Math Point

Developing Patient Problem Solvers

2011-06-15T07:14:00.000-07:00

Differentiation Card Game

2009-03-20T06:42:00.001-07:00

Differentiation Card Game

In secondary school calculus, one must be proficient in differentiating algebraic expressions. One way for students to practise the power rule and get them to learn from one another is to use a card game. I have tried this last year with my sec 4 students.

Basically, the differentiation card game consists of a stack of cards and a group of players. I recommend about 50 cards and about 4 players so that each player can have 10 or more cards. It would also be good to form the group of mixed ability so that each group has at least someone who has understood the concept of power rule.

Each card consists of an algebraic expression. For instance, I have (painstakingly) printed cards with polynomial expressions such as 1, x, x^2 and so on. besides these, we can have more complicated cards such as x^2-x+1, 1/x, sqrt(x) and so on. The objective of the game is for players to take turns to throw out one card at a time, in the order of derivative. For example, Suppose player A throws x^2-x. The next player B, will have to throw 2x-1. If player B does not have this card, he/she misses his/her turn and this will continue until someone has this card. The deck can be designed such that each set of derivatives is unique except possibly having more than one "0". The player who disposes all of his/her cards correctly wins. Once "0" is reached or once no one has a card to throw, the last player who threw his/her card will have control over the game. He/she can throw what ever card he/she prefers. Hence there is obvious strategy in the game.

At the end of the game, it would be good for the teacher to do some consolidation. What do you notice about the derivative as we progress? Students will notice that polynomial expressions will eventually have derivative 0, while rational functions such as 1/x will continue indefinitely. Such thinking and reflection will help students understand power rule better. Without a firm understanding of this basic rule, students would have problems with the other rules, such as product rule and quotient rule.

Moreover, one can extend this game by including exponential and logarithmic expressions.

Of course, we will need a few e^x in the deck!

Generalising Patterns using Multiple Routes

2008-11-15T21:08:00.000-08:00

A core focus of secondary mathematics is for students to develop inductive and deductive reasoning. In particular, this is emphasised in lower secondary mathematics where number patterns are taught. In this chapter, students learn to depict patterns using algebraic expressions and attempt to generalise. This is one route to develop inductive reasoning.

One of the central themes of the book “Developing Thinking in Algebra” (Mason, J, Graham,A, Johnston-Wilder, S, 2005) is the idea of expressing generality. Generality can be expressed in multiple ways which can help students develop algebraic thinking by approaching problems in various ways. Students often only face number pattern problems through a single approach. Here are some examples of problems that can be approached in a variety of ways:

Generalising picture sequences in the reverse manner.

A common number pattern task is to have a series of pictures and then formulate an expression for the nth picture. For instance, take a picture sequence made of sticks, perhaps in the following:

I found this example meaningful as there are many ways of finding number patterns. For instance, what is the number of sticks required to build the roof in the nth house? What is the number of sticks required for perimeter of the nth house? These questions should not be given to the students. Instead, students should come up with their own questions and try to determine the number patterns on their own. This will enable them to develop metacognition through “pattern sense-making”.

Indeed, the reverse process is even more interesting and challenging. For instance, suppose we have the sequence 2+(n-1)+(1+2n+1) and 3(n+1) for the nth house. Find ways to build structures of houses using these expressions. As mentioned by the author, this process helps to support flexibility in algebraic thinking as the student will need to interpret the algebraic expression before visualising the structure.

Cube Painting

A cube is painted on all faces. It is then cut up into 27 equal cublets by two plan cuts parallel to each face, (imagine a 1 color Rubik Cube). How many cublets are painted on how many faces? Generalise the problem.

Cube painting is but one way to expose students to inductive reasoning which is not so direct. Students may need to utilise their problem solving skills such as making a list (Creating a table for a two by two cube first, then by a three by three cube) and working backwards in order to attain the answer. Moreover, there are no clear steps to generalise the problem. Students must first think of how to extend this problem. For example, besides cutting the cube into 27 cublets, what is the next smallest number of cublets can be cut using the same operation? What if more than one color is used? If 6 different colors were to be used, like in a Rubik Cube, how many faces would have 3 colors? How many would have two? There are many questions that can be asked to extend this problem.

In summary, number pattern activities listed above require students to apply problem solving skills, metacognition, inductive and deductive reasoning. This will support students’ thinking in algebra.

Algebra and moderation

2008-10-20T07:29:00.000-07:00

Here is a simple way to assess students' understanding of algebraic functions. Suppose you gave a test and you are about to return them the paper. Ask them this:

Let's say your marks for this paper is x, and you are giving a choice to change your makes by using a certain function. Which function will you choose? You are only given one choice!

x+5
max(60, x+10)
100-x
10sqrt(x)
20ln(x)

Which one would you choose and in what situation would you choose each of them?

Of course you can extend this problem into a two variable function. Suppose paper 1 is x and paper 2 is y. Which one would you choose as your marks?

(x+y)/2
sqrt(x^2+y^2)
sqrt(xy)
max(x , y)

You can come up with many other functions to assess the students' higher order thinking skills.

Activities for TI Graphing Calculators

2008-10-19T06:02:00.000-07:00

The following website holds many mathematics activities using Texas Instruments (TI) GCs as pedagogical tools.
You can find the following topics in this website:

Algebra, Advance Algebra, Geometry, Pre-Calculus (ie Binomial Theorem, Trigonometry) and Calculus.

URL is http://www.timath.com/algebra1/archive/ti84-plus

Besides support for TI84, there seems to be more activites for TI inspire as well. TI inspire is also another great pedagogical tool for the classrooom. For more information on TI GCs, visit the following URL:

http://education.ti.com/educationportal/sites/US/productDetail/us_ti84p.html

3 Levels in Mathematics Education

2008-06-23T22:24:00.000-07:00

In a recent mathematics conference, I had the honour of attending a talk by Japanese educators. Prof Yoshinori Shimizu shared about the different stages of mathematical development and how students can master a particular topic. In particular, we can structure learning that is interesting and insightful. The following are 3 levels where we can structure a mathematic topic:

Level 1: 守 (Shu) Follow the form.

Following the form requires the student to apply standard techniques to standard questions that are probably similar to what they have encountered.

Level 2:破 (Har) Break the form.

Breaking the form requires the student to apply standard techniques to solve problems that they have not encountered before. This requires students to apply higher order thinking skills. Moreover, they must possess creativity and problem solving skills in order to apply prior knowledge to a new situation. This will assess their metacognition and habits of mind.

Level 3: 离 (Ri) Extend away from the form.

This may require students to extend what they have learnt to deeper concepts. For instance, if they have only learnt how to solve quadratic equations, students can question how they can extend this to cubic or quartic equations? Is there a standard formula to solve quartic or quintic as well? Also, another view is how students can extend what they have learnt to other disciplines and how the concepts can be applied to authentic situations.

In many cases, teachers often complain that students are competent to solve math problems that they have done before. However, when faced with new problems, students either avoid the question or simply raise their hands in defeat. By structuring lessons in the above 3 forms, students should be able to master their content using problem solving skills.

"Set" Card Game

2008-06-10T07:29:00.000-07:00

I came across this card game called "Set" which is used for cooperative learning in class. It is a simple card game where a group of students try to identify sets of cards from a deck with different attributes, such as shape and color. A "set" of cards is one where either each attributes is the same or different.

Try this Flash tutorial to understand the mechanics of the game.
http://www.setgame.com/set/index.html

There is alot of potential in this game for teaching and learning in the following areas:

Permutation and Combination
Eg: What is the total number of sets that can be formed from the deck?
Set Theory
Eg: If A is the set of red cards and B is the set of diamond shaped cards, what does the set A union B mean?
Probability
Eg: What is the probability that the game will end with 3 cards?
Problem Solving
Eg: What is the maximum number of cards that can be picked before a set must be found?

More information can be found in the following two URLs:
http://www.setgame.com/set/article_nctm.htm
http://www.setgame.com/set/index.html

Whatcha Know About Math?

