My Bicycle Commuting - Should be able to estimate the speed from video

I think the car highlighted in the video is speeding. Checking the time stamp of the frames and knowledge of the lane markings spacing, I think should be able to estimate the speeds of the cars. (30 Oct 2013)

Is this a spot the mistakes advertisement?

Sent from my Samsung Omnia II GT-I8000

The first mistake I noticed is in the units.
Then I noticed the error in 640, before the words "check again!" got my attention.
On further reading and thinking...

Maths or English?

Sometimes it is not easy to set questions that distinctively test on a single subject, especially when understanding the question that is expressed in some written words.

Here is a primary 2 question:

Natasha plants 6 trees on each side of a square garden. There is a tree at each corner of the garden. How many trees are there altogether?

Depending on different understandings, these may be the solutions given by students:
(a) 6 at each side and a square has 4 sides, 1 at each corner and a square has 4 corners. Thus, there are 6x4 + 1x4 = 24 + 4 = 28 trees

Snail up a wall

I didn't learn nor read any text regarding Model Drawing as taught in Mathematics classes in Singapore. I am trying this out. Let see if I am on the right track.

Question: A snail is climbing up a wall 20m high. Everyday it climbs up 5m & slips down 2m at night. How many days will it take the snail to reach the top of the wall?

A typical old-days approach would be: each day up 5m and down 2m giving net up of 5-2=3m. To cover 20m, it would take 20/3 = 6 2/3 days. However, this is not exactly correct. This knowledge led me to draw a diagram that gives the above wrong answer. This is a common mistake by many students

One directional correlation

When high oil prices were cited as reasons for transport fare hikes, we all thought that was logical.

When oil prices dropped but not the fares, it was explained that these two were not linked directly.

Hmmmm... linked or not linked?

It was further explained, “If there is a direct link, it must go both ways. If we have that system, there would (have been) a 40-per-cent increase in public transport fares.” What commuters got, instead, was a 0.7-per-cent average increase.

Hmmmm...

Firstly, I doubt anybody believe a "direct" link such as that described above. If there were some who believed so, then they have probably been misled into thinking so when high oil prices were cited as reasons for fare hikes.

Secondly, there is indirect link. Thus, if a 40% increase in oil prices correspond to 0.7% average increase, then when oil prices drop, there should be some decrease.

Are the audience being treated as primary school kids?

The new CEPAS-compliant ez-link card

I read a letter in TodayOnline about the new CEPAS-compliant ez-link card by LTA. The author claimed that the new card would require him to make more top-ups. I scratched my head..

In the old system, there is a S$3 deposit and when stored value (excluding the deposit) is less than S$0, one cannot board a MRT train or bus.

In the new system, there won't be any deposit, and when stored value is less than S$3, one cannot board a MRT train, while for buses, minimum stored value must be more than that required for the rest of the journey of the bus.

I really don't see this change can lead to more frequent top-ups necessary for anyone. ... why? The current top-up procedure ask how much one wants to top-up. Thus, regardless if there is deposit, or the amount of deposit, or the amount of minimum value (for MRT), there is no reason to do more top-ups. Unless, the author has been using a procedure that I have not come across where he was asked how much stored value he wants in the card, and he doesn't intend to change this amount when using the new card. However, I don't think there is such a procedure.

Even for those who mainly travel by bus, they may instead top up lesser, but by only 0.1 top-up lesser assuming that their minimum value is S$2 instead of 3 and each time they do S$10 top-up.

However, if the S$3 deposit on existing EZ-Link card will not be refunded, then yes. Similarly, it would still be just an additional 0.3 top up, using the above assumptions.

I believe there are many not so good or correct letters sent by many, but to be selected for publication is slightly more than the common excuse of human err. Instead of thinking that it was some mistakes by the editors, I suspect it was intentional to create some heated discussions. They are facing low volume of letters to Voices and needed to take articles without asking from blog. There is one from reclaimed.wordpress.com but I couldn't find the article in this inactive blog.

