Interactive Histograms with Geogebra
The new version of Geogebra has several nice new features which make it much more useful for a range of statistical uses. Firstly, it comes with a simple spreadsheet-style view, which allows you to enter and manipulate data in a grid of cells, similar to a spreadsheet. Secondly, it has a number of new statistical functions, covering a range of data creation, summation, and visualisation options.
Here is an applet which demonstrates a couple of these new features. It takes 50 random values, generated to fit a Normal(8,4) distribution, and plots a box-and-whisker plot and a histogram.
All this just uses what are now built-in features of Geogebra. My contribution is to make the histogram interactive: move the blue points on the x-axis around to alter the class boundaries. This lets you explore the ways that small changes to the class intervals can sometimes have large effects on the histogram.
Enable Java to see this Geogebra applet.(Source: adjustable_histogram.ggb.)
Double click on a cell in the spreadsheet view to change its value. Also, just as in Excel, press F9 while in the spreadsheet view to regenerate all the random numbers. You may see the purple x-axis points move: they have been constrained to always be below the minimum and above the maximum value.
Tags: applet, histograms, statistics
Not enough information?
It’s been a busy term, and it’s not over yet, but I promised that I would post something this week. So, a question for you to explore:
The second term of a geometric sequence is 2. What’s the sum to infinity?
If your first thought is ‘it could be anything’, then you’re not alone, but you’re also not correct!
If you get an answer algebraically, can you give a geometric interpretation?
Classroom Resource: Countdown Timer
This term my school is running some maths competitions involving timed rounds. The questions are in Word, and I've been getting quite frustrated with the countdown timer built into Activstudio -- in particular, it makes resetting the count very fiddly. There are a lot of countdown/timer applications around, but the ones that do what I want seem to be either stupidly overcomplicated, not free, or to crash all the time.
To get around my frustrations I've cobbled together the following simple countdown timer, written in Javascript:
Source: countdowntimer.html.
Not particularly flashy, but it gets the job done!
You can start/stop/reset the countdown, and alter the starting count in 30s increments, just by pressing buttons, which makes it easy to use on an Interactive Whiteboard. In addition, if you click on the ^^ link (at the bottom right) the timer will open in a new window, with as few toolbars as your security settings allow.
Tags: countdown, javascript, programming, resource, The Classroom
MA Conference: A circular NRICHing activity
I am at the Maths Association conference, being held in Robinson College, Cambridge. Over the next few days I’ll be writing reports and reflections on the sessions I attend.
Session title:
Embedding Rich Tasks into the Curriculum
Session speaker:
Charlie Gilderdale
What was meant to happen:
We have been working with a number of schools trialing how problems on the NRICH website can be integrated into the curriculum. Find out how this is being done and see how our mapping documents can help your school. Of course, we will do some problem solving along the way.
What actually happened:
I have been to sessions run by Charlie Gilderdale before, and they have all been great fun. He is one of the driving forces behind NRICH, a wonderful resource of over 5000 maths problems, investigations and activities (many with teacher notes and solutions). They have also recently started to build up documents which map their activities into the UK maths curriculum. This was meant to be the focus of the session, but happily we spent most of the time working on an excellent set of problems which would be a great way to lead into some circle theorems.
Charlie started by showing us the Virtual Geoboard available on the NRICH site. If you’ve never seen a Geoboard in real life, it’s just a board with pins sticking out. You can then use rubber bands to create various shapes. The virtual geoboard lets you do this… virtually.
He then handed out a few pages of grids with 9-pin circular geoboards to everyone, and asked a simple question: What angles can we find, if the only things we are allowed to draw are segments joining two points on the geoboard? In particular, can we find angles of 10 degrees, 20 degrees, all the way up to 360?
This triangle, for example, gives us 40 degree and 70 degree angles:
We were asked to assume that we had primary-level knowledge about angles. So, we weren’t allowed to use any circle theorems, but were assumed to know that the angles in a triangle add to 180, the angles around a point add to 360, that vertically opposite angles are equal, etc.
Over around 15 minutes, we managed to find all the required angles except 150 and 170 degrees — can you fill in these gaps?
We were then asked to draw as many quadrilaterals as possible using these 9-pin Geoboards (without using the central vertex). Here is one example:
This lead to a number of interesting discussions:
- When are two quadrilaterals different, and when are they they same? We settled on saying that two would be the same if we could cut one out and place it exactly over the other (in other words, rotations and reflections don’t make new shapes).
- How can we try to systematically find examples?
