Singapore MRT Puzzle

This post is dedicated to a young friend, E. P. Dubs, who is developing quite a passion for the SIngapore train network...

...and consequently got me thinking.

The Challenge

Starting anywhere you want, travel on a continuous path and go through every station on the map.

What is the least number of stations you will have to pass through more than once?

If this is too hard to read, you can probably Google your own

So, I think I have a solution but I'm not going to tell anyone.

You have to think for yourself.

To extend this idea, you could:

1. Choose a mandatory starting point, such as Changi Airport or Dhoby Ghaut.

2. Say you can only travel on the red and yellow lines in a clock-wise direction.

3. Use the maps for other underground railway networks, like London, Paris, New York, Tokyo....or anywhere you want!

Good luck!

London

Paris

Tokyo

Nextian Number Theory

Another brilliant development of mathematics, Nextian Number Theory, appears in Jasper Fforde's book, "Something Rotten.".

The artwork on the covers of this edition of

Fforde's books is exceptional

In this part of the story, Thursday is again talking with her Uncle Mycroft, the inventor. He has been thinking about working backwards from solutions to find out what the question was. He calls this "Nextian Number Theory":

'What work were you presenting to MadCon '88?'

'Theoretical Nextian mathematics, mostly,' replied Mycroft, warming to the subject dearest to his heart — his work. 'I told you all about Nextian geometry, didn't I?'

I nodded.

'Well, Nextian number theory is very closely related to that, and in its simplest form allows me to work backwards to discover the original sum from which the product is derived.'

'Eh?'

'Well, say you have the numbers twelve and sixteen. You multiply them together and get 192, yes? Now, in conventional maths if you were given the number 192 you would not know how that number was derived. It might just as easily have been three times sixty-four or six times thirty-two or even 194 minus two. But you couldn't tell just from looking at the number alone, now, could you?'

'I suppose not.'

'You suppose wrong,' said Mycroft with a smile. 'Nextian number theory works in an inverse fashion from ordinary maths — it allows you to discover the precise question from a statedanswer.'

'And the practical applications of this?'

'Hundreds.' He pulled a scrap of paper from his pocket and passed it over. I unfolded it and found a simple number written upon it: 2216091 -1, or two raised to the power of two hundred and sixteen thousand and ninety-one, minus one.

'It looks like a big number.'

'It's a medium—sized number,' he corrected.

'And?'

'Well, if I was to give you a short story of ten thousand words, instructed you to give a value for each letter and punctuation mark and then wrote them down, you'd get a number with sixty-five thousand or so digits. All you need to do then is to find a simpler way of expressing it. Using a branch of Nextian maths that I call FactorZip we can reduce any sized number to a short, notated style.'

I looked at the number in my hand again.

'So this is?'

'A FactorZipped Sleepy Hollow. I'm working on reducing all the books ever written to a number less than fifty digits long. Makes you think, eh? Instead of buying a newspaper every day you'd simply jot down today's number and pop it in your Nexpanding calculator to read it.'

'Ingenious!' I breathed.

'It's still early days but I hope one day to be able to predict a cause simply by looking at the event. And after that, trying to construct unknown questions from known answers.'

'Such as?'

'Well, the answer: "Good lord, no, quite the reverse!" I've always wanted to know the question to that.'

'Right,' I replied, still trying to figure out how you'd know by looking at the number nine that it had got there by being three squared or the square root of eighty-one.

'Isn't it just?' he said with a smile, thanking my mother for the bacon and eggs she had just put down in front of him.

from "Something Rotten" by Jasper Fforde

Nextian Geometry

I've been doing a bit of reading lately. One of my favourite author's is Jasper Fforde, very funny speculative fiction writer. I would like to publicly thank my colleague and friend Jane Stanton, school librarian extraordinaire, for pointing me in his direction. If you haven't read any of his works, do yourself a favour.

Jasper Fforde - worth reading

Anyway, I'm currently reading "Lost in a Good Book". Our hero, Thursday Next, is talking with her inventor uncle Mycroft. Polly is Mycroft's wife, Thursday's aunt.

"This is Polly's hobby, really. It's a new form of mathematical theory that makes Euclid's work seem like little more than long division. We have called it Nextian geometry. I won't bother you with the details, but watch this."
Mycroft rolled up his shirtsleeves and placed a large ball of dough on the workbench and rolled it out into a flat ovoid with a rolling pin.
"Scone dough," he explained. "I've left out the raisins for purposes of clarity. Usingconventional geometry, a round scone cutter always leaves waste behind, agreed?"
"Agreed."
"Not with Nextian geometry! You see this pastry cutter? Circular, wouldn't you say?"
"Perfectly circular, yes."
"Well," carried on Mycroft in an excited voice, "it isn't. It appears circular but actually it's a square. A Nextian square. Watch."
And so saying he deftly cut the dough into twelve perfectly circular shapes with no waste. I frowned and stared at the small pile of disks, not quite believing what I had just seen.
"How--?"
"Clever, isn't it?" he chuckled. "Admittedly, it only works with Nextian dough, which doesn't rise so well and tastes like denture paste, but we're working on that."

