Patterns in squares in Year 4

We been having a look at patterns of squares using Cuisenaire rods in Year 4, starting by just making any design featuring as square.

At first it wasn't as simple as that might sound. We were getting a lot of almost-squares. Look, for instance, at this one, created on the Cuisenaire Environment -

At first it fooled a lot of us! It's got at ten on each side...

A lot of interesting patterns emerged worth following up.

It would be interesting to think of mathematical question about these shapes. With the faces for instance, how does the perimeter progress in each of the two kinds of pattern? What area of table is visible in each?

Pascal, Paloma, Parsang and Perry Perimeter

(Also published on the Year 4 blog.)

Lego Maths with Y6

Unusually this autumn has seen more than its fair share of wet playtimes.  During these sessions, many children have been attracted to the box of Lego bricks. Watching them build and play with the bricks, they instinctively know that if they are building with 8-stud bricks, and they run out, then two 4-stud bricks will do, or a 6-stud and a 2-stud, or four 2-studs.  Arrays are everywhere.  They are doing maths all the time without realising.

Wanting to take advantage of this enthusiasm, I decided to investigate the numerical properties of the bricks with my Year 6 class to see what they came up with.  I asked them to look for numbers and (for today) discouraged building.  I encouraged recording anything they found out.

They found out evidence of factors, number bonds, arrays, divisibility, sequencing, square numbers and experienced lots of data handling and discussing the numerical properties of their bricks.

One student found out that 100 was not divisible by 8, whilst another was intent on finding patterns of square numbers using 4-stud bricks (numbers of studs and numbers of bricks).

Another group were keenly trying to find the relationship between the number of studs on top of a brick with 2 rows of studs and the holes underneath.   

They could see that the number of holes was one less than one 'line' of studs.  Once they organised their results and checked their ideas, it was only a small step for them to realise how to explain using maths vocabulary and then with a bit of help created a formula.
This sparked ideas in others who wanted to try and find the 'rule' if the flatter base (rather than a brick) had rows of 4 studs instead of 2.  Although we ran out of time, the discussion went into the next lesson as one student was determined to explain to me how to get the answer (correctly!).

He came up with the mathematical rule and formula overnight.  
Can anyone else find it too?

This year's Forest of Factors

Following on from last year's post Important Factors, and after our work finding prime numbers with Cuisenaire rods and in the playground, Year 4 had a go at making prime factor trees this year. Here's our Forest of Factors:

See more on the Year 4 blog

Our trees made a really big Forest of Factors display for the primary corridor. As the Year 2 and Year 3 children are passing it regularly, it was a good idea to explain the display to them. We did it without using the words "factor", "multiple" or "prime" so that it would be easier for them to understand:

 "Hello everyone. This is our Forest of Factors."

"1, 2, 3, 4.... 40"

"What do you notice about our pictures?"

"Number 1 is my bird. It's different to all the other numbers. If you times something by 1 it just stays the same number."

"So 1 is special, and we don't think about it with our trees and mushrooms. It's just flying up in the sky!"

"Some of the numbers are simple trees. Look at 38. 38 is 19 times 2, so we've given it two branches that finish with flowers."

"Ten is five times two."

"26 is 13 times 2."

"These ones had just two branches and two flowers."

"But some of the trees have more flowers. Like 12, and 32."

"With the mushrooms there's no two numbers that you can times together to make that number."

"Can you see a number that is a mushroom?"

"So that's our Forest of Factors. We hope you understand it a bit better now."

Finding prime numbers in Year 4

In Year 4 we've been looking at prime numbers this week.

Some of us have been out in the playground, using the hundred square there:

Here's a how we did it:

Later we had a go in the classroom, using a photo of the hundred square (16 colour bitmap format):

and, on Paint, filling he numbers in the 5, 3, 7 and 2 times table with colour:

And there we have it: the grey ones (except for 1) are the prime numbers!

(also blogged on the Year 4 blog)

How big is a million?

We've been talking about the dinosaurs in Year 4, which has involved us in some pretty big numbers. For instance, the last dinosaurs died out 65 million years ago.

Have a look on the Year 4 blog to see exactly how big a million is, and how it could be dangerous in the classroom.

Mirror Dancing

For a number of years we have run this annual mirror dance activity at the International School of Toulouse, but this year we decided to make a bit more of an event of it! I first got the idea from a Workshop run by Anne Watson at the UK based ATM conference a few years ago!

The idea is simple! In groups of two or more, students must make a dance routine lasting between 60 and 90 seconds. The routine must be based on symmetry. in the first instance, reflective symmetry where students are each others reflection either side of an imaginary mirror line. Afterwards, students can consider using two (or more) mirror lines, before going on to consider translation, rotation and enlargement! Students were judged and given a score out of 10 in three different categories!

1. Aesthetic appeal
2. Technical difficulty
3. Symmetrical accuracy

The total scores were then divided by the year group the team came from as a bit of a leveller!

More important than that was amazing creativity shown by all of the students involved. It is wonderful to see how students will play with these mathematical ideas when the right circumstances are created and we are really impressed with and proud of the work our students did on this. We are also particularly pleased with the willingness they have shown to both take part and then perform in front of each other! Well done to all students who took part!

Students have chosen their own prize and that is that they are able to throw wet sponges at the maths teachers!

See also 'Maths and Feet' from Simon Gregg based activities from http://www.mathinyourfeet.com/ from Malke Rosenfeld 

Here are some photos for now and there will be some videos to follow......

And now following the comment below, evidence of the prize giving!!!

Ancient Greek Geometry

Year 5G had a look at a great game, where the object is to create shapes with the classical "straight edge and compass" techniques: http://sciencevsmagic.net/geo/ It's author is Nico Disseldorp.

What's so good about this is that:

• It's a game! How this geometry should be. Maybe how it was for the ancient Greeks before someone wrote it all down and it had to be "learnt".
• It constrains you. You can only put your lines and circles in certain places. You have to follow the rules.
• It starts easy and gets harder, and records your progress.

We had a go with physical rulers and pairs of compasses first. We created all sorts of precise diagrams, and deviated off to some really beautiful ones that some of the children wanted to finish off at home.

by Amandine

But, with the kind permission of the people who'd done some of the pictures that weren't quite right (and it's OK to make mistakes in this classroom) we looked at how they were wrong. After all, these are not uncommon mistakes, and understanding them helps us understand something about knowledge itself. Here's one attempt to make a regular hexagon (it's in pencil so it didn't scan very clearly):

We could see here that the problem was that although the bottom points had been arrived at by finding the precise place where the circles cross, the top ones were, well, guessed at. You might call this kind of guess an informed opinion; we wanted something closer to a fact.

This one used the circles to draw all the points of the hexagon:

This is a lot closer to regular, but suffers from a lack of precision in the drawing.

So, armed with these reflections, we all managed - sometimes it took several goes, and tuition from those who got there first - we all managed to draw a good regular hexagon.

Then we went on to the Ancient Greek Geometry game / puzzle. Here we were helped to get over the problems of drawing by hand by being forced to be precise and to define points with known lines and circles.

One of the interesting things was that ten year olds can be quicker than their teacher! Jose showed Mr Gregg how to do the square, and Sophie showed him a quicker way of doing circle pack three!