Whilst some groups were finding out how much it would cost in petrol to drive to different cities in Europe from Lausanne (where we live), others were trying to find holiday destinations outside of Europe from Geneva Airport that would take less than 10 hours flying time and others were creating ratio maps, this one group enquired into how we use ratios and proportions for recipes.
We began with a pancake recipe and discussed how we can say the ingredients are ratios.
We then thought about what we need to do to make different amounts of pancakes.
I asked the group to give us 3 different amounts and we recorded this on our chart. We thought of making 3, 12 and 30 pancakes.
Asking why we chose those numbers, it was explained that we could calculate the proportions more easily because of their relationship with 6 and 6 was the original recipe we are working with.
It made sense to us.
And it gave me an indicator of where this group were comfortably at with understanding strategies and problem solving these types of questions.
Children are often better teachers for themselves than we are and this helped me see that clearly. They hadn't chosen these numbers just to make it easy because the children in my class enjoy making tricky challenges for themselves and each other. They see proposing challenges as fun- that's the type of culture we have been able to create in our classroom with maths and I think that's mostly because we have stripped away the notion that maths is about getting a correct answer and that maths is actually very serious. Our last unit exploring angles introduced the idea that maths is about playing has really resonated with a lot of children- especially those who might deem themselves as 'not so good' at maths.
These selected numbers of 3, 12 and 30 told me a lot`
So we started with imagining we only wanted to make 3 pancakes.
What do we need to do with the ingredients?
- Halve them.
- Because half of 6 is 3.
Do we have any other suggestions?
We didn't, so we halved them all and placed the pictures in the column to help us visualise it.
What if we want to make 12 pancakes?
- We should double the ingredients?
- Because when we double 6 we get 12.
Can we think of another reason why we should double them?
- Because half of 12 is 6.
Any other ideas?
We didn't so we doubled the ingredients placing the images in the column.
Let's take a close look at our pictures. Do the proportions make sense when we compare them to the original recipe? We thought they did.
Let's say we have some guests at our house so now we want to make 30 pancakes.
What do we need to do to the ingredients in our recipe?
Quite a few different suggestions were shared:
- We can't do it because doubling 12 is 24.
- We could create a ratio table and see what we get when we reach 30.
- We should multiply the ingredients by 5 because 30 divided by 6 is 5.
- We could multiply the ingredients by 10 to make 60 pancakes and then halve them to make 30.
Listening to all our suggestions, what new thoughts do we have?
We shared our thoughts using reasoning skills. Which strategies made sense to us, which seemed easier to do etc?
We decided that the multiplying by 5 made sense but we needed to prove if it really does work.
So, we went through each ingredient and multiplied them by 5.
To help us visualise what we are doing when we calculate proportions, we looked at our recipe and compared it to the 30 pancakes proportions.
When we look at this, do they make sense to us?
We all agreed they did.
Were making the quantities of 3, 12 and 30 pancakes easy or challenging for us?
We felt 3 and 12 were easy and 30 for some was a bit challenging, but not for others.
What number of pancakes would be really challenging for us to solve do we think?
- half a pancake
- a quarter of a pancake
I thought this question was an interesting way for us to think more deeply about why 3, 12, and 30 weren't that tricky and so it would help us to further understand that with ratios & proportions, we are thinking about how numbers relate to each other.
We thought the idea of making just 1 pancake was funny, so we went through each ingredient and decided what we needed to do (dividing each by 6).
Next time we explore ratios, what would you like to do?
A few of us wanted to continue with the pancake recipe and try to find strategies to make 32, 7 and 1/2 a pancake.
A few wanted to find other recipes and find ways to make different amounts of it.
Giving children opportunities to think for themselves and come up with their own investigations is really important. It gives them opportunities to think creatively and to take ownership of their own learning.
The children who were a bit challenged in this activity wanted to continue with something they are familiar with (the pancake recipe) and those who weren't too challenged wanted to try different recipes they could find on the Internet. They were ready to move on and apply their new understandings.
It made me wonder if, as teachers, we do move children on too far beyond the conceptual comfort zone and so I need to remember this strategy to help cater to the learning needs at particular times in our maths enquiries.
The children in our class are the greatest source of PD!
The following day the same group split into partners and continued the pancake enquiry this time choosing numbers they felt might be even more challenging to calculate.
They shared not only the strategy they used, but were also encouraged to share a different strategy they could have also used to solve the ingredient proportions.
Below, this pair thought they could have multiplied the ingredients by the amount it would equal OR they could have looked at ratio proportions already solved and add the together.
We liked both those strategies a lot.