Exploring Ratios Reveals the Distributive Law

We had been doing an investigation into Vitruvian Man this week. You know, the picture that Leonardo drew:

A fascinating picture with a fascinating story. 

Anyway, all the scribble around the outside of the picture is actually Leonardo's mathematical calculations based on the ratios of the human body.

Did you know that, in the "ideal" human body:
- a palm is four fingers

- a foot is four palms

- a cubit is six palms

- four cubits is the height of a man

- a pace is four cubits

- a man is 24 palms tall

- the length of the outspread arms is equal to the height of a man.

- from the hairline to the bottom of the chin is one-tenth of the height of a man.

- from below the chin to the top of the head is one-eighth of the height of a man.

- the maximum width of the shoulders is a quarter of the height of a man.

- the distance from the elbow to the tip of the hand is a quarter of the height of a man.

- the distance from the elbow to the armpit is one-eighth of the height of a man.

- the length of the hand is one-tenth of the height of a man.

- the foot is one-seventh of the height of a man.

- from below the foot to below the knee is a quarter of the height of a man.

- the distance from below the chin to the nose is equal to:

• The distance from the nose to the eye brows
• The distance from the eyebrows to the hairline
• The length of the ear
• One third of the face

Well, didn't we have a lot of fun exploring all these ratios and proportions!

We got the children to trace their outlines and then start recording measurements and ratios. At the start, they were not too sure what ratios were - by the end they had a better idea.

So the next day, I took it in another direction. We hadn't played with blocks much this year, so I got some out and asked a few provocative questions:

- Get one block
- Get two blocks of a different colour
- Get two blocks of the first colour
- Get four blocks of the second colour

Continue.

How much are you increasing by?

How many blocks of colour #2 do you need if you start with 6 blocks of colour #1?

This was to look at what happens when you use ratios to increase a given value.

Here is what the kids did:

You can see what is happening here.

1 green + 2 yellows = 3 blocks
2 greens + 4 yellows = 6 blocks
3 greens + 6 yellows = 9 blocks

and so the pattern grows...

But if we look at the greens as being our 1x tables (1, 2, 3, 4...) and the yellows as being our 2x tables (2, 4, 6, 8...) then when we add them together we get the 3x tables! (3, 6, 9, 12...)

We tried this with 1x and 3x:

1 green + 3 yellows = 4 blocks
2 greens + 6 yellows = 8 blocks
3 greens + 9 yellows = 12 blocks

And it worked. (1x tables) + (3x tables) = 4x tables

"So what might be another way to work out your 4x tables?" I asked.

Quick as a flash, one bright young thing said, "2x tables plus 2x tables."

And he was right.

The Distributive Law in action.

Start the Year with an Attitude Survey

Here in my part of Australia it is the end of Week 1, Term 1. Being a good teacher, I am always keen to get to know my new class quickly and to find out all about them.

And this often involves assessment.

Kind of sad if the first words of greeting are, "Hi kids! Did you have a great holiday? Sit down and do this test."

Really, doing tests in Week 1 is harsh. But I still want to know about my kids. 

So instead, I give my students an attitudinal survey so they can tell me how they "feel" about maths.

I am teaching Year 4 this year. They are some of the kids I taught in Year 2 back in 2018 who did the awesome patterns you may have seen on my blog from back then. So I was hopeful that there would be some pretty strong connection happening with maths. 

I don't have room to go through the whole 2 pages of the survey (if you want a copy I can email you one) but I want to highlight a few interesting questions and the results from my class.

Question 1

Put a circle around the words you would use to describe

maths:

boring     cool     exciting     fun     difficult         

hard        confusing     challenging     scary    easy

No - this is not a test to see who knows what a circle is and who can draw one. It is about tuning in to the language that the students use when talking about maths - whether talking out loud or just talking in their heads.

Results for Question 1

Students could circle as many words as they wanted to. This is how the students in 4BF responded:

Interesting.

No-one in my class finds maths scary - or maybe they're too scared to admit it.

The most frequently recorded words were "cool", "fun" and "challenging". This is very positive language. Even "challenging" acknowledges that maths can require effort but that this is seen as a challenge rather than as being simply "difficult".

The one boy who circled "boring" also circled "hard", "difficult", "challenging" and "confusing". He is not a student that I have taught before. He will be at the top of my list to learn more about and to see where we can make some positive gains.

I wonder if the students will still be using the same language at the end of the year?

