It may not seem a big deal, but if you’re a teacher of Applied Maths then worked solutions – whether they be Department of Education marking schemes or full solutions – can be a life-saver.

The official marking schemes go back to 1995 but for some reason you can only get back to 2001 on the Department’s site examinations.ie.

Over the course of the last couple of weeks one of my first-year Science students has been diligently giving up his lunch-times to scan in these earlier marking schemes which are not online (from 2000 – 1995), along with full solutions to 1994 and 1993.

These are all now available on the Applied Maths solutions page which is here.

Together with the existing older scanned in solutions it now forms almost a complete set of solutions going back 35 years.

Over the last couple of years I have converted the exam questions for all of these into Word format and these can be accessed here.

Alternatively I have incorporated the questions into a series of booklets (one per topic) where the questions are grouped by similarity. These booklets also contain introductions to the topics plus gradual lead-in questions taken from Ordinary Level papers.
Many of the questions have a ‘how-to-solve’ guide which gives you some help in answering the question without giving you the full solution (the full solution is available elsewhere).

As always, you can download these one document at a time or alternatively just send me a memory stick and I’ll copy them on and post them back.

Oh – and thanks to Eoin Robinson for all his help; If the resource is useful you might leave a comment and thank him!

Thanks to my first years for being so co-operative

Here’s another way of illustrating the idea (which we didn’t do)

Apart from the standard demonstrations using tuning forks and wine glasses we also tried out variations

1. The Chinese Bowl

2. The aluminium rod

There are different types of graph questions, and we will look at each of these different types in turn.

There is nothing scary here, and you have probably covered them all in maths anyway. It’s just that the science textbooks don’t seem to do a very good job of telling us why we have them in the first place, or why there are different types.

Why do we have graphs?

You won’t get asked this so you don’t have to learn it off by heart – I just thought you deserved to know.

There are many different reasons, but we’ll just look at two here.

Reason 1:

To see what the relationship is between two variables, e.g. between the extension of a string and the force which caused it.

Now assuming that a bigger force causes a bigger extension, the question is; are the two quantities directly proportional? i.e. if the size of the force doubles then the extension should be twice as much, if the force triples the extension will be three times as much etc.

Another way of saying this is that the two quantities increase at the same rate (as force is increased the extension increases at the same rate).

Or finally the scientific way of saying this is to say that the two quantities are directly proportional to each other (you must learn the phrase in italics off by heart because it gets asked a lot as you will see below).

To investigate this you would plot the results on a graph, and if the two quantities are directly proportional then you will find that if you draw a line through the points you will end up with a straight line through the origin (the origin is the (0,0) mark).

Reason 2:

In some graphs the slope of the line gives us some extra information (and you must know what this is).

There are only three graphs which fall into this category so make sure that you know each of them.

1. The slope of a distance-time graph corresponds to the speed (or velocity) of the moving object

2. The slope of a velocity-time graph corresponds to the acceleration  of the moving object

3. The slope of a voltage-current graph corresponds to the resistance of the resistor under investigation.

Note that for each of these graphs you will also get a straight line going through the origin, which verifies that the two quantities are directly proportional to each other.

Which brings us to our next problem – how do we calculate the slope of a line?

To calculate the slope of a line

Pick any two points (from the graph) and label one point (x₁y₁) and the second point (x₂y₂).

Make life easy for yourself by picking (0,0) as one of the points (assuming the line goes through the origin).

You must then use the formula:                                    

slope = (y2 – y1)/(x2 – x1)

Note that you can also find this formula on page 18 of the new log tables

Yo – Which axis is the y-axis?

Remember the yo-yo? It goes up and down right? Well so does the y axis (and it begins at zero) so y-zero = yo

Now that’s just freaky.

Today however I came to what I think may be my favourite – the structure of the atom. You see the concept of solidity is merely an illusion. That table appears to be solid but actually it is 99.9999999% empty space.

“But it looks solid!”
Yup – but that’s (just) an optical illlusion caused by the interaction of electrons and incoming electromagnetic radiation.

“But it feels solid – when I bang my hand off the table I feel a solid surface!”
Again an illusion I’m afraid. This time caused by the interaction between electrons in your hand and electrons on the surface of the table repelling each other.

So next time you’re sitting in your car holding on to the steering wheel, just remind yourself that the steering wheel is hardly even there – neither is the car and for that matter neither are you (I’m trying not to have an exclamation mark at the end of every sentence here, but it’s not easy).

