Wednesday, 14 February 2018

Valentine's Ratio Problem

It's St. Valentine's Day.

We began our day writing messages about what we love / appreciate about us as a community of learners. We enjoyed reading this out and discussing the ideas in a circle. 




It's always good to create a positive atmosphere and to constantly guide children to appreciate others. :)

To help feel festive about the day, we looked at the following problem:

I was hoping this would help us inquire into using a ratio table or double number line naturally as we haven't directly looked at these as a visual strategy yet. 

As per usual, we had a choice of trying to solve this visually individually or with a partner. The key word was solving it in a visual way. 

After experimenting and lots of deep dicussions- especially testing whether answers made sense or not, we discussed some of the strategies we were interested in finding out about and those children explained to us.

Here are some of the strategies:


We really liked all the different approaches of visualising and we discussed how it didn't matter whether we got the answer or not. Instead we should celebrate the creative thinking that took place. 

Obviously the double number is still a new concept and so I introduced it as a possible tool we could use in future inquiries:

Some of us liked this strategy and others preferred their own as it made more sense to them. Thoughts like that are really as mathematicians. Whatever helps us to make sense of numbers is what we should use another child remarked. And that, was a pretty perfect way to move on to our next investigation. 

Thursday, 1 February 2018

Multiplying Decimals

As part of our Unit of Inquiry exploring forces, today we were going to find out how much we would weigh on different planets- the kids love it! 

Part of that requires multiplying their body mass with decimals.

So, to better understand what we will be doing with the numbers, we began with this lead in:

We used the think-pair-share strategy.  Firstly the children jotted down different ways. They then shared with their table group and then we discussed and shared ideas as whole class.  Some of the key understandings included:

We have done this a few times over the year and each time additionally unique perceptions are shared which I find really interesting. I can see how some children's conceptual understandings and abilities to express have expanded and deepened. 

The whole class sharing raised some interesting discussions such as a child sharing how decimal numbers are like negative numbers.  

- Are they? What do we think about that?

Some of us agreed, some disagreed and many were unsure.

Another asked: Can we have negative decimals?

- Can we?

We thought of having a temperature of   - 1 . 5 ° C

Does this prove we can have negative decimals?

Does this mean a decimal is a negative number?

A student shared a theory that: 

This really got us wondering: Can a demoninator be zero?

We chatted about what this might mean- Can we have zero whole parts?  Could this represent a decimal as a negative number?

It was a very interesting thought and I explained that we will definitely come back to this in a few weeks when we explore negative numbers. 

Some of us still weren't 100% sure about decimals being negative numbers or not and that is ok. I think it's good to keep wonderings floating in our minds to think more about later. 

Having tuned our minds back into what a decimal is, we then looked at the following: 

I posed this question to help build number sense and to also value creating theories which we are always doing. 

We showed with thumps up, down or sideways what we our theory was.

There was a mixed initial thought to this.

To help each of us find out for ourselves, the following was posed: 

Asking 'How many different ways......' is a great way to encourage deeper thinking and creativity.

If we ask children to think of one way / one answer, we are limiting their potential to discover different ways of thinking.

Again, we used the think-pair-share routine. 
It was interesting to be able to walk around and chat / observe the different ways the children were trying to visualise it. 

One children shared a great hypothesis:

When we multiply numbers, the answer is bigger so my theory is that the answer to 5 x 0 . 5 will be bigger.

This completely makes sense and I love how she has drawing upon prior conceptual knowledge to create and test a theory.

After a while, the children shared with ther table partners their visual ways. Each table group were asked to select one of the visualisations that they thought the whole class would benefit from seeing. 

Together we discussed the following shared:

We appreciated all the different ways we could visualise the numbers.

We then repeated with a larger decimal:

Again, we first estimated and then tested by visually drawing what it looks like in different ways.

Some of the table sharing included:

We really loved the creative story behind the one below. He explained how there were 5 whole people at the airport and each carried a bag that was 0 . 25 of their body mass:

Valuing creativity and visualising in maths can greatly help children to make stronger number sense like these examples show. 

Lastly, the children created and visually showed their own number sums. Afterwards, they shared and discussed these with their table partners:

Some wanted to challenge themselves further by multiplying a decimal with a decimal to see what it looks like:

And this student found it very interesting to discover how small the number would be when multiplying 5 by 0 . 0000002. We giggled about it being about half the size of a ladybird's poo :P

Before moving on to multiplying the gravitational force of our mass to see how much we would weigh on different planets, I asked if we had some wonderings we wanted to explore next week and so we have the folllowing to find out about: 

I think this has helped us launch into some really interesting wonderings to explore that will further help develop our number sense with decimals and fractions. 

Monday, 29 January 2018

Tuning into Measuring Mass


To we began exploring our new unit:

Central Idea: Converting units and using decimals helps us make sense of the measurement of mass.

Lines of Inquiry:

° FUNCTION: Strategies we can use when measuring mass

° FUNCTION: How we can convert units of mass

° FUNCTION / CAUSATION: How we use decimals with mass and why

° CONNECTION: How we use the measurement of mass in our daily lives

To begin tuning into the concept of mass, as the children entered, they saw the following on our board:

As we were going to begin exploring how to convert grams into kilograms and vice versa, I thought it was important that they gain a sense of what 1 kilogram actually feels like.  

Using their arms like a balance scale, they hefted their school bag and compared its mass with the 1 kg weight in their other hand.

