Maths Games – Monopoly

When our children were small, we had a couple of games supposedly designed to help them learn maths. This really meant teaching them arithmetic and tables, which neither of them ever had any difficulty with anyway. And the games were DULL… They might have seemed fun in comparison to school activities, but they weren’t much of an attraction at home. So they didn’t get played often and long ago went to the jumble sale.

Any game involving dice helps with counting, but I particularly like playing Monopoly with kids. The rules aren’t too complicated, you have to add and subtract all kinds of numbers and work with money, you can play with three people or with lots, and almost everyone has some idea how to play. And, most importantly, it’s FUN. There’s enough going on to keep an adult interested, but it’s still worth playing if all you do is roll the dice and see what happens. Great for all ages from six (with a little help) to eighty six (or a little older), and a much better arithmetic workout than any of the so-called ‘educational’ games I know.

The main criticism that can be levelled at Monopoly is that it takes a long time. This is fair enough if you play to the last person standing, but is easily countered by having a house rule to end the game. When playing just with grown ups, we stop when the first player goes bust. With children, we usually use the dice from the ‘Lord of the Rings’ version on the standard British board and stop when the Ring reaches Mount Mayfair. (I don’t like the LotR board, or the way the Chance and Community Chest cards have been adapted. I’m not opposed to innovations – just to bad ones).

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The LotR dice. One of the 1s is replaced by the Ring symbol.  The Ring starts at ‘Go’ and moves along one property every time the Ring symbol is thrown. You could do the same thing yourself by having two different coloured dice and a rule like ‘Ring moves on red six’. I’m sure you can find a ring somewhere.

Monopoly also contains some nice probability lessons based on the distribution of scores obtained by adding two ordinary dice. There are 36 ways the dice can land [(1,1) (1,2) (2,1)…… all the way to (6,6)], and eleven possible scores from 2 to 12. The most likely score is seven which can be obtained in six ways [(1,6) (6,1) (2,5) (5,2) (4,3) (3,4)]. The next most likely are six and eight (five ways each), then five and nine (four ways each) and so on.

Now, consider which square on the board you are most likely to start your turn on. They are not equally likely, and here’s a picture to help you.

image

Yes, it’s Jail. You can end your turn (and so start the next one) there by:

  • landing there

  • landing on ‘Go to Jail’

  • throwing three doubles

  • drawing a card

Over the course of a game you are more likely to start a turn by moving off this square than any other. And your most likely throw is a seven… which puts you on Community Chest on our board. But three of the four next most likely throws put you on the orange properties:

  • Bow Street needs a six (5 ways of getting this)

  • Marlborough Street needs an eight (also 5 ways)

  • Vine Street needs a nine (4 ways)

So 14 of the 36 ways the dice can land put you on an orange square if you start from Jail. That’s a better than 38% chance… and you’ll be leaving the Jail square more than any other. So as a first approximation, it looks like the orange properties are the ones to go for when you’re trading cards.

That’s not the whole story, and you will find lots more at sites like these:

http://www.retroactive-vintage-games.com/games-articles/best-monopoly-properties.asp

https://rijusarkar.wordpress.com/2013/09/04/probability-distribution-on-monopoly-squares-using-python/

Alternatively, you could get the family together, crack open a tin of sweets and have a game…!

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Straight Lines Don’t Exist

No post for a couple of weeks ….. we’ve been busy painting the kitchen. And part of my mission was to get a straight line in the angle where the ceiling slopes down to the wall. The trick is to use masking tape, and take it off immediately after painting the line so the paint has no chance to bleed underneath. It looks all right, from a few feet away!

image

While I was doing this (and between wondering why we didn’t just paint the whole thing white), I started pondering what a straight line is, anyway.

In one sense, we all know – it’s that thing you draw with a ruler and pencil for graph axes, triangles and so on. But if your school ruler is anything like mine was, it will get tapped on the edge of the desk, used for sword fighting and flicking paper pellets, wedged into gaps just to see if it will fit… The one thing it won’t have is a straight edge.

This is true of all straight lines we draw in the real world, however careful we are and however well we look after our equipment. At some level, visible or microscopic, they are not straight. They’re just straight enough for whatever it is we want them to do.

We also talk about going straight to somewhere, which usually means we take the most direct route (and don’t stop off for coffee and cake on the way). This makes sense: the shortest distance between two points is a straight line, isn’t it?

