Let’s say your house is two hundred metres from a supermarket.
Walk, buy, return home in half an hour.
The bus stop is one hundred metres from your house.
Walk for five minutes and you arrive.
There are four bus services that lead to a nearby train station. They each take two minutes to reach there.
That is convenience.
Every month, without fail, you receive a salary for putting in your time and effort at your workplace. If your employer has an exceptional year, you get months of bonus. Best of all, they pay you a percentage of your salary into your retirement account monthly.
That is security.
You can have lunch anytime. 10.30 a.m. 11 a.m. 12 noon. Or even 1 p.m. You can take as long as two to three hours to have lunch or a short 10 minute booster nap.
That is freedom.
Convenience. Security. Freedom. If you were to give up one of them, which one would it be?
Here is a hint.
If you want your child to master something, give up convenience. Convenience leads to comfort. Comfort breeds stagnancy. Stagnancy kills mastery. This is the path of a craftsman. Think Archimedes, the one who derived the formula for the surface area and volume of a sphere.
If you want your child to excel QUICKLY, give up freedom. Ask his or her teacher to feed them with tons of practice worksheets until they are so familiar with solving the exams questions that they ace them in less than 6 months. Think sinecure.
If you want your child to find out what they love to do in life, give up security. Ask them to take the path off the proven track. This path is filled with adventures and dangers. There is no guarantee they will succeed or survive. Think Steve Jobs.
Most choose the path of a full time job in exchange for security and convenience. After all, job stability brings consistent income.
It also brings in consistent complacency. There is no reward to work hard or work harder. Or go the extra mile. You are paid the same amount, regardless of how many benefits you brought for your employer.
That could be why your child is reluctant to do more than a pass in math. Their environment is filled with security and convenience. They are rewarded for doing minimal work. Why work so hard when the reward is still the same?
Which of the three paths are you asking your child to follow?
Convenience. Security. Freedom. Choose two.
PS: Is passing math enough for your child? Read on in the guest post that I wrote for Arianna Huffington’s Thrive Global.
https://thriveglobal.com/stories/passingmathisuselessyouaresettingyourchildupforfailure/
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Schools can only measure hard skills.
Communities have to support with soft skills.
Schools can’t do everything.
Neither can communities.
Each of us plays our part.
Provide an environment of emotional support.
So that when we fall, we have pillars to support us.
It’s the roots, not the stem, that shape our resilience.
I help diligent parents and hardworking children improve their math scores.
I teach. They do the work. We discuss the responses.
Rinse and repeat for vast progress, ascension.
Ascend and see things for what they truly are.
What you seek for your children is a good education.
To solve problems. To be creative. To be imaginative.
Most of all, to be kind and compassionate.
Education is about compassion.
Treating people as people, expanding their growth, awakening their giants within.
It’s about growth. It’s about compassion.
]]>1. Get into a fight or quarrel with his or her other siblings at the slightest irritation
2. Get into a fight or quarrel in school and the discipline teacher calls you to inform you about it.
3. Stop handing in homework or worst, scribble like a ball of thread had fallen onto the pages of worksheets.
4. Argue with you just because you gently inform him or her to pack their bags for the next day’s lessons.
5. Kick the door to their rooms.
6. Study table becomes a jungle of papers and books and stationery.
7. Anything equivalent (or worst) than the above points.
And you notice these things happen ONLY when the exams are one month away. Otherwise, they are your little sweet darlings who listen to you even when you nag at them.
“What if I can’t solve it?” their cries ring in your mind when you encourage them to solve difficult questions in their homework.
“What if (fill in the blanks)?”
Translated: Your child is frightened of exams. They show such behaviour as a way of protecting themselves from the danger of exams.
In other words, they are showing the ‘fight or flight’ symptoms. Because they can’t ‘flight’ away from exams, they therefore have to ‘fight’.
Be aggressive. Show strength via their anger, whining, irritation or any form of negative emotions. Cold war is also counted.
At this stage, how do you calm your child down?
The answer: You discipline them and also counsel them.
Discipline them, because there are some boundaries in the house you have set and once crossed, they pay the consequences for their actions. I once heard a mother cane her son in a room after he gave excuses for not doing homework from his tutors and teachers.
That was uncomfortable to hear. That was also the time he woke up and began to hand in work on time.
Counsel them, because after disciplining them, you want to understand why they are behaving the way they are behaving. “You are not usually like this. What is on your mind?” is one simple question you can ask to understand them.
So, here are two simple (and probably uncomfortable) things you can do for your child.
1. Discipline them.
2. Then, counsel them.
Love is not about molly coddling. It’s about doing what is right because you want your children to have the right values to grow up with. It will be hard on you now but it will easy on you and your children later on in their lives.
As the saying goes, “parents are the first teachers to their children”.
]]>Dear parents with Primary 3 children,
You might have one or more of the following thoughts when you are guiding your children in doing their math homework.
This Ultimate Guide addresses all of the above concerns.
