HSC Mathematics Reference Sheet

As term 2 starts in NSW it is worthwhile to take a breath and consider the impact of the new Reference Sheet that will be issued in HSC Advanced and Extension Mathematics examinations this year.  Will it help students in examination conditions or hinder them? This is a major change to mathematics education in NSW. Previously, students were not given formulae. Now formulae will be provided for topics in the Advanced and Extension 1 courses (formulae unique to Extension 2 topics are not provided). Many experienced teachers are advising their students to remember formulae (as in the past) and not fall into the trap of thinking that the sheet will cure poor mathematical understanding of concepts. Teachers have also commented that students are spending too much time flicking through pages looking for formulae when a simple application of important results, such as using the trig ratios of 30, 45 and 60 degrees, is required. As well, not all formulae are given, an example being the formula for the tan of the angle between two straight lines. It is important to reinforce that examination questions require the use of mathematical techniques and systematic working. Areas such as money payment problems in the Advanced course, which are usually poorly attempted, are not provided with formulae. Treat the Reference Sheet as an aid for the memory, not as a source of mathematical understanding. Once more, clearly show all working and be careful with basic algebra. This is the key to mathematical success.

Pi......From a Student's Perspective

Examiners in the HSC Advanced Mathematics (2 unit) course have commented that many students become confused when the constant 𝜋 is used in calculations. What is it about 𝜋 that causes problems? Students from an early age learn that 𝜋 is the ratio of the circumference of a circle to its diameter and so this reinforces the geometrical interpretation of 𝜋. However, 𝜋 can be involved in problems that have no connection with circles and this is when confusion in examination answers can arise. Some students convert an answer involving 𝜋 into revolutions or degrees just because of the  connection of 𝜋 to the geometry of a circle. Pi is a number, an irrational number, and should be treated like any other number in calculations. In the 2015 Mathematics examination question 15 c involved calculating the time when the volume of water in a pool first started to decrease given the rate of change of the volume. The answer was 2𝜋 seconds and many candidates lost marks by converting this into degrees.

“The fact that 𝜋, a purely geometric ratio, could be evolved out of so many arithmetic relationships- out of infinite series, with apparently little or no relation to geometry-was a never ending source of wonder and a never ending stimulus to mathematical activity”

— Mathematics and the Imagination by Edward Kasner and James Newman, G Bell and Sons 1951, page 78

Energy and Entropy

The word energy is widely used in everyday life. "I do not have the energy to exercise today" or "the world has an energy crisis" are familiar sayings. We use the word energy so much that it is taken for granted that we know what it means (?). The first law of thermodynamics tells us that energy can neither be created nor destroyed. When hot and cold water are mixed together the heat energy lost by the hot water equals the heat energy gained by the cold water. Another word in the vocabulary of physicists and engineers is entropy. Entropy is an indicator of the disorder of a system and the second law of thermodynamics states that natural processes tend to move toward a state of greater disorder (entropy) as time passes. The entropy of a system can never decrease. Students sometimes think that entropy is mysterious since it can be created, whereas energy cannot. When hot and cold water are mixed together the entropy of the hot water decreases and the entropy of the cold water increases by a greater amount. The total entropy of the mixture increases during the process. The mixing process created the entropy. The mixture does not naturally separate back into hot and cold components as this would involve a decrease in total entropy. Entropy is nature's arrow, always increasing, pointing in the direction in which systems evolve in time.

“If, when we observed a highly irreversible process (such as cooling coffee by placing an ice cube in it), we said, “That surely increases the entropy,” we would soon be as familiar with the word entropy as we are with the word energy.”

— Fundamentals of Classical Thermodynamics by Richard E Sonntag and Gordon J Van Wylen, 3rd edition page 232

What is Physics?

It is surprising that many students who are studying physics have never really thought of what physics actually is. A common answer given is that "physics is the study of the universe". On a smaller scale the common branches of physics are mechanics, thermodynamics, electricity and magnetism, relativity, quantum physics and nuclear physics. What do these areas have in common? Each of these areas studies something happening in nature, such as an apple falling to the ground, two magnets pushing on each other or the sun emitting light. To describe how nature behaves we use the language of nature, and this is mathematics. Combining all of these threads together we can define physics as the mathematical study of phenomena occurring in nature. The great physicist Paul Dirac commented on the association between mathematics and the natural world

“The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by nature, but as time goes on it becomes increasingly evident that the rules that the mathematician finds interesting are the same as those which nature has chosen”

— Ian Stewart, Seekers after Truth and Beauty, Basic Books 2008, page 279

What is Mathematics?

