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.
"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.
"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?