How can you benefit from this guide?
Who am I?
My name is Mr Cai (Chai) Shaoyang. You can call me Brandon.
I am a former Ministry of Education primary school teacher with over 12 years of teaching experience in math. I help 7 to 12 year old children score 90% or more in their math exams and tests with minimal stress and maximal confidence.
I have used practical and easytouse learning habits that help children minimise their stress and increase their confidence (and interest) while scoring 90% or more for their exams and tests at the same time! These are what some of my parents have to say about me.
Why should you listen to me? (A 7minute read on my successes and setbacks while learning math)
When I was in Primary 4, I failed my math midyear exams. I scored 40%+.
It was discouraging. I had been passing my math.
So, my mother decided to teach me the basic concepts with…the math textbook.
Her, and the textbook.
Plus, lots of patience in explaining the concepts that look like Greek to her.
This went on for five months.
In the Primary 4 yearend exam, I scored 90%. I remembered my teacher congratulating me when I received my math exam paper.
YES, I DID IT!
This gave me confidence and belief that with patience and understanding, a score of 90% is achievable!
I carried that confidence to Primary 5 and Primary 6, learning the challenging math brimming with ‘YES, I CAN!”. No matter how tough the homework was, I had faith I could solve them correctly, even if it means taking a longer time to solve them.
Then came the Primary School Leaving Examination (PSLE). I scored an ‘A’ grade, equivalent to at least 75% to 90%, for my math.
I felt like I was enjoying myself in the Math exams! I wore a big wide smile on my face!
Secondary 1 gave me a reality check. By the end of Secondary 1, I scored 50% overall for math. I was disappointed. Maybe that ‘A’ in PSLE Math was just a oneoff achievement.
Then came Secondary 2, when I transferred to another school as my family was shifting house from the east to the west of Singapore. Confidence in learning math was at an alltime low, with doubts about my ability hovering around like dark clouds in my mind.
Oh well. I decided to go back to basics.
The basics of:
I repeated these basics daily when preparing for my class test. I wasn’t expecting much, just a pass will do.
When I sat for the test, I answered to the best of my understanding for each question. I would be happy with a passing mark of 50%. When I got back the test result, I was stunned.
I scored 43 out of 50 points (That score is 86%).
I nearly wanted to shout ‘YES!’ but held back. I realised I was still in class, not at home.
Shout or no shout, that was the turning point of my confidence in learning math.
The confidence was back!
Upon reflection, I learnt this result was no fluke but the practice of the basics of
I repeated the basics for all my math subjects from the standard Elementary Mathematics, the challenging Additional Mathematics for the Cambridge exams GCE ‘O’ and the Cambridge exams GCE ‘A’ Mathematics syllabus C (or known as H2 Math today).
Make a guess what the results look like.
‘O’ levels
‘A’ levels
The math got tougher.
I just got tougher by hitting these grades.
Life would then throw me another challenge during my teacher training days. I had a lecturer who taught calculus like a robot. When we ask him how to arrive at the solutions to his questions, he would rather we figure out the solutions ourselves…without the slightest hints from him.
I had no clue what he was teaching about. I was getting fed up with failing his quizzes.
So, I decided to spend time in the library reading his prescribed calculus textbook to prepare for his ‘lessons’. The more I read the textbook, the more I understood calculus. I was wondering why we need this lecturer in the first place.
During the lesson, he was in his robot mode of teaching. Ask questions but provide no clues to his solutions. There was an instance when he asked us to calculate the area under a curve using integration. When most of us could not answer the question, he moved on!
I had enough of this ridiculous robot. I calculated the answer correctly and raised my hand to answer the question
“Is it one quarter of π ?” I replied with a poker face.
“Yes,” he replied…like a robot.
The class went silent but he didn’t explain why it was the answer.
If he didn’t explain it, some of my classmates came to me after class to borrow my notes and asked me questions to find out how I solved his question.
The answer came from hours and effort of reading and understanding the calculus textbook.
My confidence grew over the years in the teaching and learning of mathematics. These are some of my achievements.
This is my life journey in learning and teaching of mathematics. It is a journey of:
Are you and your child ready to:
If so, great! Let’s head on to the next chapter!
Table of Contents
Chapter 1
What are the mustknow topics and concepts for Primary 3 math?
Chapter 2
How to solve word problems and nonroutine problems?
Chapter 3
How to score for word problems?
Chapter 4
How hard should you push your child?
EPIC BONUS!
Cheat sheets to help your child score 90 marks or more for Primary 3 math
Just one more thing…
Chapter 1: What are the mustknow topics and concepts for Primary 3 math?
Based on my twelve years of teaching experience in primary math, there are three major hurdles that most Primary 3 students find challenging to clear.
Worst of all, higher levels of math in Singapore schools are based upon these hurdles of math concepts.
Children who master their fundamentals proceed on to become confident in learning math.
