The Eclipse of November 14 2012

The exploratorium website said to look out for crescent-shaped shadows appearing through trees. And what do you know...?

For the time and place the photo was taken, the Astronomical Society of Australia calculated 83% of the sun to be covered by the moon. Looks about right!

Physics Doesn't Convict A Murderer After All : The Acquittal of Gordon Wood

This story takes place at The Gap, Sydney at a place called Suicide Point:-

The Gap, 188-. From the State Library of NSW 

Yesterday, after 3 years in gaol, Gordon Wood was acquitted of murder after he was originally found to have pushed his girlfriend, Caroline Byrne, off this popular suicide spot. In the original guilty verdict, the evidence relied on an expert witness called Rod Cross who subsequently wrote the book, "Evidence of Murder: How Physics Convicted a Killer." Previously, Cross had written books about the physics of tennis.

The physics of the Murder/Suicide/Misadventure is best found at mathspigs. In short, the cliff is 29m high and the body was found on the rock ledge below, 11.8m out from the cliff. This means the woman's velocity when she left the cliff was 4.85m/s. Since she couldn't have run off the cliff (there is a fence only 1.5m away from the edge), the woman was deemed to have been thrown off the edge and her partner went to gaol. Now this ruling has been overturned!

In the appeal, the judge ruled that to throw someone so far away from the edge of the cliff would endanger the life of the murderer. They figured this out by having some police officers throw others into a pool. Also, Gordon Wood was chaffeur for a famous rich man called Rene Rivkin. At the time of Byrne's death, Wood and Rivkin were being asked by the Australian Securities and Investment Commission (ASIC) about a suspicious fire and true ownership of shares in the company affected by the fire. In the murder appeal, the judge ruled the original jury should not have known about ASIC questioning Wood and Rivkin.

So, if she didn't run off the cliff, wasn't pushed, what now...?

Here is the trajectory of a body that mysteriously alighted from a cliff 29m high and landed 11.8m out from the base. Take the photo and use it with your class. CSI comes to maths!

Trajectory of a body that mysteriously alighted from a cliff 29m high and landed 11.8m out from the base.

 The equation of the parabola simplifies to:

y = - 5x²/24

If Byrne has taken her own life, she was running at 4.85m/s when she left the cliff. Could have Byrne reached the necessary velocity in 1.5m? This would require an acceleration of 7.84m/s².

Let's try it. That means you need to run 1.5m in 0.62s. Can data loggers help us accurately measure our rate of acceleration?

Nothing, Nada, Zip

I have found this the best way to teach The Null Factor Law:-

(Morbidly Seriously)

Come in quietly, sit down and open your books for a Multiplication Quiz. Anyone who gets 10 out of 10 gets a sticker. If starts off easy.

Question 1, 1 times 2.

(A few chuckles as well as suspicious looks.)

Question 2, 1 times 0.

(Some serious thinking, but confidently written answers.)

Question 3, 8 times (slight pause) 0.

Question 4, 300 times 0.

Question 5, 4 million, 600 thousand, 4 thousand, 7 hundred and ninety-nine times 0.

Someone answers, "Could you say that again?"

(We all have a giggle and I attempt to remember the number, amongst some disagreement)

(Serious Again) Question 6, X times 0.

Question 7, X + 3, in brackets, times 0. Oh, I better write that one on the board.

Question 8, (X+3)(X-2)(X-1) times 0. I'll write that one on the board too.

Question 9 ... (and so on)

This is the cool part! We revised a trinomial factorisation and had a 5 minute discussion about the resulting binomial expression when equalled to 0. Being equal to 0 meant the factorised expression was easy to solve. Then they tried some and BINGO! The success rate in the class was higher than ever before.

It seemed a very successful approach, highlighted by one person commenting, "Am I doing this right? This seems too easy."

Star Doodles and Factors

Today, we are going to investigate numbers with some doodles. Everyone look at the first side of the worksheet.

What do you think p stands for?

