘cube inside’ as being smaller only because its further away. Its really the same size.

 

 

 

To an extent we also had this problem in drawing the cube, as shown above. There we were trying to show three dimensions on a two dimension screen.  That was only a ‘gap’ of one extra dimension though, and we are familiar with what a cube looks like.

In my next post I will continue with this theme, and consider how we can show other 4 dimension shapes in two dimensions.

For more on Higher Dimensions, see my next diary entry

 

 

Is BODMAS for Life (Or just for Christmas)?

The question is –  Does the rule for order of calculation, BODMAS  always apply?

(OK so it’s a bit eccentric having a post about Christmas at the start of August, but I liked the title so I am sticking to it)

For a reminder of what the BODMAS rule is,  check here

Can we ever bend the BODMAS Rule?

Well the real answer is ‘yes’ – but should the rules should be bent sometimes?

This idea started with a post I saw on Facebook. I gave my initial answer yesterday. This morning I had to admit I got it wrong, if we follow BODMAS to the letter.

A BODMAS Example

The question is, simply, what is the value of

     

 

Add 2 + 2 to get 4; multiply by the 2 outside the bracket and get 8;  then 8 divided by 8…  we get the answer 1.

Actually, by BODMAS rules the divide should come before the multiply (D before M)   so it should be 8 divided by 2 (Giving 4)….  4 x 4 = 16

That is probably the ‘Correct’ answer and I had to accept I was wrong  – and there is nothing bad about accepting one is wrong sometimes

 

Why I might disagree?

 

But I still feel somewhat attached to my original answer!  To me,  2(2+2)  LOOKS like a single unit for calculation. If the x sign had been there between the first 2 and the ( , as below, I don’t think I’d have made the same mistake

 

For me, the () is such a powerful sign,  I see any digit next to it as ‘belonging’ to it, and hence how I did that calculation in the way I did. I can’t claim that is the official rule; just the way I read it.

So my recommendation is, BODMAS rules as they are, if you want to communicate a calculation, if there is any doubt on what you mean, include extra brackets to avoid confusion

 

The Maths of Temperature

Well, its hot here in the UK, and that may not make people think immediately of maths! But its a subject useful for every occasion. Today we look at the Maths of Temperature

So today I am writing about ‘temperature’ and

Temperature

how it helps us understand Maths.

Temperature is the sample I often use when explaining negative numbers to students.  It is one example were minus numbers are used in everyday life – but perhaps not in this weather!

 

Formulas used in temperature

After negative numbers, another interesting and useful part of the Maths of Temperature is using conversion formulas and rearranging formulas.

There are two temperature scales in common use; Celsius and Fahrenheit, both named after the scientist who invented them.

(We use abbreviations  ͦ C  and  ͦ F for these.  Some people think the first of these is called Centigrade. This is understandable because we started using  ͦ C in the UK about the same time we started using metric measurements like centimetres. But it’s more accurate to use the inventor’s name. The initial is just co-incidence)

We define both scales by the freezing and boiling points of water – 0 and 100 in Celsius; 32 and 212 in Fahrenheit. Quite why we use these numbers for Fahrenheit seems very strange, but its likely that wasn’t how Mr Fahrenheit came up with his scale.

We can use these numbers to find the formula to convert  ͦ C  to  ͦ F

Let’s say this formula is F(c) = ac + b – It is a linear function, we do know that

F(0) = a x 0 + b = 32 so b = 32

F(100) = a x 100 + 32 = 212

From this we can get 100a = 180  (I’ve simplified the a and taken 32 from both sides

This gives a = 9/5  (180/100 simplified)

So F = 9C/5 + 32 – which is the familiar formula.

Rearranging a formula.

That is useful for temperature in Fahrenheit if we know the temperature in Celsius.Thermometer

If we know the temperature in Fahrenheit but we would like to know it in Celsius, then we need to rearrange the formula. This is especially useful if we need to do this calculation several times.

F = 9/5 * C – 32

F + 32 = C * 9/5

C = (F – 32) * 5/9

This is a subject on which I helped a student once after his Science tutor said he wasn’t getting it. After working on it with me, the Science teacher noted the improvement.

In fact a study of rearranging a formula is not only useful in Maths. It is also useful in the study of the Sciences, especially Physics

This has been a short post, tidying up the last one.

Taking our Polygon formula to the limit

In yesterday’s post I considered how we can find the area of a many sided regular polygon. Today we will look at taking our Polygon formula to the limit.

Yesterday’s formula

So, given the number of sides and the length of each one.  We found the formula

Now let’s consider the Circumference

For this post I’m going to add Circumference into the mix. We know that word for circles but we can use it for polygons too – its just another word for perimeter – the distance all the way round the shape, the length L, S times

C = L x S

In the formula, we are now going to replace the L with a C, since the more sides we have, L is going to get small, and C is an easier thing to measure.

L = C/S is just a rearrangement of the formula above which we can use to replace L in our area formula

This gives a formula with C in of

A = C2/4STan(180/S)

We now have moved all mention of S to the bottom of our formula.

Now if C is kept the same but S gets bigger and bigger, what does that mean for our formula?

As S gets bigger. 180/S will get closer to 0, and so too will Tan(180/S), S will obviously get bigger, so what does that mean for  S x Tan(180/S).

What happens when S gets very big?

Now this is where I am going to cheat a little; It may be possible using mathematical techniques to see what happens as S gets bigger, but I am just going to plug some numbers in

S S x Tan(180/S)
10 3.249
50 3.146
100 3.143
1000 3.142

This shows that S has to get quite big before the pattern is clear, but it seems that
S x Tan(180/S) is getting close to a very familiar number, π.