Students who suffer from shaky fundamentals end up relying on tips and tricks to tide them through the exams (and your anxiety starts to shoot through the roof).
They often end up hating math.
Let me show you what the three major hurdles are and how you can help your children overcome them.
Hurdle 1: “More than” means adding two numbers, “lesser than” means subtraction between two numbers
Look at the diagram below.
Which basket has more circles?
Basket A has more circles than Basket B.
Which basket has lesser circles?
Basket B has lesser circles than Basket A.
Now, look at the diagram below.
Which basket has more circles?
Basket A has more circles than Basket B.
Which basket has lesser circles?
Basket B has lesser circles than Basket A.
The positions of the baskets have changed in the diagram but two facts remained the same.
Learning point:
Simple to understand, isn’t it? Let’s move on to the confusing part about “more than, lesser than”.
Refer to Word Problem 1 below.
Word Problem 1
James and Anwar have some stickers.
James has 15 stickers.
Anwar has 3 more stickers than James.
How many stickers does Anwar have?
Key question: Who has more, James or Anwar?
Anwar.
How many more for Anwar?
3 more stickers for Anwar.
Your diagram may look similar to the one below.
From the diagram,
15 + 3 = 18
Anwar has 18 stickers.
So, is it true ‘more than’ means adding two numbers?
Let’s look at Word Problem 2 below.
Word Problem 2
Jane and Lisa have some apples.
Jane has 26 apples.
Jane has 3 more apples than Lisa.
How many apples does Lisa have?
Ah! You know how to solve it!
26 + 3 = 29
Lisa has 29 apples!
Wait!
This doesn’t make sense. How can Lisa’s 29 apples be lesser than Jane’s 26 apples?!
Let’s check our understanding again. Who has more apples?
Jane.
So, in the diagram below, Jane has 26 apples and 3 more apples than Lisa.
From the diagram,
26 – 3 = 23
Lisa has 23 apples.
Yes, Lisa has 3 less apples than Jane.
Learning point:
Hurdle 2.1: Confusing perpendicular and parallel lines with each other
When it comes to understanding parallel and perpendicular lines, students mistake one for the other. What is confusing them (and frustrating you with their confusion)?
Let me illustrate to you one difference between parallel and perpendicular lines with a stone wall below.
From the diagram, we see the rectangle has four right angles.
The right angle is the point where two PERPENDICULAR LINES MEET.
From the diagram, line AB forms a right angle with line BC in the rectangle ABCD.
So, line AB is perpendicular to line BC.
Learning point: RIGHT ANGLES tell you where the PERPENDICULAR LINES MEET.
Practice time!
Name the remaining pairs of perpendicular lines in rectangle ABCD.
The remaining pairs are:
Congratulations if you got at least one correct pair!
Learning point: RIGHT ANGLES tell you where the PERPENDICULAR LINES MEET.
(Like how you form a right angle when you stand at attention to the ground)
If perpendicular lines form right angles, then parallel lines form NO ANGLES.
Let’s go back to the rectangle ABCD again.
Imagine
Both of you lift up a blue pole as long as AD as shown below.
You walk from A to B. Your friend walks from D to C.
Both of you hold the pole on each endsand walk at the same time.
You reach B and your friend reaches C at the same time. Both of you place down pole AD onto BC as shown in the diagram below.
Throughout this time of walking, you are of length AD (or BC) away from your friend all the time. So,
So, AB is parallel to DC.
What happens when you make AB become longer than DC in the diagram below?
AB is still of an equal distance away from DC.
That means…AB and DC are still parallel to each other.
Learning point: Parallel lines are of an equal distance from each other and never meet.
Practice time!
Name another pair of parallel lines in the rectangle ABCD.
Answer: AD is parallel to BC. J
There are a total of two pairs of parallel lines:
Learning point: Parallel lines are of an equal distance from each other and never meet.
Now, for the confusion between area and perimeter.
Hurdle 2.2: Confusing area and perimeter with each other
Let’s look at the rectangle ABCD again.
You see that there are many squares that fill up the rectangle. Each square is 1 square unit (or 1 unit^{2}). How many squares are there in the rectangle?
Answer: 24 unit^{2}
Congratulations! You have counted the area of the rectangle ABCD using the squares in it.
Learning point: Area is simply the counting of the number of squares in a figure (in this case, rectangle).
If area is counting squares inside a figure, what is perimeter?
Imagine you are at point A of the rectangle ABCD and you run one round around ABCD in the diagram below.
Perimeter is like running one round around the rectangle ABCD.
Learning point: Perimeter is the total length of sides around a figure (in this case, rectangle ABCD).
IMPORTANT POINT: AREA is counting the total number of squares INSIDE a figure but PERIMETER is the TOTAL LENGTH of sides around the figure.
Let’s recap the difference between area and perimeter, and parallel and perpendicular lines.
Hurdle 3: Not knowing what equivalent fractions mean (THE TOPIC that MAKES OR BREAKS your child’s primary math scores)
Imagine your Primary 3 children being able to do the following:
Or this.
