Rounding Numbers Inquiry

Rounding numbers is an important maths skills that oughtn't be overlooked each year of primary school.


When children round numbers, they are gaining a deeper sense of the value of a number and how each number can relate or connect with other numbers.

It's also an important skill for children to estimate.

Today's generation of children will, like us today, do most of the calculating using calculators on their mobiles.  They won't be the generation scratching out sums butcher's paper.

To use a calculator effectively, we need to be able to round / estimate numbers. Once we have punched in a sum into our calculator, we should think: does this answer make sense? If we don't, we could wrongly assume an answer is correct even if we accidentally typed in a wrong number.

Valuing the skill of rounding / estimating numbers is key for a learner's number sense to deepen. 



We started our inquiry sharing what we already knew about rounding numbers with our table partners. From this, theories, wonderings and reasoning skills were shared.


Which key concepts might help us the most to think deeply about rounding numbers?

We decided to use FORM,   FUNCTION and CONNECTION.


We started with FORM and used the think-pair-share routine and came up with these thoughts and remembered the importance of visualising in maths:

We then decided to use FUNCTION and again used the think-pair-share routine to come up with these thoughts:





We repeated using CONNECTION:





This lead to interesting discussions 



The big picture:

Using this understanding, we our now more appreciative of the learning experiences we will undertake to improve our rounding / estimating skills.





Related Links to Rounding:

How and why do we round numbers?

Can I prove or disprove my own theories about rounding numbers?

How do we round numbers in real life? Let's go shopping at Ikea to find out!

Decimal Numbers

Using the Key Concepts to Think About Decimal Numbers......

We have been inquiring into our base 10 number system and there has been a lot of wonderings happening related to decimal numbers on our class wonder wall (post it note wonderings the children are encouraged to add to when they want). 

From listening into groups and pair number discussions I have been able to determine that whilst the majority of us have a pretty good basic understanding of decimal numbers, there is a significant handful that show that decimals is a concept they still only just beginning to grasp.

In situations like this, the PYP key concepts can be a really useful learning tool. They can help assist those learners in foundation levels of understanding to develop more solid conceptual understandings whilst at the same time deepening the understandings of those learners who already have good understandings.

I explained how we can see that decimal numbers appear on our class wonder wall a lot and so we should try to address some of them.

I asked, 'To think about decimal numbers which key concepts do we think would be the most useful?'

We decided upon:

° Form
° Connection
° Causation

We then used the think-pair-share routine to help us deepen our understandings.


This is what we came up with as we shared together:

Whilst sharing, the children were encouraged to share with us examples of ideas they had when the need arose.  Lots of turn-and-talks took place to help build stronger understandings and communication skills as we thought about ideas.


Doing think-pair-share routines like this is a very useful formative assessment tool and it greatly helps all learners regardless of where they are at conceptually to expand and deepen their understandings.

This also helps develop our sense of being a community of mathematicians sharing theories, wonderings and understandings.

From this, I felt more confident in us moving forward with using decimal numbers and in helping each learner feel more confident in inquiring into the other wonderings we have created.

The WHYS of dividing by 10, 100, 1 000 etc

The WHYS of maths is more important than the HOWS.

Yet, it seems we tend to focus so much on the HOWS and this leaves mathematical thinking as 'doing school', I think, rather than the real learning and discoveries of maths. 

We have been exploring how our number system is based upon 10 and what that means regarding place values.

A benefit we are discovering from having a base 10 number system is that we can multiply / divide numbers by 10, 100, 1 000 etc easily.

Yesterday we explored how and more importantly WHY we can multiply numbers easily by 10, 100, 1 000 etc easily.

To find out about dividing, I wrote the following on the board and the children used the I see / think / wonder routine to record their thoughts:

We then shared what we noticed, theories we had, and our wonderings:

Most of us had discovered (a few already knew) the HOWS of dividing by 10, 100 etc, but the WHYS are more important.

Looking at our gathered wonderings, students chose a question they found interesting and spent time exploring it and then sharing their theories or discoveries with others. Even those wonderings that at first sounded 'simple' became actually quite complex when trying to understand the WHYS behind them.  

Whilst sharing, some students had used strategies which we had previously done when finding the WHYS behind multiplying by 10, 100 etc.

Some students explained to us why this works using a place value grid. They explained how we can see each digit changing its place value:

This helped us to also see why numbers grew smaller when we divide them.

Other examples were explored:

We took our thinking further by exploring how each digit changes with decimal numbers:

Another pair shared how they used our multiplication strategy with the division which helps us see what we actually do to the number.

We then used the same number and had a choice of dividing it by 10, 100 or 1 000. A few students decided they wanted to challenge themselves further by dividing it by 10 000:

By using the split strategy, we felt we could really see what we were doing when we divided by 10, 100 etc.   (Some of us used a calculator when dividing 20 by 10 etc)

When we looked at our answer, we identified the place value of each digit to see how they had changed.

Some great reasoning thinking emerged when we looked at dividing 20 by 100. A student shared how they knew the answer must be a decimal because the place values were 'used up' by the 100. we thought this was a great theory and so tested it out together when we looked at dividing the number by 1 000:

All of these great discoveries and learning was student-led and all had sprung from simply using the 'see-think-wonder' routine.

When we created and tested our partners with questions to divide by 10, 100 etc, another interesting wondering was shared:

When we write large decimal numbers, do we also use spaces or commas?

Some of us thought yes, others no and most unsure.

It was a great question so we tried to reason our theories.

We had learnt recently why we need commas or spaces when writing whole numbers, so it made sense that we should have commas with long decimals. 

( Link: Why do we need spaces or commas with large numbers)

The reasoning behind this really made sense.

