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Research tell us what to do with mathematics education crisis

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Research tell us what to do with mathematics education crisis

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Expressions
with surds in their denominators are cumbersome. The good news is that we can make the
denominators into *rational numbers*, which are nicer. **Rational numbers** those that can be expressed as a *ratio* of
integers i.e. they are (proper or improper) fractions or can be converted to
fractions. Whole numbers are also part
of rational numbers because you can always put them upon a denominator of 1; e.g.
2 = ^{2}/_{1}, so 2 is a
rational number.

The standard trick for simplifying expressions
with surds in their denominators is to **rationalise the denominator** by mutiplying the numerator
and the denominator with its conjugate surd. For example, the **conjugate surd** of Ö5 + Ö2 is Ö5 – Ö2. Just change the + to – or the
– to +. Let us see how the magic works.

In mathematics, “rationalising” does not mean you give some reason or excuse for something that you know you have done wrong. It means “make it into a rational number”. Why does

(

Since squaring “gets rid” of square roots,

* revision for GCE ‘A’ Level H2 Mathematics

* revision for IB Mathematics HL / SL

* other syllabuses that involve surds

* precocious kids who always want to learn more

This “Primary
5 mathematics” (actually an upper secondary Olympiad) logic puzzle has gone viral. It has been making its rounds in various
forums in Singapore
and overseas, stumping adults and children alike. It is actually a parody of an old puzzle. Can it even be solved? It seems that there is no information given by
each parties that we can exploit. Actually
there is! In a subtle way ...

In the
beginning, everybody knows that Albert knows only the month and Bernard knows
only the numerical day of the month.

When Albert
tells us “I don’t know when Cheryl’s birthday is, but I know that Bernard does
not know too.” he is leaking out information (from his knowledge of the month) that
the day of the month appears more than once and cannot be (June 18 or May 19). Actually, the original phrasing is more like “**If** I don’t know when Cheryl’s birthday
is, **then** Bernard does not know too.”.
The person who set this question merely
changed the names of the people and the dates, without appreciating the subtle but crucial difference between a
statement of *fact* and an *implication* (an “if ... then ... ”
statement).

Ruling out June
18 and May 19, we also know that Albert knows that the birthday month is
neither June nor May. Otherwise, how
would he have been so confident in saying that he knows Bernard would not know Cheryl’s
exact birthday? So we can eliminate
those months.

Bernard
acknowledges the above state of affairs and the embedded hint. With the choice narrowed down and with his
knowledge of the numerical date, he now knows Cheryl’s birthday. Since we know that Bernard knows Cheryl’s
birthday, we know that it cannot be a numerical date that appears more than
once (otherwise he would not have been able to know). So we can cross out July 14 and August 14.

Now Albert would
telepathically thank Bernard for this helpful hint. Because now he is able to deduce Cheryl’s
birthday with his knowledge of the month. That would mean that this cannot be a month
with two candidate dates. We blot out
the August dates and see for ourselves the only remaining possibility.

This puzzle was solved using the process of
elimination and analysing our knowledge of what each party knows and can
know. Thus we successively narrow down
the possibilities until the answer becomes obvious. Here we learn that

knowledge of other people’s
knowledge can itself give us knowledge.

This principle is actually employed in **cryptology** (the use of secret codes) which
finds applications in fields like *banking*, the *military* (cf. interesting story of how the German Enigma code was broken in WorldWar II) and *communications*. As an example, radio communication can tell the
enemy of troop positions and warn of an impending attack, and that is why *radio silence* is imployed as a
precaution. Sensitive information in certain organisations
is restricted on a “*need to know*” basis.

* other syllabuses that involve knowledge or epistemology

* application of mathematical principles in real life

* for all people interested in logic puzzles

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This is an
olympiad type of question that tests one’s use of logic. Sadly, logic is seldom taught in schools,
except maybe in courses like knowledge inquiry, theory of knowledge, or in selected topics like geometry. Mathematics
is actually very much connected with logic, but it seems to be regarded as
difficult to teach. Being able to use logic
and think critically is very important in our lives in various aspects.

