Archive for the ‘Resources’ Category
Purchasing power: the changing value of the pound
“Is he married or single?”
“Oh! Single, my dear, to be sure! A single man of large
fortune; four or five thousand a year. What a fine thing for our
girls!”
(Pride And Prejudice, first published 1813, by Jane Austen.)
If you’re like me, you’ve wondered from time to time exactly how large this ‘large fortune’ is, in today’s money. What about someone strugging on a couple of hundred a year? Several years ago I stumbled across an interesting UK government research paper which attempted to answer that question, by tracking the purchasing power of the pound through the last 250 years. It could easily be used as a rich resource for a variety of activities in the classroom.
The most recent edition of the document was published in 2006, and is called ‘Inflation: the value of the pound 1750-2005‘. It provides a series of tables which enable us to convert prices between any two years from 1750 onwards.
From the introduction to the research paper:
This paper presents a price index covering the period 1750 to 2005 and illustrates the changing purchasing power of the pound over the long-term. No attempt is made to measure changes in the external value of the currency as a result of movements in exchange rates, but changes in the prices of imported goods are reflected in the price index.
It must be stressed that, for a number of reasons, such an exercise is very approximate. Expenditure patterns have changed dramatically over the past 250 years. Many products now commonly purchased (cars, electrical appliances, processed foods, etc) simply did not exist in 1750 and, conversely, goods that consumed a large share of household budgets in the eighteenth century - candles for instance - are now an insignificant part of most families’ expenditure.
It is, however, possible to compare price levels over the long-term by linking price indices covering relatively short periods into a single series.
The heart of the document is a table giving a relative price index for each year from 1750 to 2005 (artibrarily setting the value of the pound on January 1974 to be 100), as well as the inverse of this, which they call the ‘purchasing power‘
Let’s take the ‘large fortune’ I quoted at the start as an example of how to interpret the values.
| Year | Price Index | Purchasing power | % Change |
|---|---|---|---|
| 1812 | 15.9 | 630.4 | 13.2% |
| 1813 | 16.3 | 615.3 | 2.5% |
| 1814 | 14.2 | 704.8 | -12.7% |
| 2004 | 736.5 | 13.6 | 3.0% |
| 2005 | 757.2 | 13.2 | 2.8% |
The price index for 1813 was 16.3, and that for 2005 was 757.2. So in 2005 the average price level was roughly 46.5 times the 1813 level. This means that to have the same purchasing power in 2005 as £5000 had in 1813, we would need an income of around £230,000 a year. Not too shabby!
We’d get an even larger result if we’d used the value at other years close to 1813 — the pound in 1813 was at its weakest point since 1750. It soon recovered, and wouldn’t return to that low level until 1917 — over 100 years later.
Purchasing power and inflation
This research paper highlights a key way in which the character of money has changed over time. As the charts (which are reproduced below) show, the world we live in today, where prices rise year on year, is a fairly recent phenomenon. The first world war triggered a large jump in inflation, but we find significant deflation through the 20s and 30s — a pound was worth 50% more in 1935 than in 1920. The second world war triggered another bout of inflation, and we’ve been on the inflationary roller-coaster ever since — to the extent that a pound today is worth less than 10% of a pound in 1970!
As the research paper points out, prices have risen every year since 1945. Positive inflation is now built into modern economic theory (it encourages you to spend, rather than hoard, as the hoarded money will become less valuable over time), and ‘deflation’ is now looked upon as something to be avoided at all costs. Indeed, the Bank of England is instructed to keep inflation within a band around 2.5% per year, and is required to act when inflation is ‘too low’ just as much as when it gets ‘too high’.
We can see the contrast between the periods of price stability and price inflation very clearly if we graph the price index:
The price index is essentially flat for hundreds of years, and then ‘takes off’ after the second world war.