Now there is a (very, very) small amout of ‘matter’ in you. And here’s the thing; if you somehow managed to remove all the empty space in your body you would be left with a lump of solid matter which would be smaller in size than a grain of salt. How much would it weigh you ask? Why the very same as you do now, after all we have only removed the empty space (having a real tussle with the exclamation mark key here).

Okay, but this is all theoretical right?
Well maybe. It’s theoretical for humans, but there are objects out there which do not have empty space in them and which therefore are incredibly dense. They’re called neutron stars and to quote from Wikipedia:

this density is approximately equivalent to the mass of the entire human population compressed to the size of a sugar cube.

This gives them some other unusual properties. The radius of a typical neutron star is about 12 km, and just like a pirouetting ice-skater whose rate of rotation increases as they draw their arms in, so also does the rotation of a neutron star increase as its radius decreases. So how long does a full rotation last (remember that on Earth this is 24 hours)? On average there are somewhere between 100 and 1000 full rotations per second (see, if I can’t use exclamation marks at least I can use italics).

But back to the atom. To help students appreciate that it’s not just me who is bonkers, here I invoke the assistance of some more well-renowned experts.

Professon Brian Cox

Note that it took Rutherford two years to arrive at a correct interpretation of these results. It’s not like what you see in the textbooks – describe the experiment and then form an obvious conclusion.

It kinda freaked him out – Neil de Grass Tyson (audio link)

And for a bunch of other links see the related page on my website.

And finally, while you are expected to know that the atom is mostly empty (and be familiar with the experiment that ‘proved’ it), there is no sense in either the syllabus or any textbook I have come across of the wonder associated with this crazy idea. In fact it’s normally presented as just another piece of information to be learned off my heart. And I never hear anybody giving out about this, so I save my rants for unfortunate students and the odd blog post like this.

Now that’s mad!!

Some of these are in the new log-tables, but others are not so you should really try to remember all of them because it not always easy to identify the formula you need from the list in the log-tables. Practice looking for them in pages 51 – 56.

To encourage students not to give up on these questions I have put together a document which includes all the equations on the Junior Cert Science syllabus. There are (only) ten of them and by-in-large they are very straightforward. The document includes every maths question that was ever asked on a Higher Level or Ordinary Level paper from when the syllabus was first examined in 2006 up until 2010. It also contains solutions to all these questions.

If you’re looking for a top grade in June you really should ensure that you’re familiar with all of these.

You can download this document here.

There’s also a section on Units (we tend to be a little fanatical about these in Physics) and finally a list of practice questions.

If you’re using this as a teacher it should make a nice revision class or two – good luck with it!

Drawing the graph

• You must use graph paper and fill at least THREE QUARTERS OF THE PAGE.
• Use a scale which is easy to work with i.e. the major grid lines should correspond to natural divisions of the overall range.
• LABEL THE AXES with the quantity being plotted, including their units.
• Use a sharp pencil and mark each point with a dot, surrounded by a small circle (to indicate that the point is a data point as opposed to a smudge on the page.
• Generally all the points will not be in perfect line – this is okay and does not mean that you should cheat by putting them all on the line. Examiners will be looking to see if you can draw a best-fit line – you can usually make life easier for yourself by putting one end at the origin. The idea of the best-fit line is to imagine that there is a perfect relationship between the variables which should theoretically give a perfect straight line. Your job is to guess where this line would be based on the available points you have plotted.
• Buy a TRANSPARENT RULER to enable you to see the points underneath the ruler when drawing the best-fit line.
• DO NOT JOIN THE DOTS if a straight line graph is what is expected. Make sure that you know in advance which graphs will be curves.
• BE VERY CAREFUL drawing a line if your ruler is too short to allow it all to be drawn at once. Nothing shouts INCOMPETENCE more than two lines which don’t quite match.
• Note that examiners are obliged to check that each pint is correctly plotted, and you will lose marks if more than or two points are even slightly off.
• When calculating the slope choose two points that are far apart; usually the origin is a handy point to pick (but only if the line goes through it).
• When calculating the slope DO NOT TAKE DATA POINTS FROM THE TABLE of data supplied (no matter how tempting!) UNLESS the point also happens to be on the line. If you do this you will lose beaucoup de marks and can kiss goodbye any chance of an A grade.

What goes on what axis?

Option one

To show one variable is proportional to another, the convention is to put the independent variable on the x–axis, and the dependant variable on the y-axis, (from y = fn (x), meaning y is a function of x). The independent variable is the one which you control.

