## A level lifts problems – doing what you feel

This has come up a couple of times in the past week and, as every year, seems to be more troublesome than it ought to.

So – how should you model the forces acting on a moving lift?

Newton’s Second Law tells us that if there is an acceleration, there is also a force acting in the same direction, and this is the key to the problem. Remember these three ideas:

If the lift is accelerating upwards, there is an overall force upwards.

If the lift is accelerating downwards, there is an overall force downwards.

If the lift is moving at constant speed, there is no overall force because there is no acceleration.

So far, so good…. But when do you need the tension in the cable and when do you need the reaction on the floor?

The answer is that you use the tension when you are considering the lift and its contents, and the reaction on the floor when considering the contents alone.

For the whole lift we have:

where M is the mass of the lift and m is the mass of the contents.

If the lift is accelerating upwards the overall upward force is T – (Mg + mg).

If the lift is accelerating downwards the overall downward force is (Mg + mg) – T

I’ve swapped the expressions round to make the force positive. You can, of course, stick to one form or the other in which case the force and acceleration will come out negative when the lift is accelerating the opposite way.

And at constant speed, Mg + mg = T as there is no acceleration.

For the contents, we only consider forces acting directly on them – the weight and the reaction at the lift floor (R ).

If the lift is accelerating upwards the overall upward force is R – mg

If the lift is accelerating downwards the overall downward force is mg – R

And at constant speed, mg = R

Deceleration is simply negative acceleration, so a downward deceleration is the same as an upward acceleration, and vice versa.

Still confused? Then use the way your stomach feels in a moving lift.

As you set off going up or slow down while descending, there is an upward acceleration. You feel heavier – and your stomach tugs downwards.

As you set off going down or slow while ascending, there is an downward acceleration. You feel lighter – and your stomach tugs upwards.

And at constant speed, you feel your normal weight as you are not experiencing any extra acceleration.

Gut feel – sometimes it really does give you the right answer!

## 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).

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.

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…!

## 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!

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.

## 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:

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:

which gives

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:

(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.

## Going Live With Linux

I’ve been pretty happy with Windows Vista.

There, I said it. Duck me in the village pond then burn me at the stake.

I bought a HP Pavilion dv9000 laptop with Vista preinstalled when it was brand new. I nearly had the shop give me XP instead, but then I thought about future support and upgrades and the hassle of keeping an old system going against the grain… So I accepted Vista and, after the ups and downs of the first few months, it has served me well. I’ve used the laptop for work and basic web surfing and shopping, and it’s still my typewriter of choice for large projects.

A year ago, I installed Pinguy Linux on the family desktop (in a partition alongside the kids’ old Vista machine, which nobody ever uses but I’m not allowed to get rid of) and since then I hardly connect the laptop to the internet. It’s still fine as a standalone office machine, but can be desperately slow on line. And then there’s all the endless updating. Pinguy has everything I need for day to day work, runs fast and updates in a couple of minutes if you do it regularly.

I tried running the laptop from my Pinguy installation disc in live mode. For anyone unfamiliar with the jargon, this means the operating system runs in RAM and nothing gets written to your computer drive. Great for testing. It worked, but not well. The laptop only has 2Mb of RAM, which isn’t enough for a full Pinguy system. My live/install version of Pinguy also lacks the Nvidia drivers the laptop needs, so I had to edit the boot parameters to force vesa. (I’m not going to explain this – there are lots of good threads on the web if you need help on editing the boot string in a Grub menu. Besides, the whole point of this article is to ensure you won’t have to!) I considered partitioning the hard drive, but was unable to free a decent amount of space without manually resizing partitions. I chickened out at this point – I have a clean and useful Vista system which I still use regularly for writing and things like Graph (https://www.padowan.dk/download/ ) that don’t run under Linux (I’m not a fan of Wine), and if it ain’t broke I prefer not to fix it.

At this point, I asked myself what I really wanted. The answers were:

1) Vista working on the laptop. (So leave it alone!)

2) The ability to connect the laptop to the internet without too much fuss and updating. (Won’t do this often, but never say never).

3) Getting Linux working on the laptop. (Pride!)

4) Trying a few other Linux distros. (I love Pinguy, but you can’t help being curious.) ‘Distro’ is shorthand for ‘distribution’ – in other words a particular bundle of Linux and associated software.

The obvious solution was to create a multiboot usb stick, with a selection of lightweight distros, and then have a play. The rest of this article describes what I did and compares some features of the distros I tried.