"Points," someone answers.

Not just points, but NUMBER of points. Let's do the first one together.

Choose any point. Then, going clockwise, jump 2 points from your first point. Connect this new point to the last point with a straight line. Go jump another 2 points and join this point to your last point. Keep going until you reach a point already joined up.

What do you get?

"A star."

There is a special name for this star. Does anyone know what this type of star is called?

"A pentagram," says someone.

Excellent. The secret society of Pythagoreans used this star as their secret symbol.

j stand for the NUMBER of points you jump. Write j = 2 under your pentagram. Now try the next group where p = 5 and j = 3.

What do you notice?

Now try the group where p = 6 and j = 2. Stop when you get to a point already connected.

At this point, many students were excited by the stars and completed 2 equilateral triangles to form the Star of David. That is, 2 lots of 3.

What happened?

"We got 2 lots of 3, which makes 6."

Go ahead now and experiment drawing different doodles with different values for p and j.

This is fun. Everyone thinks the stars are cool and many people decide which is their favourite.


Let's look at the group where p = 9. Can anyone predict what value of j we should use so that we connect every point before we get back to the point where we started?

"I think that will happen if we use a value for j that is a non-factor of the p number."

Non-factor; is that a real word? Sounds good to me.

A good conjecture; everyone try it.

I have some balls of wool. Let's gather into circles, pass the wool around and make some of our favourite stars.


This activity led to great discussions and fabulous fun. Terminology included factors (and non-factors), multiples, divisibility and prime numbers.
But the best thing about the lesson was everyone was able to access the work. At the other end of the spectrum, I thought I had understood everything about the lesson, until someone says...

"But what about when p = 10 and j = 4? 4 isn't a factor of 10."

P.S. Thanks to vihart for the inspiration.

Your Mind's Eye - Spatial Abstraction

Everyone stand up.

On the count of 3, point to the location I am about to name. Don't be influenced by anyone, they could be wrong.

The canteen   1, 2, 3 ...
Everyone got this right, as it was visible from the room!

The bus stop   1, 2, 3 ...
Most got this right. It wasn't visible but it wasn't far away.

The local shopping mall   1, 2, 3 ...
A few more arms are pointing the wrong way, but consensus wins over.

Los Angeles   1, 2, 3 ...
Now, being in Australia, this location was a source of great debate.

After looking at Mercator and Peters projections of the planet, everyone agreed on a direction.

Now someone says, "We could go the other way around the globe."
Easy, point in the opposite direction to the last answer.

But then someone says, "We could dig through the earth."
I think, in the end, we actually did point towards L.A.

Giving Students Autonomy

Humans are motivated by Autonomy, a sense of having meaningful input and making a difference. So my Junior Maths Class' first task is to design, in groups of 3, the seating for their learning space. I like Dan Meyer's approach to patient problem solving so I will give the class as little information as possible and start with, 'What do we need to know?'

Of course, I have ideas I want the students to learn and use, so will try to subtly direct the task, notwithstanding past experiences where I have changed my approach based on ideas from the students. Anyway, here are some things I hope to achieve:

1. Measure accurately the room, desks and chairs. We are lucky to have two types of desks in the room.
2. Complete scale drawings of the floor plan and the desks, using grid paper or Google SketchUp.
3. List human considerations for interior design, such as group work, a break-out area, working space requirements and universal accessibility, such as wheelchair access.
4. Manipulate cut-outs to look at alternate arrangements.
5. Justify, in a 1 minute video, why the final design should be used by the whole class.
6. Evaluate all designs to reach a consensus on how our Maths Class will be organised.

Watch this space for progress reports on the task:

#1
"Is this for real?" asks one incredulous student. Says volumes about the curriculum!
Another barrier, of course, is the other people who share the room. Why would you give students a say over the arrangement of the room? On the other hand, we are interrupting other people's routines. (Is that a bad thing?)

Other than that, we are all having fun, figuring out "What we need to know" and "What skills we must use"