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For younger learners

  • Early Years Foundation Stage

Counting Stick Conjectures

Age 5 to 11
Challenge Level Yellow star
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

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?

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Counting Counters

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Cuisenaire Squares

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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