Five Rectangles

In Year 4 we've been looking at a puzzle, taken from Gary Antonick's 'Numberplay' in the New York Times:

Create a set of five rectangles that have sides of length 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 units.

The really quick and easy way to explore this is with Cuisenaire rods:

After we'd found five rectangles with those side lengths, we recorded them using the Cuisenaire Environment:

Here are more pictures:


Then we started looking at the area the rectangles covered.

We got total areas of
120, 121, 123, 125, 130, 154, 161 and 184.

Next question: what is the biggest possible area?

Alicia answered this - use the biggest side lengths on the same rectangles:

Area = 190

Can you see why this is the maximum?

And then, what is the smallest possible area?

Mimi answered this one - use biggest and smallest lengths on the same rectangle:

Area = 110

Can you see why this is the minimum?

This is a great investigation - manageable, easy to understand, and susceptible to taking off in many directions. We didn't try to see how many possible ways of making the rectangles there are - that was a step too far.

But just seeing some of the ways was worthwhile. It works well because the rods and the numbers 1 to 10 are easy to grasp. And the maths it takes us into is worth it - length, area, 2 X 3 = 3 X 2, multiplication facts, addition of five numbers... And then a bit of more abstract thinking - what would the maximum and minimum be, and why.

It may be the first time a class has tackled this particular puzzle. We can certainly recommend it to other classes!

We rounded this investigation off with a small puzzle: 

Make these rectangles, and then see if you can make a square by putting them together:
 1x6 4x7 5x8 3x9 2x10
 3x6 4x7 2x8 1x9 5x10 
1x2 4x5 3x8 7x9 6x10
 1x2 4x6 3x7 8x9 5x10

Also blogged on the Year 4 blog.

More maths with Cuisenaire rods in Year 4

We've posted some of our work with Cuisenaire rods here already (see for instance our work on patterns in squares). Since then, we've used them in lots of other ways. Click on the headings to get more detail on the Year 4 blog.

Cuisenaire multiplication square

This was great fun to make, and the image of it has been a great resource to refer to:

Forgetting about square roots (not)

Not usually in the Year 4 (9 and 10 year old) curriculum, so we said forget about it, but of course this paradoxical injunction worked magic! Topped off with a little work on the differences of squares.

Algebra with Cuisenaire rod sums

This seems to be the ideal way to introduce the idea of using a letter instead of a number. (And thanks to Don Steward for the inspiration.)

Rozenn and Pythagoras

When Rozenn came in with something her dad had shown her - Pythagoras Theorem - the rods were the ideal way of quickly seeing that for the 3-4-5 triangle the theorem was right.

 Of course this is Secondary school work really, but just as in art we look at work by, say, Paul Klee, or in music we listen to adult musicians and composers, why shouldn't we look ahead and marvel a little in maths too?!

Co-ordinates games and quizzes in Year 4

In Year 4 we've discovered some islands. (Naturally they all have buried treasure on them somewhere.) We've made careful maps of them. All the maps have coordinates, and we've used those to make a guessing game about where the treasure is buried. We started our work using a letter-number coordinate, and referring to the spaces, the squares, rather than the lines. Then we moved on to a number-number system, this time referring to the lines and where they cross.

For the treasure hunt game, first of all we secretly decide on and write down the coordinates of the treasure. Our partner then has to guess the coordinates. We say "hot" if the guess is in the squares around the buried treasure - and we mark that with a red cube. We say "warm" if it's int he squares around those, marking this with a yellow cube. And "cold" if it's anywhere else.

We've also used created some coordinates quizzes using Hot Potatoes. Here's one of them:

Year 4 dodecagons

Year 4 used pattern blocks each make  a different regular dodecagons (= 12-sided shape). At first they were tricky to make, but some people like Ryan, Alicia and Gustavo really got into the swing of things and made more than one!

Tessellations in Year 4

In Year 4 we looked carefully at the six pattern block shapes.

Then we made tessellations (regular patterns that could carry on for ever in all directions) with some of them.

Here are the tessellations we created:

This post is also published on the Year 4 blog.

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?