Monday, 25 February 2013

What Is My Area?

Wait - before you say anything - I know your body is 3-dimensional and area is a measurement of two dimensions - so I probably should say, "What is the area of my silhouette?"


What is the area of my silhouette?

In the past week we have measured height and width, talked about Vitruvian Man, combined our heights, compared them with other classes.

Time to have a look at comparing areas.

My chief interest here is to look at how the kids are going to approach this question. I want to see if they can come up with any short-cuts that will speed up the calculations, so they don't have to cover their entire body in 1cm grid paper and count out each square.

Look what they did!

First job was to draw an outline of the body. It was important to get this accurate - what do you think of this attempt?

"It's never going to be accurate!" said one student. "Even with the best artist in the world doing the outline, you couldn't make it 100% because the arm is round not flat."


Once the outline was done, most of the kids got busy covering their shape with 100 squares, though not all of them recognised that they had 100 squares in them or that they were equal to 100 square cms. But we got there in the end.

We had a good discussion at the end about accuracy (again) and about those missing gaps - would that make your "estimate" under or over the actual area?

This one was interesting. I had been harping on about finding out ways to make it easier than counting up each individual square cm. The picture isn't great but you might be able to see that they have divided their body in half, deciding that the body is basically symmetrical and you only need to do half of it, then double your result.

This one shows that the student has calculated each individual finger in an attempt to produce an accurate answer. When questioned about their methodology, for example what you do about the bit not covered by the 10 block, there was some talk of fractions and halves and how they were added in to the calculation. I wasn't entirely convinced but there was some thought put into improving the level of accuracy.

Realising that the 100 square was too big for the arms, this group has opted to use 10 blocks and has drawn them in individually - a much more accurate method.

 And this was a fascinating example but again the photo is disappointing. This group decided to measure around the outline with a piece of string. The string was then reorganised into a rectangle and this was used to calculate an area. Even though the resulting rectangle was obviously several times larger than the body shape, which neatly inside it, they maintained that their measurement was accurate because they had been very careful to get the string the exact length.

So how big is a body?

Well, we got measures of between 4500 and 5500 square centimetres. There were some closer to 6000 square cms and even one of 8500 square cms - not sure where they went with this one.

And then much discussion followed about the inaccuracy of this method and how much better it would be if someone invented a machine where you could lie down and the machine could go over you and measure you in a three-dimensional way. This would be so much more accurate.

Had to break it too them - they already exist.

Top Math Movies 2012

So it's Oscars time.

And everyone is getting excited about who is going to win.

Well, I'd like to introduce a new award - 

Best Math Movie of the Year  

And here they are, in descending order:

5. Clint struggling with basic geometry.

4. Yep - it's the number after 39.

3. A basic numerical operation - the reciprocal of multiplying

2. Dividing by zero does it every time.
Look - I get an "E" on my calculator!

1. This year's winner of the Oscar for 
"Best Math Movie of the Year 2012"


And while you're thinking about movies, check out this blog:

It's a math movie pictogram quiz!

Sunday, 24 February 2013

Da Vinci and the Vitruvian Man

You've seen the picture before, on posters, t-shirts, in books and movies. It's pretty famous. I even used it last week in this blog!

It's known as the "Vitruvian Man." Da Vinci did the drawing and made some notes based on the ideas of Vitruvius, who may sound like a famous volcano near Naples but who was in fact a famous architect in Rome. He had some ideas about the proportions of the ideal man. Da Vinci was able to use these to draw this diagram inscribed inside a square and a circle.

According to Wikipedia, my first port of call when I can't read Leonardo's unintelligible scrawl, the notes by Da Vinci around the picture are a summary of the key ratios of the perfectly proportioned human body:

  • a palm is four fingers
  • a foot is four palms
  • a cubit is six palms
  • four cubits make a man
  • a pace is four cubits
  • a man is 24 palms
  • 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
  • from above the chest to the top of the head is one-sixth of the height of a man
  • from above the chest to the hairline is one-seventh of the height of a man.
  • the maximum width of the shoulders is a quarter of the height of a man.
  • from the breasts to the top of the head 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 root of the penis is at half 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.
  • from below the knee to the root of the penis is a quarter of the height of a man.
  • the distances from the below the chin to the nose and the eyebrows and the hairline are equal to the ears and to one-third of the face.

   These measurements are amazing. I could share most of them with my class at school but there might be a few giggles about using the penis as a point of reference so I might hold off on that one.

   Anyway, a useful insight into the mind of a genius. Maybe if I worked backwards, I could get the kids to discover some of these relationships for themselves....

Friday, 22 February 2013

Kindergarten Number Patterns

Working with our buddies

Every second Friday we get together with our kindergarten buddies. It is a great chance for us to socialise with the younger kids in the school and to develop a bit of leadership.