2008-06-10T06:14:00.000-07:00

Kinematics and Calculus Made Easy

2008-06-10T06:10:00.000-07:00

Geometry from the Incas

2008-06-10T05:13:00.000-07:00

This is one of the most comprehensive sets on geometry that I 've seen. It contains dozens of examples of different results in geometry from Marion Walter's Theorem to Menalus' Theorem. A wonder site for exploration in the world of Geometry.

The URL is
http://www.gogeometry.com/index.html

A paradox in logic

2008-06-10T04:52:00.000-07:00

Mr Lee, the teacher, announces to the class that there will be a surprise test next week. It can occur on any day from Monday to Friday afternoon. Alex, the smartest student in the class, stood up and pointed out the following to Mr Lee.

"Mr Lee"

"Yes, Alex?"

"Well there can't be a spring test on any day next week."

"Why not Alex?"

Alex begins to explain his argument:

"Let's assume you are telling the truth and the test is indeed a surprise. If there is no test by Friday morning, then it is no surprise that the test will be on Friday afternoon. Hence the test cannot occur on Friday, so it must occur between Monday and Thursday. However, if the test does not occur on Friday, then we would have known by Thursday morning that the test must be on Thursday afternoon, which again is no surprise. After ruling out Thursday and Friday, we can apply a similar argument to the rest of the days. Hence there cannot be a surprise test from Monday to Friday"

"Very well Alex, we shall see how it goes next week."

In the following week, a test was given out on Wednesday when the students did not expect it. It was indeed a surprise test.

Analyse Alex's argument using Paul's Wheel of Reasoning:

What is the problem in the question?
What information is given by Mr Lee?
What are the assumptions made by Alex?
What are the implications and consequences of his assumptions?
Whose perspectives should be considered?
What are the main concepts in this paradox?

Teaching Mistakes in Mathematics

2008-03-13T23:56:00.000-07:00

Perhaps the most inspiring book on mathematics teaching I've read this year is "Twenty Years before the Blackboard" by Michael Stueben.

Giving too many passive lectures
Omitting Counter examples and omitting examples of common errors.
Not offering suggestions toward efficient learning
Not giving enough problems that use mixed strategies and require recall of significant ideas
Not checking homework daily
Failing to show perspective, to show applications and to explain connections with other parts of mathematics
Not showing the internal motivation behind the mathematics and not answering the questions: "Why would anybody want to learn this?" and "What can we do with this knowledge or skill?"
Omitting discussions of the subject's history, its etymologies, and the personalities of its creators.
Omitting personal stories of my own experiences, my colleagues' experiences and my former students experiences in the study of mathematics.
Teaching only the brightest or ignoring the brightest.
Proving statements that are obvious to most students or giving proofs that few or none can follow.
Making the classroom experience equivalent to reading a textbook: definition, lemma, theorem, proof, corollary, definition, lemma, theorem, proof...
Taking no interest in students' difficulties in mastering the subject.

Interesting sites

2008-02-16T04:22:00.000-08:00

Here are two interesting websites on mathematics:

Cool Math Sites

A comprehensive list of links on mathematics. Very useful for research on elementary topics in mathematics.

Math Mistakes

A very good site that offers a variety of examples of mistakes made by students. Math topics covered includes algebra, trigonometry and Calculus.

Lateral Thinking: Math Brainteasers

2008-02-06T19:43:00.000-08:00

Here are a few brain teasers in mathematics that requires you to think creatively.

Add 1 continuous line in each question to make the following correct:

V-IV=IV
1+2=3
5+5+5=550
IX=6
5105=4.55

Shift 1 digit to make the following correct:

6. 62-63=1

Open ended Circle Problems

2007-12-16T05:00:00.000-08:00

Open ended problems generate a variety of solutions and motivates the student to suggest different creative approaches in solving a problem. Here are two related to circle geometry.

Given a circle drawn on a piece of A4 paper. How can you find the diameter of the circle using only a ruler and a pencil?

Given pair of compass and straight edge, a circle and 2 points in the interior of the circle, inscribe a right-angled triangle in the circle such that the 2 points are on different sides of the triangle.

Ideas for Math Pojects

2007-10-30T02:39:00.000-07:00

I've included a few ideas for math projects that allows students to investigate a problem. These problems are not run of mill and hence they will provide students a glimpse of how mathematical research can is done. This list is certainly not exhaustive. It is targeted to juniors only. The ideas are grouped according to major topics in mathematics.

Arithmetic/Number Theory
There are many sets of interesting numbers. Examples include polite numbers, amicable numbers and frugal numbers. Come up with your own set of numbers with some special properties and investigate.

Combinatorics
How many latin squares can be formed using a 3 by 3 grid? What about an n by n grid?
What is the answer to the 36 officer problem?

Geometry
Investigate the tiling of polygons. What polygons can be tiled and which can't?
Investigate the ways of estimating the volume/surface area of irregularly shaped object

Applied mathematics
What is the most efficient way to insert people in a duty roster whom have their own preferences?
Do you get more wet when you run in the rain or when you walk?
What is the fastest and cheapest way to get to a location in Singapore (for instance home to school) using MRT, Train and Taxi?
How do we program a lift so that it can transport people efficiently?
How can we estimate the volume of the Earth?
Why do buses sometimes come in pairs (or threes)?

Nurturing the Mathematical Mind

2007-10-13T23:21:00.000-07:00

We use mathematics in the course of their lives. However, we’ll possibly only need to use everyday math like arithmetic, percentages, ratios, etc. These are the content taught at the primary school as well as in the secondary 1 and 2 levels. But what is mathematics education about really? Surely, most of the students will not become mathematicians. How many of us are going to use logarithms in our everyday lives? How many of us would apply trigonometry identities to solve problems? And how many of us need to apply techniques in integration when many calculus problems can be solved readily with the use of computers? Mathematics education should not just be about learning content but about the thinking process. This article will discuss the many dimensions of mathematical thinking and how it applies to everyone.

Identifying Patterns

Identifying patterns is what mathematicians do. Looking at a set of mathematical objects, we can classify them according to certain attributes. There might be patterns involved that interest us and these patterns may be used to draw conclusions. At the school level, students come across many patterns in mathematics lessons. However they may not be aware of it.

Given a number pattern, what comes next?

What pattern do you observe regarding sine, cosine and tangent ratios?

What pattern do you observe in the y-values of a quadratic graph?

Observing a pattern is one thing, interpreting and drawing conclusions is another problem. For instance, given a number pattern, can we formulate a general formula for the nth term? Given a quadratic graph, how and why do the values of y change? All these questions relate to patterns.

We all see patterns in real-life. For example, stock market analysts try to identify trends in graphs to establish times to buy and sell shares. Biologists study patterns in the spread of epidemics to establish the source of the infection. Business executives study trends in the use of products in order to decide on which product to market. All these situations require the mathematical mind.

Recognizing Relationships

A mathematical mind does not only observe patterns, it is also able to establish relationships between different areas. In the classroom, students have to recognize relationships too. For instance, how does the sum of interior angles of a polygon relate the number of sides? Does changing the number of sides of a polygon change the sum of exterior angles? In medicine, a doctor might relate chocolates to the elevation of depression in his patients. How does he arrive at this claim and how does he test his hypothesis? An anthropologist studies the height of individuals over the past 100 years. He relates the increasing height of individuals to the better standard of living the world now has. All these relationships are identified via the mathematical mind.

Analyzing arguments with rigor

Mathematics as a subject is not without rigor. Proofs must be presented with absolutely no loopholes. For instance, in the classroom students learn geometry. It is apparent that when proving geometrical theorems, we cannot exemplify proofs using concrete examples but also proof in a mathematical logical and general way. This is done either by induction or deduction. Now for instance, we want to prove that the sum of interior angles of a triangle is 180 degrees, we cannot just draw one triangle, measure each angle and ascertain our result. This is an example of a fallacy in analyzing statements. One must prove in general. One way of course is to use geometrical axioms to help us establish this fact. Starting from the foundations of geometry to help us build the proof with no fallacies. When we are outside of the classroom, we should also analyze statements, assertions with our mathematical mind. For instance, there are always many reports of health issues. Should we believe them all? For example, does eating aspirin reduce the risk of coronary disease? We should use our mathematical mind to analyze and scrutinize statements to better enable us to make the right (or rather better) decisions in everyday life. The mathematics student will be able to identify the following fallacies in thinking:

Cyclical reasoning
Linking casual relationships of cause and effect
Using particular examples to draw general conclusions
Using obvious results to draw conclusions
Misinterpreting data

Synthesizing arguments with rigor

We should not only be satisfied with only analyzing statements made by others. The mathematical mind is able to synthesize arguments as well. By avoiding the pitfalls in reasoning as mentioned above, one can formulate statements, prove assertions, draw conclusions in our everyday lives and hence make the right (better) decision.