They were then asked to estimate what they thought the mass of their bag was.

As well as helping them sense a kilogram, it was also an interesting one-on-one pre-assessment as I listened to the thinking of each learner. 

Some of the thoughts shared were:

° I can feel my bag is a lot heavier than 1 kilogram.

° I think my bag is just a tiny bit lighter than 1 kilogram so I estimate it is 3/4 of a kilogram.

° I feel my bag is more than double the kilogram.

° Wow! Is this really a kilogram? It's really heavy!!

° I think my bag is about half of the kilogram. 

° Olivia said her bag was about 1 kilogram. Can I now heft her bag with mine to see if my estimate is close?

° Could I put my bag on a scale to see if my estimate was close?

° What has Louisa got in her bag? Her bag is more than double mine!

° I think its somewhere between 1 and 2 kilograms.

° I would say its less than 1 kilogram, but I'm not sure by how much.

° So my mum weighs about 70 of these weights?!?

° What do you call the things that are less than a kilogram?

As you can see, there were lots of diverse thoughts taking place. It was interesting for me to hear their reasoning skills, who was thinking of fractions, who was comparing by doubling etc. 

After each had hefted, I then asked them to show on the number line where their estimate belonged.

This also become a key informal assessment as each child shared where they would place it and what their estimate was.

I was thinking about:

° Who is using simpler masses such as 1 kg and who was confident in taking their thinking to masses such as 0. 75 kg or 1 . 25 kg etc?

° When they gave their mass estimate, I asked if they could express it in different ways?  Some could express it in grams, some could express using fractions and some could only express it one way. 

° Who was noticing the other masses already recorded on the number line and used that to determine their own estimates?

I felt I was able to gain a sense of where each child was at with their experiences and / or understanding of measuring mass and using decimals.

When everyone had hefted their bags, we then looked at the number line of estimates and I asked what does this make us wonder?

° How could we check if our estimates were close?

° Is hefting really a useful way to measure the mass of objects?

° How could we add all the decimals?

° What's the difference between the heaviest and the lightest?

Table partners then discussed these questions posed sharing their ideas and strategies.

As the purpose behind this was to tune us into what a kilogram feels like and also to gain an informal glimpse of where each learner was at with mass, we didn't go into this further. 

Sometimes, I think it is good to keep us wondering about certain things that we can come back to later.

I suggested that we use our number line as part of home learning next week, so during our day, if you have a mathematical question you think we should try to solve, jot it down on a sticky note and place it beside the number line. 

At the end of the day, decided upon four questions to solve for home learning in addition to a suggestion that we convert the kilograms to grams.

So, this is what our home learning looks like:

We will come back to the strategies they created to measure the exact mass of their bags later in the week after we have gained a deeper understanding of converting grams and kilograms.

What we did do after this, was we explored how to convert kilograms into grams and vice versa and then used this understanding to apply with our 'Travelling into Space' investigation: 

Tuesday, 23 January 2018

FInding the Fraction of a Quantity Investigation

Today we started exploring how to find the fraction of a quantity which is a key way we use fractions in everyday life. 

To find out strategies the children already knew and to give them an opportunity to create their own strategies, we looked at the following situation:

Developing number sense requires children to constantly estimate and get a feeling for numbers, so before we started working it out, the children were asked to identify their 'gut instinct' of which they feel might be cheaper and to write a star beside it to test later.  

We then had time either using strategies we already knew or trying to create strategies to help find out. 

For those children who could find a successful strategy easily, their challenge was to try to create another strategy that might also work. I think that giving children opportunities like this helps develop their creative thinking whilst also helping them to delve deeper into mathematical understandings. 

As they were creating strategies, this was a great opportunity to discuss with children their thinking and to find a few interesting startegies that could be shared with the class.

I found the following, wrote them up and then those children shared their strategy with the class who then gave feedback on what they thought of it:

After discussing these strategies, we then needed to discuss which t-shirt was actually cheaper since we had different ideas.

Some children hadn't realised they needed to subtract the fraction from the original price so listening to others explain why helped them to see the mathematical thinking needed:

Looking at the various thinking that took place can greatly help create a picture of where each child is at, whether they prefer visualising or not and any misconceptions they might be harbouring such as the last sample where the child is thinking a third equates to 25%. 

We then extended this understanding with our next situation:

Again, to help develop their number sense, before applying any strategies, the children drew a star next to their estimate or 'gut feeling' for which was greater. 

This question helped identify misconceptions some children were harbouring and we used some of those ideas to help deepen learning of everyone in our discussion.

We are constantly reminding oursleves that mistakes are great when we take the time to think why they are mistakes and how sharing mistakes can help deepen everyone else's understandings.

This was a great example:

Quite a few children had explored this as a strategy. Others were able to explain how we either need to 'stretch out' the fifths or 'shrink' the tenths so they are the same sized whole. 

Some thought of comparing the two fractions to a half as a strategy which we thought was an interesting idea: 

Students were then invited to share stratgeies of how we can find the fraction of the amount that helped us to determine which was more. 

We really loved this classmates' visual strategy:

We thought this was an interesting strategy too:

Lots of questions were asked when a student shared this strategy:

Finally, this student's strategy was chosen to be shared:

We then practised a few 'fractions of an amount' questions and reflected in our Maths Reflection Diary:  

Maths Reflection Diary Link