First a little technical caveat. My laptop is a HP Pavilion dv9000 with 2Mb of RAM, an AMD Athlon X2 64 bit processor and an Nvidia GeForce7150M/nForce630M graphics processor. And I’m an ordinary computer user, capable of reading a help thread and a manual, and typing commands at a terminal prompt, but no kind of tech guru. So there are probably lots of better ways of doing the job, and a different machine might need different things. This worked for me and I hope it works for you, but no guarantees, OK?

Choosing Distros

I wanted an operating system light enough to run in 2Mb of RAM and with appropriate Nvidia drivers installed as standard. It should allow me to connect to the internet (wirelessly if possible), check my email, edit a document locally or on the Cloud and watch video on You Tube. I expect to have to tweak a live system a little before actually using it, for example to turn on a firewall, but the less the better.

After reading lots of reviews (and this is my way of saying thanks, guys) I decided on the following:

Puppy Linux:    Puppy 5.2.8 Lucid from ibiblio.org via the link from

http://puppylinux.org/main/Long-Term-Supported%20Puppy.htm#lucidpuppy

Kanotix:   Kanotix Dragonfire LinuxTag2013 LXDE version from

http://www.kanotix.com/changelang-eng.html

PCLinuxOS.    The lxde 2014 version via the website

http://www.pclinuxos.com/get-pclinuxos/

(I used one of the Irish mirror sites for the download)

Lubuntu.     The 64 bit AMD version via

http://lubuntu.net/blog/lubuntu-1504-vivid-vervet-released

Porteus.     Porteus is a bit different. You specify in advance many of the things that other distros require you to set or install after booting. Go to http://www.porteus.org/ and follow the instructions.

I chose the LXQt desktop, Nvidia Legacy drivers and Firefox as defaults, and said ‘No’ to things like a VOIP client (for Skype) development tools and printing support. This gave me a very lean system.

A couple of general points.

I chose a light desktop – so LXDE or LXQt rather than KDE. However, the distros are so light in general that I am fairly sure you could go for KDE without having a huge impact on performance.

I saved the .iso files (the images of the OS) to a sub folder in my Downloads folder, just to be tidy. The file sizes were: Puppy Linux 139 Mb, Kanotix 785 Mb, PCLinuxOS 715 Mb, Lubuntu 724 Mb and my basic Porteus distro 260Mb.

As you can see, Puppy is the smallest by a good margin. Lighter distros exist, but I wasn’t confident they would have everything I wanted.

I have a 64 bit machine, so I chose 64 bit distros even though I only have a 32 bit version of Vista. All the above distros come in 32 bit flavours if that’s what you need.

Creating the Multiboot USB

I used Multisystem LiveUSB which has a version designed to run in Linux, as I wanted to create the boot usb on my Linux desktop machine. Other options are available if you are working with a Windows system.

(NOTE Multisystem is French software, but their website translates into pretty much any language you like, including Welsh. Da iawn a ddiolch, ffrindiau! If you don’t want to download your own distros, they will sell you a stick with a selection already installed. If you are following what I did, you won’t need to go there at all).

1) Search for MultiSystem LiveUSB.

2) Follow the link to pendrivelinux.com:

http://www.pendrivelinux.com/multiboot-create-a-multiboot-usb-from-linux/

3) The window now has all you need. Click on the download link in paragraph 1, then follow the instructions. The installation shell script will be placed on your desktop and when you run it the package is installed. This takes a few minutes. In my Pinguy setup, Multisystem ends up in the Accessories section of the menu. There is also a tool to test an iso file, which I didn’t try but might come in handy.

4) Once you have installed the software, follow the instructions in the lower half of the window to create your usb drive. Some additional pointers (please bear in mind I was doing this from Linux, not Windows):

• I had to format my new usb before Multisystem would recognise it. I used the Ubuntu ‘Discs’ utility to create an 8 GB partition on a 16Gb stick, give it a label and mount it, after which Multisystem detected it immediately. The five distros I installed, plus the associated tools that Multisystem includes, came to about 2.7Gb in total, which is 100Mb more than the sum of the .iso file sizes. So you don’t need a massive usb stick.

• The device came up as /dev/sdf1. This will probably be different for you – make sure you are installing to the correct place!

• Open a file manager alongside Multisystem, then drag and drop your .iso files from there. (There is no built in ‘browse’ facility.) This needs a root level (sudo) command so you have to enter your root password each time.