By the way, I pity the LTA and TransitLink being wrongly accused by many. From what I understand, it was the IDA initiative and directive to adopt the CEPAS standard that led to the change, while many unhappy comments were targetted at TransitLink and LTA.

Which are of more concern? Detected or undetected HIV carriers?

35 HIV cases detected. What a mild and unalerting title for a piece of Breaking News!

I loves maths, and thus, let me do some simple maths here.

Known figures are:
35 HIV cases detected in 6 months in the CHANGI General Hospital (CGH) among 3 out of 10, or among about 6500 people who opted to be tested.

What does this imply?

It means there might be in total 35 * 10/3 = 116.667 HIV positive among the 6500* 10/3 = 21666.67 patients who visited CGH.

By further extrapolation, it means there might be 116.667 / 21666.67 * 4M = 21538.46 HIV positive people in Singapore.

OK, there is a bias here ... , as patients cannot be used to represent general population. Thus, it needs to be corrected by multiplying by the proportion of people who visited hospital in the past 6 months in Singapore. I don't have it, but let use a low ratio of 1/100, there are about 215.38 HIV positive people in Singapore. This ratio of 1/100 seems too low since 215.38 is only about twice of 116.667 while Singapore has more than 2 similar or larger hospitals like CGH.

Thus, if we use the number of hospitals in Singapore of similar size, which I also don't have the exact figure, but can safely state as > 5, considering SGH, TTSH, AH, CGH, NUH, and many smaller clinics. Then, the number might be 116.667 * 5 ~= 583 HIV positive among only those who visited hospitals in the past 6 months. Note, this is only the estimated number of HIV positive patients, NOT general population.

What we need to pay attention to is not the 35 HIV cases detected. We really need to pay attention to the possibly 116-35=81 undetected HIV cases! From the above estimate, we need to pay attention to the possibly 81*5=405 undetected HIV cases! And if considering the whole population, there are possibly much much more than 405 undetected HIV people who have not visited hospital in the past 6 months, i.e. they are likely to be healthy people.

What are they doing? Without knowing that they are infected, they are living their normal life. What's their "normal" life? At least quite a few of them have a "normal" life of having regular casual sex with multiple partners, including their spouse.

After the first posting, I read this part:

...many of those who did not want the test were elderly or repeat patients who had been tested before. Tests were not offered to those under 21 years old, the age of consent.

OK, seems not as bad, but even after discounting, the estimated HIV positive, healthy and sexually people out there who don't know they have the virus are still among the hundreds!!!

Re-considering my first job - as a tutor

I have found my backup income! Quoting from the report, Singapore emerging as 'tuition nation'

Physics tutor Phang Yu Hon has nearly 90 students and earns nearly 20,000 Singapore dollars (15,000 US dollars) a month, the report said. The 41-year-old gave up his research engineering job in 1994.

Probably without word of mouth, I imagine can get ~20 students and monthly ~S$ ... 4000 is quite nice. All thanks to the high pressure by teachers, who unshamefully asking parents to get tutors for their kids. Anyone interested for a mathematics tutor may check out my answers at Yahoo!Answers.

It isn't just the money. 20 students/week at 1.33 students/session, 5 days/week and 2 hours/session means about 3 sessions/day for 5 days/week, or about only 30 hour work-week.

Be careful how you ask a question

A JC friend returned from US to this region recently for a short visit. He is into research on student's thought process when learning math. I took a quick browse of his PhD Thesis and found an experiment by Cramer, Post and Currier (1993). I tested it out at Yahoo!Answers. However, it doesn't work as expected the first time. Then, I rephrase the way I present the question, and ... it works the second time.

I suggest readers to try it out first and see the results I got from Yahoo!Answers.

Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?

Link to my experiments: Does your answer make sense? (for students)? and Quick, how many laps Sue ran?

