# Posts from the ‘KS4 (GCSE)’ Category

Dr Ron Knott in the Department of Mathematics at Surrey University is not a name I recognised, but reading his resume, I now realise that I have heard him talk a few times about Mathematics on Radio 4, both on Simon Singh’s 5 Numbers series, and in Melvyn Bragg’s In Our Time podcast.

I was looking for some information about exact values of trigonometric ratios, and came across his most informative site. I was extremely pleasantly surprised to discover that for some values the trigonometric functions give exact solutions in terms of phi, the golden ratio, among other information.

For example, did you know that the cosine of 27 degrees is exactly a half of the square root of (two plus the square root of (two subtract phi)). (One day when I finish writing my own equation display movies, I’ll write that out in a prettier way, Dr Knott’s website tries a little harder than I do). I love that the number 27, which clearly wants to be prime so much it tricks generations of children into thinking it is, the square root of two and the golden ratio are connected inextricably through the circle-based cosine function. Fantastic!

The whole page, indeed the whole of his site in general, is steeped in extremely interesting, and relatively accessible mathematics with Fibonacci numbers, Egyptian Fractions and so on and so forth. It’s mostly a site for KS4 and beyond (14 years old +), with most material for the older students. Some of it is not for the faint-hearted. However, it is a valuable resource for mathematicians of all hues, and well worth a look.

Phun is a free two-dimensional physics sandbox for Windows.

A video of it in action can be found on this You Tube. Unfortunately they don’t yet have a Mac version, so I haven’t been able to try it out myself, but the videos looks stunning.

This has fantastic potential educational value for physics and maths, but in the same way that the Geometer’s Sketchpad does – it is easy to see the potential, but rather more difficult to harness it.

There must be some middle road between the openness of this sort of ‘sandbox’, which for university students and older computer literate school students has tremendous educational value, and something more rigid that allows more nervous or younger students to engage with the simulations it offers constructively. The problem is, what is that road?

I was recently invited to do an IQ test. One of the questions was as follows: “You walk five miles north, five miles west, then five miles south. How far are you from where you started?” The answer that they were looking for was 5 miles.

Perhaps our use of maps convince us of this logic; a logic based on Cartesian 2-dimensional geometry. Unfortunately, we do not live on a Cartesian plane!

How should we understand this IQ test question when we correctly consider that we live on a sphere? A good way to consider this is to think about what happens at the two poles!

### The South Pole

Start at the south pole. Travel five miles north. Travelling west is to travel parallel to the equator, so then when travelling five miles west, you get no further from the pole. Then travel five miles south. You arrive where you started, with no distance between where you started and where you are now.

### The North Pole

The north pole example is more difficult to imagine, and some may think there’s a trick here. If you start five miles and a bit south of the north pole, move five miles north then five miles west in many tight circles around the pole ending up exactly opposite to where you were when you originally arrived near the pole. Then move five miles south. More or less, you are now ten miles away from where you started.

*The trick here is the ‘bit’ which ensures two things: firstly that you can in fact travel west; secondly that having travelled five miles west you finish up exactly opposite where you were when you started moving west. It might be argued that if you ended up exactly at the pole then you would be unable to move west at all. The ‘bit’ ensures that there is a trivially small circle around the pole that you can travel west around. It is also necessary to end up exactly opposite where you started to maximise the resultant difference between the two starting positions. If you imagine the bit as a radius of a circle around the pole, then it can be calculated as any r such that 5 miles = (2n+1)*pi*r where n is a positive integer.*

### Summary

Thinking about a problem often involves thinking around its extremes or limits. When thinking about compass bearings on a sphere, the poles offer places where their odd relationship to each other are most apparent. The IQ problem assumes that we live somewhere where the relationship between the compass bearings closely resembles the relationship between Cartesian axes. At the poles this similarity breaks down most markedly. By thinking about moving to and and from the poles, it transpires that if you move five miles north, five miles west, and five miles south, you may end up a distance of x miles from where you started, where x is such that: 0 miles <= x < 10 miles.

Now, if the IQ test was testing for this as an answer, I would have been suitably impressed!