So it seems for very large values of S

A = C2/4π

Remember that C = 2πr2 so C2 = 4π2r2

So, A = 4π2r2/4π

The 4π on the bottom cancels with elements on the top and we are left with

A = πr2

Which is the familiar area of a circle, and if you think about polygons with many, many sides you will see they are very close to being circles. That’s what we get by taking our our Polygon formula to the limit.

This is why I love maths! Everything fits together!

Area of a polygon

In today’s post I’ll be investing further the area of a Polygon. Is there a formula for the area of a regular Polygon!

What are the familiar Polygons?

If it is 3 sided or 4 sided – a triangle and a square – then we know the formula for area, but I was thinking – what about a formula that works for any regular polygon – That is to say, one with all the sides the same.

 

 

 

 

Here is a polygon and lets say the length of all the sides is L.  You can count the sides here and see there are 8 – this is an octagon  – but let this represent ‘any polygon’ with a number of sides S.

Read more about polygons and their angles here.

Area of a polygon : More general approach

I will be looking for is a formula where A = something with L and S in, as they are the two ‘properties’ of the polygon that might change in our ‘general’ polygon

The art of finding the area of any unfamiliar shape is to divide it into shapes for which you know how to find the area.  Any polygon can be divided into triangles by drawing lives from each corner into the middle.

How many triangles?  S  – One for each side of the polygon.

 

What is the area of each of these triangles?  For this we need to know the base and the height; The base is L. To find the height we will need to use some trigonometry. Half of the triangle is a right angled triangle. I am going to use the angle at the top of the half triangle, which I have labelled x

The full angle at the top is 1/S of the 360 degrees at the centre  =

360/S.   x is 1/2 of this  = 180/S

Tan(x) = L/2  /  h  (Opposite/Adjacent where the ‘opposite’ is half the base  and h is what we will call the height, for now.

Add in some detail and rearrange

Tan(180/S) = L/2h
Rearrange again to give h = L/2Tan(180/S).

Area of the small triangle is (using 1/2 x b x h)   = L2/4Tan(180/S).

This is only 1 triangle out of S triangles, so the formula for the whole polygon is

Checking my formula with some examples

Now, as I said at the start, I worked out this formula a week ago, bit I wanted to check my work because it looked a bit cumbersome….. But it does seem to stand up, and I’ll show you how.

There are two polygons for which we know another formula; The Triangle and the square (S = 3 and S = 4).  Lets see what happens if we make S= 4

A =  L2 x 4 / 4Tan(180/4)

Tan(180/4) = Tan(45) = 1 – so we can leave this term out. Also the 4 and 4 can cancel, and we get

A =  L2 – the simple and familiar formula for the area of a square!

Checking this for the triangle is more tricky because we to find the ‘normal’ area for a regular 3-polygon – or as we usually call it, an equilateral triangle.  For this we need our old friend, Pythagoras’ Theorem.

Using the theorem in half the triangle, we get

(L/2)2 + H2 = L2

H2 =  L2 – L2/4  = 3L2/4

H = √3L/2

so A = √3L2/4

Now lets see what we get from our polygon formula with S=3.

Tan(180/3) = Tan 60 = √3

A = (L2 x 3)/(4 x √3)

Remember that 3/√3 = √3

so A = √3L2/4

The same as with the direct method!

Conclusion

That, I claim, justifies my formula.  If anyone can think of an alternative way of finding the area of a pentagon or hexagon, then the formula can be checked for these shapes too.

Now, interesting things happen to our formula if S gets bigger and bigger, but that will have to wait for another post!

Just what is an external angle?

In yesterday’s post I wrote about how we can find the internal angle in regular polygons, and how the total of all the internal angles have the pattern 180, 360, 540, 720.   This patterns isn’t too hard to remember, but the pattern is even easier for the external angle.

What is an external angle?

Actually its easier to understand what is going on if we look at the external angles.

It’s important to know what the external angle is.  It is NOT the angle all the way round the outside

It is the angle between each line.. if it were drawn longer..  and the nest line round the shape in that direction.

It is useful to imagine you are walking around the shape. The external angle is the angle you  turn your body round at each corner.  Then once you have walked round the whole shape..  and ready to start again ..  you have turned the full 360°

So the size of each external angle = 360°/Number of angles.

The number of angles is the same as the number of sides, of course.

For more information on external angles, look here

Comparing the Angle Formulas

 

That is an easier formula than the one we saw for the internal angles, but I always get curious in these situations. We have two formulas…  do they work together?

For a given regular polygon lets say it has S sides (so also S corners, A internal angles and S external angles). Let A be the size* of each internal angle and X be the size* of each external angle.

*They will all be the same because this is a regular polygon. This sudden move into the language of algebra is because we don’t know how many sides our polygon has – we are looking at all polygons at the same time.

X = 360/S (Today’s formula)
A = (S-2) x 180/S (Yesterday’s formula for the size of each angle)

Also, X = 180 – A : The two angles make a straight line. Look at the diagram.

To show these formulas all say the same thing, we need to combine two of them and show we get the other one. This can be done a number of ways, but I’ll only show one here.

Take X = 180 – A  and substitute in the formula for A

X = 180 –  180(S-2)/S

X = (180S – 180(S-2))/S – I’ve made the whole equation ‘over S’ by including the first 180 in the fraction)

X = (180S – 180S + 360)/S  I’ve multiplied out the bracket on the top

X = 360/S – Because 180S – 180S = 0!
And so we get to the other formula for X