Well, yes, it is – on a plane surface (that is, a surface that does not curve in any direction). But that’s not much use to us Earthlings, inhabiting as we do the surface of a sphere (or a spheroid if you want to be pedantic about it). The shortest distance between two points on the surface of a sphere is a segment of a circle whose centre is the centre of the sphere (a Great Circle is the term used in connection with the Earth, even though it is not quite a sphere). This line is, of course, curved and is often referred to as a geodesic – a term more generally used in mathematics to mean the equivalent of a straight line in curved space.

And here’s a thought experiment. Imagine a spherical Earth, then draw a line due south from the north pole until you reach the equator. Turn through 90o and draw a line along the equator until you have gone 90o west. Then turn north again and draw a third line back to the north pole. This will make an angle of 90o with our first line. You’ve just created a triangle with three right angles – 270o! Who says all triangles have 180o? Curved surfaces are different!

If you want to know more about spherical geometry, there’s a lot on the web.  This is a good place to start:

http://euler.slu.edu/escher/index.php/Spherical_Geometry

And thinking of curves, if you are fond of calculus you might like to consider a straight line as a curve with zero curvature. In other words, the second derivative – the rate of change of gradient – is zero. And if the gradient is unchanging, the curve…. isn’t curved!

My head hurts. Here are some lines I made yesterday. They’re not straight, and it doesn’t matter in the least.

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Checking – Catching Mistakes Before They Catch You

Everybody makes mistakes from time to time. It’s one of the ways we learn. It’s also something that distinguishes us from computers. When I was a very junior programmer, one of the team leaders in the office said to me, “The machine is doing exactly what you told it to do, lad. It’s just not doing what you thought you told it to do.” I repeat, everyone makes mistakes…

This week’s post is about ways of checking your work. Some of them give you an absolute guarantee that you are right. Some of them merely show that an error exists. They’re worth doing, either way.

Mental Arithmetic

Use these when you have to work it out by hand, or as checks against typos when using a calculator.

Last digit checks can highlight errors in multiplication. The most basic comes from the simple fact that you only get an odd answer when you multiply odd numbers. However big the multiplication, a single even number makes the answer even. One factor of two is all it takes… (sounds like an advertising slogan). So if you multiply a string of odd numbers and get an even answer (or vice versa), something has gone wrong.

A more sophisticated last digit multiplication check is the units make the units. For example: 27346255346277347 x 38482942904939 must have an answer ending in a 3. This is because 9 x 7 = 63 and although the 60 gets mixed in with the other tens, nothing else can affect the 3 in the units column. Try it with smaller numbers and you will see what I mean.

Ballpark estimates are always a good idea. See if your answer seems ‘about right’ by rounding to simple whole numbers. In particular, take π as 3, and gravity as 10 ms-2. In a recent lesson we looked at a question involving a stone being thrown vertically downwards from the top of a tower with initial speed 4ms-1, and taking 2 seconds to reach the ground. The problem was to find the height of the tower.

This is easy enough using s = ut + ½at2 = (4 x 2) + (½ x 9.8 x 22 ) = 27.6 m

But if you miss out the decimal point in gravity (and use 98) you get 276m as the answer. This is a tall tower, but not impossibly so.

The ballpark check 4×2 = 8, plus ½ x 10 x 4 = 20, giving 28 will highlight the error faster than you can write it down (or the stone takes to fall).

Counting minuses in multiplication (or division) is a simple way of making sure you haven’t got the sign wrong. If you have an even number of minuses, the answer is positive. If you have an odd number of minuses, the answer is negative. Easy!

Counting decimal places is worthwhile whenever you have to multiply decimals. The rule is that you always end up with the same number of d.p. that you start with. This may include trailing zeros. Here are some examples:

0.5 x 0.5 = 0.25 2 dp on either side

0.24 x 0.45 = 0.1080 4 dp on either side.           (Calculators just show 0.108)

0.02 x 0.0004 = 0.000008 5 dp on either side

In the last example, counting the d.p. on the left hand side is a quick way of determining that we need five leading zeros to bring the total d.p. to 6.

Calculator checking

Comparing answers. If you are working from a text book and checking your answers against the ones in the back, you sometimes get a situation where the forms of the answers differs.

For example you might have 0.805612004, and the book has 3e-5 + π/4.

A quick way of checking is to type in the book version, followed by divide and ANS (the button that recalls the answer you just had). If you are right, the new answer will be 1. You could make the same test by subtracting (answer = 0 if correct) but I always use division. No particular reason.

Simplifying algebra From time to time you may have to ‘simplify’ horrible pieces of algebra. For example:

Selection_009

To check this, pick a couple of easy values for x and y (Don’t choose anything that will lead you to divide by zero!). So with x = 1 and y = 2, we have:

Selection_010which gives Selection_011

It only takes a few seconds to type this into your calculator using the fractions button. You could even do it by comparing answers by entering:

Selection_012 (there’s now a minus in the middle)

This gives an answer of zero straight away.

