Tuesday, 13 September 2016

Discovering Patterns of Multiples & Number Connections

If you're looking for a learning experience that will fascinate children and to also set a tone of creating theories and sharing wonderings as the foundation stones for an inquiry-based maths learning environment, then this is the one!

I can't remember where I found this idea years ago; it might be Marilyn Burns (?)

Being a new class of a week, I wanted to help establish what our maths learning should be like this year- student owned.

To help us continue exploring our central idea:

By understanding the properties of numbers, we can discover patterns and relationships.

Today, we began by looking at the concept of relationships in general.

We shared and discussed who we have relationships with and then thought about why:

We then thought about how we could apply the whys of relationships to relationships numbers can have with each other.

- What are two numbers that have a close relationship? 
- Two numbers that have a lot in common or have a lot of connections?

See previous learning about properties of numbers

We chose 12 and 24 and thought of their common properties:

We felt they had such a close relationship that they are like siblings.

- What two numbers would we say have a distant relationship?
- Two numbers that have little in common or very few connections?

We chose 3 and 42:

We could only think of them both being whole numbers so saw them as 4th cousins twice removed. That was how distant their relationship was based on shared properties. 

Comparing them,we could gain a better understanding of what we mean by numbers having close or distant relationships based upon their properties:

Just like our personal relationships can be based upon commonalities / similarities / connections etc, numbers also form close relationships based on those criteria too. 

We then discussed how recently we have been exploring how mathematical thinking involves visualising.

- To help us to visualise, to create theories to test out and to find out more about numbers and their reationships we are going to explore the multiples of numbers visually.

We were given a circle with the digits 0-9 spread around a circle to explore patterns that form with the multiple of 2:

We started at 2 on the circle and drew lines to the next multiple:

2, 4, 6, 8, 10, 12 etc

For each of these, we would skip count by the multiple till we had done it 12 times ( __ x 12)

( After 8, we use the zero to make 10.  To make 12, we use the 2 and so on)

With partners, we jotted down we noticed from this pattern.

A few discoveries were shared.

To guide us deeper, I introduced some guiding questions that might help us explore the patterns further:

  • FORM: What shape or pattern does the multiple make?
  • CAUSATION:If we continued counting on by 2, will the pattern stay the same or change? Why?
  • FORM: What odd / even pattern do you notice?
  • FORM: Does the pattern move in a clockwise or anti-clockwise direction?
  • CONNECTION: Do you think other multiples will have a similar pattern?

We then shared some wonderings we had and also some theories we wanted to test out.(see above)

Making theories is a safe way for children to think mathematically.

Theories shouldn't be seen as successful or unsuccessful we were reminded. Its the thinking of a theory to test out that is what we value regardless of the outcome.

After our sharing, we did the same with multiples of 3:

Our wonderings and theories showed how some of us were making connections with our prior knowledge of the properties of numbers. 

We were also really impressed and amazed that it made a star!

Because we could see a connection between triangles and 3, another wondering was shared:  I wonder if the numbers 4, 6 and 8 will make squares or rectangles?

This is the great sort of thinking that can emerge when we give children the green light to think for themselves and to encourage them to theorise and share wonderings?

After sharing our discoveries, wonderings and theories we did the same with multiples of 4:

The multiples of 5 really surprised us:

It also sparked a debate. Most of us believed it didn't make a clockwise nor anti-clockwise direction. It was thought it just went up and down.

But then someone explained how she found that it did actually move in a clockwise direction. The 5 moves clockwise back to 0 to make 10 and continues clockwise to the 5 to make 15.

Some very interesting theories developed when we saw how the multiples of 6 moved in an anti-clockwise direction:

With the multiples of 7, we wondered a lot of why 7 would make a decagonal star. We felt 7 and 10 didn't have a close relationship and yet they make a connection here. Why?!?

More patterns and theories were tested when we explored the multiples of 8:

A lot of WOWS emerged when we started making the multiples of 9 pattern:

Throughout each, theories and wonderings we had created were re-examined.  If a theory was disproven, we discussed how that is equally great as if the theory had been 'right'.  We continually discussed how it is the thinking behind the theory that is important rather than its outcome.  I've found that establishing this understanding is key to a safe and risk-taking inquiry-based maths learning environment. Using the word 'theory' gives children a safe hook to share possible ideas.  

This learning experience also really helped establish the importance of curiosity as a community of mathematicians. Encouraging children to think and share about their wonderings is the basis of the learning in inquiry-based learning, so making this drive the learning helps make the learning student owned and therefore deeper and more relevant.

Sometimes I would also share my wonderings with the class to model my own curiosity. (They weren't fake either- I still get amazed each time I do this activity with children)

I sometimes modelled how a theory I was testing wasn't proven. I shared how I thought that was fine and actually made me think more deeply about why the theory wasn't proven as I thought it might. That is where deep learning can happen.

To help us think about relationships between numbers and how as mathematicians we should try to always look for connections, we then looked at our number patterns and with our partner drew lines between them when we found connections.

We then shared some connections we had found:

We especially appreciated how a child shared that the fact that 5 and 9 didn't have a strong connection, that made them actually have that connection.

We also appreciated another student who summed it up nicely for us by sharing how they are all connected by the fact that they all make shapes within a circle.

Reflecting is a key component to learning. So, to finish up, we each wrote on a post it:

I used to think........

Now I know.........

This is one of my favourite ways to reflect- it helps children to reflect on how their thinking has changed and that is key to learning.

We shared these out loud and discussed ideas as they arose.

A lot of positive thinking about maths was shared:


  1. Love this investigation. Your students inspired this GeoGebra sketch: https://www.geogebra.org/m/nP92jBCN

  2. This is wonderful. Always enjoy your posts.


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