Top students easily solve these questions at the snap of their fingers. They coolly finish off the questions correctly like how your children would coolly finish off your icecream.
The reality is you see your children going ‘ahhhh’, ‘erhmmm’ or giving you blank looks when they do questions similar to the above. Your panic meter smashes the roof again. What can you do to help your children master the above questions?
Here is the one major concept they must master in fractions to increase their confidence (and dozens of marks) in math: equivalent fractions. Let me show you what equivalent fractions mean in the figure below.
From the figure, you see
1 out of 2 rectangles is black. So, of the two rectangles is black.
Now, let’s split each rectangle into two smaller but equal units.
From the same figure, you see
Two out of four rectangles is black.
So, of the figure is black.
At this point, let’s gather the two figures below.
You notice 2 smaller black rectangles = 1 bigger black rectangle.
So,
How is this possible?! How is the ‘1’ in the numerator the same as ‘2’ in the other numerator?
Remember, 2 smaller black rectangles = 1 bigger black rectangle.
So, you are splitting 1 bigger black rectangle into 2 smaller black rectangles.
We also have 2 smaller white rectangles = 1 bigger white rectangle
That means we have 4 black and white smaller rectangles altogether. That means…
And…
Bonus question: Can we split that 1 part into 3 smaller units?
Answer: Of course we can. In fact, you can split into as many smaller units as you want from each part. Let me show you how in the figure below.
From the figure above, you see
Three out of six units is shaded. So, of the figure is shaded.
At this point, let’s gather the two figures below.
Since both figures have the same area, you notice that 1 bigger black rectangle = 3 smaller black rectangles.
So,
So,
So, are EQUIVALENT FRACTIONS.
Also,
Instead of splitting, you join the smaller units into bigger parts. In this case, 3 smaller equal rectangles = 1 bigger rectangle.
Learning point:
You can find an equivalent fraction of any fraction by either
You are ready to solve the following questions!
Let me show you below.
Equivalent fractions also work for subtraction of fractions! Let me show you below.
Chapter 2: How to solve word problems and nonroutine problems?
You saw your children trying to solve word problems for their math homework.
You couldn’t remember the exact details.
However, you can remember that dreaded feeling of being transported back to your schooling days where your math teacher scolds you ‘stupid!’
‘Stupid!’, because many of your friends could solve the word problem on the blackboard but you couldn’t. Your answer was different because you made a calculation error.
Then, the dreaded question from one of your children arrives.
“Mum, can you help me solve this problem?”
Your heart starts to accelerate. What if you can’t solve it?
You thought of asking for help in the parents’ chat group from your child’s class in Whatsapp. Then again, the parents are like you: out of touch with the math syllabus for twenty years.
Next helpline: your child’s tutor!
You took a photo of the word problem and sent it to your child’s math tutor.
You realised one thing. It’s 8.30 p.m.
Your child’s tutor usually only replies in the morning. Your children’s homework is to be handed in tomorrow morning.
Damn! Your nightmare came true. YOU have to help your child solve the word problem.
The “What if one day my tutor isn’t available?” scenario has arrived.
Introducing Polya’s fourstep approach to problemsolving. It is easy to remember, systematic and most importantly, it helps you solve ANY problem of ANY difficulty.
Best of all, this approach is found in many primary school math textbooks and explained in detail to help students solve word problems of any difficulty systematically and with minimal stress.
Polya is a Hungarian mathematician of the early 20^{th} century who observed that those who solve complicated math problems correctly often use a system of four steps to solve them.
This is how Polya’s fourstep approach looks like.
Systematic? Check.
Simple to remember? Check.
Let me show you how to use this fourstep approach.
Step 1: Understand the problem
Here is an example of a word problem.
Jane and Mary have 180 stamps altogether.
Jane has 48 stamps.
How many stamps does Mary have?
To understand the problem, read one sentence at a time and label each sentence like the following example below.
(Sentence 1) Jane and Mary have 180 stamps altogether.
(Sentence 2) Jane has 48 stamps.
(Sentence 3) How many stamps does Mary have?
By labelling each sentence, you are chunking out the huge problem into smaller, manageable chunks.
Just like how the SMRT engineers have to replace the ageing train tracks from wooden sleepers to concrete sleepers month by month by replacing the tracks on the Red Line first, followed by the Green Line.
There is no way both lines can have new tracks in 1 week.
Now, read sentence 1 again.
Who has or have stamps? (Jane and Mary)
How many stamps does Jane and Mary have altogether? (180)
Sentence 1 now looks like this:
Next, read sentence 2 again.
Who has 48 stamps? (Jane)
Sentence 2 now looks like this:
Now, read Sentence 3 again. What does the word problem want you to solve for?
Yes, Mary’s number of stamps. That is the unknown we are solving for.
Congratulations! You have understood what each sentence means. Let’s summarise what we have understood so far in terms of known information and unknown information.