Let us
study how we can analyse
A *conditional
statement* is a statement of the form “If *H* then *C*.”
An example would be: “If the blue litmus paper turns red, then the liquid
is an acid”. The front clause *H* (the part about the litmus paper
turning red) is called the *hypothesis* (or *premise*). The latter clause *C* (the part about the acid) is called the *conclusion*. Every conditional statement has three other
related forms: the contrapositive, the converse, and the inverse.

For example, let say there is a NC16-rated movie and you can watch it only if you are at least 16 years old. So*H* :=
“watch NC16” and *C* :=
“16 years old and above”. The *conditional
statement* means “If you want to watch the movie, then you must be 16
years old and above”. The *contrapositive*
is “If you below 16 years old, then you cannot watch the movie”. The *contrapositive*
always has exactly the same meaning as the *original*
conditional statement. If one is true,
so is the other. If one is false, so is
the other.
The contrapositive is merely phrased in a different
way.

*original conditional statement* and its *contrapositive* in one camp, and the *converse* and the *inverse* in the other.

For example, let say there is a NC16-rated movie and you can watch it only if you are at least 16 years old. So

The *converse* says “If you are
16 and above, then you will watch the movie.”
The *inverse* would say “If you don’t want to watch the movie, then
you are not 16 and above.” Note that the
inverse is the contrapositive of the converse.
The *inverse* always has exactly the same
meaning as the *converse*. If one is true, so is the other. If one is false, so is the other.

The converse (and the inverse) has a different meaning
from the original statement. If the
original statment is true, the converse may or may not be true. In our
example, “If you want to watch the movie, then you must be 16 years old and
above” is true, but the converse “If you are 16 years old and above, then you
want to watch the movie.” may not be
true, as some people who are 16 years old and above may want to do something
else. Like reading a book. Or playing golf. So you see, there are really two camps: the
Let us
analyse the given problem using logic.

The
statement ‘any card with a letter **A** on one side always has the number **“1”** on the other side’ can be rephrased
as ‘If letter **A** appears, then the
other side is **“1”**’. Let us analyse each position in order. For (a), since the letter **A** appears, we definitely need to know
what is behind the card to test whether the claim is true. For (b), note that the claim does not say ‘If
the letter is not **A**, then the other
side is not **“1”**’ (*inverse*)
nor ‘If the number is **“1”**, then the other side must be an **A**.’ (*converse*). It is possible
that **A** is on the other side and we
are fine. But it is also possible to
have a non-**A** with the number **“1”**. The given condition does not prohibit
that. So, if the other side is not an A,
it is still OK. So the other side can be
anything, and it does not matter. For
(c) we definitely have to flip to check the other side in case the other side
is an **A**, then the claim would be
proven false. For (d), since the card is
not an **A**, it is irrelevant. The claim is talking about **A**. It is not talking about **B**. We can summarise the
analysis in the diagram below:-

* syllabuses that involve logic and epistemology

This is an equation involving algebraic
fractions, usually for secondary 2 (approximately grade 8) pupils in Singapore . Many students (and teachers?) like to use the
“cross-multiplying” method, as shown in Solution 1. A usually more efficient method is to
multiply every term by the *Lowest Common Multiple* (**LCM**) of all the denominators appearing
in the equation, as shown in Solution 2.

Note that division by zero is not allowed. Furthermore, in algebra, it is dangerous to
cancel or divide by an unknown quantity, because there is a possibility that
you are dividing by zero. So any
division or cancellation by an unknown quantity *must* be *justified beforehand*.
Mathematics is not a game of blind senseless manipulations. If you look at the second solution, which is
short and sweet (only 4 steps), multiplying through by the LCM of denominators
not only avoids this awkwardness, but it clears all the fractions in one fell
swoop. The solution takes only 4
steps, and it is in fact the recommended method. All students, whether “good” or “poor” in
maths, should use the second method. Teachers
who refuse to use/teach the LCM method (out of habit, or because their own
teachers taught them otherwise, or because this makes them or their pupils “uncomfortable”)
are really doing the weaker students a huge disservice. You are widening the achievement gap. The better students are better, precisely *because* they use better methods. The longer one’s working is, the higher the
chances of making mistakes and the more time is wasted. If the “weaker” pupils have to jump through
lots of hoops to achieve a certain standard before they are allowed to learn
this “advanced” method (actually it’s just the normal method), they will have
to unlearn their old method and may get confused as they learn this method. A triple whammy! All learners need to practice anyway, so one
might as well practice the correct thing right from the beginning and learn
good habits (striving for efficient, effective, elegant solutions). So please, please, please everyone: use the
LCM method!

