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. 

Maths is Everywhere- Even on Our Faces!!!

Maths is Everywhere- Even on Our Faces!!!

That's what was written on the board as a provocation starter.

What do we think this means?

- We can measure the lengths of features on our face.

- We could measure its area.

- We can compare the sizes of things on our faces.

- We could count the number of hairs on them.


Some nice ideas to warm up our minds.

To begin, we drew a 3 minute sketch of our partner's face:



In visual arts, we been exploring drawing techniques and in particular learning to draw with our eyes instead of our hands. We have been exploring how we should try to look at the object more than the paper, but when it comes to drawing a human face that seemed more challenging to do than other drawing experiences we've been doing.

Still, this served the purpose well.

What is the shape of the human head?  

- ovally

Is it wider at the top of the bottom?  

- hmmmmm

Look at your partner's head.

- It's wider at the top.

Exactly.

So we draw a more realistic human head shape.

The human face is made up of fractions, ratios and proportions.

Look at your partner's head. From the top of the skull to the chin where are the eyes placed?

- a third

- a half

- nearly a half

The eyes are half way.

So we sketched a line dividing the head in half and drew the eyes there.

We continued doing this with the nose, then mouth. Each time looking at our partner to determine the fraction or proportion.

As we continued, we discovered by looking at our partner's head each time and then sharing these other fraction /proportion relationships:

° You can fit 5 eyes across the face. So to draw the eyes in the correct place, we divide the space into fifths and then draw the eyes in the 2/5 and 4/5 place.

° To draw where the mouth ends, we draw an imaginary line a third of the inner eye. That's where the mouth ends.

° the mouth is half way between the nose ending and the chin ending

° the neck line joins where the end of the eyes are

° the ears begin at the halfway mark and three-quarter mark (where the mouth is)

In determining these, we practised closing one eye as we have discovered previously whilst sketching to help see an object in 2D rather than 3D (more maths!!!).  

We also practised holding our thumbs up as a measurement tool for sketching proportions ( MORE MATHS!!!!!)

We practised these new maths discoveries of the human head for about 5-10 minutes.

When to close, we redrew our partner's face. This time with our new mathematical understandings and compared the two drawings.

Most of us felt our second drawing was a much better likeness.

In orally reflecting:

- This year I've been discovering loads about how maths is connected to everything, but even our heads has fractions! That's amazing!

- I'm amazed that we can use ratios and proportions for so many different things.

- I want to be a professional artist when I'm older and now I know even more how maths can help me.

- I wonder if we can find similar fractions and proportions in animal faces?

Before & after mathematical understandings with practise sketchings:

 

With reflective understandings like that, we know we have had a pretty successful learning experience.

That together with observing the pride in so many of the children at seeing the progress they had made when they compared their before and after sketches!

My next mission- how to help children enquire into the relationship between maths and music........ I need to do some research.

Ratio / Proportion / Rates Investigation Sharing

Sharing our Investigations

From where the children self-assessed where they felt they were at and what they wanted to find out next regarding strategies we use for ratios / proportions /rates ( Blog Post ), I thought of some different ways they could investigate using real life situations.  Not the typical 'real life situation' word problems such as a girl has 266 yellow flowers.......found on the net that are really stretching any relevance children might have to why we learn how to use ratios.


So, I ended up creating 4 investigations of varying levels of understanding.

After the groups investigated, they then presented their findings in poster form ensuring they included the strategies they tried using.

Groups were partnered up and each had 5-10 minutes to share what they did.

Usually my students are pretty great at engaging themselves in these sharing discussions, but of course, some need extra support in how to discuss.  

So, before we began sharing, we brainstormed a list of possible questions they could ask the group teaching them.  This is what we came up with:

What could we ask?

° Which strategy did you use?

Follow up questions:

– Why did you use that strategy? 

– What did you find interesting about that strategy? 

– Would you use that strategy again? Why/Why not?

° Would you use that strategy again?  Why / Why not?

° Did you make any connections during your investigations?

° Was this all new to you or did you know some of this before?

° What important discoveries have you made?

Quite a great load of questions  we thought!

We kept this displayed on the data screen to refer to if needed.

These two partner groups imagined they owned a Mini-Cooper.  

They found out the fuel consumption of the car and then calculated how much it would cost in petrol to drive to different cities in Europe (their choice) from where we live in Lausanne.

( They were able to search the Internet to find driving distances from Lausanne)

In our self-assessment and enquiry, those children had indicated they felt they were ready to learn about rates so I thought this investigation would be an interesting introduction to the concept.







These next partners were given the information that the A380 aeroplane travels at an average speed of 900 km/hr.

If we wanted to go on a holiday outside of Europe, but we didn't want to travel longer than 10 hours, where could we travel to and how long would it take?


They didn't have access to the Internet, but they did have access to atlases.  using the atlases required them to use the scale map ratios to measure the distances to places from Lausanne. (Double ratio thinking- yes!)

Whilst sharing their investigation, this partner group discovered they had actually made a calculation error, but both they and those they were sharing with still liked how they tried the double line strategy and that was what was most important (Exactly- and without me needing to prod that idea):

Whilst these children were conducting these investigations, others were investigating how to make Pancake Recipe Proportions and others were working out strategies for Ratio & Proportion of Eggs on a Free-Range Chicken Farm




All of these different enquiries taking place in the same classroom at the same time catering to different levels and wonderings that they essentially selected by using our strategies continuum: 

( How to create the continuum   )



Finally, to help each learner conceptualise and reflect on what they had learnt from other groups, we spent some time adding more of our new understandings to our journey maps which are being used as our formative and summative assessment for our unit:

( Ratios Formative & Summative )

How do we use maths in visual arts?