Checking solutions to equations is easy with a calculator if you use the memory to hold variables. (I am assuming you have a typical school scientific calculator such as a Casio fx-83GT).

Step 1 Assign the value you want to check to the variable X by entering the value then pressing shift, RCL, ). For example:

7 Shift RCL ) sets X to 7.

You don’t need to press =. The display shows 7X at the top. Any previous X value is overwritten.

Step 2 Enter your expression, using the Alpha button whenever you want X.

Alpha ) x2 – 2 Alpha ) + 3 gives you x2 + 2x + 3

Step 3 Now press =. The calculator works out 72 – (2×7) + 3 to give 38.

Step 4 Now press AC, then enter the next value you want to test. For example:

5 Shift RCL ) sets X to 5.

Step 5 Press the up arrow. (This scrolls you back through previous actions). The expression x2 + 2x + 3 will reappear. Now press = again, to get the answer with X = 5.

You can do this as many times as you want, and if you are dealing with awkward decimals or surds it is quicker than typing them into the expression directly. It’s probably overkill at GCSE but can be a great help at A level where you often have equations with multiple solutions. (Tip If you aren’t sure of the current value of X, press Alpha ) = to display it.)

These are just some useful checking tricks. Maybe you have your own favourites you would like to share.

Calculators can bite!

Calculators have become as ubiquitous as pencils, and you have to be at least as old as I am to remember a time when they were an expensive luxury that you weren’t allowed to use in school – like mobile phones were only a few years ago! However, also like phones, most people don’t really know half the things their calculator does – and even when you think you do, they can still catch you out, as I discovered yesterday in a lesson. (Toby, if you’re out there this one’s for you!)

So here are a few things you might want to remember. They refer to the standard Casio fx-83 machines that are commonest in schools these days. (Mine is an fx-83GT PLUS).

1. The manual

Don’t throw it away with the packaging! It’s not the easiest thing to read, but it is useful, especially when your best friend has switched your calculator into a mode you don’t understand and can’t get it back.

If you have thrown it away, or the dog ate it as a side dish to your homework, all is not lost. You can download Casio manuals from here:

http://support.casio.com/en/manual/manuallist.php?cid=004

2. Modes

Most of the time, you will want mode 1 (COMP) for ordinary computation. Press the MODE button, then choose option 1.

The TABLE mode (mode 3) is useful if you have to work out lots of values to plot a graph of an equation. f(x) = means the same as y = for this. Enter the equation using the ALPHA button followed by the right bracket button to get an x when you need one. Then specify the start and end values you want, and the step value (that’s the gap between your x values), press enter and you will have a table of x and y values all ready to plot. And remember to put the calculator back into COMP mode when you’re done.

3. Setup

Enter the setup menu by pressing SHIFT then MODE. Options 1 and 2 let you choose how the calculator displays fractions, surds and so on. You will probably prefer option 1 (MthIO), followed by output option 1 (MathO). These let you enter fractions the way they look on paper, and display results as fractions, surds and multiples of Pi whenever possible. To get the decimal equivalent, press the SD button.

You shouldn’t need the other setup options for GCSE, unless somebody (perhaps a friend in the sixth form) has changed the way your calculator handles angles. You want option 3 for degrees. A letter D appears at the top of the display when this has been selected.

4. Recurring decimals

If you enter 1/7 then press = , the display will show 0.142857 with dots over the 1 and the 7 to show that these digits recur. (Pressing = again gives as many digits as the display will hold, with the last digit rounded. A third press gets you back to the fraction).

You can also enter a recurring decimal by hand, by pressing SHIFT followed by the x2 button when you want to enter the repeated digits. You get a box with the dot over it, and type the numbers you want into it. Typing 0 . SHIFT x2 142857 = will give an answer of 1/7. This is great for checking those ‘recurring decimal to fraction’ problems, but won’t get you any marks in the exam, as you have to show your working.

Very large recurring sequences may not show the right hand dot straight away. For example 1/17 displays 0.0588235294117 with a dot over the first decimal place and an arrow pointing right after the 7. Pressing the right arrow key three times displays an extra 647 with the right hand dot over the final digit.

5. Factors

You can express an integer as a product of prime factors by pressing SHIFT followed by the ‘degrees, minutes, seconds’ button underneath the square root button. For example, 47952 is expressed as 24 x 34 x 37. For some reason, the calculator only factorises answers, so you need to type in the number, press = and then factorise it.