Known information:
Unknown information:
With this information in hand, we can proceed on to step 2.
Step 2: Plan the solution
Solving a problem requires creativity. According to chess grandmaster, writer, spiritual teacher and investor James Altucher, we simply list down any ten ideas we think will solve the problem. We don’t judge if the ideas are good or bad until we have tried them out. Let the results of the ideas tell you if the ideas solve the problem.
Now, let’s list ten ideas that can help us solve for Mary’s number of stamps.
Ten ideas to solve for Mary’s number of stamps
I leave you to fill in ideas 6, 7, 8, 9 and 10.
(Remember, focus on generating ideas, not judging if they are right or wrong. Getting the ideas out of your head onto paper gives you more clarity on what might work instead of wondering in your head if the ideas work.)
Congratulations! You have now thought of ten ideas to solve for Mary’s number of stamps!
We can now proceed to step 3.
Step 3: Solve the problem
Now, let’s choose one idea.
Let’s pick idea 1. We randomly plug in numbers (or “tikam”, which is Malay for guess).
Could Mary have 200 stamps? Let’s try this number out.
Attempt 1
Mary: 200 stamps
Jane: 48 stamps.
Let’s add the total number of stamps they have altogether.
200 + 48 = 248
248 is not equal to 180. That is way too many! Haha! Cross out 200.
Let’s try 150.
Attempt 2
Mary: 150 stamps
Jane: 48 stamps
150 + 48 = 198
198 is not equal to 180. You are getting closer to their total of 180 stamps. Sounds good!
Let’s try 130.
Attempt 2
Mary: 130 stamps
Jane: 48 stamps
Let’s add the total number of stamps they have altogether.
130 + 48 = 178
Wow! You are getting close to their total of 180 when you guess Mary’s number of stamps to be 130.
And you thereby conclude one thing from your two attempts at guessing.
Mary has 133 stamps!
Wait. Is that the actual number of Mary’s stamps?
Let’s go to Step 4.
Step 4: Check if the solution solves the problem
Let’s recap what we have found out so far.
Jane: 48 stamps
Mary: 133 stamps
Total stamps: 180
Let’s add 48 and 133 together.
48 + 133 = 181
Oh! The total is 1 more than 180. I can’t go on randomly plugging in numbers like this.
Let’s try subtracting 48 from 180 to find the number of Mary’s stamps.
180 – 48 = 132
Ah! I found it! Mary has 132 stamps.
And to verify if it is 132, you add 132 to 48.
132 + 48 = 180
YES!
Final answer: Mary has 132 stamps!
There you have it. A 4step approach to solving a word problem.
You realise that no word problem can be solved until you understand what the problem is. You also realise that there are known and unknown information in the word problem. The key to understanding is to use the known information to solve for the unknown information.
To put it simply, if you don’t understand the word problem, you can’t solve the word problem.
It is like you asking for a table but given a lampshade instead.
You might also have other questions in mind.
Will drawing a diagram help us solve the question faster?
Isn’t acting out the problem even easier?
My answer to these questions is that you test them out. After all, no idea is good or bad until you try them out.
Let’s recap Polya’s fourstep approach again.
These 4 steps provide direction to solve any problem of any difficulty level. Should you be stuck at Step 2, you can always go back to Step 1. Should you complete Step 2, you can always go to Step 3.
It is like your child having a compass to find his or her way to their destination.
Compass = Polya’s fourstep approach
It wouldn’t be easy but it would be simple to get there.
Action step
Take a pocket size notebook and generate ten ideas a day on solving anything. This helps you sharpen your creativity levels and also have fun with making mistakes and learning from them.
As James Altucher says, there are no good or bad ideas, just ideas. Don’t worry if the idea doesn’t work. The whole objective is to be creative. Being creative helps you solve problems.
So, what happens when the nonroutine problems become routine?
By experience and intuition, you and your children realise there are only a few types of word problems that keep recurring.
You will also realise randomly plugging in numbers is going to be tedious when the numbers used in the word problems reach tens of thousands or millions.
In other words, you can’t solve different problems with the same skill of random generation of numbers. It is similar to using a hammer to solve every problem in your house.
Here are FIVE common tools of making solving routine problems efficiently and enjoyable (and also gradually hand over the problemsolving process to your children instead of hawking over them so that you can go for a swim, watch Netflix or have ‘me’ time).
Tool 1: Bar models
Let’s refresh your memory with the earlier example of Jane and Mary.
(Sentence 1) Jane and Mary have 180 stamps altogether.
(Sentence 2) Jane has 48 stamps.
(Sentence 3) How many stamps does Mary have?
You would recall we use guessing to solve for Mary’s number of stamps.
I am going to be honest here. Random guessing can be very painful when the numbers get larger.
So, we take the guesswork out by drawing 180 stamps.
And after drawing 4 stamps, you are probably wondering,
“Is there a faster way to draw 180 stamps? This is tedious and takes too much time!”