Another question that I use asks the students to rate their own ability at maths by putting a cross on a line. It looks like this:

Question 4

This example is my "boring" boy from the question above. Definitely need to get involved here.

The length of the line is 15cm (or 150mm). With this question, I measure the position of the cross in relation to the left hand end of the line and give a score our of 150 (because the line is 150mm long, so 1 point equals 1mm). 

Results were:

So the boys rate themselves pretty highly at maths, even given the "outlier" (above) who scored himself at about 25 out of 150. And the girls have a (significantly) lower opinion of their abilities. Do the boys have an inflated sense of how "good" they are? Are they being realistic? Are the girls too modest? Are they being realistic? Do they doubt themselves?

One thing that does stand out - I need to work on attitude. 

This quick survey gave me a lot of information to think about - probably more than a skills-based number test would have. There were 10 questions in all. Some questions were similar in format to the above, others were short answer and required sentence responses. I can highly recommend that you have a go at doing something similar to start your year. It will tell you a great deal about the students in your care.

All 5 of our Year 4 classes did this survey. Hopefully in the next few days I will have data from all the classes that I can share.

Have fun!

miniMaths Book 2

The Background

Following our successful application in 2018 for an ACT Government Nature Play grant, I submitted an application on behalf of the Canberra Maths Association for another grant in 2019. As we had planned in 2018, we again proposed to:

- write 10 inquiries/tasks/lessons/resources to be used outside in the natural environment
- run a series of workshops with preschools to share these resources

I am very pleased to announce that we were again successful in our application and got the grant. A group of about 10 interested preschool teachers met together one afternoon earlier in the year at Radford College, my school, and brainstormed some ideas. It was a productive time. I went away with pages of notes that I wrote up as the miniMaths Book 2. 

The 10 tasks we came up with were:

1. Which hand? - an exploration into probability
2. Hands and feet - measuring with informal units
3. Symmetrical mandala - building a symmetrical pattern
4. Leftovers - what to do with remainders when dividing
5. There and back - giving directions to describe a journey
6. Pattern dance - using symbols to represent the parts of a dance
7. Flower count - collecting data about flower colours and representing the data as a "graph"
8. Shape detective - finding shapes in nature
9. Treasure sort - sorting objects based on shared and unique features
10. Fill the pot - how many cups of sand does it take to fill a saucepan?

A beautiful flower mandala created by the wonderful students 

at Orana Steiner School

The 10 tasks make up Book 2 of the miniMaths program. Every preschool and early learning centre in the Australian Capital Territory will receive copies of the book early in 2020.

The 10 tasks have also been uploaded as a website that is available to anyone, not just ACT teachers:

http://www.minimaths.com.au 

Over the next week or so, I will post a bit more detail about each of the 10 tasks. Maybe you will find them useful. I would be keen to hear any feedback.

And any Canberra teachers - keep an eye out for the miniMaths workshops rolling out over January and February.

Regards.

1 10 100 1000 10000 100000 1000000

Let's have a look at place value with Year 4.

The Australian Curriculum says:

Recognise, represent and order numbers to at least tens of thousands (ACMNA072)

So we did.

I got some 1mm grid paper and we started off with one tiny square.

Then we cut out a row of 10.

Then we cut out a square of 100.

Then a row of 1000.

It started getting a bit technical here. It seemed like I had hit a developmental barrier - a place that required some additional cognitive effort. I was a bit surprised but we worked our way through. 

Obviously (to me at least) 10 000 was going to be 10 rows of 1000. It didn't appear to be that obvious to the students. Why was there this hesitation? Perhaps this is why the Australian Curriculum identifies this as the target size to go for in Year 4. However, once we started counting by groups of 1000, we soon got the idea of what 10 000 looked like.

One of my ambitions with this task was to get a visual experience of what happens when we start multiplying by powers of 10. Also to get the connection with the decimal place value system we use - more of this later.

Once we had 10 000 sorted, it was a smaller step for mankind to decide what needed to be done to represent 

100 000. It's 10 of the 10 000 squares. 

At this point, a few students began to see a pattern forming in the way we were representing the numbers:

1 = square

10 = line

100 = square

1000 = line

10 000 = square

100 000 = line

1 000 000 = square...

Well, we haven't quite finished the 1 000 000 square but we are nearly there.

And the classroom is a bit of a mess - just the time when a member of the executive chooses to walk in...

This is the room after we did a bit of a tidy up.

