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.

Optimal Cuboid

Year 7 students had fun with this idea last week. We have been exploring the idea that there are different nets for a cube and going on to look at the nets of cuboids and the ideas of surface area and volume. So students were given a piece of A4 card and asked to draw the net, cut out and build a cube 5cm x 5cm x 5cm. Then with the remaining card they have to build the biggest cuboid they can. There was lots of thinking about this to do before they cut anything out! How do I draw the net of the cube that leaves me the most card left over? How can I make the cuboid as big as possible? What does 'big' mean in this context? How can we decide who is the winner? Who has built the biggest cuboid? Fun was had, cuboids were created and thinking was done!

More can be read about the thinking behind this activity here!

Mobius Mess

A small group of year 6 and 7 students gathered after school today to make a proper 'Mobius Mess'. With paper scissors and glue, we explored all the different features of the wonderfulMobius Strip. How many faces does it have? How many edges? What if you twist it twice? Three times? What if you cut them all in half along their length, what happens then? It would be great to write all about that, but that would spoil the fun for those of you that havent tried it. There are lots of sites out there on this topic and we used this one to get us started. We made some a bit likethese too! There are lots of surprises and fascinating things going on! We recomend you try yourselves, but the video below might give you a quick overview and you might understand how we made such a 'Mobius Mess!'

Posted by Jim Noble, Curriculum leader for secondary mathematics, International School of Toulouse, www.intst.eu, Co-author www.teachmaths-inthinking.co.uk

     

The Big Breakfast Venn

It started out as a simple idea. IB Maths studies students drew a giant Venn diagram on the playground with chalk to do a survey of what people had for breakfast. They stood where they belonged in the diagram and they drew little stick figures of themselves to serve as a record. At break we invited other staff and students to put themselves in the diagram, then lots of classes from the primary school came over and before we knew it we had created 'The Big Breakfast Venn'. The drawings got more and more elaborate with the art department steeling the show! It was a simple idea just to play with different sections of a Venn diagram before going on to some more complicated problems and it turned in to a great whole school activity. Hopefully below you can see some of the great pictures we got to keep a record!

Posted by Jim Noble, Curriculum leader for secondary mathematics, International School of Toulouse, www.intst.eu, Co-author www.teachmaths-inthinking.co.uk