[Pri20150302PBW] Pebbles on a Road Backwards

Question

A jar contained some pebbles. Nigel took out half of them plus 2 more for display. Jenny took out half of the remaining pebbles plus 1 more. Finally, Alice took out half of what remaining plus 5 more for her project. In the end, there were only 8 pebbles left. How many pebbles were in the jar at first?

Solution

We can trace what happens to the remaining pebbles at each stage, and represent the information given in a sequence as above.  Note that when “Nigel took out half of them plus 2 more for display”, we would have half and then two less the number of pebbles.  Similary for the Jenny’s and Alice’s stages.

                            

Now we just do the opposite of all the operations.  The inverse operation of ‘–5’ is ‘+5’, the inverse operation of ‘´ ½’ is ‘´ 2’ and so on.  Working our way backwards, we arrive at 112.

Ans: There were 112 pebbles at first. 

Commentary

     Remember: In mathematics, there is no such thing as "the only way" to solve problems.  There are many ways to skin the cat, as they say.  Other methods to solve this question include: using algebra, and “the branching method” (taught by some tutors/teachers) which tracks both the remaining pebbles as well as the ones taken away.

     If you do use algebra (of some sort), remember to use different letters e.g. v, w, ... etc to refer to different things.  Likewise, do not just write  ‘½’  to represent ½  a unit, otherwise it could be misinterpreted as literally the number ½.  I suggest you can surround the ‘½’ with different shapes (e.g. circle, square, triangle) .

Precision is very important in maths, as well as in life. You don't want an imprecise person to be your aircraft designer, accountant or doctor, do you?

Heuristics Used

H05. Work backwards

Suitable Levels

* Primary School Mathematics

* other syllabuses that involve whole numbers and fractions

* any learner who is up for a challenge

[Pri20150302MCP] Multiple Choice Questions with Penalty

Question

Solution

The ratio of the number of words spelt correctly to that of words spelt wrongly is 3 : 1.  I use triangle to surround the ratio units.  Multiplying by 10 and 4 respectively, we obtain 30 and 4 units respectively.  The score accrued is 30 – 4 = 26 units which corresponds to 130.  Dividing by 26, we get 1 triangle unit corresponding to 5.  And that is the answer required because it 1 triangle unit represents the number of words spelt wrongly.

Ans: 5

Check: 15´10 – 4´5 = 150 – 20 = 130.   Yes!  Woo hoo!!

Reflection

This question is similar to this fish and chicken cost problem, but there are no monetary units (e.g. dollars or cents) involved.  Also, because of the penalty for wrongly spelt words, we subtract instead of add.  We need to be mindful of the context of the question and to apply the appropriate arithmetic operation.

[Pri20150302FCW] The Cost of Fish and Chicken

Question

Solution

The ratio of the cost of fish to that of chicken is 1.5 : 1 or 3 : 2.  I choose this nice whole number ratio and write them in triangle units.  You can choose draw any shape you want to surround the ratio unit actually.  Multiplying by 3 and by 4 (the numerical dollar costs), we obtain 17 triangle units equal to 76.50.  From here we can divide by 17 to get the equivalent of 1 triangle unit, and then multiply by 3 to get the equivalent of 3 triangle units.

Ans: 1 kg of fish costs $12.50

Commentary

Once again my Ratio Units method solves this problem quickly and you can use this method to also present the answer in a matter that even the teacher who is marking this can understand J.  

[Pri20150302BGP] Boys, Girls and Party

Question

A group of pupils met at a party. Each child exchange phone number with everyone else. Melissa exchange phone number with 5 times as many boys as girls. Peter exchange phone numbers with 4 times as many boys as girls. How many boys and girls were at the party?

Solution

In the set up stage, we read the question carefully and transfer the information into a good representation.  I am using my Distinguished Ratio Units method i.e. using different shapes for different types of ratio units.We set up as follows:  One row for boys (B) and one row for girls.  Reserve one column for ‘original’.  In the first scenario, if you subtract 1 (Melissa) from the girls, you get 1 unit (let’s use a circle to mark it) and then the boys would be 5 circle units.  In the second scenario, if you subtract 1 (Peter) from the boys, you would get 4 units (let’s use triangle) for the boys as compared to 1 triangle unit for the girls.

Going back to the ‘original’ column, we use circle 5 to represent the number of boys (as this was unchanged in the first scenario), and we use triangle 1 to represent the number of girls (as this was unchanged in the 2^nd scenario).