Jusper Fforde from "Lost in a Good Book"

There is more to come. When I find the other mathematical ideas (there are several) I will post them as well.

In the meantime, get down to the library and borrow a few of Fforde's books to read for yourself.

4-Steps with Polya

Here is an example of problem solving from a school in SIngapore. The school uses a 4-step problem solving model based on the work of George Polya, a common model in Singapore schools.

George Polya - Hungarian born mathematician

interested in problem solving

The 4-step model goes like this:

Step 1 - Understand - what I know

Step 2 - Devise a plan

Step 3 - Solve the problem by carrying out the plan

Step 4 - Check to see if you have in fact solved the problem

Solving a Problem

So, here's the story. Carol and Dina start with the same amount but at the shops Carol goes crazy and splurges to the tune of $345. Dina, a veritable model of restraint, only spends $80. 

Our student uses the 4-step model beautifully. 

Step 1 - here's what I know, the facts from the story

Step 2 - really interesting that our student decides to use a "before and after" representation. This is a nice way to represent the story given that it is told in a "before and after" style. Nice strategy.

Step 3 - is the calculations in the "Model" column but also represented in the diagram in Step 2

Step 4 - yep - student has checked by substituting back into the story. By working backwards they end up with the starting amount - all good!!

Same story, new student. 

Note that this student goes straight for the jugular. Not needing to show the starting point, this student focuses right on the question of how much cash is left?

In fact, they didn't really need to know how much Dina and Carol started with. The calculation in Step 3 got them there:

$53 x 6 = $318

The other bit in Step 4 is probably superfluous.

But some very nice application of the 4-step model.

So What?

I am getting the feeling that having a system for solving problems is a useful tool. 

Polya's 4-step idea is not that dissimilar from the 9 steps I saw in the school in Japan. 

The beauty of both models is that they are general enough to provide a framework that is useful in a variety of contexts but not so general as to lack practical application.

Stay tuned - I feel a "Ferrington" model can't be far away....

Find the Volume of Irregular Objects

This was my last day at Aoki-Chuo school in Kawaguchi. My visit has been brief but has had a huge impact on my thinking - in fact I'm still thinking about my thinking, if that makes sense.

Anyway, today I was privileged to sit in on several fantastic lessons but I really want to share this one with you.

Topic - Finding Volume of Irregular 3D Shapes

Link to Previous Learning:

The class had been looking previously at how to work out the volume of regular cubes and rectangular prisms. The teacher presented this diagram:

There was a brief discussion about this shape along the lines of how it was similar to and different from the previous shapes that the class had worked with. Everyone seemed happy with what they could see.


Problem Solving Strategies:

Then the teacher asked them for some strategies to deal with this shape.

Two students had ideas:

1. You can separate the shape into smaller parts

2. You can add different bits of the shape together

Even though expressed differently, I think both ideas showed that the kids were looking at the composition of the shape and demonstrated their knowledge of how smaller shapes can be combined to make bigger shapes. This is something that they had done previously in Year 4 with 2-dimensional shapes.


Working Independently:

The class quickly got down to work. To save time, the teacher had prepared multiple copies of the diagram, drawn to a scale of 1:1. This was really helpful - no time spent on doing activities that were not related to the topic. Yes - drawing shapes is a great skill to learn - but not in this lesson. The focus here was always going to be on finding the volume.

The teacher circulated and talked with different children. Once they had a solution, they were encouraged to get another copy of the diagram and work it out another way.


Presentation:

The teacher selected several children who had different strategies and asked them to prepare a presentation for the class. This involved getting some A3 paper with a copy of the diagram (already prepared by the teacher) to write up their solution.



Method 1:

A simple cut divides the shape into two manageable pieces. Works well and gets an accurate solution. 

Note the use of colour in the diagram to highlight the relevant calculations. This was done by the teacher after the presentation when she was clarifying with the students what was going on.


Method 2:

In this solution, the student decided to chop off the top part and add it to the end of the remaining rectangular prism. This works neatly because the base of the chopped off part is 4cm x 6cm, same as the end of the bigger rectangular prism. Neat match! 

There was some confusion and almost disbelief - some students wanted clarification on how this worked.

So the teacher pulled out a model of the shape to show them.

I think she was holding back on showing this model because she didn't want to shape the children's thinking. Personally, I think I might have got the model out earlier or at least got the kids to make the model for themselves BUT making models wasn't the point of the lesson - everything points back to the topic: "Finding the volume of irregular 3D shapes."




Method 3:

This method starts by working out the volume if there was no missing part and subtracting the piece that is gone. Notice once again the use of colour (by the teacher) to show which part of the calculation relates to which part of the diagram.


Same and Different:

This part of the conversation is always interesting. 

The two things that children identified were:

1. You had to use multiplication

2. You had to find rectangular prisms


Generalisation:

"You can find these volumes by looking for cubes and rectangular prisms." 

