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Counting Stick Conjectures

Age 5 to 11
Challenge Level

Have you ever seen a counting stick? You might have one in your classroom at school. Here is a photo of one:

What do you see?
What do you notice?
What would you like to ask?

Here is a simple picture of the counting stick:

How many rectangles can you see? (The rectangles may be different sizes.)


Once you've had a think about how many rectangles there might be, click below to see what Zoya thought.

Zoya says:

"I can see some small rectangles, which are either yellow or blue.
I can also see some bigger rectangles, made of two of the small ones. And some that are even bigger too.
I tried to count all the rectangles but I got very confused."

What might you suggest that would help Zoya? 


Again, have a think and then click below to see what Max suggested.

Max says:

"Perhaps it would be easier if we started with a smaller counting stick.
We could try a counting stick with just two of the smallest rectangles, like this:


I think that has two small rectangles and one larger one, which makes three rectangles altogether.

Then we could try a counting stick with three smallest rectangles, like this:

This time I can see three very small rectangles, two blue and one yellow.

I can also see a rectangle which is made up of two of these small ones, which I've outlined in red:

And there's another one like this:

And then there is one large rectangle:

So, I think that makes six altogether.

If we do this a few more times with different sized sticks, perhaps we'll see a pattern."

Can you see how Max worked out the total number of rectangles each time?


Try out his suggestion. How many rectangles are there on a counting stick which is made up of four small squares? Five squares? Six...?

What do you notice? Can you see a pattern?
Is there a quick way to work out how many rectangles there would be, for a counting stick with 100 sections? Or 1000? Or...

How can you be sure that what you have noticed will always be true?


Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true.
Can you provide an argument that would convince mathematicians?