• The copy runs in a terminal window, and what you see varies from distro to distro. BE PATIENT – it will take several minutes for each distro.

• Just quit at the end. There is no ‘finalise’ step – so I suspect you can add more distros later if you want, although I haven’t tried this.

Booting from the live usb drive

Insert the drive, then start your machine. As it begins to boot, press F9 (on my two computers) a few times to open the boot menu and choose your boot device. On my desktop, it is recognised as ‘USB Pen Drive’ but on my laptop it is ‘USB Hard Drive’. Consistency, thy name is Hewlett Packard! If you are going to be booting from the stick often, it would be worth editing the BIOS to choose the appropriate drive before the hard disc, but I advise checking first via F9. (I edited the laptop BIOS in advance to choose pen drive, which didn’t work. Took me a while to figure that one out).

You will now see the Grub menu (something like the photo above), giving a selection of Linux installations and some other system tools. Simply pick the one you want and you’re in business. Lubuntu gives you the choice of booting live or installing the software (choose ‘Try’ to begin with), the others just boot up. If you like what you see, you can install it later.

NOTE The photo shows Kanotix, Lubuntu, Puppy Linux, and PCLinuxOS as the first four entries.  Porteus is displayed as Syslinux. Select this to get another screen, then press Enter when Porteus is highlighted on the top line. My installation prompts for ‘Always fresh’ (i.e. don’t save anything) during startup. There is an option to preselect this on the Porteus menu screen, but when I tried that the boot ran part way, then hung. So just press Enter.

As you would expect, the lighter distros boot faster, but none of them takes more than a minute for me except PCLinuxOS, which requires about 2.5 minutes. They all shut down very quickly, unless you have applied an option to save settings.

First Impressions

First up, they all work. You get a functioning Linux system that you can adapt to your needs. They all connect to the internet if you plug them in. And they are FAST – especially compared to a creaky old Vista system. Puppy is so fast that you might have to reduce the mouse sensitivity to stop it selecting things all by itself as you pass over them.

I’m not going to say much about appearance, as you would want to customise all of them for extended use. PCLinuxOS, Lubuntu and Porteus have modern minimalist desktops, all nice looking and easy to use once you learn where everything is in the menu structure. Puppy Linux puts icons on the desktop for the most common tasks which I found helpful to begin with. Kanotix reminds me a little of older versions of Windows, or the desktop on my Raspberry Pi, which is also based on Debian. (I think the others are all Ubuntu based.) Lubuntu is the prettiest at first sight but I could live with any of them.

I had no problems with video drivers on my laptop. Everything is clear and sharp and works as you would expect. When I try the live usb systems in my desktop, which has a Radeon graphics card, I get slightly different results. SO… if you are thinking of using a usb live system to move between several machines, you will probably have to do some tinkering to get everything just the way you want it.

The sound is good on all the distros except Kanotix, which is a bit crackly. That may be because I haven’t spent enough time adjusting settings, but as my aim is to find a distro that needs minimum tweaking… I’m typing this on one machine while listening to Cream on headphones through You Tube running on Puppy on the other. Awesome!

Word Processing

If you are planning to do much typing, you will want a word processor on the computer.

PCLinuxOS, Lubuntu and Puppy Linux come with Abiword included. This seems fine on first trial, although if I was installing to the hard disc rather than running from the live usb I would definitely add Libre Office. Puppy has a menu item to download this. Kanotix and Porteus only come with the Leafpad text editor. (I could have sworn I asked for Porteus to include Libre Office. Either I clicked the wrong button, or the build didn’t work.)

Alternatively, you could use something like Onedrive, which gives you MS Word on the Cloud. I occasionally do this if I want to share a document between my iPad and another computer, and it works very well. (Other Cloud options are available!)

Internet Connection

All five distros work perfectly with the internet if you are plugged into your router.

Only Puppy and Porteus give simple access to our wireless network at home – select service, then enter password. The other three distros require you to set up the router connection from scratch. This is not hard, if you haven’t lost all the information that came when you bought the router, but I wouldn’t want to do it every time I logged in.

I used the ‘save settings’ option when shutting down Puppy, and after this could connect wirelessly without entering the password again. I’m sure the same would be true if I made any of the other distros persistent after setting up wireless connectivity. (See below for more on persistence).

None of the distros activates a firewall automatically, but it’s pretty easy to do once you find where it is hiding. On line help is a wonderful thing! After I saved my settings, Puppy started the firewall on the next boot.