### Philosophical Footnote

The earliest known argument against the earth being flat comes from Aristotle, who argued that the shadow that the Earth casts onto the moon during lunar eclipses is always circular. The only object which casts a circular shadow irrespective of its orientation is a sphere, and since night and day convince us that the earth does not have a constant orientation with respect to the moon and the sun, the Earth must be spherical. (Aristotle *De Caelo*, 297b31-298a10)

Our natural instincts about the world are perhaps that it exists on a plane that is looped; the outside of a cylinder with the poles at the top and bottom of the cylinder. This is a practical simplified model of the earth because until you get into the arctic and antarctic more or less, two people moving north are moving more or less parallel to each other, and the consequences of longitude and latitude working on quite separate principles need not be considered.

We could of course change the way that north and south work, and make them akin to east and west. Perhaps the great circle through Grenwich could be the East-West equator, as in some senses it is, and we could therefore define an east and west pole, somewhere on the equator! It is an interesting thought experiment. Would our concepts be more easily understandable if we did this? Why did North and South become defined as it is?

In Our Time is a BBC Radio 4 presented by Melvyn Bragg. It is an intellectual talking-heads discussion programme about philosophy, science, mathematics and so on. This week the discussion was aobut the Fibonacci Sequence. The podcast can be downloaded here.

For maths teachers, KS3 and KS4 students will enjoy and learn from the discussion between Melvyn Bragg, Professor Marcus du Sautoy, and others. It is well worth a listen.

The Education Guardian reports that the Specialist Schools and Academies Trust urge school exam overhaul. It is encouraging to hear another voice added to the growing clamour for change.

The SSAT argue that the government has “consistently exaggerated the technical rigour of national assessments and the GCSE“. They argue that by changing the curriculum and therefore changing the content that is being tested, it becomes extremely problematic to maintain and compare standards.

The SSAT also argue that there are testing cause a degree of stress and that the level of continued stress that students are exposed to has become unreasonable and counter-productive. In place of the SATs they suggest using sample testing of randomly selected pupils to monitor performance.

The response from the DCSF is staggering: “… we are not looking at sample testing of randomly selected pupils … It is hard to see how any sample of children could be truly representative of one school … the idea that children are over tested is not a view that the government accepts … we don’t believe that in this day and age parents can be expected to have hidden from them the real achievements of their children at school.”

If the governmental body responsible for our curriculum do not understand sample testing, then I am deeply concerned with the science curriculum; if do not understand how sample testing can give representative data, then they do not understand science. Science is based wholly upon the statistical analysis of sample data. Given a sample and the overall population size, we can very accurately calculate how representative that sample is. Simply, this argument is nonsense.

The original premise for introducing SATs was as a means of measuring schools performance. The DCSF statement concedes that they have now become GCSEs for younger students – performance assessments for the students and their parents to measure themselves with.

Despite this the government does not accept that children are over tested. I am absolutely and utterly convinced that the government is wrong about this. Sadly, there is no easy way of measuring what level of stress is acceptable to expose children to. However, I would have thought that until tests GCSEs, the natural inclinations of all parents and teachers would be to minimise unnecessary stress. This is not the government’s inclination.

## Sketchpad Files

Geometer’s Sketchpad is one of my favourite programs but extremely annoying to author in. Here I have created a zipped archive of some of my past Sketchpad efforts for anyone who wants them. I have included a brief description of each file below.

If you don’t have Geometer’s Sketchpad then these will not be very useful to you. You can get hold of a copy from The Curriculum Press in the UK.

#### ConstructingAnEllipse

An extremely simple file exhibiting the construction of an ellipse defined by two centers and a point. The problem is that the “Locus Constructions” that are revealled don’t tell the whole story; how did I arrive at choosing those two circles to construct the circle with? This requires thought about the definition of an ellipse and the nature of construction.

#### ConeFromCircle

If you cut a sector from a circle (the bright-red sector) you can join the two radial sides together to form a cone. What is that cone’s volume? How do you maximise the volume of the cone? This sketch allows an exploration of the structure of the problem prior to doing the maths on it. It is a good exploration for construction of graphs at lower levels, or using calculus at higher levels.

#### Projectile Motion

A useful sketch for teaching calculus, start with the initial position of the projectile on the ground. Some of the ‘features’ on this sketch are odd and may require some exploration before they make sense!

#### ProjectiveCuboid

An example of how to construct a cube using two-point perspective

#### DoQuadrilateralsTessellate

I believe that this sketch comprises a proof that all quadrilaterals do in fact tessellate. It is an interesting question to whether it does in fact constitute a proof (what is a proof anyway?!) Choose your quadrilateral and click the buttons in order to follow the argument steps.

