Independence Day!

This blog post is about an exercise that IB Maths studies students have been through to think about how the chi squared test of independence works! It is quite a sophisticated statistical concept designed to test if there is any significant relationship between two non numerical sets of data! In this case the aim was to consider if there was a relationship between Gender and political persuasion. We started by looking at this by imagining what survey results we would expect to get is we assumed that there was absolutely no relationship. This would mean that the likelihood of being left wing is the same for men and for women. Here are the tables we worked with...

Filling in these numbers helps students understand the idea of 'Expected Frequencies' through building on their intuitive understanding if the idea!

Now we went on to consider how the numbers in the table might look if we imagined quite a strong dependence between the two variables. Try these....

What is nice about approaching things this way is that the essential concept behind the statistical test has been explored before any mention of the technique and analysis itself. The whole idea depends on the difference between what we would expect to happen if there was no dependence and what actually did happen. With this established we can go on to analyse those differences. 

Of course, to do that, we need data about what did happen and this is where today's connected world has a real edge! We put a quick survey together using a google doc and decided we would collect some age data as well.... (Please feel free to add your own responses below) We then employed the power of the social network to get some responses. Students posted it on their facebook pages and we used what ever means we could to get some responses. We hadd 100s in no time and were able to use some real live data to perform and independence test on!

IB Maths Studies

The blog post is just to give an idea of some of the things IB Maths Studies students do here at IST. The theme here is 'Enquiry based learning' and by that I am talking about the sorts of activities that provide opportunities for students to make discoveries on their own through engagement induced enquiry! At least that is the aim! Let explain the point with the following examples The title each links to a fuller, resourced, outline of the activity.

Probability trees - This activity is about bridging the gap between the intuition of sample space diagrams and the efficiency of tree diagrams. Students will look at a problem from the two points of view, play with multiplying and adding fractions and hopefully see how tree diagrams are a more efficient way of doing the same thing! The activity is good for group work and physical manipulation, although it could be completed on computers by individuals if required. It may well take 2 to 3 hours to complete all of the tasks, but at the end, the hope is that students have a strong understanding of how tree diagrams work that they can apply to different problems.

Scattertastic - This activity makes use of two excellent virtual manipulative that are freely available on the web. The activity helps students to begin understanding the concepts of correlation, lines of best fit and degrees of correlation through the use of these manipulative.

Meeting Functions - Challenge students to really understand the concept of a function. Match a set of input values with a function and a corresponding set of output values. There are eight sets of three to make and only one correct solution. This activity is 'old meets new'. Students work with cut out bits of paper but can use calculators/computers to help them solve the puzzle!


Quadratic Links - This activity is about linking the graphing of quadratics with the equations themselves by looking at their key features. Students match pieces of information with different graphs using logical deduction. This practical group activity leads to being able to sketch graphs from their equations.


Making Cones - Explore cones by making one! This activity helps students understand where the formula for the surface area of a cone comes from and play with the associated mathematics. A great practical task that seems easy and works out to be more of a challenge. In making the cone students will confront some great mathematical reasoning and maybe even some algebraic proof! 


Which Rule - This activity is designed to help students solve trigonometry problems by encouraging them to 'Speculate' about what might be possible. Students are asked to state different truths or complete different equations for a given diagram without being told what to solve for. Having completed the equations they are asked to think about which of them is most useful for solving for a particular variable. So often students feel that they must know 'the right thing to do' before they proceed and are afraid to try things out to see what happens. Yes, it is possible to learn how to recognise certain types of problems but it is equally important to learn that problems can be solved by trying to use the different pieces of knowledge you have to make new ones. The sine rule and cosine rule can both be applied to any given triangle it is just that often only one of them generates an equation that can be solved. We can either learn to spot types of problems or to speculate with both. In practice, one often leads to the other and then we are better equipped to solve more problems. 

Geoboards

5 X 5, and even 3 X 3 geoboards are great for provoking some careful geometric thinking. We've been using them in Year 5. (Once again there are online environments to use, like this one - http://nrich.maths.org/2883 but we didn't use these this time.)


We used the 5 X 5 ones for investigating possible squares:
How many different-sized squares are there? At first no-one could find any other square than these ones:

Then I gave a clue: I twisted my board round 45°. Immediately, lots of the class saw some others:

and then we went home. It was only when I was checking out a book by Caleb Gattegno, who invented and popularised the geoboard, that I realised we'd all missed a few more. Can you see what they are?

To answer that question, we can label the pegs like I did with Year 5. Then we can name the square by the letters at the four corners:

A	B	C	D	E
F	G	H	I	J
K	L	M	N	O
P	Q	R	S	T
U	V	W	X	Y

The next question was: for all of those possible squares, how many different positions are there?

and, most interesting...
If the smallest square of nails has an area of 1, what is the area of the other squares?

We also asked the same questions of the triangles possible on the 3 X 3 geoboard (there are more).


That bottom right triangle, what is its area? How do you work that out?


Different students had different methods. 

Here are some students talking about their thinking:




One pupil made an observation on method. She said, "If you don't know how to do something, split it into bits." Useful advice in this case.

What's the Measuring Story?

Here’s the graph – what’s the story?

The Great Fractions and Decimals Debate!

In year 10 maths we had a great couple of lessons debating the following question.....

What is easier? Changing a fraction to a decimal or a decimal to a fraction?

This blog entry tells us only a little about the debate because we are inviting readers to cast their votes using the quick poll below. Rather than tell you what we concluded, we will explain how we approached the debate!

Firstly, we asked everyone for their gut reaction and to cast a vote straightaway. The group was a bit unevenly split, so we asked a few people who were wavering to change camps and then set the two groups the task of preparing their cases and preparing to counter anything put forward by the other team. Students decided to make each of their points with an example. We also agreed that following each point we would let the opposing team set a similar example that they thought would be harder. The great part of the discussion was teams discussing what they thought the other team would throw at them and anticipating the other teams arguments so they could think of examples to throw back.

After this we had a good go at a formal debate on the topic, with each side taking turns and responding to the others before putting the question to a vote based on the evidence presented. Perhaps we can add to this entry after some more people have cast their votes. Perhaps some people will continue the debate with some comments. Students would love to see some comments coming from different places! At this stage though, we will just say that this was great fun and an excellent way to bring out the many associated mathematical issues and generate a need for techniques and discussion. A great debate was had!

Cast your vote and continue below....

Maths by Measure

This week in Primary we are celebrating Maths Week at IST through measures and measuring.  As a taster, do you know what each of these are called and what they measure?

Paper Protractors

Inspired by this video from Vi Hart on 'Angle - a trons', year 10 students today set about making some of their own paper protractors! It is a fascinating exercise and a brilliant way to look at geometrical reasoning and proof. No measuring is done here, just reasoning. For example, starting with a square piece of paper and by folding, how many different angles can you make? How can you prove they are what you say they are? What do you have to assume as true to start with? It is surprising what can be achieved by combining a simple set of logical steps. For example, folding an angle in half or in to thirds will create that fraction of the angle. Once you know some angles you can figure out the others using a series of 'If Then' statements. A running theme through the exercise was the notion of considering how we go from axioms to theorems and how each theorem is dependent on the original axioms used to build it! Lovely, practical, engaging, and fun!

This video helped us get started,

And this link from the exploratorium was useful too!

Some pictures to follow!