In summary, mathematics education is not just about learning mathematics concepts and techniques. It is about nurturing the mathematical mind which lies dormant in all of us. By learning mathematics, we learn to awake the mathematical mind and help us identify patterns, recognize relationships, analyze and synthesize arguments.

Tips on Math Oral Presentations

2007-09-16T01:04:00.000-07:00

After attending a workshop on Project work oral presentation, I've summarised the main learning points and added a few that are related to oral presentations that involve mathematics.

Preparation for Oral Presentation (OP)

It is important to organise the content into an order that flows. It depends on what you're trying to present. For instance, for an OP on history of mathematics, one can organise the presentation in terms of chronological flow. For group presentations, ensure each speaker has an equal amount of content to deliver. Sometimes, the first speaker who introduces the subject is more interesting than the middle speakers. The last speaker who summarises the OP would also have a better job. Hence make sure each speaker has enough interesting points to deliver. The presentation will flow well when the transition between speakers is smooth. Try to lead the discussion on to the next speaker by giving a short prelude of how the next speaker will add on or continue with the topic at hand. This will allow the audience to anticipate what the next speaker will say, and hence makes the presentation clearer.
Once the presentation material is organised and allocated to each speaker, it is important for the group to rehearse. Rehearsals serve many purposes. Firstly, rehearsing helps the individual speakers learn their speech. Secondly, it provides an opportunity for practising speaker transition, powerpoint transitions and other forms of coordination among the group. For instance, a flip chart needs to be flipped at a particular instant when the speaker arrives at a point in his delivery.
For each individual, cue cards should be made. Cue cards are small cards that can be bought at any stationery shop. The latter is extremely useful for the speaker to state what he wants to say during the OP. Writing on the cue card itself provides a way for the speaker to organise his thoughts. Moreover, priority of points can also be made with a highlighter on the cue card. For example, the speaker can highlight all important points that he must deliver in yellow, and any other less important points in green. Orange highlighted points might mean miscellaneous that are saved for the Question and Answer (QnA) session.
In short, if you fail to plan, you plan to fail, especially when it comes to oral presentations.

Audio Visual Materials in OPs
Here are a few pointers when using AV materials during presentations.

Keep Powerpoint slides simple and sweet. Use concise words and avoid long and cluttered sentences. It is very hard to read and follow a speech when there are too many words on the screen. Try to be as concise as possible.
Use only pictures and animations that add value to the OP. Avoid using animations that serve no purpose but to decorate the slide. This may distract the audience and hence hamper them from paying attention to your delivery.
Coordinate font, font size, backgrounds and color combinations. Try to have a good contrast for words and background. For instance, if you have a dark background on you master slide, try to use light colors for the words. Otherwise the audience may not be able to see the words clearly on the screen. Always rehearse with powerslide done. What you see on your PC monitor may not turn out as well on the projector screen.
Limit videos and music files to 1- 2 minutes, unless it has a very important and direct impact to what you want to deliver. A video which is too long may bore the audience and cuts down on the time you have to speak. Remember, your delivery is the most important grading criteria, not your resources. Use MS moviemaker to crop clips down to the bare minimum.
Always test out all equipment such as PCs, projectors etc prior to the presentation. You might never know what might break down at the eleventh hour (Murphy's law).

Public Speaking Skills
When it comes to public speaking, the most important point to take note is to always practise. Practise makes perfect. There is no one who is born to be an effective speaker. One needs to learn and practise speaking in order to be a good oral presenter. In order to be a good speaker, one needs to engage the audience. Here are some pointers to note:

Understand the background of your audience. This should already been acknowledged when planning for the OP, since, for example, it is fruitless to explain in depth how to tune a viola to a group of students who have no prerequisite knowledge about string instruments. Hence the speaker must first bridge the gap with the introduction of what a viola is, its parts before moving into the body of his speech.
Interact and engage the audience. There are many techniques of doing this. Although I've mentioned that one can refer to cue cards during OPs, it is important to look up and make eye contact with the audience. Try to look at various people to ascertain whether they can understand what you're saying. If they are nodding to certain points, they are probably following. But if they are frowning or not paying attention, perhaps you are going too fast or they cannot follow your speech. There are other ways to engage the audience. Questions can be posed to the audience, a poll can be taken so that the audience can participate in your OP. A bit of humour can also liven up your speech. Also, one can move a bit around the room to exert your presence. However, do not stroll too much into the audiences' personal space, which would make them uncomfortable (e.g. breathing down their necks).
Speak in a varying tone. Practise on your intonation. Nothing makes a OP more boring than a speaker who speaks in a monotone, nonchalant way. Be enthusiastic and speak with suitable degrees of pitch, tone and volume. When rehearsing, get feedback from your group mates. Are you speaking at the right volume? Are you using repetitive words like "ok" or "basically" in the speech? Lastly, display the right body language. It gives the most impactful impression to your presentation. For instance, always stand straight and not one one leg. Use hand gestures if necessary.

A word on Mathematics Oral Presentations
Very often, students present long mathematical working during their short OPs. It is important for the speakers to understand that the audience cannot grasp all the mathematics in the short period of time. Personally, one should try to give clear outlines of the working and present main ideas of proof. It is not effective to list all the working down on slides either. Just present the main ideas of the proof. But more importantly, discuss the result and conclude on how the mathematical result has an impact to the topic being delivered. Start and end off with this in mind so that the audience can follow more easily. This applies to any math presentation, be it a noble prize lecture or a short classroom presentation.

In conclusion, I have briefly gone through three important aspects of OP. A good OP cannot be done without sufficient preparation. Practise the OP as many times as possible to ensure all elements are in place. Plan your audio visual materials in an effective way. Remember, powerpoint slides are used to enhance your OP, not to distract the audience or confuse them. Lastly, practise on effective public speaking skills. Engage the audience with your body language and voice tone.

Parallel lines and optical illusions

2007-08-26T05:40:00.000-07:00

Look at the 2 lines above, are they parallel or bent?

Look at the diagonal lines above, do they look parallel?
Optical illusions occur in everyday life. Sometimes things do not look as what they are supposed to be. It is important to illustrate this point to students learning geometry, that one should not assume some properties of the object by visual inspection.

In short, for questions not drawn to scale, one should not make unnecessary assumptions. So if you're given two lines, how can one determine whether they are parallel?

Mathematics and Careers

2007-08-25T05:26:00.000-07:00

In the book "Careers for Number Crunchers and Other Quantitative Types" (VGM Career Books) , Rebecca Burnett explores various careers that require a certain aptitude in mathematics. Here is a list from the book:

Tracking of financial records

Accountants
Book keepers

Managing Money and Cash flow

Chief Finance Officer (CFO)
Office Manager
Bank Teller

Guiding Investments

Mortgage Officer
Financial and Securities sales representative

Buying in the workplace

Purchasing Manager

Marketing and Selling in the Workplace

Sales Service Representative
Insurance agent
Marketing

Quantitative Thinking Careers

Engineers
Computer System analysts
Calculating Risks and Probabilities
Actuaries
Operation Research Analyst
Statistician

Research in Natural Sciences

Biological scientist
Meteorologist
Physicist
Mathematician

Education

Teacher

The book also lists some real life examples of the responsibilities in each career. For instance, for an actuarial analyst such as Brian, he uses computers to do mathematical analysis on databases to help him make recommendations about raising and lower insurance rates and changing policy coverages for insurance companies.