So now we have models of 1, 10, 100, 1000 etc.

A valuable experience.

But the other connection I wanted to make was with the decimal place value system. Because with my models, I can now choose any of the samples to be the unit - it doesn't have to be the tiny little square = 1. It could be any of the other models we have made. The very biggest square, that we thought was 1 million when we made it, may in fact become 1 unit, or 1 thousand or 1 anything. 

And if that is the case, then what is the value of the other numbers? And where does that decimal point need to go?

Time to start playing...

miniMATHS - 7. Make a Star

This is my favourite.

I know - you're not meant to have favourite children.

But this one is where it all started. This was my first idea for the miniMaths series of tasks. So I think I had given it more thought and probably used it as the benchmark for what I imagined the other tasks would look like.

And this one has real maths behind it. As you build up your star, you are adding a constant value to your pattern - one rock for each arm of the pattern. It is symmetry. It is an arithmetic sequence. It is pre-algebra. It is the connection between repeated addition and multiplication. And it is fun. It makes a cool looking pattern.

This task is linked to EYLF - Outcome 4, focusing on the disposition of the learner. The task itself is open enough to allow creativity and curiosity. It can be done as a collaborative and cooperative task. It can help students develop confidence and perseverance. 

Give it a go - it is a great task.

miniMATHS - 6. Stacking

This is fun. I know - I could spend hours doing it. In fact, I used to get kids to do it when I was on playground duty and I would challenge them to get 5 rocks stacked on top of each other.

But where is the maths?

Well, there are a couple of ideas here that need to be explored.

1. Have you ever worked with kids (probably Year 2 or Year 3) who had the idea that when constructing a "sum" (or algorithm - addition or otherwise) you had to start with the big number and then perform the subsequent operation on it? And isn't there an addition strategy for counting on that says you need to start with the big number first? Is this somehow related to stacking up rocks or other items where it is a good idea to put the big one on the bottom of the stack? Please note, I'm just asking the questions here - not providing the simple answers.

2. Or could this be a revelation of the mathematical concept of combining different values to create new one?

3. Or is it an exercise in balance, similar to an equation where one side has to equal the other?

4. Or are we playing the mathematical idea of prediction - what happens if...?

Perhaps the EYLF will give us some insight:

The EYLF certainly talks about "balance" as an important element of growth. This task provides a neat entry point for this conversation.

This task will also provide an opportunity for students to develop perseverance - to try to get that stack higher and higher. It will also encourage students to reflect on their unsuccessful strategies and modify and improve on the performance. 

So, just as in the "Shadows" task, this activity may not appear at first glance to be explicitly "mathematical", there is great capacity for students to explore deep mathematical thinking by attempting the stacking task.

If you haven't seen it yet, there is a miniMaths website:

http://www.minimaths.com.au

And for people based in the Canberra region, I am running some workshops in preschools and early learning centres over the next six weeks. Details can be found on the miniMaths website.

See you there. 

miniMATHS - 5. Shadows

Sorry about the delay of a few days there - we went back to school last week and I have been a bit distracted getting the class into shape..

So here we have possibly the first major deviation in the miniMATHS resource tasks. This task, "Shadows", moves away from activities that have direct mathematical concepts as a focus, such as extending patterns or investigating area, to a place where we are investigating a natural phenomenon, such as shadows, using mathematical skills and knowledge. No longer playing directly with maths - now we are using maths as we play with something bigger.

Shadows are fascinating. There is so much that we can learn by observing them. They are a great leaping off point for exploring time; they are an explicit example of change; we all have that follows us around - and there are lots of games you can play with them.

All of the things that are suggested as activities associated with this task can be investigated using mathematical language. This is a key opportunity to develop our use and understanding of words that describe position, shape, movement and change.

This task is a great way to engage directly with the natural environment in an explicitly mathematical way. We can record changes over time using the outlines of shadows drawn on the ground. We can discuss the position of the light source relative to the object casting the shadow and the shadow itself. 

The EYLF talks about knowledge of and respect for the natural environment. This task will give students a greater understanding and appreciation of what happens in the world in which they live, helping them to be connected with it and to learn respect for it.

Is it "maths by stealth"? Sneaking a bit of maths into a fun exploration of a cool phenomenon? 

Well, I would say no - this is what real maths is: using our skills and knowledge to make sense of the world around us. And hopefully we can start this in early year education with young kids, engaging them as mathematicians to learn more about their world.