Archive for the ‘Lessons Taught’ Category
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
Visualisation: More rotating squares
While we’re rotating polygons, here’s another nice visualisation, this time of a linkage of squares that make up a larger square, and rotate around without self-intersecting as you move the slider.
(Source: rotating_square_linkage.ggb)
Just as with the previous visualisation, you can see this just as a source of pretty pictures, like
or you can start asking yourself questions. For example:
- What other ways can we find to connect the small squares together so that they will expand and collapse back into a square without any of the small squares intersecting? How many of these have rotational or reflectional symmetry?
- How can we describe the movement (the locus) and the amount of rotation of the different squares?
- Can we do something similar with other polygons?
- More generally, can we do something like this to move smoothly between different shapes? And what does ’something like this’ mean?
For the second question, we can use the ‘trace’ feature of Geogebra (and all other dynamic geometry programs) to follow the path of particular points, which might give us some idea about how to derive the equations of the points.
On the last point, there are a large category of dissections which involve taking an object, slicing it into a finite number of shapes, and reassembling them into a different shape. There are several in Amusements in Mathematics, a puzzle book by Henry Dudeney from 1917 which I had a part in digitizing for Project Gutenberg. See this section, for example, for a great introduction to dissection puzzles.
Moving further afield, we can do something very similar to this example to demonstrate Pythagoras’ Theorem. That visualisation may appear later this week :).
Tags: Books, dissections, geogebra, visualisations