With the set-up done, now my plan is to try to match up one of the types of units (either circle or triangle units).  I choose to match up the triangle units.  Take the relation indicated in yellow for Melissa that says ‘1 triangle unit minus 1 gives 1 circle unit’.  When we multiply this by 4, we get ‘4 triangle unit minus 4 gives 4 circle units’ and then we link this up the the existing triangle 4 (highlighted in yellow on the right).  We see that from circle 5, when you subtract 1 and then subtract 4, you get circle 4. 

This means that 1 circle unit is 5.  Adding 1, we see that 1 triangle unit is 6.  5 circle units makes 25.  The total number being 5 circle units and 1 triangle unit, we add up 6+25 and get 31, which is the answer. 


Commentary

In this word "problem", you start of with two original quantities and when you add/subtract something from one of the quantities, you get a certain ratio, and when you add/subtract from the other quantity you end up with another ratio.  This is a typical difficult type of question that stumps many students and many parents who trying to help them.

Many word "problems" in Singapore's Primary School Mathematics curriculum are just linear simultaneous equations in two variables in disguise.  The pupils will revisit this sort of maths problems in their secondary school years under algebra.  For primary school, simple algebra is taught but not emphasised.  Singapore is famous for her "Bar Model Method".  This method is good for visualisation for pupils at the lower primary levels.  However, at primaries 5 and 6 (~ roughly equivalent to grades 5 and 6), the numbers get bigger and the types of problems get harder.  You often have to cut or draw many bars.  It is especially challenging to draw bars to a suitable estimated length and if you draw the bars wrongly, you often have to erase the whole thing and redraw.  In their year 6, pupils take the Primary School Leaving Examination (PSLE), a high-stakes examination that determines the secondary school and stream (track) that they will go to. .  Many anxious parents and tutors cope by teaching the kids algebra.  However, algebra seems to be socially frowned upon and used only as a last resort.

Last year, reflecting on the above-mentioned challenges, and upon realising that many people use the same letter "u" (for unit) or same bar to represent different types of units and end up getting confused, I cooked up my Distinguished Ratio Units method.  It is powerful and avoids the need to draw and redraw the diagram, hence allowing the pupil to concentrate on the thinking and solving, rather than wasting time with drawing.  I admit that my method is actually algebra in disguise.  The bridging tactic where you equalise one unit to allow the other type of unit to connect is like substitution and elimination.  One triangle unit and one circle unit, for example, can be interpreted as variables x and y, preparing the pupils for secondary school.  The triangle and circle units also link with the concept of ratios which they have learnt in their middle primary years.  By the way, you do not need to use triangles and circles.  You can use squares, diamonds, heart shape etc.  Who says you cannot be creative in mathematics?

I hope you like my new method.

Here is a related problem.

H2Maths VJC/2007/P2/Q9 The Power of Rephrasing

Introduction

     Today we discuss a question taken from one of the top junior colleges in Singapore’s 2007 preliminary examinations (final school internal examinations before the actual A-levels).  The seemingly difficult problems dissolve quickly using the right heuristics (rules-of-thumb / guidelines / problem-solving tactics).  Once the correct approach is determined, the calculation is simply a matter pressing buttons on the graphing calculator (e.g. TI-84 Plus or Casio fx-9860G).  This question can be tackled using heuristics.

     Try to guess what heuristic(s) will be useful in solving the above problem.  In a typical examination question, they will not tell you what topic or concepts are being tested in that question.  Try to determine what concepts are involved.

     Let’s tackle the question part by part.  As usual, we use the 5 step problem-solving process with metacognition (self-prompting, self-monitoring).

Part (i)

 

Part (i) Stage 1:  Understanding the Problem

What concept(s) is being tested?

     That seems to be something to do with probability …

Do you understand the question?  Can you put it in your own words?

When Ai Wan (爱玩 “loves to play”?) rolls the die for the eight time, he gets exactly three ‘6’ to win the prize.  What are the chances of this event happening?

Part (i) Stage 2:  Planning the approach

Is there a heuristic you can use?

     How about “9. Restate the problem in another way” …

So how do you rephrase the question?

     “win on 8^th roll” means “two ‘sixes’ on the first 7 rolls” and “a ‘six’ on the 8^th roll”.

How will you proceed?

Break down the problem into parts (“11. Solve part of the problem”).

     (1) find probability of “two ‘sixes’ on the first 7 rolls”

     (2) find probability of “a ‘six’ on the 8^th roll” (this is easy.  Obviously  ¹/₆)

Then multiply them together, since these are independent.

How do you write out probability of “two ‘sixes’ on the first 7 rolls” in symbols?

P(X = 2)

Think backwards:  What is X?  How do you define it?

X  is a random variable.  It is the number of ‘sixes’ in the first 7 rolls.