Many students enjoy doing maths due to the satisfaction of agreeing with the answer when they turn to the back of the book. A sense of achievement comes from using thought processes to go through a series of steps to determine the correct result. Mathematics is a work-out for the mind just like a visit to a gymnasium is a physical work-out for the body. Harvard physicist Lisa Randall was drawn to mathematics "because it had definite answers....I liked the certainty of knowing which answers were right". Mathematics develops mental agility, logical thinking and systematic problem solving techniques. But what actually is mathematics? Algebra, calculus and geometry are branches of mathematics which operate using a common thread. Mathematics is the manipulation of patterns according to a defined set of rules or ideas. Three classic books that discuss the nature of mathematics  are What is Mathematics by Richard Courant and Herbert Robbins, Mathematics and the Imagination by Edward Kasner and John Newman and A Mathematician's Apology by G H Hardy. The Cambridge mathematician Hardy on page 84 gives the following insights into mathematics

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

A New School Year

As a new school year opens it is timely to start a conversation on the fundamentals for successful learning in physics and mathematics. Many students say "I do not understand physics" or "I cannot do maths". What can be done to remedy this? The solution is to go back to basics and pursue steady step by step practice in each subject with regular testing. If a test reveals that a physics concept is not understood or a mathematical technique is not correctly applied then we zoom in on this area and practice it again until it is mastered. Learning involves dialogue between teacher and student with encouragement to aspire to new levels. This is the heart of education. The tutorial method had its origins with Socrates in ancient Greece and is followed today at the best universities in the world. The journey to academic success comes when students embark on a regular program of study and revision and the best time to start the journey is now.

A Boy Playing on the Sea-Shore......

To Physicists and Mathematicians this title invokes images of one person. Isaac Newton. In a farm house in Colsterworth near Grantham in Lincolnshire the principles of natural philosophy were born. Newton's central problem was this: Given that the path of a planet about the Sun is an ellipse and that the radius vector from the Sun to the planet sweeps out equal areas in equal times, what must the law of attraction be? 

I do not know what I may appear to the world, but to myself I have only been like a boy playing on the sea shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undiscovered before me

To Michael Faraday and James Clerk Maxwell

There is no better way to finish the first year of conversations on relativity and electromagnetic topics than to talk of the contribution of Michael Faraday and James Clerk Maxwell to the study of electricity and magnetism. The Royal Institution of Great Britain in Albemarle Street in London has an Electrical and Magnetic Museum devoted to Michael Faraday. Faraday performed his investigations at the Royal Institution and his original equipment is on display. Faraday's lines of electric and magnetic force permeate the thinking of physicists and electrical engineers. While at nearby King's College James Clerk Maxwell used Faraday's lines of force in his Dynamical Theory of the Electromagnetic Field, published 150 years ago. Maxwell's four equations, and their solutions, give us a complete description of the electromagnetic field. As an example, imagine that in a rectangular region of space an electric field E exists whose x-component is constant and whose y-component is proportional to the x coordinate. Many physics and engineering students do not realise that a perpendicular time varying magnetic field B must accompany this E field. Why?

As the Earth Turns

The Science Museum in London has a marvellous long swinging pendulum that appears to change its direction during the day. Some books say that the pendulum keeps swinging in the same plane in space. Is this true in London? How do we explain the shift in the position of the pendulum during the day?

“At the north pole the direction of the swing of the pendulum does not change in space, so the pendulum shows the Earth turning beneath it in 24 hours. Here in London it is not so simple; the direction of the pendulums swing moves around the circle in about 30.5 hours. As we travel towards the Equator the direction of swing moves more slowly, until at the Equator no movement is seen. In the southern hemisphere the direction of swing moves in the opposite direction.”

— Description of Foucault's pendulum at the Science Museum, South Kensington, London

Acceleration

Physics and Mathematics students use the equation

x = ut + 1/2at²

to calculate x the distance travelled by an object moving with acceleration a during a time interval t when its initial velocity was u.  The equation tells us that the distance travelled when an object accelerates is equal to the distance travelled when there is no acceleration plus the distance it would travel if it accelerated from rest. The relationship between distance and time for accelerating objects was first deduced by Galileo using the assumption that the passage of time is the same in all reference frames. Special relativity teaches us that at very high speeds compared time intervals are different in the moving and stationary reference frames and so Galileo's equation cannot be used to calculate the distance travelled. Let us suppose that the acceleration continues for a very long period of time so that relativistic speeds are reached. A common misconception is that special relativity cannot be used to solve problems involving accelerating objects. It can, provided we refer the acceleration to the instantaneous rest frame. Imagine that an object accelerates from rest uniformly at 9.8 m/s/s. What distance does this object travel in the laboratory reference frame in 10 years of laboratory time?

