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.