[S1_20160117AFNS] Much Ado About Nothing?

Problem

 

Introduction

     Assuming no typing errors, this is a tricky Secondary 2 question involving an equation with algebraic fractions.  How to solve it?  How to present the solution?

Solution

Remarks

[1]   Once the LHS expression has no meaning, it would not even make sense to continue.

[2]   This is a proof by contradiction type of argument.

It turns out that not all algebraic equations are soluble (or solvable).  This problem is one case in point.  The “unknown”  m  cannot be ⁵/₂ because that would make the expression undefined.  But if you substitute any other value, you always end up with nonsense like “15 = 0”.  So no matter what, there is no solution.  In other words, there is no value of  m  that you can substitute into the equation that makes it a true statement.

Heuristics Used

H05. Work backwards

H08. Make suppositions

Suitable Levels

* Lower Secondary Mathematics (Sec 2 ~ grade 8)

* GCE ‘O’ Level “Elementary” Mathematics

* other syllabuses that involve algebra

* any learner who is interested in algebra

[EM_20160204PBIE] “Hillarious” Mathematics of Politics?

Problem

Six coins are tossed to decide a result for either “C” or “S”.  Assuming that the coins are fair, and that the results of the tosses are independent, calculate the probability that all the tosses are in favour of “C”.

Introduction

     Hot in recent news is the story of the purported six coin tosses that were needed to determine certain county delegates in the race between Hillary Clinton and Bernie Sandersin the state of Iowa.  All six coin tosses were in favour of Hillary Clinton, and the result is so improbable that some people said it was “hillarious”.
     How is the probability calculated?  This is an example of mathematics in real life events that has the potential to affect the United States of America, and the whole world (including Singapore).

Solution

Discussion

     In order to make the calculation, we make two assumptions: 

          (1) that the coins were fair, and

          (2) that the coin toss results are independent.
     So, what does it mean that the coins are “fair”?  It means that the probability of getting a “heads” is the same as the probability of getting a “tails”, which means ½ for each.
     Coin toss results can be “heads” or “tails”.  These are examples of events.  An event is something that can happen or not happen, and we associate a probability with it.  The probability is a number that indicates how likely the event happens.  It is between  0  and  1  inclusive.  Zero probability means a practically impossible event.  A probability of  1  means a practically certain event.  [The reason for me using the word “practically” is technical, which I shall not discuss.]  If the events do not affect one another (i.e. in our case, the coin tosses are not affected by the other coin tosses) then the events are said to be independent.  If the events are independent, then we can simply multiply the individual probabilites together, as above.  If the events are dependent, the calculation would be more complicated.

     So are the coin toss results valid?  I do not know.  All I can say is: improbable does not mean impossible.  Mathematics cannot tell whether the above assumptions (1) and (2) are correct.  But at least I “lay all the cards on the table”, so that astute students of probability know the basis of these calculations.  It is up to you to decide, but at least you would have made a mathematically-informed decision.  There could be other twists to the story, which is beyond the scope of this article.  This is one of the reasons why you need to learn mathematics carefully and think critically, whether or not you would become a mathematician,engineer, teacher or have a mathematics-intensive career.

Heuristics Used

H08. Make suppositions

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels

* GCE ‘O’ Level “Elementary” Mathematics (Number patterns, with algebra)

* other syllabuses that involve probability

* anybody in the whole wide world!

[EM20151129CGCB] Bisector of a Chord in a Circle

Problem

Introduction

     This “elementary” mathematics question poses a challenge because it actually testing Coordinate Geometry and Circle Geometry.  In the setting of tests and exams, there seems to be a trend of combining topics.  To solve this problem successfully, students need to know that when a chord is bisected (cut into two equal parts), the line segment joining its mid-point to the centre of the circle will be perpendicular to the chord itself.  Thereafter, we can proceed with Pythagoras’ Theorem.

Solution

Remarks

     There is actually no boundary between topics and even subjects.  Things to be learned are separated into topics only to facilitate teaching of the material.  Students are encourage to adopt a more wholistic view of knowledge.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

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

* GCE ‘O’ Level Additional Mathematics (revision)

* other syllabuses that involve geometry and coordinate geometry / analytic geometry
* whoever is interested

[S2_20151122IXBQ] A Bonus for your Index Fun?

Question

Introduction

     This is a “bonus” question which most likely came from an Integrated Programme (IP) school.  It really tests the wits of students’ knowledge of indices and problem solving tactics.  I present two solutions without the use of logarithms.  The first solution uses reciprocal indices and the matching of the base ^a/_b.  In the second solution, we match up the indices to  xy.

Review of Important Laws of Indices

Solution 1

 

Solution 2

Remarks

     No logs from any forest were harmed in the process of making this blog post.  J

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* challenge for Lower Secondary Mathematics (Secondary 2)

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

* other syllabuses that involve algebra and indices

[EM_20151120QGXS] Quadratic Graphs by Completing the Square

Question

Introduction

     This “Elementary” Mathematics question is usually pitched at secondary 3 (» grade 9).  It is quite a standard type of question, but from my experience, many students do not know how to begin.  The wording of the question (especially part (a)) may throw off some people.  The first stage in solving any mathematics question is always to make sure you understand the question, to know what is expected of you.  Once this is done, this question is actually quite straightforward if you know what to do and know where to look.  The coefficient of  x²  is  a = 1, so the question is not that tricky.  Part (a) is simply instructing you to complete the square.

Review

 

Solution

Tip:  You could get the y-intercept by using your fingers to cover the x² and x terms.  There is an even faster way: just look at the constant term!  You get the answer  5  immediately!

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H11. Solve part of the problem

Suitable Levels

* GCE ‘O’ Level “Elementary” Mathematics

* any syllabus that includes quadratic functions and parabolas

* anyone who is interested 

[S2_20150513IXS] The Power of Repeated Chunks

Question

Introduction

     This is a question on indices (taught in secondary 2 or 3) that looks daunting.  There is a way to unravel it.  Look at the equation carefully.  Can you see anything that looks like it is repeated?  What about that small little “+1”?  How do you deal with it?

Reminders and Tips

    

Solution

Summary

     The key to solving this question is to apply the laws of indices and recognising repeated chunks.  This allows one to see the overall structure of the expression and this often leads to a simplification of the expression involved.  It is good to recognise the powers of small prime numbers.  Like atoms, prime numbers are the building blocks of all whole numbers.  Using prime bases allows us to compare indices more easily.

Heuristics Used

H04. Look for pattern(s)   e.g. repeated chunks

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Mathematics (usually Sec 2)

* “Elementary” Mathematics or “E Maths”

* Additional Mathematics (revision)

* other syllabuses that involve indices

[S1_AFMLCM_20150412] Conquering Algebraic Fractions

Question

Introduction

     This is an equation involving algebraic fractions, usually for secondary 1 (approximately grade 7) 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.

Discussion

     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!

Suitable Levels

* Secondary 1 Mathematics

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

* other syllabuses that involve algebraic Fractions

* precocious children who want to learn algebra