9.1 light-years

Mass of Light

One concept that confuses Physics students is whether light has mass and momentum. In everyday life we observe the effects of the energy carried by light. A famous piece of Physics demonstration equipment is the Crookes' radiometer. The vane of the radiometer spins when the radiometer is exposed to sunlight. Does this demonstration show that light has mass and or momentum?

Some Mathematical Meandering III

In our last mathematical digression we studied the Snowplow Problem of Ralph Palmer Agnew, the solution of which using the logarithmic function appears on page 39 of his book Differential Equations (McGraw-Hill 1960).  Let us make the problem even more interesting following a suggestion by Murray S Klamkin that is given in Bender and Orszag's Advanced Mathematical Methods for Scientists and Engineers. Here is the Great Snowplow Chase.

One day it started snowing at a heavy and steady rate. Three identical snowplows started out at noon, 1 pm and 2 pm from the same place and all collided at the same time. What time did it start snowing?

11:30 AM

Mass

We often find in the relativity chapters of Physics and Mathematics books the statement that "the mass of an object increases as its speed approaches the speed of light". Is this correct, does the mass of an object actually increase, or is this a means of describing what is happening to a very rapidly moving object in Einstein's four dimensional space-time in terms of familiar quantities using Newton's laws of motion?

Changing E and B

In this conversation we will attempt to go into the depths of students' understanding of electromagnetism and try to unlock, or at least loosen, a common mindset of Science and Engineering students. We have already had a conversation on electric, E, and magnetic, B, fields. Taking this to the next stage, a common sentence found in many Physics and Mathematics textbooks is  "a changing electric field creates a changing magnetic field and vice versa". Is this correct? Does the changing electric field itself actually create a changing magnetic field (and vice versa) ?

E and B

Physics students learn about the electric field vector E and the magnetic field vector B. In most courses these vectors are taught separately with E introduced first as being produced by a stationary charge and B introduced later as being produced by a moving charge. A common mindset is to consider these fields to be different, just because of the order in which the courses are taught, but fundamentally they are not. They are both aspects of the electromagnetic field and become active in particular situations. In SI units E and B have different measurement units and this tends to cloud their difference. In CGS units E and B have equivalent units and so this system is more suited to electromagnetic calculations as the numerical values of the fields can be directly compared. Many physics students do not realise that electromagnetism is a direct consequence of Einstein's special theory of relativity, as Einstein's 1905 paper on relativity was entitled 'On the Electrodynamics of Moving Bodies'. There is no better way of expressing the relationship of electromagnetism to relativity than the following  marvellous quote given by Leigh Page, then professor of mathematical physics at Yale, in an address to a meeting of the American Institute of Electrical Engineers in 1941.

“The rotating armatures of every generator and every motor in this age of electricity are steadily proclaiming the truth of the relativity theory to all who have ears to hear”

Some Mathematical Meandering II

Here is a beautiful problem. This problem starts with a minimum of assumptions and soars to great heights in its simplicity of solution using high school calculus. It is called a "marvellous problem" by Carl M Bender and Steven A Orszag on page 33 of their legendary book Advanced Mathematical Methods for Scientists and Engineers. Now be warned. Once you are captured by this problem its beauty could lead you on mathematical travels to distant places. Here is the Snowplow Problem.

One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing?

11 :23 AM

A New Twist on an Old Problem I

Physics and Mathematics Extension students are taught that, provided resistance forces are neglected, an object released from rest undergoes the same vertical displacement in the same time interval as an object projected horizontally. Now let resistance forces be included. Would it ever be possible for the horizontally projected object to have a greater vertical displacement in the same time interval than the object released from rest?

Einstein's Happiest Thought

A common phrase often seen in textbooks is "Einstein's greatest blunder". Many Physics and Mathematics students are not aware of "Einstein's happiest thought". What was it?

“In 1907 Albert Einstein was sitting in his patent office in Bern Switzerland when he had what he later described as “the happiest moment of my life”-that a person who falls freely under gravity feels weightless.”

— From the Einstein Display at the Science Museum London, December 2015