Indeed, even in normal life events not necessary mathematical, the way we ask does affect the way the other party responds.

Standing in Yahoo!Answers

I am recently given the "Top Contributor" title after having 30% of my answers voted or selected as the best answer, and earned enough points to Level 4!Too bad no $ attached :(

I should consider solving maths problems as my hobby :)

Easy = Difficult

Recently weeks there are discussions on a difficult mathematics examination paper in this year PSLE.
It seems that some questions in it were difficult.

To me, there is not much different if an easy or a difficult question is set. If it is easy, then most will probably score. If it is difficult, then most will probably not score. Overall, it doesn't really affect my score/grade. Rather, I dislike easy question because I must make the efforts to do and get it right, whereas for a difficult question, I may choose to skip it but my score for it may be the same with many who spent time doing it yet didn't get it right.

Furthermore, I am one of those [selfish ones] who when spotted a faulty question will keep quiet and move on, since eventually the teachers have to void that question. I hate it when at later times, especially near to the end of the given time, someone raise the issue and the teacher announce an correction to the question. I think that is being unfair by making correction half way through an examination.

What matters is when a question is easy or difficult only to you or a small group of students. Otherwise, no problem.

Thinking further, I suspect it was the nice but unrealistic pictures that parents/teachers have given to students that causes some of them to be annoyed by not being able to score 100% due to some out of scope difficult question. I think this is an issue that should be addressed. The over ideal system in school is harmful in terms of getting students ready for the world. People who grew up in system-less environment probably have much higher ability to adapt and excel in the real world.

Did the shop boy cheated or earned the money?

Another of my answer chosen as the best by the asker:

Asked by ashishv_patel
Three friend deside to give a one presant to common friends so they had puchase one item by 75 $ by contribution of 25 $ but shopkipper deside to give a discount for 5$ so shop boy gone for to return 5$,in between shop boy had think to take 2 $ from 5$ at last he give only 3$ to that three friends & they had distribute 1$ each
now costing of presant is
3 friend * 24 $ = 72 $
shop boy had taken + 2 $
72 + 2 = 74
but actualy frist they had expence 75 $ - 74 $
so where is 1 $ ? ? ? ? ?????????????
Best Answer - Chosen by Asker
A variant of another version where hotel room rate at $10.

Lets try the accounting method:
Balance sheet for the 3 friends:
Debit: $25 * 3 = $75 to buy the present
Credit: $3 from the shop boy
Balance: spent $72
Asset: the present that costs them $72

Balance sheet for the shop:
Credit: $75 revenue
Debit: $5 discount
Balance: $70
Asset: less a product that sold at $70

Balance sheet for the shop boy:
Credit: $5 from shopkeeper
Debit: $3 discount to that 3 friends
Balance: $2

Thus, cheated $2 from the shopkeeper and that 3 friends... errr, or did he earned it?

For Primary levels mathematics teachers

I'm delighted to find this site, which in many ways have similar views as I on problems and misunderstanding when learning mathematics.

abstractmath.org

Pattern-recognitional mathematics

Another best answer by me :)

---

Asked by quantum_dot: Why do children make zero and decimal position mistake while dividing?
For e.g.: 21 divide by 2 is 10.5, but many times children end up with 1.5 as the answer
---

1 year ago - 6 answers
Best Answer - Chosen by Asker

I suspect this error is possible because the child is not learning maths, but merely pattern recognition. If the child learns numbers and calculation the mathematical way, she (from my experience, higher chance is a she) will know 1.5 is very unlikely to be the answer, since it is so much smaller than what would be expected for when a value near to 20 is divided by 2.

However, if she were pattern recognizing, 1.5, 10.5, 15, 15.0 are similar patterns.