What distribution does it follow?  How do you know?

It follows a Binomial Distribution: Because there are only two outcomes: either you get a ‘six’, or you don’t.  There is a fixed number of trials and these are independent, giving a constant probability. 

So what’s the number of trials (n)?

There are 7 rolls, so seven.  Each roll is a trial.

What is the probability of “success” (p)?

One out of six.

Is this a probability distribution function (p.d.f. ) or a cumulative distribution function (c.d.f.)?  How do you calculate it?

This is just probability for a single value of  X, so it’s a p.d.f.  It can be found from the graphing calculator.  [ Or use the formula ⁿC_x p^x q^n–x = ⁷C₂ (¹/₆)²(⁵/₆)⁵ ]

Part (i) Stage 3:  Executing the plan

Part (i) Stage 4:  Evaluating the answer

Does this answer feel correct?  Is it believable?

     Yes.  It’s quite small, which tallies with my intuition, as getting three sixes in 8 rolls is highly unlikely.  If I get an answer like 0.4 something, I might smell a rat.  If I get a probability that is less than 0 or more than 1, then I know for sure it’s definitely wrong.

Part (ii)

Remark

     This problem is related to the Geometric Distribution, which is not in the syllabus.  Nevertheless it is solvable using the knowledge contained in the syllabus, so it is within the student’s reach.

Part (ii) Stage 1:  Understanding the Problem

What concept(s) is being tested?

     probably probability again … maybe Binomial or something related

Do you understand the question?  Can you put it in your own words?

Ai Ying (爱赢 “loves to win”?) already has got 2 ‘6’ and 2 ‘1’.  That means she needs just one more ‘6’.  She needs to throw exactly four more times (no more and no less).  What are the chances of this happening?

Part (ii) Stage 2:  Planning the approach

Is there a heuristic you can use?

     “9. Restate the problem in another way” … but the problem still seems complicated because of the initial conditions (two “sixes” and two “ones” ) …

Is there a way to simplify the problem or another way to think about it?  Is there a feature about the situation … ?

     Ah!  Since all the trials are independent, what happens in the past does not affect the future.  This is the memoryless or forgetfulness property of the (Bernoulli) trials.  That means I don’t need to worry about the two “sixes” and two “ones”.  I can forget the past!  It is as though I can just start from a clean slate!

So what does it mean to say “Ai Ying needs to throw exactly four more times”?

     It means after throwing three more times, she does not get any six, and then she gets a six on the next throw.

How do you write this out mathematically?

P(S’, S’, S’, S)

What do you mean by S?  What is S’ ?

S  is the event of getting a ‘6’ in a roll of the die.  S’  means not getting a ‘6’.

Now, how do you proceed?

The chain of events can be broken down.  (“Split the problem into smaller parts”)

P(S) = ¹/₆.  P(S’) = 1 – ¹/₆ = ⁵/₆.  Since the four events are independent, I can just multiply their probabilities  all together!  This is easy!

Part (ii) Stage 3:  Executing the plan

 

Part (ii) Stage 4:  Evaluating the answer

Does this answer feel correct?  Is it believable?

     Yes.  Again the answer is quite small, which tallies with my intuition.

Part (iii)

Part (iii) Stage 1:  Understanding the Problem

What concept(s) is being tested?

     Probability, Binomial Distribution and … er … rephrasing?

Do you understand the question?  Can you put it in your own words?

Ai Du (爱赢 “loves to win”?) cannot win in eight rolls, so he needs to roll some more.  What are the chances of this happening?

Part (iii) Stage 2:  Planning the approach

Is there a heuristic you can use?

     “9. Restate the problem in another way”

So how do you restate the question?

 “requires more than eight rolls of the die to win a prize” means “after 8 rolls, he does not get three ‘sixes’”.

Which means?

Which means “after 8 rolls, he gets only 2 or less ‘sixes’”.

How do you write the required probability mathematically?

P(D < 2)

Think backwards:  What do you mean by D?

D  is a random variable that counts the number of ‘sixes’ within 8 rolls.

What distribution does it follow?  Why?

Binomial Distribution, again.  Same reasons as in part (i).

So what’s the number of trials (n)?  What is the probability of “success” (p)?

n = 8,  p = ¹/₆.

Is this a probability distribution function (p.d.f. ) or a cumulative distribution function (c.d.f.)?  How do you calculate it?

Because it is a “<” probability, it’s a c.d.f, which can be found from the graphing calculator, or checking tables.  [ Calculation by hand is possible, but tedious.  Also, there is a recurrence formula that can help a bit, but this is out of syllabus. ]

Part (iii) Stage 3:  Executing the plan

 

Part (iii) Stage 4:  Evaluating the answer

Does this answer feel correct?  Is it believable?