For an operations research analyst, one must apply quantitative thinking to help refine and solve work flow problems in a company. For example, an analyst in a manufacturing facility might deal with modifying production lines for new products, while an analyst employed by a hospital might monitor the use of pharmaceutical and laboratory services. They also also employed by airlines to help build a system for scheduling airline staff.

A good book to read for those who are interested in a career that deals with numbers.

ThinkQuest Website on Mathematics

2007-08-03T02:45:00.000-07:00

The following thinkquest website is well worth visiting. It contains abundant information on Algebra, Geometry, Calculus and Trigonometry. This will prove handy in O level mathematics.

http://library.thinkquest.org/10030/math.htm

Java Applet on Algebraic Tiles

2007-08-01T17:11:00.000-07:00

A useful online algebraic resource can be found in the website highlighted below:

http://nlvm.usu.edu/en/nav/vlibrary.html

Go to yr 9-12>Algebra>Algebra Tiles

Accompanying the java applet are instructions and activities that can be used in class to teach algebra. This java applet can be used to illustrate many algebraic techniques including expansion and factorisation.

An Inspiring Quote

2007-07-22T07:20:00.001-07:00

"To teach well is easy. To teach well enough for students to understand is not difficult. But to teach so well that the student wants to learn more is the challenge."

Are you up for the challenge?

(Courtesy of Samuel)

Geometry Java Applets

2007-07-12T06:23:00.000-07:00

The following IES website contains very good java applets for geometry investigations:

http://www.ies.co.jp/math/products/geo1/menu.html

http://www.ies.co.jp/math/products/geo2/menu.html

There are many geometric figures that can be manipulated with questions for students to think about. There are also applets on other topics such as calculus and trigonometry. The main site can be found here:

http://www.ies.co.jp/math/products/

Introduction to Coordinate Geometry

2007-07-08T18:30:00.000-07:00

In the learning of coordinate geometry, alot of emphasis has been placed on the more abstract concepts of the topic. For instance, students are required to formulate the equation of a line from 2 points, or the gradient of a line. However, students may not have adequate understanding about the concept of the properties of a straight line. For example, what is the significance of gradient and y-intercept? Why do we need to formulate equations of lines? But most importantly, what is the use of graphs? An a example is given below to illustrate the significance of gradient and y-intercept in everydaylife.An experiment was conducted to find the speed of tennis ball in air and a distance time graph is found using experimental tools like a data logger:

From the graph, we see that the distance travelled by the ball is linearly related to the time taken. Consider the gradient in this case. How would the graph of the slow and fast moving tennis ball differ?

The gradient of the graph represents the speed of the ball, as the gradient measures the change in distance over the change in time. In other words,

gradient = rate of change of distance= speed

Hence, there is a physical intepretation of gradient of the line in this case.
From the graph, we also see that the ball does not start from rest. What is the distance travelled by the ball when the timer was started? In this case,

y-intercept = distance travelled by ball at the 0 sec

Hence there is also a physical intepretation for the y- intercept of the line.

Flatland the movie is out!

2007-07-02T07:33:00.000-07:00

I received the following news on 28 June 2007.

Dear Flatland Movie Enthusiast,

We are writing you because some time ago you registered to receive information about Flatland: The Movie on our website. Flatland: The Movie is an exciting animated film adapted from Edwin Abbott’s classic novel, starring the voices of Martin Sheen, Kristen Bell, Tony Hale, Michael York and Joe Estevez.

We are pleased to announce that you can now order the DVD directly on our website! You may place your order securely online at the Flatland Store

http://store.flatlandthemovie.com

Currently we are taking orders and now shipping the Special Educator Edition of Flatland: The Movie. This edition is primarily intended for educators, teachers, schools and institutions. The purchase of the DVD includes a perpetual school site license, a bonus featurette discussing the 4th dimension, and math worksheets for use in the classroom.

If you are interested in the DVD primarily for home or personal use, not to worry. We are accepting pre-orders for the Home/Personal license DVD edition of Flatland: The Movie. The private home edition of the DVD naturally costs much less than the school site license. This edition will ship in the fall and you can place your order now to be the first in line to receive the movie when it ships. Your credit card will not be charged until the DVD actually ships.

Thank you for your continued interest in Flatland: The Movie!

Warm Regards,

Seth

Algebra Scale Balance

2007-07-02T00:52:00.000-07:00

In learning algebraic manipulation of linear equations, it is important for students to try and grasp the basic concepts of balancing the equation on both sides. For instance, in the equation 2x+1=7. Students must understand why "bringing over the 1" to the other side turns it into -1. ie 2x=7-1. Students need to understand there is balancing process here so terms like "bringing over" and "cross-multiplying" should not be used first until they get the concept right. There are 4 basic balancing methods for an algebraic equation. They are as follows:

Subtracting both sides by the same number, Eg: 2x+1-1=7-1
Adding both sides by the same number, Eg: 2x+1+(-1)=7+(-1)
Multiplying both sides by the same number
Dividing both sides by the same number

I would recommend not lumping 1 and 2 as some students may not see when and what to add or subtract on both sides. An example should be given for each case.

Ample practice should be given for students to try out manipulating algebraic equations, with a mixed variety and ultimately questions that require all 4 techniques to be used successively. For 4, it should be emphasized early that dividing both sides can only be done for non-zero numbers. This can be done by the fallacy given in previous post.

A very good manipulative for this topic is the algebraic balance. A virtual one can be found here, by Utah State University.

Index Site URL: http://nlvm.usu.edu/en/nav/vlibrary.html

List of online resources

2007-06-09T01:40:00.000-07:00

The following URLs are taken from the book "Real Life Math: Everyday use of mathematical concepts" by Evan M. Glazer and John W. McConnell (2002). It is a book on real-life applications of math sorted in alphabetical order. Below are the links I have selected from the book after checking authenticity. Some links differ from the book as they were moved. Do take time to look through some of the topics, especially complex numbers where both teachers and students have a hard time coming up with examples of complex numbers in action. The website by British Columbia Institute of technology provides many interesting mathematical applications to technology.

Topic	Description
Angles	A study of angles of light in a diamond. Deals with reflection and refraction. Good for the topic of angles as well as reflection and refraction in physics.
	The mathematics of rainbows. Uses concepts of reflection and refraction as well.
	Instructions on how to throw a boomerang with reference to angles and vectors.
Asymptotes	The basics of food cooling. Uses log function and exponential functions
Coordinates	Finding and plotting celestial coordinates
Circles	Arches in architecture
	Tree rings
	Another comprehensive site on tree rings
Complex Numbers	A forum page that discusses the use of complex numbers in real life.
	Application of complex numbers in electric circuits
	Using complex numbers to generate fractals.
Exponential Decay	Carbon dating
Exponential Decay
Exponential Growth	Population growth.
	Savings and compound interest. The main site contains other uses of math that are applicable to teens: Click here
	The exponential spread of chain letters and scams
	The Charles Ponzi Scam
	Exponential growth of the Internet
	Pricing diamond rings
	Inflation calculator calculates the same amount of money you have in 2 different years. Calculates inflation between the years 1800 to 2006.
Logarithms	Ph Factor in Chemistry
	Ph chart
	Table of decibel levels
	Excellent list of applications in technical areas of technology
Matrices	Cryptology and coding (Describe the benefits of this form of coding instead of direct substitution)
	Another website on cryptology and coding using matrices
	Matrices in balancing chemical equations
	Various matrix related activities. Lesson plans included.
Polar Coordinates	A Programme can be downloaded to plot polar pictures
	Sun position in polar coordinates
	Polar coordinates in Robotics
Polynomials	A website that describes how to build boxes using rational and polynomial functions
Polynomials	Polynomial equations in drag racing
Quadratic Functions	Science of cycling
	Investigating Fluid flow using algebraic equations
	Using equations to determine the minimum surface area of a cylinder
Rate, Ratio and Proportion	Gear ratios of bicycles
	Ratios in musical scales
	Ratios in musical scales, tuning the piano.
	Proportion and ratio in cooking
	Eratosthenes using proportion to estimate the radius of the earth
	Proportion in medical imaging
	Exchange rates for international currencies
	Yerkes-Dodson law of Stress (Rate of change)

BayWatch Part 2

2007-05-16T22:24:00.000-07:00

From the previous episode on "Baywatch" (see previous post), you realise after you reach the swimmer at A, you realise there is another swimmer who is drowning at B. As a life guard, you can only bring the swimmer at A to shore at some point C so that he can be attended by paramedics, before you can swim towards B. This is illustrated by the diagram below.