You feel like pulling out your hair or yelling at me.
Let me show you how you can draw 180 stamps in five seconds in the diagram below.
The rectangular block you see in the diagram above is like the M&Ms chocolate wrapper. The 180 stamps are like the cute colourful little chocolate pieces in the wrapper.
We call this rectangular block a bar.
We use the bar to model (or show) the key information in the word problem in pictures. Or, bar modelling in simple terms.
Let me show you how to use bar modelling to show the other key information in the stamps question (and radically increase your children’s happiness and confidence in solving a word problem correctly).
From the bar model above, you observed that Mary’s number of stamps is calculated by the working below.
180 – 48 =?
So, according to Polya’s 3^{rd} step,
SOLVE IT!
180 – 48 = 132
Now, let’s check the accuracy of the answer by working our calculations backwards.
132 + 48 = 180
Yes! Mary has 132 stamps.
Bonus question! How many more stamps does Mary have than Jane?
Let’s recap the bar model we used to solve the stamps problem.
How can we use the above bar model to solve the bonus question?
Here’s a hint.
To eat the chocolate in the M&Ms packet, you must cut/tear the wrapper to allow the chocolate to land in your hand…or mouth, if you prefer it that way.
So, like the M&Ms wrapper, tear the bar model apart as seen below.
Aha! Now, we can solve for how many more stamps Mary has than Jane.
132 – 48 = 84
Let’s check the accuracy of the answer.
84 + 48 = 132 (Yes!)
Mary has 84 stamps more than Jane.
The bar model above is called the comparison model. We use it to compare two or more blocks of different values.
In summary, we use bar modelling to show key information quickly and accurately so that we can identify quickly what is to be solved.
Practice time!
Hamid and Alan have 256 marbles. Hamid has 217 marbles.
(Cover up the answer key below when you solve this practice question)
256 – 217 = 39
Answer: Alan has 39 marbles.
217 – 39 = 178
Answer: Hamid has 178 marbles more than Alan.
Tool 2: Listing
You might have seen the following question in your Primary 3 or 4 child’s math homework.
Florence thinks of a 2digit number.
This number is more than 20 but less than 28.
It gives a remainder of 2 when divided by 3.
It gives a remainder of 5 when divided by 6.
What is this number?
You shudder at the overwhelming information. You and your children can’t solve it with bar modelling.
However, you can solve it with listing. To list is to write out all possible outcomes that can solve the word problem.
In this case, you list the numbers from 21 to 27 because they are more than 20 but less than 28.
21, 22, 23, 24, 25, 26, 27
Next, divide each number by 3 to find out which gives a remainder of 2.
21, 22, 23, 24, 25, 26, 27
(0, 1, 2, 0, 1, 2, 0)
Aha! Only 23 and 26 are left to solve the problem!
Now, divide both numbers by 6 to find out which gives a remainder of 5.
23, 26
(5, 2)
So, 23 is the 2digit number.
According to Polya, you need to CHECK if the number solves the question. So,
23 ÷ 3 = 7 remainder 2
23 ÷ 6 = 3 remainder 5
Therefore, 23 is the 2digt number.
Practice time!
This number is more than 30 but less than 40.
It gives a remainder of 3 when divided by 4.
It gives a remainder of 4 when divided by 7.
What is this number?
(Cover up the answer key below when you solve this practice question)
Answer key
Numbers: 31, 32, 33, 34, 35, 36, 37, 38, 39
Respective remainders: 3, 0, 1, 2, 3, 0, 1, 2, 3
Numbers with a remainder of 3: 31, 32, 33, 34, 35, 36, 37, 38, 39
Respective remainders: 3, 0, 4
Numbers with a remainder of 4: 31, 35, 39
Answer: 39
Tool 3: Work backwards
Sometimes, you don’t have the information you need at the start of the problem to solve. However, you have the known information at the end of the problem.
In this case, you work backwards to solve the problem.
Let’s look at how we can work backwards to solve the problem below.
Sentence 1: Jim baked some tarts.
Sentence 2: He ate 5 tarts and gave 9 tarts to his sister.
Sentence 3: He had 15 tarts left.
Sentence 4: How many tarts did Jim bake?
Step 1 of 4: Understand it (Polya)
We know the following:
What we don’t know is the number of tarts at first. So, we work backwards by starting from the 15 tarts left.
Step 2 of 4: Plan it
Let’s use a bar model to represent 15 tarts (or you can draw 15 tarts if you wish). The bar model looks like this below.
Next, recall that Jim gave 9 tarts to his sister. That means the 9 tarts were part of the number of tarts baked at first. So, we add 9 tarts to the number of tarts left. The bar model looks like this below.
Lastly, recall that Jim ate 5 tarts that were also part of his number of tarts baked at first. So, we add 5 tarts to 15 tarts and 9 tarts. The bar model looks like this below.