This linear graph does, however, hide some of the quite severe price fluctuations that occured even during the period of price stability. We can get a better idea of the relative changes in the value of the pound if we use a log scale, rather than a linear scale, for the price index:
Remember that both of these charts show the value of the price index — the higher the number, the less the currency is worth. In a sense, the value of the pound is the inverse of this price index. A (logarithmic) plot of the value of the pound over the last 250 years (scaled to make the value in January 1974 equal to 100) looks like this:
We can see very clearly here the fact mentioned earlier — that the value of the pound in 1813 was at a low point which would not be reached again for over 100 years. We can also see the deflationary period after the first world war, and the constant loss in value of the pound ever since 1945 (with a particularly steep loss in value during the 1970s — the period of double-digit inflation rates).
Further thoughts
In the middle of the 19th century, according to this site,
Servants, who had all living expenses taken care of, earned as little as £10/year, and the sign of being (or having become) a member of the middle class was having at least one servant. Some poor vicars at mid-century earned as little as £40-50/year.
Would you be able to survive on the modern equivalent of the poor vicar’s wage?
An interesting further investigation would be to get hold of some information on the wages of various occupations, and try to translate them into modern equivalents — or, equivalently, to try and convert modern salaries into those of Victorian England. Many interesting difficulties await!
Tags: inflation, Maths, money, purchasing power, research paper
Visualisation: Six moving lines
As promised, here is an animated Geogebra visualisation. Click ‘Start’, and six moving lines will appear.
- What behaviours do the lines exhibit?
- What relationships are there between the different lines?
- How are these lines defined?
(Source: movinglines.ggb and movinglines.html.)
Click on ‘Show Values’ and three values will appear — all six of these lines are generated from these three values. If you click on ‘Stop’, and move the sliders, then the lines will be automatically updated to reflect those new values.
- Can you make any conjectures now about how the lines are defined?
- What values would we need to make one (or more) pair of the lines
- perpendicular?
- parallel?
- vertical?
- horizontal?
- Can we generate any line we wish by setting the values appropriately?
I managed to captivate both a lower school class and an A-level class with this visualisation for quite a while last year, and the A-level class made good progress in making and justifying conjectures about what was going on.
Technical Note
One of the few downsides of Geogebra compared to other dynamic geometry software, is that it contains no in-built facility for animation. Thus, to create any animations using Geogebra, you have to create the animations manually in Javascript. It’s not particularly difficult to do, particularly if you know some Javascript, as you can see if you view the source of the .html file above. Putting the ability to animate inside Geogebra itself is, though, my one big wish for future versions.
Tags: animation, geogebra, lines, visualisation
Taught: Using Excel to calculate Mean and Frequency
While on the subject of statistics, here is a resource which can help with teaching a useful facet of spreadsheets — calculating the mean, median, mode, etc. of data.
(Source: mean-and-frequency-in-excel.swf)
This was created using the excellent (and free!) piece of software Wink.
Wink is basically a free version of software like Turbodemo (or for the more Web 2.0 people out there, Jing), which allows you to create ’screencasts’, capturing screenshots and turning them into standalone animations (in Wink’s case, Flash .swf files). These are used in the help files of many pieces of software, like the graphing package Autograph. As well as capturing screenshots, and keypresses, you can add annotations, and link forward/back to different sections of the recording.
Screencasts like this have great potential for demonstrations, not just of technology, but also as ways of recording how to solve maths problems. They also let a teacher run through a problem without having write material on the board constantly. The downside, of course, is that they do require a significant amount of effort to produce, but sites like MathCasts are beginning to offer a number of premade screencasts, which I need to look through at some point!
Tags: Excel, ict, Maths, screencasts, statistics, wink
Taught: Can you generate Binomial data?
Background
This is an way of developing the use of the chi-squared distribution, which can also be used to test whether your students can remember what the Binomial distribution looks like!
Start by challenging everyone in the class to generate some data which they feel could be modelled by a Binomial(5,p) distribution, for some value of p. Calculate the value of chi-squared for the frequencies entered (which involves reviewing how to estimate the mean, and what the formula for Binomial is), and then compare that with the critical value needed for the data to be a ‘good fit’ (to, say, a 5% level).
