Archive for August, 2008
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
Hinged square dissection
In several recent posts I have referred to ‘linkages’ (which should more properly be called ‘hinged dissections‘. One good recent book on these is Hinged Dissections: Swinging and Twisting, by Greg Frederickson, but there are several classics out there that discuss hinged dissections, including Amusements in Mathematics by H. E. Dudeney.
Dudeney was responsible for one of the most well known hinged dissections, of which this is a simplified example. It converts a square into, well, something else.
The white circles are the hinges — move the coloured circles to move the corresponding parts of the square.
Can you predict what the end result of the transformation is? Can you prove it?
(Source: dudeney-dissection.ggb.)
The Haberdasher’s Puzzle
Dudeney’s classic dissection, published in his ‘Canterbury Puzzles‘ in 1907, is a slightly altered version of this, which allows you to transform a square into an equilateral triangle. You can download a program which will allow you print out a template for this here.
Constructing Dudeney’s dissection takes a touch more effort than the dissection illustrated above, but the process is described incredibly well in this lesson plan, which demonstrates how to make a model of the dissection using foam rubber. I haven’t tried it yet, but it may make a tempting break from lesson planning next week!
Tags: dissection, Dudeney, geogebra, linkage, puzzle
A quickie on quadratics
Fix two points in the plane, and consider all the quadratics which go through those two points.
Find the locus of the stationary points of these quadratics.
The following Geogebra worksheet might help. If you select “Show stationary point” and move the blue point, it will trace out the locus.
(Source: quadratic_stationary_points.ggb.)
Thinking about this question kept me from going to bed until far too late on Thursday. My answer, and other thoughts, will appear tomorrow.