Numerous teachers of mathematics will tell you that x/x = 1, (x+1)/(x+1) = 1, (x-7)/(x-7) = 1, etc.
This is not quite correct. If x = 0, then x/x does not equal one. There is no inverse for zero and hence 0/0 has no answer.
In the second example above if x = -1 we again have 0/0. In the third example if
x = +7 we again have 0/0.
Hence, (x +9)/(x+9) = 1, but x cannot equal -9 is the complete answer.
Suppose you graph y = (sine x)/x. When x = 0 we have y= (sine x)/x = 0/0. Hence at x = 0, y = 0 on the number plane we put in a discontinuous point.
Monday, 3 February 2014
Sunday, 2 February 2014
RECYCLED WATER
Some communities in Australia refuse to drink recycled water. Some people say that they will never drink recycled water.
About 2000 years ago a Roman soldier dies in a secluded part of the North African desert. His body slowly decomposes and his water molecules evaporate into the atmosphere. From there his molecules become part of the world's water cycle. After 2000 years his molecules could be in the Indian Ocean, the Pacific Ocean, your local lake, your local water supply, etc.
One day you decide to drink a litre of water from your kitchen tap (this tap could be "anywhere" in the world). What is the most probable minimum number of water molecules that you will drink which were once in this soldiers body?
Most of my students usual guess between zero and one hundred. Onto the whiteboard I put up 0, 10,100,1000,...................1,000,000,000,000. I put the numbers up in powers of ten with the last number being ten to the power of 12. My students are asked to guess the closest number to the estimated answer.
The most probable minimum number is about 20,000,000 (20M). Hence, ten to the power of seven is the closest number which they can guess.
We ALL drink recycled water. The next time you drink a can of coke (which is mainly water) try estimating the minimum number of molecules which came from that Roman soldiers body!
About 2000 years ago a Roman soldier dies in a secluded part of the North African desert. His body slowly decomposes and his water molecules evaporate into the atmosphere. From there his molecules become part of the world's water cycle. After 2000 years his molecules could be in the Indian Ocean, the Pacific Ocean, your local lake, your local water supply, etc.
One day you decide to drink a litre of water from your kitchen tap (this tap could be "anywhere" in the world). What is the most probable minimum number of water molecules that you will drink which were once in this soldiers body?
Most of my students usual guess between zero and one hundred. Onto the whiteboard I put up 0, 10,100,1000,...................1,000,000,000,000. I put the numbers up in powers of ten with the last number being ten to the power of 12. My students are asked to guess the closest number to the estimated answer.
The most probable minimum number is about 20,000,000 (20M). Hence, ten to the power of seven is the closest number which they can guess.
We ALL drink recycled water. The next time you drink a can of coke (which is mainly water) try estimating the minimum number of molecules which came from that Roman soldiers body!
Sunday, 1 December 2013
Christopher Pyne
It does appear that Christopher Pyne might have finally woken up as to why the "Gonski Program" is woefully inadequate. The program does next to nothing to address the major problems which exist within our school educational systems. Namely:-
1. The chronic shortage of properly qualified and highly motivated teachers with excellent
classroom management credentials.
2. The ever growing problem of violence especially within the classroom.
3. The continuous watering down of curriculum to meet the needs of struggling teachers.
4. The lack of leadership skills exhibited by many teachers in management positions.
1. The chronic shortage of properly qualified and highly motivated teachers with excellent
classroom management credentials.
2. The ever growing problem of violence especially within the classroom.
3. The continuous watering down of curriculum to meet the needs of struggling teachers.
4. The lack of leadership skills exhibited by many teachers in management positions.
Tuesday, 12 November 2013
Diagnostic Testing
Educators suggest that diagnostic testing in secondary schools is very important. I agree. However, this is difficult if not impossible for many teachers of science and mathematics to undertake. Suppose a science teacher asks his/her class what is light. Assume the student replies by suggesting that light is a steam of particles. How can that teacher continue unless he/she is familiar with the two major models for light? Suppose a science teacher asks his/her class what is heat. Assume a student replies by suggesting that heat is related to how fast a particle moves. How can that teacher continue unless he/she is familiar with the two major models for heat? I suggest that numerous teachers of science and mathematics are reluctant to ask such open ended questions because their knowledge of the subject matter in these two disciplines is lacking.