Step 3 of 4: Solve it
Observe that the number of tarts baked at first is made up of 15, 5 and 9 tarts.
So, we add 15, 5 and 9 tarts to solve for the number of tarts Jim baked at first.
15 + 9 + 5 = 29
Step 4 of 4: Check it
Let’s check if 29 is the number of tarts Jim has at first.
Subtract the 5 tarts Jim ate from 29 tarts.
29 – 5 – 9
= 24 – 9
= 15
15 tarts is the number of tarts left.
It matches correctly with Sentence 3 of the word problem.
Answer: Jim baked 29 tarts at first.
Practice time!
Sentence 1: Sharon had some stickers at first.
Sentence 2: She gave 25 stickers to her friend and lost 17 stickers.
Sentence 3: She had 42 stickers left.
Sentence 4: How many stickers did Sharon had at first?
(Cover up the answer key below when you solve this practice question)
Answer key
42 + 17 = 59
59 + 25 = 84
Answer: 84
Tool 4: Patterns
Some problems appear in the form of patterns. Let’s look at the pattern below.
A, B, A, B, A, _ , _ , _ , A, _ , …
What are the missing letters?
Step 1 of 4: Understand it
First, we observe there are only the letters ‘A’ and ‘B’ that make the pattern.
Next, we also observe that
Step 2 of 4: Plan it
Aha! You have found three rules for this pattern.
Step 3 of 4: Solve it
Let’s fill in the pattern based on Rule 1 and Rule 2.
A, B, A, B, A, B , A , B , A, B , …
Step 4 of 4: Check it
Let’s check if the pattern follows the 3 rules.
Yes, the missing letters fit the pattern!
Answer: A, B, A, B, A, B , A , B , A, B , …
Bonus question: What letter appears in the 165^{th} position of the pattern?
Answer: ‘A’, based on Rule 1.
Practice time!
Look at the pattern below.
A, B, B, _, B, _ , A, B, _, A
What are the missing letters?
Answer key
Rule 1: There is a set ‘A’‘B’‘B’ that repeats itself.
Rule 2: ‘A’ appears in the first, fourth, seventh and tenth positions.
This means
Answer: A, B, B, A, B, B, A, B, B, A
Bonus question: What letter appears in the 110^{th} position?
Answer: B
Tool 5: GuessandCheck
As much as the title suggests, there is no guesswork. Instead, we use a table to check for patterns what the solution is to a problem. Think of it as how the librarians arrange their books based on the Dewey classification system.
Let’s look at how a guessandcheck question looks like below.
There are 12 chickens and dogs in a farm.
A chicken has 2 legs.
A dog has 4 legs.
They have 32 legs altogether.
How many dogs are there?
Step 1 of 4: Understand it
Known information:
Unknown:
We don’t know the number of dogs and chickens but we know their total is 12.
We don’t know the number of dogs’ legs and chickens’ legs but we know their total is 32.
So, let’s just start with the total number of animals.
Step 2 of 4: Plan it
We draw two columns below. One is for the number of dogs and the other is for the number of chickens.
Total: 12 animals 

Number of dogs  Number of chickens 
Next, we draw two more columns to the table. One column is for the number of chickens’ legs and the other is for the number of dogs’ legs.
Total: 12 animals 
Number of legs for 

Number of dogs  Number of chickens  Chickens  Dogs 
Lastly, we draw one more column to check if the total number of legs altogether is 32.
Total: 12 animals 
Number of legs for 
Check: do they have 32 legs altogether?  
Number of dogs  Number of chickens  Chickens  Dogs 
Lastly, draw five more rows to the table.
Total: 12 animals 
Number of legs for 
Check: do they have 32 legs altogether?  
Number of dogs  Number of chickens  Chickens  Dogs  
You will notice that there are 5 columns altogether. This is the usual number of columns for a GuessandCheck table.
By starting with the total number of animals, I can focus on listing the various combinations of number of dogs and chickens to generate their total number of legs.
This is where the ‘Guess’ part comes in. ‘Guess’ the number of dogs. Let me show you how you can guess the number of dogs in just four rows.
Step 3 of 4: Solve it
Let’s start with zero dogs. That means we have twelve chickens.
Fill in ‘0’ in the table below.
Total: 12 animals 
Number of legs for 
Check: do they have 32 legs altogether?  
Number of dogs  Number of chickens  Chickens  Dogs  
0  12 – 0 = 12  
Next, calculate the number of chickens’ legs and dogs’ legs.
Total: 12 animals 
Number of legs for 
Check: do they have 32 legs altogether?  
Number of dogs  Number of chickens  Chickens  Dogs  
0  12 – 0 = 12  12 x 2 = 24  0 x 4 = 0  
Last but not least, at the total number of legs altogether.
Total: 12 animals 
Number of legs for 
Check: do they have 32 legs altogether?  
