[AM_20160115SEAV] Simultaneous equations? Absolutely!

Problem / Question

Introduction

     This question was from a Facebook group not dedicated to “Singapore math”.  The thing is, mathematics is really an international experience, especially with modern social media.

     Simultaneous equations can pose a challenge to students, but this one is absolutely more challenging, because of the absolute value.

The Absolute Value

Strategy

     The problem is: we do not know whether each of  x  and  y  is negative or otherwise.  That potentially raises complications.  There seems to be  4  cases to check.  However, by making assumptions  [H08]  separately and checking for contradictions [H07], we can narrow down the possibilities.  [H10]  Finally, we can simplify the problem to a regular pair of simultaneous linear equations   [H10]  and solve it by the method of elimination.  [H10, H11]

Solution

Heuristics Used

H04. Look for pattern(s)        [deciding what to eliminate]

H07. Use guess and check     [is x negative?  is  y negative?]

H08. Make suppositions        [is x negative?  is  y negative?]

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Olympiad

* GCE ‘O’ Level Additional Mathematics, “IP Mathematics” (challenge)

* other syllabuses that involve simultaneous equations and absolute values

* any precocious or independent learner who wants his/her mind tickled

[EM_20150412SCA] Is your Airport Design career “taking off”?

Question

An Airbus 380 has constant acceleration of  1 m/s2.  Its takeoff velocity is 280 km/h.  How long must the runway be at a minimum to allow the plane to take off?

Introduction

     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/s².

Solution 1 [“Elementary” Mathematics, speed-time graph]

     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.

Solution 2 [Physics, constant acceleration]

     We use the important formulas  v = u + at  and  s = ut + ½at².  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.

Heuristics Used

H02. Use a diagram / model

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels
* GCE ‘O’ Level “Elementary” Mathematics
* GCE ‘O’ Level Physics
* other syllabuses that acceleration, speed and distance
* precocious kids who always want to learn more

[S1_20150408DST] Funky Bus between Towns

Question

Introduction

     This is quite a convoluted word problem for a Secondary 1 (roughly equivalent of grade 7) pupil in Singapore.  There are a lot of data given and lots of unknowns.  How shall we cope with this?  One good way I recommend is to use a table to organise the information.  [H02]   Many primary school teachers teach the Triangle Mnemonic for the relationship between Distance, Speed and Time.  This continues to be useful in secondary school.

To get Distance, cover “Distance” with your finger, and you get Speed ´ Time.

To get Speed, cover “Speed” with your finger, and you get Distance ¸ Time.

To get Time, cover “Time” with your finger, and you get Distance ¸ Speed.

Solution

Remember: Tables are very useful to help to organise information before you try to solve the problem.  They also help you to formulate the algebraic equations correctly and quickly.

Heuristics Used

H02. Use a diagram / model

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels
* Secondary 1 Mathematics
* precocious primary school pupils
* anyone, young or old, who enjoys an algebra challenge

[OlymPri20150402RLD] Peter, James and the Dog

 Question

Introduction

     This is a cute primary school olympiad type of question.  I vaguely remember reading a similar quiz question from Readers’ Digest (?) many years ago about some bumble bee flying between two trains going towards each other.  Or something like that.

     If you try to solve this using “advanced mathematics” like using the sum of two infinite geometric series you can get the answer, but it is quite a swirling mess, like this:-

Another way to slice the dice

     Is there a short cut?  Yes!  That requires thinking about the problem in another way.  Notice that the dog’s speed is the sum of Peter’s and James’ speeds.  Imagine ... if James is not moving, but Peter is running at  3 ms^-1,  from Peter’s point of view the Earth would be pushed backwards at 3 ms^-1  and James would appear to be going towards him at  3 ms^-1.  But James is running towards Peter at  2 ms^-1, so from Peter’s point of view, it seems that James is coming towards him at  5 ms^-1.  Likewise, from James’ point of view, Peter appears to be coming at him at  5 ms^-1.  This is the concept of relative speed.  Notice also that the relative distance (gap) between James and Peter is reducing at this speed.  This is because at Peters’ end the gap is reduced at 3 ms^-1  and at James’ end the distance is reduced by  2 ms^-1, giving a total gap-reduction speed of  5 ms^-1.  With these perceptive observations, the answer falls straight out.

Solution

     Since the dog’s speed (5 ms^-1) is the sum of Peter’s speed (3 ms^-1) and James’ speed (2 ms^-1), it is always covering a distance at a speed which is the same as the speed of the closing of the gap between Peter and James.  Hence the total distance travelled by the dog must be the same as the initial gap, which is  1 km. 

Moral of the Story

     Sometimes, you do not need advanced maths, but acute observations.

When two entities are moving towards each other, their relative speed is the sum of their speeds.  This is also the same as the rate at which the relative distance (gap between the two) is closing.

Suitable Levels
* Primary School Olympiad
* anyone who is interested in creative maths problem solving