Where should point C be in order for you to reach the other swimmer at B in the shortest length of time? Assume you swim at a constant speed throughout course of action.

Baywatch

2007-05-16T22:11:00.000-07:00

Have you ever watched Baywatch? It features David Hasselhoff and Pamela Anderson, 2 life guards at a Miami beach. Imagine you are one of the life guards at point A on the beach (shown below), and a swimmer needs your help at point B. How would you reach the swimmer?

There are obviously 2 choices:
1) Take the shortest distance between A and B. Just run straight and swim straight towards B
2) Run at an angle before swimming perpendicular to the shoreline. This ensures that you swim the least.

Note that you can run faster on land than you can swim in the water.
Which way would you take? Is there a faster way?

Skill mastery v.s. Conceptual approach

2007-05-03T07:35:00.000-07:00

We all learn arithmetic in Primary school using the standard multiplication and division algorithms. NTCM has reformed their curriculum using 8 Focal points (see previous post) to develop mastery of these skills by the 5th grade. However, there are many books which focus on problem solving rather than skill mastery. The video below shows out two reference books for elementary school show alternate approaches to arithmetic, namely multiplication and division.

Math Education: An Inconvenient Truth

A very informative video of different approaches and techniques to solve the same problem. Ultimately, this is a question of efficiency vs understanding. Stay tuned till the end of the video for a surprise.

Driving revisited

2007-04-20T05:16:00.000-07:00

From my earlier post on Quadratic Equations of driving, we can establish another proof of why the breaking distance of a car increases to about 4 fold if the speed of the car increases by 2. This proof relies on two facts:

1) For a velocity-time graph, the area under the graph is equal to the displacement.
2) For two similiar figures, the ratio of their area is proportional to the square of the ratio of the corresponding lengths.

For instance, suppose a car travels at 50 km/h, he sees a boy and breaks. The reaction time is, say 1s and the car decelerates uniformly ( it need not be uniform. for simplicity, let us assume that it is uniform). one can now find the distance travelled by looking at the area under the speed time graph.

Now if he were to travel at twice the speed, at 100 km/h, how would the area under the graph look like?

From the graph, we see that the figures that form the areas for the triangles are actually similar! Let A1 and A2 be the breaking distances when inital speed is 100 km/h and 50 km/h respectively.

Then A1:A2=4:1

Thus breaking distance is increased four-fold instead of two-fold. However, we do note that in reality, the deceleration is not uniform in both cases. Nevertheless, the breaking distance will increse by more than two fold.

Similarity and Optics

2007-03-18T01:56:00.000-07:00

It is purely geometry that we can determine properties of convex lenses and their images. In this case, it suffices to use congruency and similarity to show the relationship between object distance u, image distance v and focal length f. From there we can also determine the height of object h and height of image k .Consider the ray diagram below.

Triangles OCD and OBE are similar, hence h/k=u/v .
Also, triangles AOG and BEG are similar, hence f/(v-f)=h/k=u/v.

But h/k =u/v. Substitute this into the second equation will result in the neat formula:

1/f=1/u+1/v

Confusion over abbreviations in kinematics

2007-03-17T08:09:00.000-07:00

Consider the following abbreviations, s,d,v. What does each one mean in kinematics?
Firstly, students are often confused over "s". They might assume that "s" stands for speed, rather than displacement.

But really, why does "s" stand for displacement?

If anyone has an answer to the above question, do let me know.

As for "d", it would mean distance. For "v", it might mean speed or velocity depending on the context. When doing questions involving speed and velocity, these abbreviations would generally not lead to any confusion. Here is a question to think about:

Given a "s-t" graph, how do we know that it is a distance time graph or a displacement time graph?

Similarly, given a "v-t" graph, how do we know v stands for speed and not velocity? Why is there no ambiguity in either cases?

Light and geometry

2007-02-18T20:26:00.000-08:00

Do you notice any similar shapes in this picture?

LCM question using Astronomy

2007-01-31T05:26:00.000-08:00

Who says teaching Arithmetic is boring? Perhaps having a context will help students get more interested in finding the lowest common multiple (LCM) of numbers and highest common factor of numbers (HCF), and perhaps understand the importance of these 2 concepts. One area of science where LCM can be applied is astronomy as illustrated by the following:

Suppose Jupiter takes 12 years to make on revolution around the sun, and Saturn takes 30 years. If these 2 planets were to line up in the Earth sky, how long would it take for them to line up in the Earth sky again?

Adapted from Teaching Secondary Mathematics through Applications (2nd Ed.), Herbert Fremont. A highly recommended read for any one who would like to find interesting applications of high school mathematics.

Flatland the movie

2007-01-30T05:22:00.000-08:00

In 1884, Edwin Abbott wrote the novel Flatland depicting the adventures of a square in a two-dimensional world. It is an extremely useful novel to teach geometry. Topics that can be taught include polygons, circles, plane geometry and triangles.

You'll be interested to know that Flat Land the Movie is coming out this year. Check out its trailers at www.flatlandthemovie.com

2 paradoxes

2007-01-10T04:46:00.000-08:00

Algebraic manipulation can lead to certain fallacies can give raise to surpising results. Here is a sample of 2 of them taken from the book "one equals zero" by Nitsa Movshovitz-Hadar and John Webb (1998). These paradoxes can be used in class for students to think about.

1) suppose we have the equation x-x^2=1.

Since 0 is not a root of this equation, we can divide both sides by x:

x+1/x=1

x-x^2=x+1/x

subtracting both sides by x,

-x^2=1/x

which gives

x=-1

But -1 isn't the answer. What went wrong?

2) Given the equation

log(x-1)^2=2 log 3
2log(x-1)=2log 3
log(x-1)=log 3
x-1=3

so x=4

Is this the only solution? If not? What went wrong in the working above?

Overall, the book proved very interesting for me. There is a variety of paradoxes ranging from algebra to statistics. It also includes calculus and geometry.

The mysterious 6174 revisited

2006-12-24T05:26:00.000-08:00

From my earlier post on 6174 comes a simpler 3 digit variant. Select 3 integers which are not all the same. Arrange these digits to form the largest and smallest numbers and subtract them. Repeat this process with the result. For instance,

981-189=792
972-279=693
963-369=594
954-459=495

You will end up with 495!
The following gives a proof that all 3 digit numbers which are not 0,111,222,333,...,999 will end up with 495. This proof can be adapted to 4 digits as well, as in the case of 6174.

Let x,y,z be 3 digits such that z is the largest while x is the smallest, and x,y and z are not all equal. Then if we subtract the largest number formed from the smallest number formed, we have

zyx
- xyz
ABC

Since x < z
C=10+x-z,

Also,

B=10+y-1-y

And

A=z-1-x

Hence B = 9 = z since z is the largest digit. We want A and C to be either x or y. Solving, we see that x=4 and y=5. Hence 495 is the only number that repeats itself after an iteration.

But do any set of 3 randomly selected integers not all equal will lead to 495 after a finite number of iterations? First we see that for x,y and z,

z(100)+y(10)+x-(x(100)-y(10)-z)=(z-x)(99)

Hence the resulting number after any iteration is a 3-digit multiple of 99.
Hence we only need to check 9 possible numbers: 99,198,297,296,495,594,693,792 and 891 (10(99)=990 which is the same as 99)

Since all the above numbers lead to 495, QED.

Surprisingly, 495=5(99) which is in the middle of the pack of multiples of 99!

Can you see if you can generalise this result to 5 digit numbers and above?