#### HexagonalSlideTessellation & HexagonalRotationalTessellation

Dynamic examples of how to distort tessellating hexagons in such a way to ensure that the result also tessellates.

#### Egg

An interesting construction of an egg using four arcs of four circles, it questions students to consider the relationship between mathematics and aesthetics.

How many spots are there in a complete set of dominoes?

It’s a lovely problem that offers the opportunity to discuss methods of calculation, and methods for avoiding mistakes. It links to combinations and permutations, as well as triangle numbers, and is a really simple question to pose. An intermediate question is to ask how many dominoes there are in a set, and establish that solution before continuing to the original question.

At University I was introduced to a really interesting CD called Problem Pictures. It had some fascinating images that all had some mathematical significance. I printed out about fifty and they made the walls of my classroom bright and appealing. It also had questions related to each photo, but I found that I rarely used them, preferring to create my own questions and ideas.

Enter Flickr, a great online photo sharing tool. Though you will not get questions related to each photo, the database is huge. Typing into the Flickr search engine ‘geometry’ turns up 10,810 photos at the time of writing. Most of them are really relevant; some of them are utterly spectacular. Here is a random sample:

#flickr_badge_source_txt {padding:0; font: 11px Arial, Helvetica, Sans serif; color:#666666;} #flickr_badge_icon {display:block !important; margin:0 !important; border: 1px solid rgb(0, 0, 0) !important;} #flickr_icon_td {padding:0 5px 0 0 !important;} .flickr_badge_image {text-align:center !important;} .flickr_badge_image img {border: 1px solid black !important;} #flickr_www {display:block; text-align:left; padding:0 10px 0 10px !important; font: 11px Arial, Helvetica, Sans serif !important; color:#3993ff !important;} #flickr_badge_uber_wrapper a:hover, #flickr_badge_uber_wrapper a:link, #flickr_badge_uber_wrapper a:active, #flickr_badge_uber_wrapper a:visited {text-decoration:none !important; background:inherit !important;color:#3993ff;} #flickr_badge_wrapper {} #flickr_badge_source {padding:0 !important; font: 11px Arial, Helvetica, Sans serif !important; color:#666666 !important;}

I have found that putting an image full screen on an interactive whiteboard when students come into the room is a great way of capturing the students’ attention straight away.

This activity is for a first introduction to expansion of brackets or for a review activity.

## Bracket Expansion Excel Spreadsheet

Imagine telling a class that 4(x+3) = 4x+12 and them asking you “why?”. What if instead you presented the information to them in a way that they generated the idea that they were equal and the explanations for their ideas themselves?

This sheet generates pairs of equations (for just expressions, read on)*, one with the bracket in place, one with it expanded. It is a very simple spreadsheet. It contains simple macros that allow either equation to be hidden. Though you can use it however you like, my suggestion follows…

I would use the sheet with both equations or expressions* visible and change the values a few times. I would ask students to try to explain the relationship between the two expressions.

Most likely the first thing that they notice is the methodological relationship the product of the two numbers in the first expression is the last number in the second expression. A brief glance at some other variations will demonstrate that this is the case.

However, they are unlikely to have grasped the numerical relationship; that they each have the same value, irrespective of the value of x. This will probably require some prompting. You could ask each student to choose a value of x and substitute it into the two values. What do they notice. Choose random numbers a few more times. Repeat the activity. Once they have established that they are always getting the same numbers then you can move on to the big one: the conceptual relationship.

The concept we’re aiming at is surely for the students to grasp that the link between the two expressions they first noticed is necessary for them to have the same value. It will be difficult for this activity alone to convince them of that, but I would at this point urge them strongly to consider whether the two things they’ve noticed are related. Since they’re expecting a relationship, they may trust that linkage. I haven’t linked it for them however, at least not in their minds.

I would then remove the expanded expression and ask them to construct it themselves, making sure that for whatever number they make x the two expressions match. This is a crucial check that should be encouraged early.

* The ‘y=’ on both the equations can be easily removed to just give the expressions. To be honest, that’s my preferred usage, because of the problems students have understanding when we’re using equality as a question signifier and when we’re using it as a binary relation between the two expressions on either side. To get rid of the ‘y=’ click on the ‘control’ tab and delete G4 and G5, where the ‘y=’ text is seen. Go back to the ‘display’ tab and they will have gone.

Also note that you must have macro security set to medium and allow macros for the buttons to function. Please disable macros and view the macro code if you are unsure whether this is safe, or if you do not understand code, ask someone who does.