Interactive Binomial Fitness Calculator
I thought I would set myself the challenge of converting this activity into a form which could be directly placed on a Webpage, like this one. After a few days messing around (and a morning wondering why Wordpress didn’t like my Javascript), I can present the following:
Can you generate data which can be modelled well by a binomial distribution?
Try entering frequencies below for data which can be modelled well by Binomial(5, p), for some p. After entering the numbers, click 'Calculate', and the computer will assess how well your data fits a Binomial by performing a chi-squared test.
| Successes | Frequency |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 |
(Source: testbinomialmodel.html.)
Note that the condition it is using for goodness of fit is the 5% critical value for chi-squared with four degrees of freedom (6 - 1 because we know the total frequency - 1 because we’re estimating the probability). Note also that it does not combine cells.
Uses
Beyond an initial check of how good students are at modelling a Binomial distribution, this interactive tool can also be used as a tool to explore the Binomial and chi-squared, by systematically altering values and seeing what happens. We can also do something similar, but fix a particular value for the probability of success — this makes it easier to improve by ‘trial and error’ toward a fixed destination, and would also allow us to discuss whether a fit can be too close.
Tags: chisquared, javascript, Lessons Taught, statistics
The Joy of Hex
(Update: This post is featured in the 39th Carnival of Mathematics. Check it out!)
Last summer I developed a bit of an obsession with hexagons. More specifically, with the patterns you can produce from tiling multiple copies of a single, simply decorated hexagonal tile:
I cut out and laminated around 100 of these tiles, and spent several happy hours moving them around into new patterns, trying different types of symmetry, and experimenting with different layouts.
If you’d like to follow me down a similar path, here’s a sheet with six of these hexagon tiles to print and cut out: hexagon-tiles-122.pdf (svg source), made using the free vector drawing program Inkscape.
Background
Although I wasn’t aware of precursors at the time, I later found this shape in several places: it is one of the tiling generators you can buy from the ATM, and it’s one of the tile shapes in the game of Tantrix. I recommend you browse the ATM’s store if you are a maths teacher — many excellent things await!
I also recall reading a very interesting web page which found this exact pattern being used as a paving slab in Spain, and developed quite an in depth mathematical discussion, but sadly this page now seems to have disappeared into the æther.
Although less common than they used to be, hexagons were often used for tiling and paving. I particularly like this hexagon paving tile designed by Gaudi, and used all over Barcelona.
Returning to the particular hexagon above, we get the following when we tile it:
The fun comes when you rotate some of the hexagons. Even with just six patterned hexagons you can find a range of different interesting patterns… but it’s much more fun to explore with a large pile of them!
In My Classroom
At the start of last year, the tiles appeared on my classroom walls in the place now occupied by the Pascal’s Triangle wall display. As with Pascal’s Triangle, the tiling wasn’t static, but slowly changed form and shape over time, and several interesting questions were raised by some of my groups (several of which couldn’t believe at first that all the different patterns were generated by a single type of tile).
Eventually the hexagons migrated over to the other side of the classroom, and took the form revealed in the very first post I made on this site:
Read more for: more examples of patterns you can produce from these tiles; some questions for you to explore; my thoughts on the mathematical content of this ‘pattern space’; and source files for all the diagrams.
I suggest you pause here, print out the hexagons, and have a play before continuing.
Some Patterns
Here are eight patterns made with the hexagon tile, on a 6-by-6 grid:
(click on the images for a better view). These by no means exhaust the patterns you can generate with this tile.
Questions Raised by the Patterns
I believe that these tilings form a rich and non-trivial environment for pattern spotting, and the generation of mathematical questions and conjectures in the classroom. Here, for example, are some questions which occurred to me as I was exploring the ‘pattern space’:
- Several of these patterns have rotational symmetry of order three. What other rotational or reflectional symmetries are possible? What about if we allow ‘holes’ (i.e. we don’t have to place hexagons in every grid position)?
- The last of the 6-by-6 patterns demonstrates glide reflection. There are seven possible frieze patterns; can we generate examples of all of them?
