That's Perplexing

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BOMDAS 2 PEMDAS

Recently, I have been moving the students from BOMDAS to BIDMAS, so indices are included in order of operations. After this TED-Ed animation though, I am using PEMDAS.

First, show the video.

Then, discuss, asking what is the video about? Will students say Order of Operations? Will anyone know PEMDAS? Will anyone recall BOMDAS, or BIDMAS?

Students watch the video again, this time writing down expressions that are shown. Do they really equal 0?

Students make up five of their own expressions that equal 0, the more complicated, the better.
Use calculators to check expressions equal 0.

As extension, spot the mistake in the video. At 2:33, the expression closes first a multiplication which is to the right of a division, even though this order of operations has the same result as the technically correct order. Can anyone create an expression where you must complete a division before a multiplication?

Isn't the dragon cool? Why not draw a different mythical creature composed of digits?

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"

Pythagoras' Theorem - From Concrete to Abstract

I have found this the best way to teach Pythagoras' Theorem. The focus of the lesson is actually on showing the students how to go from the concrete to the abstract.
The lesson takes about 100 minutes, a double where I am teaching.
Students have commented on what fun they had during the double.

First, engage the students with Donald Duck in Mathemagic Land.
The first 5 minutes or so talks about the secret sect of Pythagoreans and their contribution to the understanding of music and the golden section.

Don't forget to discuss with your students that the Pythagoreans allowed women in their sect, while women were still not classed as citizens in ancient Greek society. The Pythagoreans were ahead of their times!


Next, introduce the theorem.
It has lasted over 2500 years, although there is evidence that the Babylonian society understood the relationship between the squares of the sides of a right triangle way before the Greeks.
People think that knowing Pythagoras' Theorem makes you smart. Check out the brainless scarecrow in the Wizard of Oz who, when granted a brain, recites a version of Pythagoras' theorem using an isosceles triangle, instead of a right triangle!

Now,

The square on the hypotenuse = the sum of the squares on the other two sides.

Then have your students explore a proof using a great interactive applet available at

http://www.ies.co.jp/math/java/geo/pythasvn/pythasvn.html

Then, give the students a worksheet showing a right triangle and the squares on the three sides. Once they have found a proof using the applet, they cut out the two smaller squares and glue these inside the large square.

Here is the worksheet.

Time to go to the abstract.

Have the students draw a 3cm, 4cm, 5cm triangle as accurately as they can, then draw a square on each side.

Then ask them to find the area of each of these squares.

Now, the abstract connection. Interpret Pythagoras' Theorem and replace the words with the areas of each square. At this point in the lesson, you will here gasps of understanding - the Ah Ha! moments.

Conclusion

I conclude by giving some homework that involves finding the unknown hypotenuse. The worksheet I use is from a set of books called, "Math-O-Magic". The worksheets have corny jokes but the students like doing them. The books are available at http://members.iinet.net.au/~markobri/mom.html

I am sure you will enjoy this lesson. Have fun!