* Secondary 2
Mathematics (“Algebraic Fractions”)

* GCE ‘O’ Level (“Elementary”)
Mathematics Revision

* other syllabuses that
involve algebraic Fractions

* precocious children who want
to learn algebra

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An Airbus
380 has constant acceleration of 1 m/s
^{2}.
Its takeoff velocity is 280 km/h. How long
must the runway be at a minimum to allow the plane to take off? |

A practical
question for airport design, perhaps? I
present two solutions. The first uses a
graphical method (*speed-time graph*) which is in the (“Elementary”) Mathematics
syllabus and the other uses formulas for motion under *constant acceleration* taught
in Physics. Whichever method is used,
remember to convert from km/h to m/s.
The target velocity is ^{700}/_{9}
m/s, and the time to achieve this
is ^{700}/_{9} s starting from rest, since the acceleration is 1 m/s^{2}.

The *speed-time
graph* is very useful because it is able show the acceleration (as the *gradient* or *slope* of a straight line) and at the same time the *area under the graph* gives the numerical
value of distance travelled. If the
acceleration is constant, we usually we get a trapezium. But since the aeroplane starts from rest, we
get a triangle (see diagram below). All
we need to do is to calculate the area under the graph and get the answer.

It is interesting to observe that if we used the average speed^{700}/_{18} m/s, we would
also get the answer because the area under the graph (yellowish green
rectangle) is the same as the area of the triangle. This trick works for constant acceleration,
but it may not work in other situations.

It is interesting to observe that if we used the average speed

We use the important
formulas *v* = *u* + *at* and *s* = *ut* + ½*at*^{2}. In
our example, *u* = 0 because the initial
velocity is zero (the airplane starts from rest). This makes our calculations very easy. If we compare the two methods, you find that
the calculations are very similar, and we get the same answer. Remember that speed = |velocity|
the magnitude of velocity. In this
relatively easy problem, the *velocity*
means the same as *speed* because we
are going in a straight line and in one direction only. In other situations, this may not be so.

H02. Use a
diagram / model

H05. Work
backwards

H09. Restate
the problem in another way

H13* Use
Equation / write a Mathematical Sentence

* other syllabuses that acceleration, speed and distance

* precocious kids who always want to learn more

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1^{2}
= 1 ´ 1 = 1 Ö1 = 1

2^{2}
= 2 ´ 2 = 4 Ö4 = 2

3^{2}
= 3 ´ 3 = 9 Ö9 = 3

4^{2}
= 4 ´ 4 = 16 Ö16 = 4

5^{2}
= 5 ´ 5 = 25 Ö25 = 5

A number like 4
can be represented by a real square
whose sides have length 2 units.
Note also that if you take the square root of a perfect square, you
always get a nice whole number.

How do you
deal with numbers that are not perfect squares? You factor out as many perfect squares as
possible. This would eventually lead to surds
with small numbers, which are more manageable.
Here are some examples:-

With this weapon in our hands, let us kick some butt.

Using
perfect squares reduces your square roots to surds involving square roots of
prime numbers, which are easier to combine or cancel. This gives a short and sweet solution. In mathematics, always try to do things by
the cleanest way (if you can).

H10. Simplify
the problem

H11. Solve part
of the problem

* revision for GCE ‘A’ Level H2 Mathematics

* revision for IB Mathematics HL / SL

* other syllabuses that involve surds

* precocious kids who always want to learn more

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