Welcome Welcome to the thirteenth Autograph Newsletter! Each jam-packed edition looks at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better.
 Introduction Probability is (probably) my favourite topic in mathematics. I love the often counter-intuitive results, and how you can use knowledge of the subject to get one over on uninitiated “victims”! A key component of the topic is Probability Distributions, and I have found it frustrating in the past that students have found this particular area rather dry, abstract and uninspiring. I believe that is due to the fact that the calculations involved tend to be rather onerous, and distract from the crucial understanding behind the underlying concepts. Fortunately, Autograph can come to the rescue. The excellent statistics engine allows you to generate, sample, plot and investigate every probability distribution you could imagine, allowing your students to visualise, speculate, test and most importantly understand exactly what is going on.
 Diagnostic Question Diagnostic questions are ideal to use at the start of the lesson to enable you to get a quick and accurate picture of your students’ levels of understanding. They are designed in such a way that common misconceptions that your students may hold should steer them to one of the incorrect answers, thus allowing you to learn where the problems lie from their responses. Typically I give my class 30 seconds thinking time and then ask them to hold up their fingers: 1 for A, 2 for B, etc.
 Free Online Autograph Activity Standard Normal Distribution Calculator Use this handy tool to check and visualise calculations using the standard normal distribution. These Autograph activities do not require the full version of Autograph to run them. You just need to install the free Autograph Player (you will be guided through how to do this), which means you can use these activities in the classroom or set them for your students to do at home.
Ideas for Extension
The following ideas for extending the topic of constructions require the full version of Autograph. Click on the image to download the individual Autograph files.
Note: To create each of these files, please ensure you open a new  Statistics Page and you are in Advanced Mode.
Idea 1 – Constructing a Probability Distribution
Autograph allows you to generate user defined probability distributions and take samples from them. Here we will construct a biased dice.
 • Click on Enter Raw Data • Choose Select Distribution, click on User (discrete) and click OK • Now click on Edit Distribution • Here we can specify the proability of getting a score of 0, 1, 2, 3… etc • We want a biased dice, so type in 0, 1/9, 1/9, 1/9, 1/9, 2/9, 3/9 and click OK • Now let’s generate a sample, so change the sample size to 1000, click Create Sample and then OK • What do you think the graph of 1000 rolls of this biased dice will look like? • Select Dot Plot and ensure both spacings are set to 1 • Click on Default Scales if your graph goes off the grid • What do you think the mean and standard deviation of this distribution are? • Click on View Statisitics Box to find out • To change the distribution or sample size, simply double-click on Raw Data 1 at the bottom left of the screen • What would it look like using only 10 rolls of the dice? • Can you construct a symmetrical probability distribution for a biased dice? • We will be using this file for Idea 4 below
Idea 2 – Binomial Distribution
Here we will look at carrying out calculations and investigating the shape of the binomial distribution
 • Click on Enter Probability Distribution • Choose Binomial and click OK • Set n (the number of trials) to 8 and p (the probability of a success) to 0.5 • What do you think this distribution will look like? • Click OK • Click on  Probability Calculations and click OK • You can drag the ends of the band at the bottom of the graph over the region you want to calculate the probability of • Click View > Status Box to keep an eye on the probability calculations • Work out the proability that r is between 4 and 6 and then check it • Double clicking on the Discrete Probability Calculations allows you to change to cumulative probability measures, such as r is less than or equal to 4 • What do you think would happen to the shape of the graph if we changed the probability of success to 0.8? • Click on one of the bars of the graph to select the distribution • Click on  Animate and experiment by increasing and decreasing the value of p • You can also change the value of n and observe the effect on the shape of the graph and the probability calculations
Idea 3 – Standardising the Normal Distribution