Mumnet

Circle theorems

Jan — Wed, 11 May 2011 08:24:32 +0000

There are seven theorems for you to discover and prove.

Go to this page to discover seven hypothesis.
Go to this page to see the seven theorems summarised.
Come up with seven proofs and put them, with your name, as comments below.

Shear transformations

Jan — Tue, 10 May 2011 21:43:26 +0000

There are two kinds of shears.

1. Horisontal shear.

All points are moved horisontally, i.e. the y-value does not change. How much a point is moved horisontally is proportional to its distance from the x-axis.

In short: (x, y) -> (x + ky, y).

k is called the shear factor. The x-axis is called the invariant line. This means that any point on the x-axis does not move: (x, 0) -> (x + k*0, 0) = (x, 0).

2. Vertical shear

All points are moved vertically, i.e. the x-value does not change. How much a point is moved vertically is proportional to its distance from the y-axis.

In short: (x, y) -> (x, y + kx).

k is called the shear factor. The y-axis is called the invariant line. This means that any point on the y-axis does not move: (0, y) -> (0, y + k*0) = (0, y).

Question:
What kind of shear maps triangle X to triangle S? What is the invariant line? What is the shear factor?

Eating your cake and having it – Part 2

Jan — Mon, 09 May 2011 12:32:11 +0000

Investigating:

By choosing a suitable unit you should be able to find how big each piece is.

——

Compare what you found with the image below.

Can you solve the mystery now?

Eating your cake and having it – Part 1

Jan — Mon, 09 May 2011 10:17:04 +0000

The problem:

James Tanton has made a video where he eats a piece of a cake. The white square below.

Carefully rearranging the pieces that are left he manages somehow to make the cake whole again!

Voila!

Please explain how this is possible.

Khan x 2

Jan — Mon, 09 May 2011 09:38:02 +0000

Khan interviewed by Charlie Rose last week.

On the usefulness of Khan’s videos.

The Englishman Who Went Up a Hill But Came Down a Mountain.

Jan — Sat, 30 Apr 2011 13:01:45 +0000

“The Englishman Who Went Up a Hill But Came Down a Mountain” is a 1995 film … based on a story heard by Christopher Monger from his grandfather about the real village of Taff’s Well (Ffynnon Taf in Welsh), Rhondda Cynon Taff, Wales and its neighbouring Garth Hill.

The movie is set in 1917 (with World War I in the background), and revolves around two English cartographers, the pompous Garrad and his junior, Anson. They arrive at the fictional Welsh village of Ffynnon Garw (“Rough Fountain” or “Rough Spring” in Welsh) to measure its “mountain” – only to cause outrage when they conclude that it is only a hill because it is slightly short of the required 1000 feet. The villagers, aided and abetted by the wily Morgan the Goat and the Reverend Jones (who after initially opposing the scheme, grasps its symbolism in restoring the community’s war-damaged self-esteem), conspire with Morgan to delay the cartographers’ departure while they build an earth cairn on top of the hill to make it high enough to be considered a mountain.” (Wikipedia)

It can be seen on YouTube.

After showing parts of the film, ask the students for math related questions it arises.

If they don’t suggest any questions wait for five minutes in silence. If nothing appears after five minutes wait another five. And so on.

Sooner ot later the students will realise that their task is not to guess the questions you find interesting, but to think about what they find interesting.

Here are the questions I find interesting:

When does a hill become a mountain?
How may the height of a mountain be measured?
What is the volume of a cone of soil h feet tall?
How many buckets are needed to create the cone just mentioned?
How much time is needed to make the cone?
The hill is measured using the previously measured heights of two other hills. One of the men in the village asks, ‘how were the heights of these hills measured?’ and gets the answer ‘using two previously measured hills’ to which he replies ‘how was the first hill measured?’. This is a good question in itself and is analog to the interplay between axioms and theorems.