(In passing, it mystifies me that every calculator I have owned in the last 20 years has the ‘degrees, minutes, seconds’ button, which I have never used (not once!) and yet requires me to press SHIFT every time I want the Pi button. Sharp make calculators with a top level Pi button, but for some reason Casio choose not to…)

6. Squaring negative numbers

Calculators do BIDMAS! That is, when you enter something they look at brackets first, then indices, then division, multiplication, addition and subtraction.

This means that when you enter -3 then press x2 = you get an answer of -9.

You did know that’s wrong, didn’t you? (A minus times a minus makes a plus?)

It only does this when you actually type in the negative number. (It ‘knows’ the sign of the answer to something like 4 – 7 and will square it correctly.) So to make sure you aren’t embarrassed by a negative square:

a) Type in the number, press =, then square it.

b) Put brackets round the number, then square.

c) Don’t put the minus sign in at all!! After all, you know the answer should be

positive. Save keystrokes, save energy.

7. Mixed fractions

The best way to enter these is to press the shift button followed by the fractions button (in MthIO mode), then type the numbers into the boxes.

e.g for 2 ½ press shift, fraction, 2, right arrow, 1, down arrow, 2

This gives you the correct decimal value of 2.5 when you press = and SD.

The way NOT to do it is to type the fraction, then backspace and enter the whole number to the left of it.

e.g fraction, 1, down arrow, 2

then either right arrow twice or left arrow 4 times to get the cursor on the left side,

then 2 for the whole number

If you now press =, the answer is 1.

The calculator has interpreted what you did as 2 x ½ !!

I have no idea why this happens, but we had a fun time in yesterday’s lesson figuring out exactly what was going on….

Don’t do it!

Surds are pointless, right?

September: season of mists and mellow whatsisname… And surds revision for students beginning A level maths.

GCSE pupils hate surds, and are horrified to discover that once you are in the sixth form you are expected to use the blasted things all the time. How ridiculous is that? I mean, who cares that 12 = 23 ? Who would ever use something that stupid in real life? They’re worse than fractions!

It’s a fair argument. We have phones that work as calculators these days, and tablets that work as calculators, and computers…. oh, and calculators as well. If I need the value of 12, a few keystrokes give me 3.464101615, which is more than enough precision for any practical purpose. So what’s the point of surds?

The point is, of course, that there is no point. No decimal point. 12 is an irrational number, which means that it can’t be written as a fraction and its decimal expansion goes on forever without repeating. So 3.464101615 is only an approximate value for 12, good enough for government work (to quote the great uncle of a friend who surveyed railroads in the Amercan west) but WRONG! If you square it on your calculator, the answer will be 12 (it is on all mine), but this is because the calculator is rounding 11.999999999045608225 to 10 significant figures.

(If you want to try full precision multiplication like this, a good resource is at https://www.mathsisfun.com/calculator-precision.html Or you could try it with a pencil and paper…)

Using surds saves us from unnecessary rounding errors. Suppose we approximate 12 as 3.464 to save space.

Then 312 x 712 = 3( 3.464) x 7( 3.464) = 251.985216

But 12 x 12 = 12, so the correct answer is 3 x 7 x 12 = 252….

AND it’s actually easier to work this out than to get the approximate answer with a calculator. And less error prone – the more buttons you press, the more mistakes you make.

KEY FACT

12 is the number which when squared gives us 12.

It doesn’t matter that we don’t know exactly what the value is, because whenever we square it the surd vanishes and we get a proper number, with NO errors.

We also get better answers by rationalising the denominator of a surd fraction.

For instance: Selection_188 is nasty to work out as it stands.

We have a square root which we can only evaluate approximately and, worse, we have to divide by it. That’s two potential sources of error.

By rationalising the denominator to obtain Selection_189

we still have the surd on top but now we only have to divide by 4, which gives us Selection_190as the answer. Easier and safer.

(If you don’t know how to rationalise the denominator, there are lots of good videos on You Tube, such as https://www.youtube.com/watch?v=T_cZqeLNCHM to show you how it works.)

The same principle applies to ‘exact values’ calculations involving π. If you ask most children what π is, they are likely to give the approximate value 3.14. In fact:

π is the number you get when you divide the circumference of a circle by its diameter.

If we leave the symbol for π in our calculations, and we get lucky, it will cancel out. At worst, we carry out one computation with 3.14 (or whatever) at the end. Either way, we simplify the calculation hugely. Again, fewer keystrokes = fewer errors.

So learn to love surds, and be right more often.

(PS  I’m new to WordPress.  I’m sure there has to be a better way of incorporating fractions that bringing them over from Libre Office as images.  I’ll look into it…)