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!

Pattern Blocks

Another maths "manipulative" (as they called them in the US - I'm not sure what the equivalent British term is). This time the pattern block, invented in the 60s, if something so simple can be said to be invented:

We got them out. Not just ours, but everybody's on the primary corridor. We'd been looking at Islamic patterns, and a good way to create our own was to use the blocks.

Luckily - because it would be hard to do it on paper - there's an online way to record, and extend what was done by hand with the blocks, and to save the images and publish them to the web.

We made the distinction between pattens that could potentially continue for ever, and ones that are growing from a centre but which perhaps can't be continued outwards indefinitely. There's something very compelling about those mandala-like patterns that grow from a centre and some of the class were reluctant to switch to a repeating and infinitely repeatable tessellation.

Next, we had a look at a few of the dodecagons it's possible to make with pattern blocks. Everyone made their own "badge", their own dodecagon.

(I liked the idea of making physical badges, and if the school had a badge maker I would have done it.)

We paused to look at the symmetries in the dodecagons. They have different numbers of lines of symmetry. Some with no reflective symmetry have rotational symmetry.



Three of the pattern blocks have areas that are multiples of the triangle's area. So that leads on to considering fractions. It's also worthwhile to look at the angles, seeing how many of each shape can fit around a point.

And finally, we touched briefly on the idea of a geometric proof. As these two dodecagons below have the same area:

how would you break the first up and remake it as the second? You'd have to swap a square for two thin rhombuses. So we've demonstrated the relative areas of these two.

Thank you Monsieur Cuisenaire!

In Year 5 we've been getting some of the classic bits of maths equipment off the shelves and dusting them down. Things like the cuisenaire rods, invented by the Belgian teacher Georges Cuisenaire in 1931 and recommended to the world by Caleb Gattegno in the 1950s. They seem to get talked about less these days, as a Google Ngram check shows:

But even if they're less fashionable, it's still worth seeing what they can do...

In Year 5 at IST we used them to investigate the "walls" of numbers. So, with ten for instance, what layers of the wall can you make out of just one colour? In this case there's the orange ten itself, the yellow fives, the red twos and the wood-coloured ones. The layers, of course, are the factors of the number. And it soon becomes apparent that some numbers only have themselves and one as factors: the prime numbers.
In fact using the cuisenaire rods now makes more sense than it did in the past. You can easily make a record of what you've done with the rods online. The cuisenarie environment here does the job very well.

If you're at IST you can see a video of some of the Year 5 children explaining prime numbers using the cuisenaire rods as examples here (you'll need to sign in to your Google Apps account).

As a footnote, it's fascinating watching how Caleb Gattegno used Cuisenaire rods in this film of a model lesson from 1961:

Prime Problem?

This morning our Year 6 class 'Magimixer' gave us 41 to make using the numbers 1,2,3,4 and 5. 

(The idea is rather like the 2012 game - you use any of the 4 operations and must include all numbers in order to make the target.)

Normally we use our multiplication and factor knowledge to help us.  Straight away some of us saw that it was a prime number.  This generated such tries as (4 x 5 x 2) + 3 - 1 = 42 and others.
Try as we might, we couldn't make it unless we were a bit 'sneaky' and put numbers together e.g. 45 - 3 - 2 + 1

This left us asking the question, 'Is it impossible because it is prime?'
Maybe someone can tell us.
In the meantime we'll try some other primes with the same numbers...

The 2012 game!

Students at the IST have started the new year with one of our favourite number challenges - the 2012 puzzle. In this puzzle students have to try and calculate all of the numbers from 1 to 100 using the digits 2, 0, 1 and 2 only.  Each of the 4 digits can only be used once and they must all be used. Add, subtract, multiply, divide etc are all allowed along with some other operations like square root, inverse sin, cos and tan, recurring decimals and factorials! Full details of the game and a game sheet can be found at the link above to the game.

This is a fantastic puzzle because of they way it invites students to think laterally when they get stuck! Everyone can get a start and everyone has a challenge. Students have been working on their own solutions, but have been sharing them on a more public display! Solutions can now also be submitted through the website! One student has been given a commendation for finding solutions to 84 out of the 100 numbers and whilst some of these still need checking, this is looking like the best entry so far! How far can you get? Do you think any of the numbers are impossible? Comments, thoughts and contributions are welcome below! Happy puzzling!

Probability Scales

Following from some of our work on Probability, year 7 students have been having some fun putting events on probability scales! It's a lovely activity because making decisions about how likely things are to happen inevitably leads to some kind of numerical or fractional reasoning! This is being posted from an iPhone as a kind of test, but we will add a gallery when we can.

Guess whats in the bag!

Probability in year 7 - So we put 15 different coloured sweets in a bag and then pulled 1 out at random and noted the colour and then put it back. We repeated this exercise a number of times and got the tally chart above. Each student was allowed 2 guesses at the contents of the bag with the first successful guess winning the sweets! The challenge was 'what' to guess and 'when' to guess! Guess early and you get the first chance at winning the sweets, but you have less information to help you. Wait too long and somebody might get there first. Some tricky decisions! The whole point is of course that the more times you repeat the experiment, the more likely the tally chart is to show the right proportions. What colour do you think the sweets were? Could there be a purple one in there somewhere? Are there definitely more greens? We had a lot of fun. These and similar games for playing with this idea are published here - Guess my colour, Roll'em and In a spin.

Coming up...

This is just a short entry to show some things that are coming up with my classes next week in mathematics here at the International School of Toulouse!

Year 7s are getting stuck in to a new unit on probability (link for IST members). There is lots of game playing and experimenting planned as we try and challenge our intuition with some good logical reasoning!


Year 10s will be looking in depth, backwards and forwards at percentages. We will have some fun looking at percentages in the media and trying to get to the bottom of the headlines like 'Jedi Knights are the worlds fastest growing religion.'


Year 11s will continue their work on a statistical analysis of UK number singles using this 'NumberOnes' activity that includes a cool database of the whole history of number one singles, how long they were at number one, how long the songs are and what decade the songs were released in. There are some great patterns emerging!

Year 12 Maths Studies students will be revisiting right angled trigonometry and using dynamic geometry software to build their own 'trig ratio calculator'. This, in an attempt to see how trig ratios arise from naturally observed phenomena!


Year 13 Maths Studies students are moving on from sets and logic to probability. ( I love how year 13 are studying the same topic as year 7) We will try and marry what we know about tree diagrams, and , or etc with the work we have done on sets and logic, starting with this 'Probability Trees' activity.

Making 3D shapes!

This is a display made by year 10 students learning about the surface area and volume of prisms, cones and spheres! This quickly moves from cuboids that are quite approachable to shapes with curved surface areas likes cones and spheres. We had particularly good fun making cones (follow link for more details). The challenge was to make a cone with a base radius of 10cm and a perpendicular height of 24 cm. Students were given an A2 piece of card and those measurements. Firstly they had to figure out what the net of a cone looks like then figure out what the measurements needed to be to make sure the cone met its requirements! There is no substitute for having to build shapes from nets to help understand how the net relates to the shape. By mistake, one group made a fabulous 'Bar of Gold' shape which is a truncated rectangular based pyramid. Anyone know another name? Some students discovered 'antiprisms' and one group made a beautiful shape that we dont yet know the name of - see the picture below. A special mention is reserved for the students who printed the proof of the surface area of a cone on the T-Shirt (one for the display and one for me)! Great, practical, investigative fun all round!

The unnamed shape!

Some more photos of our display below.