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:

Is zero a number?

On my way to school this morning, I saw this great Tweet from @RichardsonLiza who is a really inspiring primary teacher in Perth (she is a great follow!)


I thought the children in my class would like to help her class out since we looked at zero quite a lot in the beginning of the year.

We have also been exploring how we can action so responding to her students' question seemed like a great, natural link to taking action ourselves.

Having children connect across the world with learning is such an amazing thing we can do these days!

We wrote our 'short' ideas down and then discussed them as a whole class debating different theories we were harbouring.

Some of our ideas:

Yes, it is:

° Zero is a number that represents nothing. 

° There is no scientific evidence to show that zero exists, therefore zero has no value. My theory is that zero does exist because it is the value of zero and it is everywhere. 

No, it isn't:

° Zero is like a bookmark because we put it amongst numbers to hold or create their value so that the other digits don't get mixed up.

° Zero isn't a number. It is used only to create place values in numbers such as 10, 100, 1 000, 105 etc

° Zero might not be a number because a number must have at least one unit in it to exist.

° Zero isn't a number but it does separate numbers into positive or negative numbers.

° Our base 10 number system would be very different without zeros, but they are just place value holders instead of being a number. 

We thought that this Tweet from students all the way across the other side of the world really helped us to think deeply about our number system and zero. We also hoped our ideas would help those students with their enquiries too!

Does Zero Exist?

One of the greatest attributes of enquiry-based learning is how a student's wondering can lead us all to a fascinating learning journey.

We had been theorising which number was the base for our number system ( Link: Why is our number system the way it is? ) when one student wondered if it was zero..........

We thought about this idea and are wonderings grew.

One student then theorised that it cannot be, because zero is just nothing.

Is it?

We shared our theories about this and how we could explain what zero was. Some fascinating ideas emerged.

I then shared how to this day no mathematician nor scientist in history has been able to prove or disprove the existence of zero.

- But how can that be?

- If we use zero, surely someone must know what it actually is?


I then remarked that if you were able to prove or disprove the existence of zero, you would likely receive a Nobel Peace Prize and become one of the world's most famous mathematicians and scientist. 

- Wow! Really??????

- Why?

- Let's do it! 


Someone excitedly suggested we should group together and see if we can prove it.  Such a great idea and enthusiastically they grouped and thought deeply. The discussions were rich with creating hypotheses and testing each other's philosophies. 


We then shared some of our thinking:

I was completely blown away by the thinking being generated!!

I particularly liked how one student proved their theory by drawing the symbol for zero and explaining how there is nothing inside it. 

And though it sounds rather simplistic, the thought shared that 'It is the whole of everything' gets more and more profound the more you think about it.

Some thought how Roman numerals didn't have a zero. Oh! So it must be a NEW number!   Wow!  But, is it a number?


We discussed that every number CAN be proven.  We can prove 2 exists because 2 is 2 ones OR 1 split in half to create 2.  Every number therefore CAN be proven, but zero?  How can we prove nothing exists?  And where did 1 come from? How can 1 come from nothing?!?!?

Someone then connected that's like the Big Bang theory - everything in the universe came from nothing suddenly! Wow!  Others then sparked from this connection and suggested zero could be like a black hole. Another thought zero is like God who must have come from nothing because He created everything. 

Woah.....talk about deep thinking (and bare in mind these are 10 year olds coming up with these amazing thoughts!)

Someone then mentioned zero can be proven because I am a 1. I am a 1 person and when I die, I don't exist anymore so that is like zero.    But does our body really stop existing we wondered?  When it decomposes it just changes into different smaller materials so the body hasn't realély ever stopped existing, it just changes it's existence!  Again - 10 year olds thinking like this!!! Amazing!

As the bell rang for recess it was suggested we try to keep thinking about this next week and the week after.  Wow! Love this so much!!  Totally engaged, really passionate and directing / taking ownership of what we should be doing in our classroom- brilliant!

To keep this fresh in our minds to keep thinking about, here is how our classroom door looks now as we enter:

Later in the day, after lunch, a student came in with her note book sharing that she might have cracked the existence of zero.  She figured that 1 + 1 = 2. Therefore 1 - 1 = 0.
She thought about positive 1 and negative 1 and that if the zero place wasn't there the positive 1 and the negative 1 would be arguing for that prime position of zero.   Does that beginning thinking help us to prove zero exists?  I'm not sure, but what I do know is that she is really thinking deeply about place value and the thoughts the students generated as a whole class, individually and in pairs about our place value system could not occur had this been a worksheet place value activity.


Who says 10 year olds can't have deep philosophical discussions about maths?

Proving & Disproving Own Theories with Rounding Numbers

Investigating:

Can we use the rounding strategy to check answers for all four number operations (add, subtract, multiply, divide)?


We began by discussing our initial theories to this questions with our table partners to help start tuning in to the sort of thinking we will be doing.

Next, we had 5-10 minutes to investigate addition. 

° Could we always use rounding to check if our addition sums were correct or not?

° Are there some numbers where the rounding strategy doesn't work? If so, why?

° Can we challenge our thinking by exploring this with decimal numbers?

After we had time to investigate, we published our findings / discoveries and some children shared with the class discoveries they had made. Some really interesting theories started to emerge with kids wondering:

 ° Do odd or even numbers makes a difference?

 ° If the difference in changing the numbers is greater than 5, the strategy doesn't work, but if the changed numbers totals less than 5, it does work.  Is this really true? (see below)

We talked about how proving a theory we are testing is just as successful as disproving a theory. Some of us wondered if this is what professional mathematicians do?

We then repeated our rounding investigation with subtraction. 

A few students came up with an amazing theory and excitedly tested it out. 