I recommend more focus needed on the 'feeling' of values and their representations using Arabic numbers. I think more practices on estimation might help. E.g. ask them to estimate, using their perception on the values that are represented by numbers to give rough descriptions of the answer. E.g. before applying method to exactly calculate the value of 21/2, ask them to describe the answer. Will it be bigger than 21? Will it be smaller than 2, will it be near to 0, 10, 20, 30, etc.? Will there be decimals? Etc.

Another advice, forbid the use of calculators! Lower the importance of getting the exact answer.

Just to share, I was amazed when a 20+ year-old working adult couldn't figure out why 3/2=1.5, but he can tell you in split seconds that a meal that cost $3.00 shared by 2 persons means each has to pay one dollar and fifty cents, or one-fifty, which in numeric form is $1.50! Why, 3/2 is so different from 1.5, and he probably find it complicated to rationalize the pattern-transformation in order to transform 3/2 to 1.5, but $3.00 and $1.50 are money to him, not numbers, and he has good sense of money :)

When brain is too free..

An interesting puzzle posted in Yahoo!Answers 
that I managed to solve, and gotten chosen as the best.

===
Let a, b, c be the lengths of the sides of a triangle. 
Show that:

a/(b+c) + b/(a+c) + c/(a+b) < 2

Now see if you can describe the shape of a triangle 
for which the above expression is very close to 2.

=== Best Answer - Chosen by Asker

Stop here for a moment, as this is your chance 
to solve it before reading my answer below :)


Somehow, I started with the last part first, 
when it is close to 2.

So, using limits, try a triangle 
with one infinitely short side.
lim_a->0 (a/(b+c) + b/(a+c) + c/(a+b))
= lim_a->0 (0/(b+c) + b/c + c/b)
but in such a triangle, where a->0, b->c
it is actually 
lim_a->0_and_b->c (a/(b+c) + b/(a+c) + c/(a+b))
= lim_a->0_and_b->c (0/(b+c) + b/c + c/b)
= lim_a->0_and_b->c (0/(2c) + c/c + c/c)
= 2

Ok, now for the main question, 
to prove a/(b+c) + b/(a+c) + c/(a+b) < 2
Let a < b < c,
a/(b+c) < 1/2 because b+c > 2a
c/(a+b) < 1 because in a triangle, 
            sum of lengths of two sides > the other side.

What about, b/(a+c)? a < b, c > b but c < a+b, 
so c < 2b, so a+c < 3b, and thus b/(a+c) > 1/3 
(Oh oh.. no use)

Probably have to consider the first two terms together, 
we have
a/(b+c) + b/(a+c) = (a^2+ac+b^2+bc)/(b+c)(a+c)
= (a^2+ac+b^2+bc)/(ab+ac+bc+c^2)
< 1
because ac=ac, bc=bc, a^2 < ab, b^2 < c^2

So, (a/(b+c) + b/(a+c) + c/(a+b)) < 2

Must Pythagoream Theorem be used?

This question is found in the Yahoo!Answers as one of the question designed to test knowledge of the Pythagoream Theorem.

1. Sam hiked 48 miles directly east and then 36 miles directly south. Find the shortest distance in miles from the point where he ended up to his starting point.
A. 60 miles
B. 36 miles
C. 48 miles
D. 72 miles

For me, MCQ question should not be solved by calculation because answers are suggested and the little time allocated, it is designed not to be calculated.

If it is used for training, it is better to demand students to think and explain briefly on all the 4 suggested answers, instead of ignore them, just work out an answer and search to see which of the suggested answer matches it.

Here, obviously, B and C are wrong because from the question, Sam travelled 48 miles in one of the direction and both directions didn't make an angle of less than 90 degrees.

Thus, left with two possible answers A and D. Since 72 can be nicely expressed as 2x36, and if it is D, then probably you should have learnt another theorem about this nice feature. Thus, it is unlikely D.