Someone else though had recently seen the number pi and explained that she didn't see commas or spaces in it.

Why not?

We didn't come to a conclusion on this, but it is on our wonder wall for us to find out........

How to read large numbers

We began our day looking at the following key concept question:

In pairs, the children wrote some large numbers and explained to each other how they read them and why they wrote them the way they did. This helped them to start thinking about our question and also helped me to see where each child was at by the type of numbers they wrote and how.


This year, one of my teaching goals is to try to use provocations as much as possible in maths learning. We know the value of using provocations in our Units of Inquiry and so i want to experiment with using them in maths more often too. 

To provoke our thinking, I then showed them this number and asked them to find a way to read it:

Lots of different strategies were used. Some children identified the place value of each digit, others put commas in the number and others tried to expand it.

One student's strategy:

We shared the different strategies we used and then had a go trying to read it. Loads of different numbers we read with lots of different strategies. We valued each of those shared for having a try and willing to take a risk. 

What made reading this number difficult?

- There were no commas!

Do we have to use commas? Can we use something else?

- We can also put spaces inbetween. 

How do spaces or commas help us?

We then looked at the number with spaces us to help us:

- They help us read the number more easily.

How?

- They help us see what place values the digits are.

How?

- The spaces tell us a place value word.

Oh really? Tell us more.

The student then came up and explained how each time we see a space or comma, we need to say a place value word.

871 BILLION 560 MILLION 378 THOUSAND 420 point seven five.


What do we think of this theory?

Why do we need to remember to put spaces or commas in large numbers?


We then practised creating large numbers with spaces / commas for our partners to read out.


This mightn't have been the most amazing provocation, but I think even simple provocations like this have a deeper learning impact than simple teacher-directed explanations. 

Life without a Number System

Today we began our enquiring into place value / number systems with this lead in provocation:



Using the think-pair-share routine, students first had time to make a list of their ideas. They then shared in small groups and then we discussed as a whole class.


Here were some of the ideas we generated and discussed:

Loads of interesting theories, wonderings and debates took place. As we explored the ideas being shared, some of us debated and others added on ideas tomato them perhaps more correct. 

For example, it was shared that we wouldn't have houses. We wondered about that and then came to the conclusion that we wouldn't have modern houses since we wouldn't have formal measurements. But we could still have simpler houses.

Another in depth discussion we had was from the idea that there wouldn't be rich or poor people since there would be no money. We wondered about that idea. Some challenged our thinking by suggesting someone could still own more land to grow food than someone else. Would that make that owner richer?

After our discussion, I asked: What did this provocation make us think?

Here were some of our reflections:

° Our lives would be extremely different.

° Numbers ave helped people to progress over time. Without a number system we would still be living very simple lives.

° We can take having a number system for granted when actually so much depends upon it.

° It makes us appreciate how and why we have a number system.


Rich, philosophical discussions like this enrich student engagement in mathematical thinking. Thinking about a provocation like this helps us deepen our understanding and perceive maths in a different way we might not have before. It also provokes lots of interesting wonderings we want to explore in our unit:

° Who invented our number system?

° Why was it invented that way?

° What did people do before numbers were used?

° Could we survive today if suddenly everyone forgot our number system?

° Were there other number systems in the past?


These are great wonderings which have helped formed our new maths enquiry unit.

Properties of the Number 12- visualising & creativity in maths thinking

We have been exploring some of the fabulous Weekly Inspirational Maths ideas on youcubed.org.

We have been fascinated by the idea of mathematical thinking requires creativity and visualising.

To help us explore this further, and to gain some key number sense terminologies and number talk beginning as mathematicians, today we explored the number 12.

The children had a few minutes to jot down anything they knew about the properties of 12. This in itself, proved to be a very informative informal assessment as I noticed those who had the language and understandings to discuss numbers and those that haven't yet.

We used the think-pair-share strategy so after sharing with their table partners what they knew, we shared as a whole class:

Collectively, we knew quite a lot. As children shared what they knew about 12, we paused and discussed what each means. Our understanding of what makes a number even was a bit shaky, but together we were able to find a deeper understanding rather than a number that ends with the digit 0, 2, 4, 6, 8. This in itself helped us to think about how this year in maths learning, we should really focus on the WHYS of maths rather than just the WAHATS or HOWS. We discussed how deep understanding and more interesting wonderings develop when we think of the WHYS.

To help us investigate further how mathematical thinking is about visualising and creativity, table groups were given counters and explored different ways we can visually represent the number 12.

Some interesting investigations took place:

This student explained how she thought of the factors of 12.

You have to love this type of thinking:

After groups had time to explore, we did a gallery walk to see what others had thought about. During the gallery walk, the children shared their thinking and were encouraged to share their successes but equally things they had tried, but hadn't worked out as planned. It is important that children see the value in attempts in maths as equal to successes.

What ideas could inspire us to take our thinking further?

Being inspired, the table partners returned to their counters and explored further.

We then had cuisenaire rods to help us creativity represent the number 12. It was apparent that for most of the children, they hadn't used these for a long time. But, it made the learning richer as they slowly began discovering for themselves that each rod could be given a numerical value. Once they were discovering these, their thinking really took off!

Some samples of the number 12:

During our next gallery walk of sharing and seeking inspiration, we appreciated the thinking of those who thought in 3D and also the group below who created these addition combinations below:

After sharing, we shared some wonderings that this learning experience invoked that we would like to explore:

° How can visualising numbers help our learning in maths?

° How many possible ways could exist using the counters?

° Which numbers would be easy or challenging to visually represent?

° Could we find just as many ways with an odd number?

and one of my favourites:

° Can we do this again tomorrow?!?!?

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?