And it is fun.

Finding patterns

In her book "The I Hate Mathematics Book", Marilyn Burns revealed a great secret and mystery...

"The password of mathematics is pattern." (pg 6)

And it's true, isn't it? If you can find a pattern, you are half way to finding a solution.

Today, the kindergarteners were finishing off some inquiry into patterns. They had used a great variety of objects to build their patterns - plastic toys, feathers, hats, blocks. 


We were able to sit with them and help them record their learning, writing down their patterns and having them explain what was being repeated.

It would be really cool to get the kindy buddies outside and hunt for some patterns out there in the big wide world. I wonder what they would find?

Four young men stepping out into the world to discover some mathematics.
Any patterns out there?

Tuesday, 19 February 2013

How Wide is Your Class?

In Year 6 we are currently inquiring into "Who We Are", a unit of inquiry that is giving us lots of room to work with measurement.

Last week we explored "How tall is your class?" Today we decided to think laterally - and see how wide our class is.

What did da Vinci say?

Seen this picture before? It is known as the "Vitruvian Man", drawn by da Vinci based on the work of Vitruvius, a Roman architect. (This picture is fascinating - a quick google gave me enough data to write a separate post - stay tuned!)

Anyway, one of the points of this diagram is to show that the height of a correctly proportioned man is equal to the span of his arms.

Is that true? And would it be true of everyone in our class? And would it be true of the other Year 6 classes?

We already knew how tall we were from our previous investigation.

Time to get inquiring...

So how wide are we?

Using a similar method to last time, we first measured each other individually and added together our measurements.

Then we went outside and lay down on the oval again - I know it looks like the kids are just lying there in the sun but really they are flat out doing some hard maths.

Interestingly we found that the distance from finger tip to finger tip was less than when we lay end to end.

Year 6 working hard again

Total height = 3714cm

Total width = 3686cm

Not a big difference but enough to ask a few questions.

Why are the two numbers different?
Was da Vinci wrong?
Were we wrong/inaccurate?
How would we test our accuracy?
Is there a reasonable margin for error?
Will it change over time?

So many questions!
Where will we go next?

Thursday, 14 February 2013

How tall is your class?

Well, a new year, a new class, a new grade. What new adventures will befall us?

This year I have moved up to the big kids - Year 6!

And like a lot of teachers I suppose, we on Year 6 decided that it might be good to start the year with a review of number skills.

And the first one we chose was addition.

So, how can you do an inquiry into a review of addition?

Making Authentic Links

Our year group was starting the year with an inquiry into the transdisciplinary theme of "Who We Are", focusing on the physical, mental and social changes of adolescence.

As part of our exploration of physical change, I thought we could measure the combined heights of all the students in the class, then compare each class to see which was tallest.

This would involve lots of addition.

And some mathematical thinking.

Then if we repeat the process in about 6 or 7 weeks, we can see if there has been any change. Maybe we could even do it again at the end of the year.

Tuning In With Some Basic Addition

We started our brains up, or began the tuning in, with a 10 minute challenge. Students were provided with a series of questions that focused on addition skills:

i) What do I need to add to these numbers to get to 50?
ii) What do I need to add to these numbers to get to 100?
iii) 2-digit vertical addition
iv) Simple word problems involving addition of 2-digit numbers

In days gone by, this would have been the end of my maths lesson - do a page of algorithms - mission accomplished.

I still use this type of material - and the questions we used were from a textbook - but it is as a starting point for further investigation, not as an end in itself.

Thinking Mathematically

Now our brains were engaged and our skills were tuned in, we began the "finding out" part of the inquiry.

I asked the kids to list questions that they would need to answer in order to direct their inquiry.

Here's a few of the questions they came up with:
  • How many people are in our class?
  • How many classes are we comparing ourselves to?
  • How tall is each person?
  • What is the average height of each person in our class?
  • How can we make it a fair comparison between classes?


So, how were we going to measure the height of a class?

Well, the kids came up with three suggestions:

1. We all stand on each others shoulders and then hold a tape measure up to see how tall we are

2. We all lie on the ground in a long line and use a tape measure to see how long the line is

3. We measure each individual and then add them together

We took a vote and decided on Option #3 so we all got to measure each other and came up with a total (well, we came up with several totals and then needed to go back and check that we were measuring accurately and adding correctly).

Our total was 3637cm.

 How we measured the height of our partner

Sorting it all out in a table

And so that we didn't miss out on the fun, and also so that we could try out another methodology, we all went outside and lay down on the grass and used Option #2.

Yes! Looks like 6BF is taller than 6TL

Rolling out the tape measures

Interestingly, when we measured the length of our line of bodies outside, we got 3714cm. This led to much discussion about the accuracy of each measurement, how mathematical we had been and which answer to the question was right.