Posts Tagged ‘Lessons Taught’
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
Taught: Exploring and Generating the Mandelbrot Set
For most of the final term of last year, I had a simple display on one of my walls showing a journey into the Mandelbrot Set (the pictures for which were taken from this Wikipedia collection). Several of my groups had asked me what the pictures were, and how they were derived, so I decided to spend a lesson with several of my A-level groups in the last week of term exploring this mathematical object.
I created the following Geogebra file to help me:
(source: mandelbrotorbits.ggb)
This shows the first thirty iterations of the map zn+1=zn2+c, where c is the value represented by the blue point. As you move the blue point around, the other points are automatically updated.
For some values of c the values will stay within a bounded region of the origin, no matter how many iterations we try. For others, the values will ‘escape’ towards infinity. It can be shown that if the modulus of the points ever becomes more than 2, then the points will eventually become unbounded.
The Mandelbrot Set is the set of values of c for which the values never ‘escape’.
My A-level sets had not met many of these concepts before (they’ve never met a complex number before, for example), so were introduced to the ideas geometrically. We generated some iterations by hand for c=0.6i, and then, using an Excel spreadsheet to help us with the calculations, created our own very low resolution Mandelbrot set on the interactive whiteboard, using counters.
It made a good end of term lesson, even if we couldn’t cover all the required mathematics in great detail. If nothing else, I find it quite absorbing just moving the initial point around, and seeing the patterns that emerge in the iterations… particularly when you notice that different ‘buds’ of the Mandelbrot set exhibit different behaviours.
Tags: geogebra, Lessons Taught, mandelbrot set, Maths