Quadratic Equations and Driving

2006-12-24T00:43:00.000-08:00

Many drivers make the assumption that braking distance would be directly proportional to their speed. We shall investigate this by using some simple quadratic equations.

From our kinematic equation

v^2=u^2+2as,

we want to estimate our braking distance d, hence v=0. If we include reaction time t, we get

d=u^2/2a+ut

Assume a= 10m/s/s and t = 0.75s for an average driver. Then we have

d=u^2/20+(0.75)u

If a vehicle speeds at 100km/h (27.8m/s), d=59.5m

If a vehicle slows down to half that speed, ie 50 km/h (13.9m/s), d=20.1 m, which is roughly one third of the original distance, and not half of the latter!

Quadratic equations apply to our everyday lives, and this being one example. Students can be given tasks such as investigating how reaction time affects the braking distance, and why it is important for drivers to slow down in congested areas, and why drivers should never tail gate or use their handphones while driving (distractions can increase their reaction times to 3s or more.)

More can be found in this link:
http://www.science.org.au/nova/058/058key.htm

The mysterious number 6174

2006-12-16T05:34:00.000-08:00

Think of any 4 non-zero digits, where not all four digits are the same. Arrange the digits in descending order and ascending order and compute the difference. For instance, if the digits are 3,3,2 and 1,

3321-1233=2088

Repeat the above step by rearranging the result into ascending and descending order and compute the difference.

8820-0288=8532

Reiterating,

8532-2358=6174,

Strangely,

7641-1467=6174

The process terminates when the number reaches 6174. We can use another example,

9750-0579=9171
9711-1179=8532
8532-2358=6174

and Viola! We get the same result again! Can you figure out why this happens?

Tiles

2006-11-29T21:54:00.000-08:00

tiles 1

tiles 2

Above are 2 pictures of tiles taken at Hong Lim food centre and Jurong East Regional library respectively. These pictures can be used to teach mensuration, similiarity and tiliing properties. For instance, in the first picture, what is the area covered by the circular shaded region if each big tile is a 1 m by 1 m square? (Ignore the shaded boundary.)

In fact, a similar tile pattern can be found at Marina Square first floor.

Mathematica 5

2006-11-21T23:22:00.000-08:00

I've just attended a talk on Mathematica 5, a software created by Wolfram Research. It is comparable to MatLab, but there are several differences in the interface. Mathematica is very verstatile and especially useful in visualisation of 3 dimensional plots. This is useful for visualising graphs found in calculus. Moreover, with a little programming language, one can create a beautiful template for manipulating plots. It has been an eye-opening experience. Apart from visualisation, Mathematica is a very powerful computing tool. It can compute anything up to any precision, provided your computer can cope with it. Moreover, it can simply solutions to any simplified form, including all functions that are accessible in its library. The interface looks simple, while the progamming language is easy to grasp, especially if you know a VB, C++ or Java. I was amazed how little language was needed to design the many wonderful templates used in the talk. These include fractals, 3 Dimensional elliptic functions and graphical plots of partial differential equations. Lastly, there is an interesting feature in the upcoming Mathematica 6. The user can use a USB Joystick or JoyPad together with the programme. One can move along a 10 dimensional manifold or space using the joy pad, which is much simplier than using the mouse or scroll bars due to the high number of degrees of freedom the space can take. This might come in useful in the classroom, where students can use joypads to manipulate some plots. This helps to create interest in the subject.

The talk comes with a free trial CD of Mathematica 5. I've talked to the marketing executive, and they are ready to promote Mathematica to secondary schools (although mathematica has been around for more than 10 years, it does not seem to be widely avaliable in schools). I've yet to find out more about the liscensing costs, but you may approach the group for more information:

OTTE Internationl
1 Serangoon North Ave 5
#04-05A Singtel Building
S(554915)
Tel: 64833323
www.ottegroup.com

The following websites contain more information of Mathematica and additional packages for the specific usage:

www.wolfram.com
mathworld.wolfram.com
functions.wolfram.com
integrals.wolfram.com
www.mathmicentral.com
gallery.wolfram.com

Pythagoras Theorem

2006-11-08T04:32:00.000-08:00

Proof Without Words

Courtesy of Steph

Mathematical Card Tricks

2006-10-31T04:21:00.000-08:00

Here is a mathematical card trick taken from "Mathematics Magic and Mystery" by Martin Gardner (Dover books, pp 7):

A spectator shuffles the deck and places it on the table. The magician writes down the name of a card on a piece of paper and places it face down without letting anyone see it.

12 cards are dealt on the table, face down. The spectator is asked to touch any four. The touched cards are turned face up. The remaining cards are gathered and returned to the bottom of the deck. Assume that the picked 4 cards are 3,6,10 and a King. The magician states that he will deal cards on top of each four in the following way.

If the card is 10 or any court card, do not deal anything on them.
If the card is not the above, deal a number of cards till they count to 10. For instance, if the card is 3, deal 7 cards by counting: 4,5,6,7,8,9,10. if the card is a 6, deal 4 cards counting: 7,8,9,10.

The values of the original 4 cards are added. In this case 3+6+10+10=29. The spectator is handed the pack and asked to count to the 29th card from the top. This card is turned over. The magician reads his prediction and discovers it the chosen card.

Using only sec 1 algebra, can you figure out how this is done?

Math cartoons for lessons

2006-10-16T05:19:00.000-07:00

Here is a cartoon website where you can find an entire list of cartoons ranging in various subjects. Under the "M" section, you have find many math related cartoons that can be used in lessons. However, most of these cartoons are not related to any particular math topic, so there is a need to be selective.

The link is:
www.cartoonstock.com

A challenging secondary school Geometry problem

2006-10-13T22:58:00.000-07:00

Let ABC be an isoceles triagnel with equal angles 80 degress at B anc C. Cevians BD and CE divide angles at B and C into 60+20 and 30+50 degrees respectively. Find Angle EDB.

geometry problem

What color is the bear?

2006-10-13T22:53:00.000-07:00

A hunter walks 1 mile south. He then turns left and walks 1 mile east, then turns left again and walks 1 mile north. He ends up back where he started and spots a bear. What color is the bear?

The answer is white, as he is at the north pole. What do you see in this question?
As the hunter walks 1 mile south , 1 mile east and 1 mile north, he has formed 2 right angles. But yet he was back at the same spot as he started. This means the triangle formed from the hunter's path contains 2 right angle.

Why is this possible?

Do Statistics Lie?

2006-09-27T08:13:00.000-07:00

This is a post from Samuel, a good friend of mine. The following describes how statistical data can be decieving.

"....A/P Chua Tin Chiu quoted some eye-catching headlines from newsapers,
...."You have 40% chance of living up to 100 years old if you are the first born"
Then he asked: Are you sure about the conclusion statements?..... he said, "The survey has been done in a village in china, where the study was done only to people who live up to 100. However, the number of first born will always be more than the number of 2nd born, who will always be more than the number of 3rd born etc (haha...for obvious reasons right? You need a first born before you can have a 2nd born) and so the number of first born who live up to 100 should also be more than the rest....

A/P Chua quoted more newspaper reports with regards to statistics in newspaper, and asked, "Why do you think percentages have been quoted in areas where direct figures might be more accurate, and vice versa? Isn't it the way the media (or the government) want to bring across a particular idea?

Another wonderful headline that students will love to quote if true,

"Study shows More tuition = poorer grades"

Well, the study was concluded based on a huge pool of students who wrote down their grades, as well as the number of hours they had for private tutoring. But the BIGGEST question is, "How will the students who have private tuition fare, if they do not have private tuition? Isn't it true that those who need private tuition are generally those who may not cope that well, and so extra help to be given?"

Curriculum focal points in the States

2006-09-16T03:49:00.000-07:00

In today's Straits times (16 Sept), there is an article on page 36 entitled: "Maths teaching in US moving back to basics." Here the article points out that "new ways of teaching maths, introduced in the US almost two decades ago have not added up much." These new ways refer to the constructivist approach where "children learn what they want to learn when they're ready to learn it". Instead, the National Council of Teachers of Mathematics(NCTM) urge that the teaching of maths in kindergarten through eighth grade focus on a few core skills. These skills, which they term as the curriculum focal points, are classified under each level*:

• Pre-K (under 5 years old): Develop an understanding of whole numbers and how to count and compare them.