They discovered that if you can round numbers successfully with addition, then those same numbers cannot round successfully with subtraction. Why not? 

They tried to find this out and theorised that it might be because addition and subtraction are opposites.  Such a great idea!

They were encouraged to further test numbers to test their new theory. 

One student with a really strong number sense decided to investigate whether we can round numbers to check answers with positive and negative numbers. He discovered it does work with most negative numbers and shared his theory with us:

This is one of the great advantages to enquiry-based maths.  Those children who need extending have the opportunity to take their thinking to deeper places that typical teacher-directed maths activity simple can't do.  Equally children in the class with not-so-strong number senses could also explore numbers that they were comfortable with investigating. 

We repeated our investigations with multiplication and finally with division. 

Some children started expanding upon each other's theories to test and help deepen them which was amazing to observe.  The same students who had come up with the 'if it works for addition, those same numbers won't work for subtraction' theory were excited to see if this also applied to multiplication and division.  When they were testing this out, another student noted that perhaps it doesn't work with numbers that can be square-rooted.  When they tested that theory out, they started to discover it does actually make a difference! 

Kids sharing discoveries and theories they created and are testing out:

Some students wondered if it made a difference if we rounded the number we were dividing by or not?

These students shared their theory that if the number was below 5, we couldn't check using the rounding strategy, but if it was above 5, it would work!

This got others investigating if this would work with larger numbers....some wanted to see if it also worked with decimal numbers too!

All of these great theories to test out were student generated.  They were arousing wonderful curiosity with each other and building upon their own theories. Quite the buzz of mathematical thinking!

Kids sharing their discoveries:

A sample of published discoveries:

Another student sample:

Another sample:

It was one of those great activities where kids could take their learning to such different aspects based upon their number sense and they all felt really proud of what they had discovered. 

Are all their theories accurate? Maybe, maybe not.

I don't think that is so important for them to know.  What IS important is that they gained a real sense of enquiring into numbers and feeling successful at it. They were beaming with pride with theories they were formulating and completely engaged in trying to prove or disprove them.  That was the key essence of the learning experience.  They gained a real sense of what mathematicians do and regardless of their maths number abilities, they all felt like really successful mathematicians - because they were!

Highlighting who was 'right' and who was 'wrong' with their discoveries would have been a counter-productive thing to do. What we want, is to inspire children to create their own deep thoughts about how numbers relate to each other, formulate their own theories and test them out.  The 'right / wrong' approach would have killed that enthusiasm and curiosity that was flooding our room today. 

Maths learning is too often drilled into children as a 'right / wrong' learning area.

It's not.

Maths is a science and should be presented as such. Just like scientists, mathematicians formulate theories and test them out.  We need to give kids these sort of opportunities to allow them to discover how maths can be interesting and enjoyable and dare I say......actually rewarding!

Shopping at Ikea (How we round numbers in real life)

Shopping at Ikea:

To help us further understand how we round numbers in our lives, we imagined we were university students moving into our first flat and needed to furnish it from Ikea. 

We have a budget of 5 000 francs / euros / dollars to furnish our flat. 

I gave the children a selection of Ikea catalogues based upon some countries we come from, currencies we are used to using, and their mother tongues.

This year, we have a Japanese student in our class still quite new to Switzerland so I also included the Japanese version of the Ikea catalogue.  We used an online currency converter to find out his budget would be 600 000 yen.  This caused a lot of interesting discussion about currency values!

They are given the Ikea catalogue via a google doc: Ikea rounding activity 

As they 'buy' things from the catalogue, they copy and paste them on their google doc.  They record the price and what they round that price to (see examples below)

They need to round each item they buy and record on paper the rounded amounts just like we might do if we really were shopping there.

They completely LOVE this activity so much and are constantly asking each other how much they should round a particular item to. This peer-teaching helps solidify their understandings far more effectively than if I help them decide.

What the Google doc looks like:

                                 

Ikea catalogue in English      ( Dollars )

Ikea Catalogue in French      (Swiss Francs  = CHF)

Ikea Catalogue in German    (Swiss Francs =  CHF)

Ikea Catalogue in Spanish     ( Euros  costs in Spain)

Ikea Catalogue in Japanese   ( Japanese Yen)

You are about to move into your new flat and will be living alone.   

You have a budget of CHF 5 000

= $5 000 if you are using the English catalogue

= 5 000 euros if you are using the Spanish catalogue

= 600 000 yen if using the Japanese catalogue

to buy all the furniture you may need for when you first move into your home.

1. Look through the Ikea catalogue and select furniture you will buy.

2. Copy the image of each piece of furniture and paste below.  

3. Beneath each piece of furniture, type the cost and then round the cost to what you think it ought to be rounded to so they can be mentally added easily.

4. After you copy and paste an image, shrink its size so your document is easy to read.

For example,

Cost:  CHF 369;  Round up to  CHF 400

Cost:  CHF 349;   Round up to  CHF  350

Cost:   CHF 669;   Round up to CHF 700

Cost:   CHF 12.95; ROUND to 13.00

4. As you are shopping, keep adding your rounded purchases on scrap paper.

5. When you think you have reached approximately CHF 5 000 (or $5 000 or

5 000 euros or 600 000 yen), stop shopping.  Then use a calculator to add the exact total.

6. Answer these two questions:

   a)  Was your rounding estimating close or not?  Why or why not?

   b)  Was rounding the prices a useful strategy or not?  Explain why or why not.

Cost: 3,99  euros               rounded:4,00  euros

This helps them discover a real life example of how we round numbers and the children eagerly wanted to continue doing this at home which makes it even more perfect. 

Kids begging to be allowed to do maths learning at home? 

Can it get any better than that?!? :)