Counting Stick Conjectures
Counting Stick Conjectures
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?
Why do this problem?
This task encourages learners to connect geometrical and numerical ideas, and not to be put-off if a problem seems unmanageable at first. In this case, simplifying the task, generating different examples and then looking for patterns will give children the opportunity to make conjectures and develop mathematical arguments to prove or disprove them.
Possible approach
This problem featured in an NRICH Primary webinar in March 2022.
If you have a counting stick, you could use it to introduce the task, but it is not essential. Whether using a real stick or the photograph, give the whole group time to talk in pairs about what they see and what they might like to ask. Gather everyone together to share some of their observations and their questions, encouraging learners themselves to respond to the ideas of others. Try to welcome all contributions, even those that seem less mathematical!
If it hasn't come up naturally, lead into the idea of the stick being made up of rectangles by showing the image of the whole stick in blue and yellow (as opposed to the photograph). Ask the group how many rectangles they can see, and give them time to talk in pairs. Draw out the idea that there are many differently sized rectangles represented in the image and then allow time for pairs to count the total number. They may like to keep track of their thinking on a whiteboard or on paper, and it might be helpful for pairs to have a print out of this sheet which contains several copies of the whole counting stick.
The intention here is that learners struggle to keep track. This is deliberate so don't leave them floundering for too long, instead draw everyone together and invite comments. Listen out for those who are trying to make sure they find all the rectangles, but who are perhaps struggling to work systematically. Ask the class what we can do about this. There seem to be so many rectangles...
The idea of simplifying a task may not be one that is familiar with your learners - it will depend on their experiences. You may like to reveal Max's suggestion on screen and read it out loud. Give everyone time to take in Max's strategy and encourage pupils to ask questions if they are not sure what Max means. You may like to distribute the work amongst the class so that some pairs work on sticks that are four small rectangles long, while others work on five/six. You can encourage pairs to check with another pair that they agree on the total before recording on the board for all to see. It may be useful to agree a common language for referring to the different sticks (e.g. a stick which is 1 long, or 2 long, meaning that it contains one very small rectangle or two very small rectangles).
Once you have agreed totals for four, five and six length sticks, ask the class what they notice. It may be helpful to tabulate the results so that they are in order (it would be useful to include the total number of rectangles for the smallest possible counting stick which only contains one small rectangle). Learners might express the pattern in different ways. Some may say that the total goes up by 1, then 2, then 3, then 4 etc. Others may notice that this could be expressed as 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5... Others may recognise the totals (1, 3, 6, 10...) as triangular numbers. Use the pattern to predict the total for the full length counting stick (of ten small rectangles). You may like to refer to Carl Gauss' method for adding consecutive numbers quickly, see our article Clever Carl.
Explain that mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true. Challenge the class to try to come up with a mathematical argument that would convince mathematicians. This is certainly not straightforward, but as children discuss their thinking, listen out for learners who have an understanding of the fact that each stick contains the previous stick, plus one more little rectangle. This means the total number of rectangles will be the total of the previous stick, plus one more smallest rectangle, plus one more rectangle that is two long, plus... Help all learners get to grips with this, if appropriate, by sketching on the board.
Key questions
How many of the smallest rectangles are there in this stick?
How many rectangles are there which are 'two long'?
How are you keeping track of the total you've got so far?
Can you explain the pattern you can see to someone else?
Possible support
Encourage learners to use any manipulative that they find helpful to access this task. Some may want to be able to physically 'break up' the stick into the different sized rectangles to help in the process of working systematically.
Possible extension
A few learners might like to consider what happens when you consider the total number of squares on a chessboard. Can they simplify that problem in a similar way?
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