Westpac
Most people are educated to believe that banking interviews, to some extent, are confidential. I had an interview with an employee of a branch of a Westpac bank, in Wagga Wagga, on 11/11/2013. Later I came to the conclusion that information at that meeting may have been leaked. I attended another meeting, at the same bank, the next day in order to clarify the bank's policy on confidential interviews. I say that the senior Westpac employee was very evasive and did not wish to discuss the serious matter which I had raised with him. It seems to me that Westpac does not have a policy related to confidentiality.
Tuesday, 17 September 2013
Telling Stories (1)
There are managers in our schools who direct teachers to stick strictly to the syllabus. How boring! How unrealistic their ideas are about the meaning of education.
Telling stories was / is an ongoing part of my teaching practice.
Many years ago I was in the school library when a fourteen year old girl (let us call her Mary) approached me. It was obvious that she was distressed and had been crying. She asked if I was
Mr Todd. I said yes and asked her what the problem was. Mary gave me a brief account of her life of misery which was being caused by a boy in her year (let us call him George). I suggested to her that she approach her year adviser. Mary quickly indicated that such an approach would prove useless. I asked her if she would like me to try and find a solution to her problem. Her eyes lit up and I then realized why she had sought my help.
Our meeting took place in the morning. A short while later I pulled George out of class and spoke with him.
Later, in the afternoon, I was doing after school duty just inside the school boundary fence. Mary rode past on her bicycle. As she past me Mary looked back and cried out at the top of her voice; "I love you Mr Todd, I love you."
I never spoke with Mary again. George never bothered her again. When I tell this story to my students they invariably ask me what it was that I said to George. I tell them that I made him an offer he couldn't refuse.
Telling stories was / is an ongoing part of my teaching practice.
Many years ago I was in the school library when a fourteen year old girl (let us call her Mary) approached me. It was obvious that she was distressed and had been crying. She asked if I was
Mr Todd. I said yes and asked her what the problem was. Mary gave me a brief account of her life of misery which was being caused by a boy in her year (let us call him George). I suggested to her that she approach her year adviser. Mary quickly indicated that such an approach would prove useless. I asked her if she would like me to try and find a solution to her problem. Her eyes lit up and I then realized why she had sought my help.
Our meeting took place in the morning. A short while later I pulled George out of class and spoke with him.
Later, in the afternoon, I was doing after school duty just inside the school boundary fence. Mary rode past on her bicycle. As she past me Mary looked back and cried out at the top of her voice; "I love you Mr Todd, I love you."
I never spoke with Mary again. George never bothered her again. When I tell this story to my students they invariably ask me what it was that I said to George. I tell them that I made him an offer he couldn't refuse.
Thursday, 29 August 2013
I.Q.
In Australia it is quite reasonable to assume that the average intelligence quotient (AIQ) of school students has not changed over time. In my opinion it is more than reasonable to assume that the AIQ of their teachers has dropped significantly.
Australia is now in the unenviable position of having significant numbers of our brighter students being taught mathematics and science by teachers who struggle to understand the subject matter they are required to teach.
I suggest that the various State Authorities have been aware of this situation for a number of years and have responded accordingly. To begin with they have dumbed down the subject matter. In doing so early introduction of difficult concepts is no longer viable in many of our schools. Errors in prescribed texts are not readily picked up and these errors are often reinforced when teachers attend staff development days. The abolition of external examinations in junior high school has helped to conceal the lack of quality subject matter being taught.
It is well known that students in years 11 and 12 are spurning mathematics, physics and chemistry in both public and private schools. The brighter students know the capabilities of the teachers at their schools and respond accordingly!
Australia is now in the unenviable position of having significant numbers of our brighter students being taught mathematics and science by teachers who struggle to understand the subject matter they are required to teach.
I suggest that the various State Authorities have been aware of this situation for a number of years and have responded accordingly. To begin with they have dumbed down the subject matter. In doing so early introduction of difficult concepts is no longer viable in many of our schools. Errors in prescribed texts are not readily picked up and these errors are often reinforced when teachers attend staff development days. The abolition of external examinations in junior high school has helped to conceal the lack of quality subject matter being taught.
It is well known that students in years 11 and 12 are spurning mathematics, physics and chemistry in both public and private schools. The brighter students know the capabilities of the teachers at their schools and respond accordingly!
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