# Posts from the ‘KS3 (11-14)’ Category

## Task

Choose any 4 digit number except for 1111, 2222, 3333, 4444. *I’ll choose 1502*.

Rearrange the digits to give you the biggest and the smallest numbers you can. *5210 and 0125*.

Find the difference of these two numbers. *5210 – 0125 = 4995*.

Repeat for the new number you get (using zeros to supplement any missing digits if necessary; you must always have 4). *9954 – 4599 = 5335*.

Keep repeating until you have a good reason to stop.

## Example

*5210 – 0125 = 4995
5533 – 3355 = 2178
8712 – 1278 = 7434
7443-3447 = 3996
9963 – 3699 = 6264
6642 – 2466 = 4176
7641 – 1467 = 6174*.

I’m not going to repeat any more because 6174 => 7641 – 1467 which is the calculation I just did!

## Discussion

- So every number we’ve tested came to 6174. Do you think all numbers will come to 6174? Why? In maths we can’t say we’re sure about something unless we have a good reason, so unless you want to go through every number and check it, we can’t say that we know that every number sequence we choose will go to 6174! (This is how mathematics differs from science, we don’t just make hypotheses and wait until the next time they break, we find ways to be certain!)
- Would we have to check every different number to be certain or are there shortcuts we can take? (should we check both 1234 and 1243 separately?)
- My rules stipulated that you couldn’t choose all the same number: 1111 or 2222. What would happen if you did use those numbers?
- What about two digit numbers or three digit numbers?
- Could we frame the problem as (1000a + 100b + 10c + d) – (1000d + 100c + 10b + a)?
- What about five or six digit numbers (or more…)? You may need a computer to help!

## Thoughts

I just came across this interesting bit of mathematics via Twitter: http://plus.maths.org/content/os/issue38/features/nishiyama/index. It offers the opportunity for a beautiful open-ended task for secondary maths classes of all abilities. The initial task is easy and produces a startling result that feels like a trick; that hook can lead to discussion and further work about probability, permutations, algebra and programming.

An incredible rendering of the MandelBrot Set, which drives home the notion of infinite complexity!

As the comments underneath the video allude to, there is some utility here to attempt to consider the size of magnification that occurs during a portion of the video, and how to express magnifications of such a magnitude.

Another interesting website, this one visualises flags as pie-charts that reflect the proportions of the colours in their flags:

It would be a fun and worthwhile activity to use this site to exercise KS3 students’ abilities to mentally construct and destruct pie charts in a game of ‘guess the flag’

I stumbled across this lovely page from the University of Utah today, which is extremely simple, but nevertheless well done:

From a scientific perspective it is interesting to reflect upon the relative sizes of elements, from a Coffee Bean to a Carbon Atom (through various things such as a human Ovum, Sperm, various viruses, compounds and so on).

From a mathematical perspective it is also interesting for the way in which the relative scales are measured in the top-left. Exploring the different notations for small sizes would be a useful exercise in place-value for all levels of Key Stage 3 and 4.

This is a delightful way to revise Venn Diagrams with older students: A Venn Diagram of mythical creatures.

The new Bowland Maths Website is the website of a new project which seeks to ground maths in an explorative, problem solving environment.

Bowland Mathematics seeks to develop meta-cognitive skills and promote an analytical, quantitative attitude towards problem solving. These goals are worthy, and important life skills, but they are difficult to measure cleanly. With curricula that separate the strands of mathematics in a way that encourages their their teaching to be separated also, and with testing that aims at accountability over intelligence, school mathematics has become ever more piecemeal and disconnected with reality. Bowland is an important project that seeks to reclaim some of the lost ground.

I urge, in the strongest possible terms, that anyone involved in mathematics education take this initiative seriously. I have no vested interest in the scheme, but simply I believe that it is crucial that initiatives such as this succeed and are built upon.

In this month’s Mathematics Teacher magazine is an article about prime factorisation by me. It discusses an idea for teaching and learning about prime factorisation that minimises ‘telling’ and maximises students mathematical exploration.

Dave Hewitt, a lecturer in Mathematics Education from the University of Birmingham (and my PGCE mentor a few years ago), has written a series of articles about separating the arbitrary (or contingent) and the necessary and mathematics, and teaching by ‘telling’ only those things which are arbitrary. The idea is that students need to engage and discover for themselves the necessary connections and patterns in mathematics, but the arbitrary are not discoverable in the same way, and so need to be told to people.

This position has influenced my thinking about mathematics education, particularly with respect to algorithms. My conjecture is that using an algorithm involves no mathematical thought; at best, it is an exercise in arithmetic. However, where an algorithm exists there is likely to be a kernel of really interesting mathematics, and creating an algorithm to perform a particular function involves a great deal of mathematical thought. The goal of the investigation was to capture the interesting maths in an interesting way, that the students can engage with, and which they can learn from.

The accompanying software can be found in the primitives section of this website.

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.

Launchball is a game produced for the Science Museum website. It is an excellent and well thought out little game that has highly transparent educational content. Despite this, it it fun to play.

Most of the puzzles deal with the concepts of power and force, both in terms of their generation and their effect. The aim is to make a little (metal) ball reach a particular goal. It can be done by using wind power to blow the ball, magnetism to attract it, or ‘rollers’ to move the ball along. Some or all of these effects require power, and the different mechanisms for generating and transferring power are really interesting and innovative.

This game is a wonderful way to introduce physics.

Firaxis Games, the makers of one of the greats of computer gaming *Civilisation*, discuss on their website the growing trend for computer games to be used in the educational arena. It is encouraging that educators are starting to understand the potential of technology to educate, though I suspect that the use of commercial games as educational tools is an transitional step before bespoke educational games begin to be produced with production values that begin to approach those of commercial games.

One of Firaxis’ contributors Kurt Squire proposes Civilisation as a good model for learning about World History. There is an interesting tension here. On the one hand, a game like Civilisation engages students in such a way that they build a sophisticated model of the game in order to succeed at it. That is good educationally to the extent to which the game models genuine historical processes. It is not clear that the ‘history’ that Civilisation presents is particularly convincing.

While Kurt Squire argues that Civilisation “represents world history not as a story of colonial domination or western expansion, but as an emergent process arising from overlapping, interrelated factors”, it does still give an essentially American – or at least New World – view of history. Land is virgin territory until moved into by the great civilisations; pre-colonial Afrians, native Americans, native Australian Aborigones do not have a story. Intellectual and technological progress happens linearly; the Middle ages and the loss of Roman and Greek learning cannot happen. There is no potential for a European type of historio-political scenario; states are the size of continents.

On the other hand, if one ignores the problems with the historical model, it does offer a ‘big picture view’ of history. Could such a grand model of historical processes be so readily expressed without the means of technology? Certainly the answer is yes, though it would take an extremely talented teacher, and those are notoriously thin on the ground.

The pipedream is for someone to create a game with production values on a par with Civilisation, but which takes as its starting point an historical model that aims at accuracy. This of course, is rather like desiring an historically accurate documentary that looks and sounds like a Hollywood movie, but there will surely be moves towards higher production values in educational software in the future.