Here are some ideas for answering these questions I would think about before presenting the task.

relevant factors: objective: height, slope, subjective: local use, elevations nearby
pedometer (mentioned in the film), clinometer, barometers, triangulation
“the surveyors used a clinometer and stepped-off distances, from which the elevation was calculated. They acknowledged this method was lacking in accuracy. In the second measurement they used barometers. This was less than successful because a low-pressure front flowed through the area, causing the base barometer to change while the observing barometer was carried to the top. The third measurement was made by triangulation with two nearby peaks for which the elevation was known. I wonder how accurate that was.” (source)
what is the slope of the cone? what if sand was used instead? when you know the slop how do you find the volume, how good is a cone as a model for the real mound?
what is the volume of a bucket? lower and upper estimate? soil is not water, how much soil will go in a bucket with given volume?5. how many buckets can be poured on the top of the mound in an hour? how can it be done? how many people are needed? what is the distance to the place where the buckets are filled?

There are enough questions here for several weeks of work. I suggest one should vote on the questions the students propose and do them in order of popularity. To promote discussions the most popular questions should be done by all. Later the students should be allowed to investigate and report back to the class what they found, how they found it, what further question arose.

Mathbits has made a worksheet for the film, but I would rather ask the students to find questions.

Curtains up for an exciting math topic

Jan — Fri, 29 Apr 2011 12:44:53 +0000

“[what is needed for a good mathematical task is a] clear premise in its first act, obstacles, conflict, and tension for your classroom heroes to resolve in its second act, and a cathartic resolution in its third act that leads naturally and necessarily to more mathematics in its sequel.” (more)

The quote is taken from Dan Meyer’s blog. The words in brackets I added.

What does “pattern” mean?

Jan — Wed, 27 Apr 2011 07:22:49 +0000

Whenever I use a word Jan Thomas (5 years) doesn’t understand, he will ask. Yesterday, he asked “What does ‘pattern’ mean?”.

My answer, which I don’t think was very successful, said something about not being random, having a rule, a system, pointing to the stone pattern in the pavement we were walking on.

Today, in New York Times, I found Leanne Shapton’s “Wednesday’s Patterns” I will show him today. The pattern in the image above is from ‘a towel left behind by house guests.’

Free videos for AS and A2 math

Jan — Sun, 10 Apr 2011 05:41:27 +0000

http://www.khanacademy.org/ and http://patrickjmt.com/ have many videos relevant to AS and A2 math. If you feel a topic is missing in the lists below, post a comment or send me an email (jannordgreen@gmail.com). Likewise if you feel one of the videos should be replaced by a better one. Before you do, however, please have a look at the free videos for IGCSE listed here.

Note 1: You can sign in to the Khan Academy to test your skills and see recommended videos. If you want me to see how you are doing add me as one of your coaches by entering my email address (jannordgreen@gmail.com).

Note 2: If you find a video you want included, please email me.

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Mathematical videos not directly related to the syllabus

Paper 1 – Pure math AS

Paper 3 – Pure math A2

Polynomials
Dividing polynomials
The modulus function (absolute value function)
Absolute Value Equations 1
Absolute Value Equation Example 2
Exponential and logarithmic functions
Differentiating trigonometric functions
Trigonometry
Differentiating trigonometric functions
Differentiating products
Solving equations numerically
The trapezium rule
Parametric equations
Curves defined implicitly
Vectors: kines in two and three dimensions
Vectors: planes in three dimensions
The binomial expansion
Rational functions
Complex numbers
Complex numbers
Complex numbers in polar form
Integration
Differential equations

Paper 4 – Mechanics 1

Paper 6 – Statistics 1

Free videos for IGCSE math

Jan — Thu, 07 Apr 2011 20:37:00 +0000

http://www.khanacademy.org/ and http://patrickjmt.com/ have many videos relevant to IGCSE math. If you feel a topic is missing in the list below, post a comment or send me an email (jannordgreen@gmail.com). Likewise if you feel one of the videos should be replaced by a better one.

Note: The list below is a modest start. It will be added to shortly.

Note 2: You can sign in to the Khan Academy to test your skills and see recommended videos. If you want me to see how you are doing add me as one of your coaches by entering my email address (jannordgreen@gmail.com).

Note 3: If you find a good, free video you want included, please email me.