• Kindergarten(5-6 yrs old): Use numbers to solve quantitative problems, count numbers in a set, and create a set within a given number of objects.

• 2nd Grade(7-8 yrs old): Learn how to count in units and multiples of hundreds, tens, and ones; understand multi-digit numbers in terms of place-value, and how to compare and order numbers.

• 4th Grade(9-10 yrs old): Develop understanding of multiplication, including “quick recall” of multiplication and division facts; select correct methods to make mental estimations and calculations.

• 6th Grade(11-12 yrs old): Know the meanings of fractions, multiplication, and division; understand relationships between decimals and fractions, and how to multiply and divide them, using multistep problems.

• 8th Grade(13-14 yrs old): Use linear functions, linear equations, and their understanding of the slope of a line to solve problems; understand verbal and graphical representations of functions; describe how the slope of a line and the y-intercept appear in different verbal, graphical, and algebraic representations.

* Taken from NCTM website :http://www.nctm.org/news/ext_articles/2006_0912_edweek.htm
Read more about this math education reform via the above URL.

There is also an interesting article entitled the 10 myths of NCTM learning where university professors detail some problems that face math education in US. The URL is:
http://www.nychold.com/myths-050504.html

My final comment on the "10 myths": A very informative read and thought provoking. Any math teacher or anyone who is interested in mathematics reform should read it. The curriculum reform as stated above is actually in response to the "10 myth" article.

Learner.org

2006-09-11T07:04:00.000-07:00

This is a very useful website for teachers. It contains tonnes of resources and videos on teaching high school mathematics. There are many interesting teaching methods for various subjects such as algebra and statistics. Many of these ideas are conveyed in videos. Although the videos are streamed, they have good resolution.

In sum, this site is well worth visiting.

Click here to access the site

Numb3rs

2006-09-07T04:55:00.000-07:00

Numb3rs is a CBS production of how FBI solves crimes with the help of a mathematician. In each episode, a different concept is introduced to aid the detectives to identify killers, track down suspects and solve robberies. Accompanying the show is a website that provides mathematical activities for the related episodes.

The URL is

http://www.cbs.com/primetime/numb3rs/ti/activities.shtml

For more information about the show, go to

http://www.cbs.com/primetime/numb3rs/index.shtml

This show is a hit, and hopefully it gets shown in Singapore. Clips from the video can be used to teach math in class.

Polya's "How to Solve It"

2006-09-06T06:13:00.000-07:00

This problem was taken from the above mentioned book, which is considered a classic on concepts mathematical problem-solving skills. It is a good example of how algebra can be applied to geometry, and how the heuristic "working backwards" (and many others) can be applied to solve the problem.

Given all quadilaterals with the same perimeter, which one would have the biggest area?

Thanks to CJ for the correction.

An Arithmetical fallacy

2006-09-06T06:07:00.000-07:00

Here is an interesting clip avaliable at google that shows 25 divided by 5 equals 14.
What is the common misconception in the 3 different workings?

http://video.google.com/videoplay?docid=7106559846794044495&q=ma+and+pa+kettle

5 problems

2006-08-25T19:55:00.000-07:00

These problems were selected from the book "In Polya's Footsteps, Miscellaneous problems and Essays" by Ross Honsberger. Each problem requires different problem solving skills. It will be good to note down what was the thinking process behind each solution.

1) 4 consecutive even numbers are removed from the set A={1,2,...n} If the average of the remaining numbers is 51.5625, which 4 numbers are removed?

2) Suppose u and v are real numbers such that

u+u^2+...+u^8+10u+^9=v+v^2+...+v^(10)+10v^(11)=8

Which is bigger? u or v?

3) Suppose a quadilateral ABCD has a circumcircle with centre O and incircle with centre I. Let E be the intersection of the diagonals AC and BD. Show that O, I and E are collinear.

4) Consider an acute triangle ABC. Suppose the median, angle bisector and altitude from A to BC cuts the angle BAC into 4 equal angles. What is the angle BAC?

5) Show that for any positive integer, there exists n consecutive integers such that none of which is an integral power of a prime.

Logical thinking in mathematical problem solving

2006-08-25T19:21:00.000-07:00

This example requires some logical thinking:

4 boys

4 boys A,B,C,D sit in a straight line, facing 1 direction. They cannot turn their heads or leave their chairs. A wall seperates A and B. They are told that 2 black and 2 white hats are placed on their heads.

A and C are wearing black hats, while B and D are wearing white hats.

D can see B and C's hats, C can see B's hat. B is facing the wall, and A cannot see any hat, since he is on the other side of the wall.
They are only given 1 chance to get their own hats right by shouting out the answer. After 1 minute, someone calls out?

1) Who called out? How can he be 100% sure?
2) Explain how each boy managed to figure out their hat's color.

( Problem courtesy of Cassandra)

Spot the difference

2006-08-21T00:58:00.000-07:00

AMTUVWY
BCDEK
NSZ
HIOX
FGJLPQR

The alphabet is grouped into 5 rows as above. How are these rows classified? What is the difference between each row?

Evil numbers

2006-08-07T02:39:00.000-07:00

In arithmetic, there are common terms such as rational numbers, irrational numbers, whole numbers, natural numbers etc. Here are some less known numbers:

1) Evil numbers: Numbers with a binary representation with even number of 1s. An example of an evil number is pi
2)Happy numbers
3) Unhappy numbers, ie numbers which are not happy.
4) Wasteful numbers: If x is a wasteful number, its prime factorisation has more digits (including the powers)than the number of digits of x. An example is 4,6,8,9,12,18 etc.
5) Frugal numbers: A non wasteful number and 1.
6) Economic numbers: Frugal numbers excluding 1.
7) I like this one the most: Lucky numbers

There are many other types of numbers. Here's a nice article on number gossip.

Constructing square root 2 for lower sec

2006-08-02T20:19:00.000-07:00

Squares

In the secondary 1/2 syllabus for math, we talk about numbers. For instance, rational numbers and irrational numbers. Now how can we show the students a line segment of length square root 2? Moreover, these students need not have prior knowledge of pythagoras' theorem. So showing them an isoceleles triangle with 2 sides 1 and hypothenus sqrt2 may not be the best way to go.

Let us consider 2 squares each of length 1. So each has area 1 (unit, can be cm, m or anything else to make things concrete, I'll just discuss in abstract) Question: From these 2 squares, can you construct a square of area 2?

The above diagram represents the solution. Cut the 2 small squares into half along the diagonal and piece them together to form the square on the right. That square will be of area 2. So the length of the square is sqrt2. And hence we have demonstrated the construction of a line segment of sqrt2 without using any fancy maths.

Math Olympiad Question

2006-08-02T08:25:00.000-07:00

From a math olympiad question targeted at lower secondary students:

Suppose a number x can be expressed as a sum of squares of the 4 of its smallest divisors, what is the highest prime number that can divide x?

For example,
130=(1^2)+(2^2)+(5^2)+(10^2)

Stars

2006-07-29T04:18:00.000-07:00

Star

We all know how a star looks like. The interior of a star is actually a pentagon. Let us now look at the vertices of the star. What is the sum of all the angles at the vertices?

We can generalise from there: Given a 2k+1 regular polygon, k>1, extend its edges to form a "star" pattern. What is the sum of the angles at the vertices?

Geometric properties of a circle

2006-07-26T20:29:00.000-07:00

quad

Consider the first picture, suppose the quadilateral as shown, with one vertex at the centre of the circle. Then we all know that the angle at the centre is twice the angle at the top vertex. Now suppose we have a quadilateral as shown in the second picture. How can we prove the converse? i.e. there exists a circle such that 3 of the corners cut the circle, and the centre of the circle coincides with the remaining vertex?

Relative velocity

2006-07-23T02:34:00.000-07:00

This is a pretty interesting question. I've heard from my student that none of his classmates could solve it. Even some of his teachers had problems with it. It did make me think for a while, but once you have the correct diagram, it should not be too difficult.