Option two

If the slope of the graph needs to be calculated then we use a difference approach, one which often contradicts option one, but which nevertheless must take precedence. In this case we compare a formula (the one which connects the two variables in question) to the basic equation for a line: y = mx.

See if you can work out what goes on what axis for each of the following examples (they get progressively trickier):

1. To Show Force is proportional to Acceleration
2. Ohm’s Law
3. Snell’s Law
4. Acceleration due to gravity by the method of free-fall
5. Acceleration due to gravity using a Pendulum

There is usually a follow-up question like the following;

“Draw a suitable graph on graph paper and explain how this verifies Snell’s Law”.

There is a standard response to this;

“The graph of Sin i against Sin r resulted in a straight line through the origin (allowing for experimental error), showing Sin i is directly proportional to Sin r, and therefore verifying Snell’s Law”.

If you are asked any questions to do with the information in the table, you are probably being asked to first find the slope of the graph, and use this to find the relevant information.

(i)     Draw a fully labelled diagram which includes all essential apparatus (have you included the apparatus necessary to obtain values for both variables?).

(ii)   Be able to state how the two sets of values were obtained (this is a very common question).

(iii) Describe what needs to be adjusted to give a new set of data

(iv) Write down the relevant equation if there is one associated with the experiment.

(v)   Be able to state how the data in the table will need to be adjusted.

(vi) Know what goes on each axis.

(vii)           Know how to use the slope of the graph to obtain the desired answer.

(viii)         Be able to list three sources of error/precautions.

Misc Points

• The graph question is usually well worth doing.
• Learn the following line off by heart as the most common source of error: “parallax error associated with using a metre stick to measure length / using a voltmeter to measure volts etc”.
• Make sure you understand the concept of percentage error; it’s the reason we try to ensure that what we’re measuring is as large as possible.
• There is a subtle difference between a precaution and a source of error – know the distinction.
• When asked for a precaution do not suggest something which would result in giving no result, e.g. “Make sure the power-supply is turned on” (a precaution is something which could throw out the results rather than something which negates the whole experiment).
• To verify Joule’s Law does not involve a Joulemeter
• To verify the Conservation of Momentum – the second trolley must be at rest.
• To verify the laws of equilibrium – the phrase ‘spring balance’ is not acceptable for ‘newton-metre’.
• To measure the Focal length of a Concave Mirror or a Convex Lens.
  Note that when given the data for various values of u and v, you must calculate a value for f in each case, and only then find an average. (As opposed to averaging the u’s and the v’s and then just using the formula once to calculate f). Apparently the relevant phrase is “an average of an average is not an average”.

I have a document here which containts exam questions on every experiment which has ever appeared on a past paper from 2002 to 2010 (Higher Level and Ordinary Level) – this should be your bible for Section A over the coming weeks. Solutions are also included.

Now get back to work.

More to come.

This is a similar image of the proposed new syllabus.

Notice the new focus on the words ‘learners’ and ‘learning’.

Imagine if a syllabus had as its most common words the following:

Engage

Passion

Awe

Curiousity

Inspire

Un-nerve

Emotion

Story

Creativity

Wonder

If any of this was a priority then chances are that Particle Physics wouldn’t have been removed (and with it Pair Annihilation, Anti-matter, Neutrinos, Fundamental Forces, etc.).

Chances are that Cosmology would also feature strongly in the new syllabus (Black Holes, Quasars, Pulsars, Big Bang, Neutrinos (again), Dark Matter, Alien Life, etc. etc.). It doesn’t.

Maybe it’s just me.

There are different types of graph questions and you have probably covered many of them in maths anyway. It’s just that the science textbooks don’t seem to do a very good job of telling us why we have them in the first place, or why there are different types.

The graphs you need to be familiar with:

1. Distance-time graphs
2. Velocity-time graphs
3. Voltage-current graphs
4. Force-extension graphs
5. Cooling curves
6. Solubility curves

As you can see these are almost all from the Physics section of the course, and unless you have practiced answering exam-questions they can seem rather intimidating. The link below is to a document which contains all recent exam questions on graphs at Higher Level and Ordinary Level and also includes the solutions. The nice thing is that once you’ve done one or two questions on a given graph they all tend to repeat after that.

There is lots of other advice on answering the graph question, but it’s all in the document itself.

Hope it’s useful!

Revision – Graphs