- Various motifs recur in different patterns: a small circle using three hexagons; an oval using four hexagons; several braids; a large circle and a trefoil using six hexagons. What other closed loops can we make using only a small number of tiles? Also, are there any forbidden lengths, with no examples of loops of that size?
- Fix a small grid size (for example, a 2-by-3 rectangle). How many distinct patterns can we make?
The meaning of ‘distinct’ here is something that would need to be negotiated with the students. If we take a pattern and rotate or reflect it, should we count that as a new pattern?
I think all these questions could be generated, explored, and altered quite productively by students at a range of levels. They also give students an opportunity to demonstrate several key reasoning skills, such as systematic exhaustion: how can we be sure that we’ve found all the possible patterns of a given grid size, or loops of a given length?
Complexity
In some sense, all the questions in the previous section were combinatorial. We had to count the number of ways that something happened, or generate examples.
Here’s a deceptively simple question which leads into an investigation of another sort:
Which of the patterns above is the most complex?
We might perhaps say that simplest patterns are those which we can describe in the shortest amount of space. The very first pattern, for example, could be described by saying ‘all the tiles are this way up’. How could we describe how to generate some of the other patterns?
Given this view of complexity, what do complex patterns look like?
What do simple patterns look like?
This seemingly naive and simple view of complexity, when formalised, leads to algorithmic information theory, and in particular to the concept of Kolmogorov complexity. This also has connections to an immense body of work in computer science, including compression algorithms.
Transformations
Moving away from complexity, let’s now consider what happens when we start with a pattern, and want to alter (transform) it in some way.
It is obvious that we can transform one pattern into any other pattern by rotating each tile in turn — but what happens when we impose constraints on the ‘moves’ we are allowed to make? For example, suppose we had a rectangular grid of tiles, and were only allowed to rotate the tiles a row/column at a time.
Starting from the basic pattern we saw right at the start:
we could rotate the second ‘column’ one step anticlockwise:
and then the third row:
and then the fourth column:
and then the second row:
Could we generate every possible pattern this way?
This is an interesting question, but probably a little difficult to approach at the secondary school level. So, let’s be even more restrictive:
Suppose we rotate all the tiles at the same time. What happens?
To be more specific, let’s start with this pattern:
What do we get if we rotate all the tiles one ’step’ clockwise?
Now is an excellent time to print out some hexagons and find out!
What about if we rotate again? And again?
What happens with different starting patterns?
What is preserved by this transformation, and what is not? If one pattern repeats every three hexagons for example, then the other one will as well. But what about symmetry? What about the ‘loops’ formed by the lines?
Notice that we are exploring, in an accessible way, several advanced concepts: applying a transformation to something; preservation of features; iteration of transformations among others.
Where’s The Maths?
Usually, in school mathematics, we only consider functions which transform numbers into other numbers — even transformations such as rotation, reflection and enlargement are almost never talked about as functions which can be combined, or reversed, or iterated. This naturally makes it harder for students to ’see the maths’ in situations which don’t directly involve numbers.
One way to help students to become comfortable with these ‘non-traditional’ areas could be to improve the emphasis, throughout their school careers, on key concepts and questions, like iteration (what happens if we do something many times?), inversion (how do we undo what we just did?) and iso- & homo-morphism (what has changed, and what has stayed the same?). These are applicable at all levels of mathematics:
What happens if we add the same number lots of times?
How do I undo a multiplication?
What properties of my triangle stay the same when I enlarge it?
What happens when I differentiate a polynomial lots of times?
How do I undo exponentiation?
What properties of number stay the same if we allow the square root of -1 to be a number, and which have to change?
The same types of question recur throughout the mathematical development of a pupil.
Conclusion
Even something as simple as a hexagonal tile can lead to some really deep mathematical questions, and thinking about them can develop skills which are applicable in many more traditional areas of mathematics.
It also lets you use the pun in the title — and we don’t get the opportunity for puns often in maths!
[As promised, here are the source files for many of the above diagrams.]
Tags: geometry, hexagons, investigations, Maths, patterns, tiling, wall-display