- a very rough translation from the Japanese

So the students now had a strategy to deal with similar problems. The focus wasn't on getting the number answer correct, in fact it was only peripheral to the conversation.

AND

THE WAS NO MENTION ANYWHERE OF A FORMULA.

I think this was pretty significant - that's why I put it in caps

Finding Area By Combining Shapes

I have been visiting Aoki-Chuo school in Kawaguchi, Japan this week as part of the "World Tour of Maths".

It has been an amazing experience. I have been treated like a rock star. The staff and students have been so friendly and helpful. What a great school!

Cut To The Chase

Anyway, one of the lessons I was watching involved Year 6 working out the area of shapes by looking at the shapes they are made up of.

Now, we are not talking about two triangles make a square here.

We are looking at finding an area like this:

Problem Solving the Japanese Way


One of the things I was seeing in Japanese classrooms was the way the students were encouraged, even expected, to find multiple ways to solve a problem.

Here are four ways that one student demonstrated that found the area of the shape:

Method 1

He works out that a quarter of a circle overlapped on another quarter circle makes the required overlapped shape.

So he calculates that two quarter circles have a combined area of 157cm².

If you subtract the area of the square, you will have the overlapping shape remaining.

157cm² - 100cm² = 57cm².

Method 2

In this one, he works out the area of the entire circle, subtracts this from the area of the bigger square and gets the area of the four corner pieces. This he divides by 4 to get the area of one corner piece. Then he multiplies by 2 and subtracts this total from the area of the smaller square to get the required solution.

Method 3

This time he uses a triangle and sees that the difference between the area of the triangle and the area of the quarter circle will give him half of his required shape. 

Method 4

His final method is to subtract a quarter circle from the square to get the outside corner piece. This he doubles and then subtracts from the square to get the internal shape.

What next?

Well, after they had time to work independently on their solutions, the teacher selected a few students to come and present their ideas to the class. The presentation is a very important part of the lesson and the children take it very seriously. The get a few minutes to draw their solution on some A3 paper and then stand up and talk about it. At the end they say something like, "This is what I have found to be true." and the class responds, "I agree." Then they ask for any questions, which they answer. Then they are thanked by the teacher and get a round of applause from the class.

After all the selected children had presented their solutions, the teacher left their diagrams on the board and asked the class to find similarities and differences between the various methods. This was important because the final step was to make a generalisation, a statement that could be used to help solve similar problems in the future. It is like the class summarising their learning for the lesson.

So what?

I was blown away. In the space of 45 minutes the class answered one question.

It wasn't 57 different questions from the textbook. It was one question.

But the depth of learning was very impressive. 

And at the end of the lesson, the purpose was manifestly clear - it was all about thinking.

Quality not quantity.

Pythagoras? When Will I Ever Use That?

You've all heard kids say it before in a lesson.

"When am I ever going to use that in real life?"

And this applies to lots of things where there's a formula or a bit of algebra and some abstract thinking.

Well, yesterday was Sunday and I was looking to find a church somewhere in San Francisco that I wanted to visit. I found it on a map. It is called the City Church on Sutter St - great church by the way if you're looking for somewhere.

Anyway, looking at the map, I had a few choices. I could go straight down Jones Street until I hit Sutter Street and then turn right (the red line).

Or I could take the hypotenuse (the green line) - much shorter!

This wasn't some clever mathematical calculation - it is just common sense. The hypotenuse is going to be shorter.

But how much shorter?

Well, if each of the other sides is about 2.5kms, then....

a² + b² = c²

2.5² + 2.5² = c²

c² = 6.25 + 6.25

c² = 12.5

c = 3.5

So I saved myself about 1.5km but cutting along the hypotenuse. 

Of course, I couldn't exactly go straight along the green line - there were houses and things in the way. My path looked a bit more like this:

Still, pretty sure it saved me some time. I got there 45 minutes before it started - which was good because I hadn't written down the exact address so I needed to walk around a bit to find the actual building. 

Well, who would have thought to look in the Russian Centre building?

POSTSCRIPT:

After posting this story, there was a certain frenzy on Twitter with several astute minds asking about the conclusions I had drawn - thanks @mathinyourfeet @ddmeyer @nathankraft1 @smithnj @tieandjeans @emilyrolsen @mauidoe @fawnpnguyen @ajm2035 and any others I forgot to mention. 

The concerns raised were specifically:

• Was the green route actually any shorter? - because it looks like it has the same vertical and horizontal distances as the red lines
• If I was driving, did I take into account the number of left hand turns I would need to do that might slow me down
• Were there any red lights I had to stop for?
• And how much stopping and starting would you do if you had to make all those turns?
• And what would that do with your mileage? 

Well, I was walking and got to cut a few corners but I take the point. Even though the green path is fractionally shorter when you walk it (by cutting the corners etc), it certainly isn't the same as going in a straight line.

And I certainly didn't save 1.5kms by doing it.

Oh, for the wings of a dove....