To find out a way we use mathematical thinking in artmaking and to connect with our enquiry into ratios / proportions / rates, we began learning how to enlarge or shrink drawings using scale ratios.

I explained how back in the day before laptops, we at school really needed to master this strategy whenever we wanted to draw a picture of something large on posters.  Today, of course, we just enlarge or shrink an image from Google and print n cut.

But, despite scale drawing being a 'dying' skill, it greatly helps develop stronger drawing skills and has a practical connection with how we use ratios in real life situations.

We began by finding a simple object and drew it to its exact scale (or size) and noted this is written as:

Scale  1 : 1

What does that actually mean?

- 1 cm in real life = 1 cm on the paper

Exactly.

Some chose to draw paintbrushes, cups, glue sticks, pens, coins etc. The simpler the object to draw, the easier it will be for our first try but feel free to challenge yourself.

After our initial drawings, we drew grids over the objects and then decided whether we wanted to enlarge or shrink the object.

Some of us chose to double the object, triple it, quadruple it!

Others chose to halve it and one student challenged himself to try to shrink the object to an eighth of the original. 

This is the type of great learning that happens when we give children ownership of their learning rather than dictating what they should do.

So, to enlarge or shrink the image, what do we need to do to the grid on our next page?

This is an interesting problem solving activity so I didn't explain what to do. Instead, the children needed to reason what would make sense to do. 

Of course, for some, this was an easy thinking task, but others did need to experiment with different ideas and then step back to analyse if what they did made sense.

Showing mine as an example, I did though explain how we use the grid to create our images.

The hint I gave was to always look where that each line in the object interepts the line on the grid.

Mentally use fractions.

Does the line intercept the grid at the halfway mark?  at a quarter mark? Just below the halfway mark? etc

By using mental fractions as a strategy, we will be able to create more accurate enlargements of shrinkings.

From this we learnt further how we use mathematical thinking in art.


We are still in the beginning stages, but eventually when we complete our images, we will think about the scale ratio our drawing is and add it to them. 


I'll post more as our enquiry continues........

Pancake Proportions

We split into different groups today each looking at a particular way we use ratios / proportions / rates.

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.

Why?

- 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?

Why?

- 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?

- 32

- 7

- 1

- 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. 

Creating Strategies for Ratio Proportions

A small group and I continued to explore our central idea:

We use ratios and proportions to help see the relationship between values in real life situations.

We began by discussing what we had already discovered about our central idea and what we were still curious about.

One of our wonderings was if there are different strategies we can use and so this tied in well with what we were about to do.

We looked at the following scenario:

A farmer had a free-range egg farm.

  

       Free-range Eggs                                  Caged Eggs

In each section of her farm, she wanted to keep the ratio of hens to roosters as 8 : 2

Use the table strategy to work out possible proportions of hens to roosters:

Hens	8	4	16		32
Roosters	2			10
Total	10

Use the images to show the proportions in each section of her farm:

= hen         = rooster

Section 1 had 10 chickens in total:	Section 2 had 5 chickens in total:
Section 3 had 30 chickens in total:	Section 4 had 20 chickens in total:

Using Google docs is a great tool to help children see what ratios and proportions look like.  

After using the table strategy to find different proportions, the children could then copy and paste the hens and roosters to visualise the proportions like so:

Section 1 had 10 chickens in total:	Section 2 had 5 chickens in total:
Section 3 had 30 chickens in total:	Section 4 had 20 chickens in total:

We looked at the proportions created and one student commented on how she could see the reason why we might want to simplify ratios.  When we simplify a ratio, it makes more sense to us! - Exactly!

I then presented the following question and in partners or alone they tried to create more than one strategy to solve it:

On Monday, she had collected 150 eggs, 25 of which were brown. The rest were white.  

What is the ratio of white to brown eggs?

1.   1 to 3
2.   1 to 4
3.   1 to 5
4.   1 to 6
5.   2 to 5

Create a similar word problem based upon the number of eggs (any amount) for others to solve using the same strategy. Ensure you have the answer.

Both and I and the children like it when we try to create more than one strategy. It helps us to realise that maths isn't about getting the answer compared to creating effective strategies to help get an answer.

After some time, we shared and discussed strategies we had created:

This student wanted to share what she was thinking about and asked us why she was getting stuck. This is exactly the type of great shared learning we want to generate. We talked about the beginning of this strategy and why it wasn't really helping us to make sense of the ratio proportion and so that's why it came to a dead end. Mistakes are wonderful ways to stretch our minds another student reminded us and so we appreciated this sharing because it helped stretch all of our minds whilst think about it:

We did the same learning activity of creating different strategies to solve this:

The farmer looks at all the eggs in her barn.

There is a white to brown ratio of 8 : 11.

If there are 88 white eggs, what is the total number of eggs?

1.  121
2.  128
3.   152
4.   176
5.   209

Again, we explored all the different strategies shared and reflected on those we felt made more sense, those we felt were the quickest and those we felt best helped to visualise the question.

We didn't reflect on who got the correct answer or even gave that much thought to what the correct answer was and that is what we should hope to instill in mathematical learning. Maths is about creativity and by creating different strategies rather than focusing on getting answer, we are broadening our maths understanding and creating a learning environment where mistakes are valued.