Suppose a plane flies North at 100 km/h. The wind seems to be blowing at a bearing of 030(degrees). If the plane reduces its speed to 50 km/h, the wind seems to be blowing at a bearing of 050. What is the actual speed of the wind?

zero divisors

2006-07-23T02:32:00.001-07:00

This is a common mistake students make. For instance,

x^2-x=0.

Some will divide both sides by x, and then obtain x=1. But this will result in a loss of a solution to the equation. Weird things start to happen when we divide through by a root (or zero). Perhaps we confuse students in the following case:

sinx =cosx, hence tan x =1.

The student will be confused. Why is it in the latter case we can divide both sides by cosx, yet in the former we cannot divide both sides by x?

Hence I think this example will show to students what is going on.

Why all numbers are equal to zero.

Let x=0. Dividing both sides by x, we have 1=0. So multiplying both sides by 2, we get 2=0, similarly 3=0,4=0 etc, hence all numbers are equal to zero.

Let the class figure out the mistakes themselves.

TAFN

The case of the missing 10 cents

2006-07-23T02:32:00.000-07:00

Suppose Ali, Bala, and John buy a ruler that costs $2.50. They each have $1, and they pool a total of $3 to purchase the latter. So the change is 50 cents. Now, if we give each boy 10 cents back (since we cannot split 50 cents evenly among the 3 boys), we would have 20 cents left. But by doing so, each boy would have contributed 90 cents. In all, they contribute 90 X 3 = $2.70. If I add back the leftover 20 cents, we would have

$2.70 + $0.20=$2.90.

So where did the 10 cents go?

Angle bisectors in triangles

2006-07-23T02:31:00.000-07:00

Figure 1

Geometry is indeed a fascinating topic. However, many students tend to have alot of misconceptions. One type of fallacy is to assume some property of a figure by looking at the diagram. The diagram, sadly is not drawn to scale. Hence even though a picture is worth a thousand words, but it takes a mathematician to explain it completely.

Take for instance figure 1. Given that the cevian AD bisects the angle at D, is it true that it also bisects the length AC?

Some students I've taught felt that it is true. But of course, it is not. A bit of simple mathematics will prove otherwise. I'll leave this to the reader to show the following:

If AD bisects the angle at A, then the ratio AD:DC=AB:BC.

TAFN

H1, H2, H3 syllabus fo A level mathematics

2006-07-23T02:30:00.001-07:00

As you'd have known by now, A level syallbus for Mathematics has been revamped. There is no longer any mechanics in A level math. In short, H1 Syallbus is for people who only need math for certain areas of application besides engineering. H2 is mainly concerned with science and engineering applications, while H3 is for research and varsity mathematics.

The outlines of the new syllabus can be found here:
H1
H2
H3

Interestingly, H3 syllabus consists of Geometry, ODE, Graph Theory and Combinatorics. Geometry is similar to the NUS course "Geometry: From ancient to modern", where one learns Ceva's and Menelaus' theorem. Furthermore, H3 exam questions will consist of proving statements given.

H2 syallbus is, in my opinion, Maths C plus a quarter of Further Maths. There is no longer any mechanics section, and even the statistics portion is reduced. Most notably, one is no longer expected to derive continous random variables, and t-test involving difference of 2 sample means is no longer included. The strange thing about the syallbus is that partial fractions is not mentioned at all. However, one would still need to use techniques involving the latter to do other topics such as series and integration.

The use of graphing calculators reduces the workload of the student. There will not be much questions on curve sketching, but rather more on questions on the behaviour of a function. For instance, finding asymptotes. Finding the inverse of 3x3 matrices is done by the calculator as well, and hence solving a system of 3 linear equations would not require one to use row operations anymore.

TAFN

Applications of maths

2006-07-23T02:30:00.000-07:00

I just started tutoring and my student is quite inquisitive. For example, he likes to ask what's the use of learning this math concept etc. I find this very encouraging, and I would like to share some application of the topics. But somehow, it's not as easy as I thought.

Take for instance, today I was going through functions. He asks me, so what's the use of learning inverse functions? I was quite stumped for a moment. Obviously, all these things are very important and apply to all sorts of situations. But to think up of one on the spot can be a daunting task.

But later on I realise I've been thinking in the wrong direction. Each concept has its purpose and applies to many areas of our lives. But not every concept can be substatiated by giving a real life example. Most math concepts are more likely to be used as a tool, or rather, a tool of a tool, to approach a problem. For instance, learning about what are injective functions or surjective functions seem rather meaningless for a first-time learner, but it is important in the study of graphs. For instance, we use such properties to interept the behaviour of a graph. And graphs are important in empirical data in many fields of study: be it in physical sciences, economics, engineering, statistics etc.

So that is what I mean by somethings being a tool of a tool. Concepts that seem to be meaningless are actually the foundation to build our "weapons" to handle real life problems. If weapons are indeed used to handle problems, then mathematical concepts we learn from 'O' and 'A' levels serve as the parts of a bullet.

However, I do lament that not enough real life examples are given to students to cultivate their interest in maths. Indeed, there are areas where applications immediately follow theory. For example, any student who has learnt differentiation knows how it applies to optimization problems. But there are many other areas left unexplored. For instance, how does complex numbers apply to real life situations?

If I put myself in a teacher's shoe, I do understand the predicament. It is often difficult to talk about an application without going in-depth into other fields. In other words, one would be inviting alot of "bombastic" terms and terminology in order to discuss an application. Alternatively, one can always just remark that such a concept is applied in a specific area of engineering say, but no one will truly be satisfied with this answer.

I still feel optimistic since the world is so vast, surely it is possible to collect some solid yet simple (to understand) examples to present to our students. Maybe in the event that it is truly difficult to find a simple example, it is best to quote a area of study as above, and guide the student in the right direction to find out more about it himself.

Perhaps it is due to my ignorance that I have not come across any good example of how complex numbers are used to real life. The example I used is geometry. I explained to my student about geometry and transformations. Then I compared cartesian geometry and looking at the same shape on a Argand Diagram. By comparison, and by applying some of the concepts he learnt earlier, he was astonished to see that transformations can be done so easily on the argand diagram, just by mere multiplication of a complex number to the orignal function. And I left it to his imagination in what disciplines such knowledge would be important.

If anyone has any good examples in any area of mathematics that we learn in 'O'/'A' level, please share with me. I would like to accumulate a wealth of information to help other teachers as well. Ultimately, the students will benefit the most.

Sudoku

2006-07-23T02:27:00.000-07:00

I've been thinking of 2 math questions recently. Let me just post it here.

The first one is an interesting question. We have been bombarded with Sudoku recently, you know, the addictive puzzle which we need to fill up 9 latin sqaures, with the additional restriction that every row and column cannot have a repeated number. I've been trying to compute the total number of puzzles that can be generated. It turns out that it is quite hard. Moreover, I found on a site that, according to this fellow, the total number of puzzles that can be generated is:

6,670,903,752,021,072,936,960

Note that this number is the number of puzzles which can produce a unique solution, not the number of solutions to Sudoku, which is obviously less. (perhaps someone can come up with a number?)

One more thing, it appears that Solving Sudoku puzzles is an NP problem. That is, no efficient algorithm can be used to compute an arbitrary Sudoku Puzzle.

All this information can be found here.

The second math issue is a really fundamental one, and it was pointed out to me by Jasper. Consider this trigonemetric identity:

cot (x)=1/tan(x)

Sounds familiar right? But did you notice something wrong with it? It appears that this isn't really the definition for cotangent fuction. It lies in the fact that when we subtistitute x=pi/2, or in fact take x to be any odd multiple of pi/2, we get something funny:

cot(pi/2)=0 yet, 1/tan(pi/2) is not defined. Note that tan(pi/2) is undefined. The left and right limits do not match. Although 1/tan(x) as x tends to pi/2 yields zero, this is just the limit and is not an equality. Hence this definition is false!!

The true definition for cot (x) should be

cot(x)=cos(x)/sin(x)

TAFN