So, assuming the MCQ is not faulty, and by considering the time and marks allocated ratio, it is wise to choose A and move on. When all is done, then can return to do checking with the left-over spare time. If the MCQ is faulty, it doesn't waste your time too since none will get it correct :)

Now, for the purpose of learning, it is better not just to calculate, but to figure out any nice pattern if the distance were 72 miles. By Pythagoream Theorem,
a^2 + b^2 = (2a)^2 ==> b^2 = 3 x a^2 ==> b = sqrt(3) x a ~ 1.7a > 1.5a ==> 1.5a < b, but
1.5 x 36 = 54 > b, thus distance cannot be 72. Alternatively, similar can be derived in a circle when a chord has same length as the radius, the chord that joins the other two ends of this chord and of the diameter that touches of this chord, is of length sqrt(3) x radius. Lastly, just a quick note, when using the standard way, it is faster to use mental calculation: 36^2 + 48^2 = (12x3)^2 + (12x4)^2 = 12^2 (3^2 x 4^2) = 12^2(5^2) = 60^2.

Non-mathematical skills

Some non-mathematical skills taught in mathematical lessons

1. cancellation
2. change the sign
3. move the decimal points
4. move the term to the other side of the equation

Why mathematics teachers teach non-mathematical stuffs?

Recently, I gave tuition to secondary 1 students, and I think nowadays mathematics teachers are skipping steps, while last time only students skip steps.

For example, 3x + 6 = 18, and the next line we put 3x = 18 - 6.
Mathematically, we subtracted 6 from both sides because we want to single out the variable x.
However, this is usually not being taught, or it was just mentioned without any intention of teaching by assuming that students won't understand. As a results, I saw in student's workings this: above the first equation, there is an arrow from above the number 6 going across to the right side of the number 18. That is, what students have learnt was they just "move"/"throw"/"bring" the number 6 over to the other side, and "change"/"convert" the plus sign to a negative sign. What is this!!! Learning mathematics, or learning magical or pattern rules of manipulating symbols?

There are many more such non-mathematical stuffs that are taught in schools!

Fairness in exam

Recently there is a hooha in the primary 6 Science examination in Singapore (PSLE). Some students sob while doing the paper with ‘difficult’ questions as complained by these students and their parents. For me, there is nothing much to worry about when there is a very difficult question in the examination paper, because probably most will lose the marks for the question. Similarly, there is not much to cheer about when there is a very easy question because almost all will score the full mark for that one. Thus, I suspect that the problem lies in the attitude that they believe and expect full marks if they have put in the sufficient amount of preparation. Thus, for those who have put in more than sufficient efforts and who believe that they are entitled to full marks if there is no fault from them during the examination, they probably have felt being cheated or the situation is very unfair to them.

Should the examination system be perfect with total fairness? I prefer not. Considering the education in school as the preparation process for students to survive in society, it should not differ from the actual world too much. The real world is not fair, surely not in perfect state. Although the school should provide a protected environment for students to experiment and explore and learn, it should not be too protected and over perfected such that students will find it quite tough when they eventually have to live in the outside world.

pattern recognition or maths

Recently a friend says that his six year-old child knows the answer for 100+1, 100+2, 100+8, etc. but doesn't know the answer for 99+1. It seems that the child can recognize the pattern when a '1' follow by two '0's, a '+' symbol and a number of 1 or 2 digits, the answer can be formed by replacing 1 or 2 zeros from the right with the 1 or 2 digits on the right of the symbol '+'. Then, it became clear why he doesn't know the answer for 99+1 as this is a totally different pattern. This seems to be an unknown/unsolved problem for many students, who may score high grades for mathematics in tests and examinations by similar pattern recognition as above, without proper understanding of the mathematics concepts.

From my tutoring experiences, girls tend to use pattern recognitions more than guys. If this is not corrected, they will find mathematics very difficult by the time they reach secondary 2 when the patterns become much more complicated and thus requires many exceptions to their pattern recognition rules.

I am quite against the use of phrases based on pattern recognition, such as “canceled off the x in the numerator and denominator in a fraction” and “bring the x over to the right hand side”.