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.

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.

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.