     Yes.  The answer is high, which tallies with my intuition.  Since it is hard to win, my gut feel is that you would very likely need more rolls of the die to win.

Stage 5:  Reflection

What did you learn from solving this problem?

     I learned that the memoryless (forgetfulness) property of Bernoulli trials (i.e. the type of trials involved in the Binomial distribution) is useful to allow me to disregard past events and simplify the problem.

     I learned that whenever I encounter a maths problem that seems difficult, I should not despair.  I should try the following heuristics:-

     * rephrasing the problem (heuristic 9)

     * breaking the problem into smaller parts (heuristic 11)

     * writing in mathematical notation (heuristic 13)

     * thinking backwards (heuristic 5)

     * simplify the problem (heuristic 10)

Rephrasing is particularly useful, as it helps to tackle all three parts of this exam question.  Even though part (ii) was on the fringes of the syllabus (just barely in the syllabus), we could still solve it by using heuristics, metacognition and using what we know already.

Conclusion

     Heuristics and metacognition will guide you when you seem to be in uncharted territory.  Use what you know (basic probability) to tackle what you do not know.  Do not be afraid.  Have faith.  May the Heuristics be with you!

 

JCCDQBHWHCB037(ii) Combinatorics : Grouping and Insertion Method

Introduction

     This question is from a source that does not credit the original source.  In the original question was poorly worded.  It did not have the words “the letters of the word”.  Instead of “each vowel must be separated”, it said “a vowel must be separated”, which is might mean there is just one such instance.  This is ambiguous.  I have taken the liberty to rephrase some parts of the question to make its meaning clearer.  In this article, I shall discuss only part (ii) of this question.

Stage 1:  Understanding the question

What is the given in the problem?  Can you organise the information?

     Though not absolutely necessary, it is helpful to draw a diagram that separate the letters into vowels and consonants and write the stack up the same letters in columns.

     There are  5  vowels (of which  O is repeated)  and  6  consonants (of which  N  and  S  are repeated).

Can you explain the problem in your own words?

     The letters of the word ‘CONNOISSEUR’ are re-arranged, which means that all the 11 letters are used.  Each vowel must be separated from another with exactly one consonant, which means that the letters must contain the pattern  “v c v c v c v c v” (where v = vowel, c = consonant).  Important: Note that the question does not say that the first letter must be a consonant.

Stage 2:  Planning

Have you seen a similar problem before?

     Yes, but this looks a bit more challenging.  There are more possibilities as first letter need not be a consonant.

What heuristics can you try?

     ·  Solve part of the problem

     ·  Split the problem into smaller problems

What topic-specific tactics can you try?

     The “v c v c v c v c v” pattern can be treated as a group (Grouping Method).  Since there are six consonants, there are two more “c”s (consonants) in the full pattern.  This looks like a problem that can use the Insertion Method.

Stage 3:  Execution

The number of ways to insert the group = ³C₁ = 3

     [these are the patterns “ccvcvcvcvcv”, “cvcvcvcvcvc” and “vcvcvcvcvcc”  ]

For each pattern,

     the vowels can be arranged in  ^5! / _2!  ways (division because there are 2 ‘O’)

     the consonants can be arranged in  ^6! / _{2! 2!}  ways (division because of 2 ‘N’ and 2 ‘S’)

Hence the total number of ways is

Stage 4:  Evaluation

Is the answer correct?

     Yes, the answer is correct.

Stage 5:  Reflection

What have we learned by solving this problem?

     We have learned once again that heuristics and metacognition are useful in solving mathematical problems.  Specifically, we have used the following heuristics:-

          ·  Drawing a diagram

          ·  Solve part of the problem

          ·  Split the problem into smaller problems

     We have also used the following techniques that are useful for combinatorical problems:-

          ·  Grouping Method

          ·  Insertion Method

          ·  Division Method (for dealing with repeated letters)

     It is also important to understand the problem correctly and not make unfounded assumptions.  If the wording is not clear, you may want to rephrase it.

JCCDQBHWHCB014 Combinatorics : Permutations & Multiplication Principle

Suggested Approach and Solution:-

     Since the order is important, we use permutations (which are just multiplications) instead of combinations.  No over-counting or division is involved.

 

The guys can be chosen and arranged in ¹⁰P₃ = 10 x 9 x 8 = 720 ways.

The gals can be chosen and arranged in ⁶P₃ = 6 x 5 x 4 = 120 ways.

Total number of ways = 720 x 120 = 86 400 ways.