- How do we record the examples that we find?
- How we could convince ourselves, and others, that we had found all the quadrilaterals?
Having found these quadrilaterals we were then asked to calculate their angles. Again we only used late-primary maths. We found a number of ways to calculate the angles, and particularly noticed how symmetry helped to make the calculations easier.
Charlie then pointed out that the results obtained from this activity could be used to get students to notice that the opposite angles in a quadrilateral added to 180. Would this happen if we changed the number of dots around the circumference? He handed out some 12- 15- and 18- pin Geoboards, and asked us to check, and then to try and develop a proof.
We soon found a very nice visual proof that the opposite angles added to 180 when the centre of the circle was ‘inside’ the quadrilateral, and were asked to try to develop a similar visual proof for the other case. He noted that this could be a great way to introduce this circle theorem to children, without having to immediate descend into the murky world of algebra, but also without copping out and just getting students to measure angles with protractors.
(Incidentally, there are a variety of other Geoboard-based investigations on the NRICH site. To find them just search for geoboard.)
In the last couple of minutes of the session we moved onto the mapping documents, which can be accessed from this section of the NRICH site. They are mostly useful for teachers in English schools, but could also be useful for teachers outside the UK, to help navigate around the multitude of problems on the site.
In summary, then, a fun hour, working through an investigation that was accessible, enjoyable, and has extensions leading in a number of different directions.
Thoughts:
It is not hyperbole to say that NRICH has developed into one of the best free maths teaching resources on the internet. Every month they present a selection of their problems, arranged around a particular theme, and at a variety of difficulty levels. They encourage students to write in with their solutions, which may get added to the site — several of my students have submitted solutions, although none of them have been featured so far.
The addition of the mapping documents has made it even more useful for those of us in the UK, because we no longer have the danger of spending a very enjoyable couple of hours on the site without actually coming out with anything we can use in a lesson!
I really liked the particular activity that was introduced in this session, and am looking forward to trying it out with the Primary Maths Club that I’m running this year.
Tags: circles, geoboards, lesson idea, nrich
Clumping and clustering
I've had a fun weekend playing with Processing and improving the clumping visualisation I posted earlier this week. The new version has been uploaded to OpenProcessing.org, a site which contains over 1000 visualisations, many of them containing beautiful and interesting mathematics:
A brief reminder of the situation: We have a grid of squares, of several different types. These squares are happy when they are surrounded by enough squares of the same type as them, and unhappy otherwise. We take two unhappy squares, swap them over, and repeat until bored!
There are two key parameters here: the number of different types of cell/person/house/whatever these squares are representing, and the happiness threshold: how many neighbours need to be the same as a cell for it to be happy.
Let's look at the types of behaviour that emerge for different values of the parameters:
A behaviour catalogue
| Neighbours needed for happiness | ||||||
|---|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | 6 | ||
| Types | 2 |  |
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| 3 |  |
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| 4 |  |
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| 5 |  |
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| 6 |  |
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You find that it takes longer and longer for the state to settle down as you increase the number of types of cell. With 6 cells, and a happiness threshold of 4, for example, you can see from the table that it does eventually settle into a fairly stable clustered pattern, but it takes a very long time. After 50,000 swaps the grid looked like this:
Apart from the special case of two types, with happiness threshold five or six, the cells always seem to settle into a relatively small small clustered pattern (with small threshold), or to not stabilise at all (large threshold).
2 types; threshold 5: this is the case explored in the earlier post on this clustering behaviour. The cells cluster together fairly quickly, and then over the next 20,000 swaps or so the clusters join together into larger features, with the borders being long stretches of horizontal/vertical/diagonal lines.
2 types; threshold 6: this is perhaps even more interesting then the previous case. We do get some clustering, but the clusters don't hold together, but fray, and move around.
As with the previous post on this topic, I have wrapped left-right and top-bottom. This doesn't change the behaviour significantly. If you'd like to explore what happens without this wrapping, then go get the source code and dive in!
Emergent Behaviour.
This 'clumping' behaviour is just one of many examples of emergent behaviour, where small-scale local actions add up to a sometimes surprising global behaviour. Sometimes, as the old aphorism says, the whole is more than the sum of its parts.
One of the best known examples of this emergent behaviour is flocking, where groups of birds fly together in flocks that seem to require some collective intelligence, but are actually generated from purely local actions. This behaviour was modelled in Boids, a classic Artificial Life simulation from 1986. Here's an excellent 3D Boids simulation.
Tags: clustering, processing, visualisation