[OlymPri_20150524MSCR] Round and Round, Twice or Thrice?

Question

Introduction

     If you think the answer is  2 cm (since the wheel rotates two rounds), you are wrong!  This tricky question is taken from a previous year Asia Pacific Mathematical Olympiad for Primary Schools competition.  In the original question, there was no colouring.  I added colours just to make the distinction between the wheel and the track a bit clearer.

Visualisation

     There is some subtlety in this question.  It is something we often do not notice unless we really think hard about it.  To get a handle on what is really happening, let us imagine we made a mark on the wheel in its original position, indicated by a blue dot at a twelve o’clock position.  Let us imagine rolling the wheel clockwise.  Note that the centre of the wheel will move in a bigger circle in an anti-clockwise (American: counter-clockwise) direction within the orange circular track.  Actually it does not matter which way you roll, the centre of the wheel goes round the centre of the track in a direction opposite to that of the turning wheel, which always remains in contact with the track.

Since the blue dot on the wheel turns two rounds, by the time the wheel reaches the bottom half, the blue dot must again be on top or at a twelve o’clock position.  Note however, the point of contact between the wheel and the circular track (indicated by a green dot whenever possible) is not the same as the blue dot!  The green dot is actually a dynamic dot (it is not always the same point on the wheel) whereas the blue dot is always the same dot on the wheel but the wheel is being rotated.

Note that at the halfway point, the green dot is at the bottom of the wheel while the blue dot is on top of the wheel.  Notice also that, relative to the wheel, the green dot goes in the opposite direction as the blue dot and they actually crossed over somewhere along the way!  Actually the green dot has made one-and-a-half turns with respect to the wheel.  Remember that the green dot is not a fixed point on the wheel, but it measures how much the wheel and the track have been in contact.

As the wheel continues to roll back up, the green dot makes another  1½  rounds.  So altogether the green dot moves through  3  rounds while the blue dot rotates only  2  rounds around the centre of the wheel!  The green dot is the one that matters.

Solution

     circumference of track (measured by green dot) : circumference of wheel = 3 : 1

     Since radius is proportional to circumference,  the radius of the track is  3  cm.

Remarks

     Yes, the solution is that short.  But the thinking behind it is profound.  But do we need to draw all the diagrams as in the visualisation above?  I did that to explain to you.  Actually I imagined it in my mind.  If imagination is difficult, you can act it out by drawing a big circle and then using a small coin to simulate the rotation around the track.  I actually drew a rough sketch by hand  to convince myself that my thinking was accurate.

     Notice that I used a proportionality argument.  If you know how to use the concept of proportion, you can make the working short and sweet.  There is nothing wrong in using the formula                     circumference = 2p ´ radius.

This formula just says that the circumference of a circle is proportional to its radius, and the constant of proportionality is  2p.  Your working would look like this

                                  circumference of track = 3 ´ circumference of wheel

                                      2p  ´ radius of track = 3 ´ 2p  ´ radius of wheel

After cancelling out the  2p,  you would get

                                       radius of track = 3 ´ radius of wheel = 3 cm

You get the same conclusion, but using the proportionality method, you do not need to bother about the  2p.

     By the way, the centre of wheel traces out the path of a circle of radius  2 cm.  See the red arrow in the first diagram on top.  The path traced out by the blue dot looks like an  epicycle.  In the old days,people thought that the sun, moon and planets rotated around the earth in epicycles.  This also reminds me of Spirograph,which is a toy that allows you to use your coloured pencils to create very beautiful patterns.  [Click to search for images of Spirograph and patterns produced.]

Heuristics Used

H01. Act it out

H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H06. Use before-after concept

H09. Restate the problem in another way

Suitable Levels
* Primary School Mathematics Olympiad
* anybody game for a challenge relating to imagination, circumference and lengths

[Pri20150303SJM] Race to the Bottom

Question

Thinking / Planning

Noting that Sarah spent all her money in both scenarios and that 22´(18´something) = 18´(22´something) , I am going to first guess that Sarah takes 18 days and 22 days to spend all her money in the 1^st scenario and the 2^nd scenario respectively. [ Heuristics: H07. Use guess and check & H08. Make suppositions ]  Then I try to adjust my educated guess.

Johari spent more money in the second scenario.  The difference in his spending among the two cases is 6, so we need

                10´(number of days₂) – 12´(number of days₁) = 6

Solution

If Sarah takes 18 days and 22 days respectively to spend all her money, then

the difference in Johari’s spending among the two cases is

                10´(22) – 12´(18) = 4

´³/₂:         10´(33) – 12´(27) = 6

Ans (a): Sarah has $22´27 = $594.
Ans (b): Johary has $(21+12´27) = $345.

Commentary

If the amounts of money involved were in the trillions, we would have thought that Sarah and Johari are certain countries, wouldn't we?

Anyway, a parent from a Facebook parent-support group posed this question asking for a simple solution without using ratios.  This question seems to be the equivalent of a system of simultaneous equations in four variables.  A few of us tried various methods to solve it, but all were quite complicated.  I knew this can be solved using algebra, but struggled for some time to give a simple solution.

I guess all solutions (even the one above) would have some notion of “ratio” hidden in it.  The reason is: every time you multiply or divide by something, there is actually a ratio involved.  But I hope this solution is “easy” enough.

P6HQ2000S46 Ratio, Proportion and Percentages

Introduction

     This is a challenging question on the topic of ratio, proportion and percentages.  I shall again illustrate the process of solving this mathematics question with metacognition and heuristics, which are applicable for all levels, and all topics.  The Bar Diagram Model is a very well known heuristic.  But there are many other good problem-solving heuristics as well.  There is a key to solving this kind of question and in order not to spoil the fun, I shall let my readers have a go at it first.  Please think a bit about this problem, and then read on.