Web Browser

Lubuntu and PCLinuxOS come with Firefox. Kanotix has Iceweasel, which is very similar. When I created the Porteus .iso file, I requested Firefox (Chrome and Opera are also available as defaults). You can, of course, add any other browser you like once the system is loaded.

Puppy does things differently, providing the lightweight browser Dillo by default and offering a choice of Firefox, Seamonkey, Chromium and Opera via the Quickpets package manager on the desktop. This is a simplified package manager that gives access to the packages you are most likely to need at the outset. As Dillo did not work straight away, I went mad, installed all four bigger browsers and spent a happy hour playing with them. They only take a couple of minutes each to install, so it would be no great hassle to download one every time, but in the end I restarted and loaded Chromium before saving my settings. If you are running Puppy on a very old machine, you might need Dillo but you will have to spend some time setting it up. It works well enough on my Raspberry Pi.

I watch music videos quite a lot, and get pretty fed up with the Flash plug-in message. All the systems here played You Tube videos without requiring any tweaking. (Lubuntu and Kanotix requested the plug-in, then worked anyway without me doing anything.)

NOTE If you are doing serious work on your system and saving stuff, you will want to make sure your browser has the latest version of Flash (which I think means using Chrome or installing the Flash plug-in to Chromium) to avoid security vulnerabilities. Alternatively, only play video that doesn’t require Flash or install a ‘video without Flash’ add-on to Firefox. Having said that, running from a live usb means that everything gets lost at shutdown (as long as it’s NOT persistent!) so you don’t really need to worry too much.

Internet history is saved in Puppy once ‘save settings’ has been turned on. If you don’t want this, you can configure the browser not to keep history in the usual way

Customization

My aim was to find a distro that ran from a live usb on my laptop with the bare minimum of adaptation, so I haven’t spent any time finding exciting software and adding it. However, here are some observations:

• Puppy makes it very easy to find the things you might want to add at the start via Quickpet, and it’s so small that you can try various options quickly, then reboot to get rid of them. There is also a Setup menu option which takes you to a list of wizards for customizing the sound card, printers and so on. I didn’t need this, as the tasks I absolutely had to do were prompted when I first loaded Puppy.

• Porteus requires you to log in as root for most system related work. The password is ‘toor’.

• ALL Linux systems are almost infinitely customizable (I think that’s a word…), so whatever you start with you can create the system of your dreams. At least in theory.

• Bear in mind that if you really want to customize your system, you will have to make it persistent or install it to your hard drive.

Persistence

Persistence preserves your settings (including firewall), internet connection details and internet history. I was also able to save files to the usb or the hard drive, and I don’t think persistence affects those one way or the other.

I made Puppy persistent as an experiment, but I’m unsure whether I would bother with a persistent live usb otherwise. The idea that you can transfer your system and settings to any machine is great. Unfortunately, once I had saved my Puppy settings on the laptop the system didn’t work as well on my desktop as it had the first time round. (Something happened to make the screen very dark). And I have a nasty feeling that correcting this could easily have resulted in those settings being saved, giving me a problem the next time I boot the laptop into Puppy. So it’s probably best NOT to make your live usb system persistent unless you only ever intend to use it on one model of machine.

Ask yourself why you might want persistence. If it’s because you have to have your system just so all the time, then you might be better off with a permanent installation. The great advantage of a Linux live disc or usb is that you can use it as a starting point on any machine, quickly add the packages you need to get running, do what you need to and move on. I imagine tech support people have dozens of distros in almost every version possible, and know from experience exactly what to use where. They may want persistence now and again, and they are expert enough not to screw things up. I’m not sure I trust myself that much!

Finally, saving settings takes time. Non-persistent live Linux systems shut down in a few seconds as nothing is being saved. Puppy Linux with ‘save settings’ takes a minute or two.

Conclusion

As you have probably gathered, I’m very impressed with Puppy Linux, and it will be my distro of choice when using the laptop on the internet. I’ve made it persistent, but I can live with that. When Vista finally turns up its toes or I don’t want it any more, I will install some form of Linux on the machine and it might well be Puppy.

A very strong second choice would be Porteus. Fast, good looking and detected our wireless router straight away. I like the way you can choose features before creating your .iso file.

If you’ve got this far, you are probably considering trying Linux on an old laptop. Any of the five distros I have discussed would be a good starting point. Try them on a live stick – and if you don’t like them, download something else instead.  Good luck!

## 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:  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

we still have the surd on top but now we only have to divide by 4, which gives us as 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…)