Which Team is the Fastest?

A QUESTION OF TIME..?

As part of our IPC topic on the Olympics, we decided to run a relay race around part of our school grounds to see which team was fastest.  We have four house teams, Emerald, Ruby, Sapphire and Topaz.  Each student ran the 360m course and 'tagged' the next member of their team.  Individual times were recorded by hand using digital stopwatches.  Two teams ran at a time so that the others could do the timing.

However, despite our organisation, one person's time was not recorded.  In addition, the teams that didn't have 5 members had to have someone run twice which resulted in a much slower time.  This then led to thinking about what we should do to put this right.  Should we run again another day, or could we use mathematics to come up with a solution?  

So our problem is this, how do we make it fair for the teams with 4 members to compete with a team of 5?

As if by magic, a right angle appeared...

Year 5 looked at a certain kind of triangle. It's one inside a circle where the long side of the triangle is the diameter of the circle. The opposite corner is on the circle too.

At first we looked at these triangles on a big scale. The school pitch, like every football pitch, has a very convenient circle, with a straight line right through its middle. If that line is the long side of a triangle, what angle would the opposite corner have?



Next we used a pair of compasses and a ruler to recreate the situation:

We've also started using GeoGebra  in Year 5. It's great for showing how the angle is always 90°:

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com I didn't take it on to this with Year 5, but there is a fairly simple proof that this will be always the case...

Hexagonal Tiling

Trying Some Tiling

On a recent trip to Barcelona, I thought these lovely hexagonal Antoni Gaudi tiles in La Perdrera would generate some tiling investigations.

Reproductions are also used on the pavements of the city centre.
On returning to school I showed this photo to my students and we discussed the tiles, their shape and design and wondered if we could make some of our own that fitted together in the same way.  We noticed that 3 different ideas centred on alternate vertices which when they were put together made 3 different yet complete designs.  The bottom left tile bothered us somewhat though, as it didn't seem to fit the pattern.  Would ours do the same?  
So we investigated, first for a homework and then with the help of the photocopier and some good old cutting and sticking.  More of an art activity than maths, it could be argued, but the end results were very satisfying and made a stunning display.

First, some that didn't follow the above pattern:  

Then, others that did!

The bottom left tile on the original didn't bother us any longer!  Well done Year 6!

Will an A380 fit on our Pitch?

During a recent Maths Week measuring event, this question was posed to groups of children in our primary school (Y1-6 grouped vertically).  Luckily, days before, the school had been given three beautiful scale Airbus models, so we decided to find out.  

After a short discussion as to how we should work this out, the children measured the plan's dimensions and did the calculations together (the scale of 1:100 made it nice and easy). 

Next, armed with trundle wheels, we braved unusual sub-zero temperatures to measure the pitch.

Interestingly, the groups came up with different answers - counting the clicks of the trundle wheel proved trickier than expected - which in itself brought up questions about accuracy.  7 out of the 8 groups found out that the A380 would fit on the pitch, despite their conflicting original estimates!


We could have completed this activity in the comfort of our classrooms with computer facts and Google Earth at our finger tips, but for this time at least, we solved a problem practically and had some fun at the same time!
Other investigations and questions now await the other models...


Will an A380 fit on your school field or in your playground/school yard?

Mathematics and displays

This post is about our recent adventures with display. At the International School of Toulouse, we are luck enough to work with some people who are brilliant at making displays and making their classrooms and the school an inspirational place to look at. As a maths teacher of nearly 15 years I must confess it has never been my strong point. So, in response to this, we decided to set ourselves a challenge - to fill the schools reception with great display work about things that go on in mathematics classrooms here and show off some brilliant student work. We had a 'Maths Week' in which we scheduled an evening for parents where we would exhibit the displays and have a fun competition which we called - 'So you think you can count?' - I will blog about that in another post!

Anyway the pressure of having a date and an audience prompted us to really work on mathematics displays and the following is some evidence of what our school reception looked like!

A walkthrough - this video is a walkthrough of the displays

Some Pictures - here is a slideshow with some images of what we got up to!

 

The activities - Here is a list and some links to some of the activities that gave rise to the displays.

The Art of Fractions - Olympic Circles - Prism People - The Rice Show - The 2012 Game - Paper Baubles - Human Loci - Volumes of Pyramids - The Wisdom of the Crowd - Rectangular relations - Visualising Indices

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...