Stage 1:  Understanding the Problem

Can you explain the problem in your own words?

     Siti paid \$ 24 for some towels at a discount of 20%.  Now, with the discount, she got three more towels than without the discount.

What concepts is this problem testing you on?

     Ratio, proportion, percentages, price, unit price …

Can organise the given information?

     We can put the information into a table, like this:-

Figure 1.  Heuristics: ‘Use a table’ and ‘compare before/after’

What are you asked to find? 

(a) the number (quantity) of towels Siti bought with the discount

(b) the price of each towel (the unit price) before the discount

Stage 2:  Planning the Method of Attack

How are things related?

     Total Price = Quantity (how many she bought) x Unit Price (how much each cost)

This relation is similar to

     Distance = Speed x Time

Have you solved a similar problem before?  How did you solve it the last time?

     Yes, a Distance-Speed-Time problem.  We used a Triangle Mnemonic.

Figure 2.  Triangle Mnemonics and Analogous Thinking

How is this problem different?

     Instead of ‘Distance’, we have ‘Total Price’

     Instead of ‘Speed’, we have ‘Unit Price’

     Instead of ‘Time’, we have ‘Quantity’

     Ah!  Maybe we can use a similar Triangle Mnemonic for Total Price, Unit Price and Quantity.

Remember: What are you trying to find?

     The number of towels - the Quantity.  OK … so … if we use a finger to cover ‘Q’ we see ‘T’ over ‘U’.  That means

     Quantity = Total Price ÷ Unit Price
     But this problem looks more difficult, because there are so many unknowns …

Don’t give up.  Try some heuristics.  (emotional management)

Can you restate the problem in another way?  (using a heuristic)

     The discount of 20% means she paid 100% – 20% = 80% = ⁴/₅ of the original price.  Without the discount, she either would have paid more for the same number of towels, or for the same amount of money, she would have gotten less. …

Can you simplify the problem?  (using a heuristic)

     Suppose the unit price was halved.  Then Siti would get double or 2 times the number of towels.  What if … the unit price was ¹/₃ of the original?  Then she would get thrice (3 times) the number of towels.  If the price is ¹/₄ of the original, then she would get 4 times …  It looks like the cheaper the things get, the more you can buy … there is a pattern … the numbers seem to go the opposite way … the fraction seems to be inverted (the reciprocal) … why? … why? … why?  Oh!  It’s because

     Quantity = Total Price ÷ Unit Price

The division ‘÷’ causes the fractions to ‘turn over’.  I see!  J  So, since the price is ⁴/₅ of the original, Siti would have bought  ⁵/₄  of the number of towels without discount!

Can you draw a bar diagram for this?  (using the famous Singapore heuristic)

     Yes!

Figure 3. Before-After Comparison Bar Model

Now, can we solve the problem?

     Yes!  Yes!  Yes!  It is now very easy.

Stage 3: Execution

(a)      1 part    =  3 towels  (after discount: 3 more towels)

          5 parts   =  3 ´ 5  = 15 towels

(b)       4 parts =  3 ´ 4 = 12 towels  (before discount)

          12 towels  ¬¾®  \$ 24

              1 towel  ¬¾®  \$  2

Answer:
(a)  She bought 15 towels.

(b)  Before the discount, each towel cost  \$2.

Stage 4:  Evaluation

     Let us check the answer:-

Figure 4.  Checking the answer

     Are we correct?

     Yes!  The numbers fit nicely.

Final Presentation

       The final presentation of the solution can be rather succinct.  The foregoing discussion seems long because all the thinking process are explained in full detail.  The checking of the answers can be done in pencil as rough work.

Figure 5. Final Presentation

Stage 5:  Reflection

What did you learn by solving this problem?

     Although all stages are important, the hardest part was stage 2, the planning stage.  This involves trying various heuristics to look at the problem from different angles.  [This is the real mathematics.  Patience, observation and creativity are needed.]   The key to solving this problem was to note the inverse ratio relationship between quantity and unit price (given a fixed total price).  Once we managed to find the key, we draw the appropriate diagram and the rest of the calculations are pretty straightforward.

In future if you encounter a similar problem, how would you solve it?

     If there is a similar problem in future, I can use the 5 stage problem-solving process:-

     1. Understanding

     2. Planning

     3. Execution

     4. Evaluation

     5. Reflection (learn and transfer for use in future)

Use metacognition (thinking about my own thinking, ask myself questions) to guide myself through the five stages.  Manage my own emotions.  If I get stuck, do not give up.  Look at the problem in different ways.

I can use the following heuristics:-

     · Bar model diagram (yes, but there are others …)

     · drawing a table

     · comparing ‘Before vs After’

     · using a Triangle Mnemonic

     · thinking of a similar problem

     · simplifying the problem

     · rephrasing the problem in another way (e.g. convert percentages to fractions)

Dear blogger: why don’t you just present the answer?

     Many book authors and teachers already do just that.  But do pupils’ actually improve much?  Mathematics still remains a mystery to many pupils.  They think that certain people are born geniuses and that there is always a way that these geniuses somehow have the knack of finding the right steps straightaway (and they themselves cannot).  All they do is try to memorise the algorithm and reproduce it, regardless of whether they understand it.   There is research to show that using worked examples alone does not improve pupils’ mathematics much, and that heuristics and metacognition are actually very important in mathematical problem solving.

     “Give a person a fish; you have fed him/her for today.

       Teach a person to fish; and you have fed him/her for a lifetime”

     If you take the time mimic these heuristic and metacognitive processes, your maths will definitely improve by leaps and bounds.  You still need to know the basic concepts, which most teachers teach, but you need to learn to put them together.   I hope you get empowered in this process.