Number of dogs  Number of chickens  Chickens  Dogs  
0  12 – 0 = 12  12 x 2 = 24  0 x 4 = 0  24 + 0 = 24 L 
From the table, zero is not the correct number of dogs. So, let’s try one dog for one row and two dogs for another row. Then, fill in their respective rows accordingly. You will get the following table.
Total: 12 animals 
Number of legs for 
Check: do they have 32 legs altogether?  
Number of dogs  Number of chickens  Chickens  Dogs  
0  12 – 0 = 12  12 x 2 = 24  0 x 4 = 0  24 + 0 = 24 L 
1  12 – 1 = 11  11 x 2 = 22  1 x 4 = 4  22 + 4 = 26 L 
2  10  10 x 2 = 20  2 x 4 = 8  20 + 8 = 28 L 
This seems discouraging. However, look at the relationship between the number of dogs and the total number of legs. What can you infer about their relationship? Take 1 minute to infer about this relationship.
Yes, as the number of dogs increases, the total number of legs increases. Well done!
Let’s get specific on the numbers.
As the number of dogs increases by 1, the total number of legs increases by 2.
(In simple language, add one dog, get two more legs in total.)
Yes, a pattern is seen in the relationship!
And your table is now reduced to this.
Number of dogs  Check: do they have 32 legs altogether? 
0  24 L 
1  26 L 
2  28 L 
So, how many more legs to 32 from 28?
32 – 28 = 4.
So, 28 + 4 =32
Number of dogs  Check: do they have 32 legs altogether? 
0  24 L 
1  26 L 
2  28 L 
28 + 4 = 32  
Since the total number of legs increases by 2 each time, let’s find out how many times we increase the total number of legs.
4 ÷ 2 = 2
2 times.
Number of dogs  Check: do they have 32 legs altogether? 
0  24 L 
1  26 L 
2  28 L 
28 + (2×2) = 32  
Therefore, we add 2 to the 2 dogs in the table below.
Number of dogs  Check: do they have 32 legs altogether? 
0  24 L 
1  26 L 
2  28 L 
2 + 2 = 4  28 + (2×2) = 32 
J  J 
We have 4 dogs!
Step 4 of 4: Check it
Let’s check if 4 dogs is the correct number of dogs.
Number of chickens: 12 – 4 = 8
Number of chickens’ legs: 8 x 2 = 16
Number of dogs’ legs: 4 x 4 = 16
Total number of legs: 16 + 16 = 32
Yes, 4 dogs is correct!
Practice time!
There are 20 cars and motorcycles in a car park.
A car has 4 wheels.
A motorcycle has 2 wheels.
There are 64 wheels altogether.
How many cars are there?
Answer key
Total: 20 
Number of wheels for 
Check: do they have 64 wheels altogether?  
Number of cars  Number of motorcycles  Motorcycles  Cars  
0  20  40  0  40 L 
1  19  38  4  42 L 
2  18  36  8  44 L 
2 + 10 = 12  8  16  48  44 + (2×10) = 64 
J  J 
Chapter 3: How to score for word problems?
A typical Primary 3 math exam paper consists of 3 question types.
They are:
Multiple Choice Questions or MCQ (1 to 2 marks each)
ShortAnswer Questions or SAQ (Openended, 2 marks each)
LongAnswer Questions or LAQ (Also known as word problems, 3 to 5 marks each)
Examiners set a fair exam paper to find out which students show
in their math concepts to solve challenging word problems.
Challenging word problems are more risky to solve. They require a deep understanding of the problem. So, their difficulty level is the highest among SAQ and MCQ.
BUT…those who solve them also get more marks!
This is put in a simple graph below.
No understanding, no marks, even if your children found the answer by fluke.
That is how examiners mark students’ solutions to word problems.
You now wonder if your child struggling with math can still score maximum points for their math exams.
The answer is…Yes! Let me sum it up in two steps: Start from solving the easiest word problems first, then move on to solve the challenging
Step 1: Solve the easiest word problems first
The easiest word problems are usually worth 2 to 3 marks each. These questions are very similar to the word problems in your children’s math workbooks.
These math workbooks are approved by the Ministry of Education. The approval seal is shown on the book cover.
Examples of approved math workbooks are:
Children who understand how to solve the word problems in these workbooks will become confident to solve challenging word problems.
Confidence drives the children to 90 marks.
Step 2: Now, solve the challenging word problems
Examiners are acutely aware that very few children will slow down to
These four steps take a huge load of time, energy and effort, especially if your children are tight on time solving unfamiliar word problems.
In biology terms,
Children have it two times worst. So, most choose the easy way out by randomly plugging in numbers and operations instead of slowing down their thinking to understand, plan, solve and check if their solutions are correct.
Translated: Exams are dangerous! I must survive!
Rushing to solve = zero marks
Therefore, train your children to read sentence by sentence for each word problem. This is to break down the huge chunk of complicated information into manageable chunks.
Manageable chunks = less stress and strain on the brain to think clearly
Understanding even one manageable chunk enables your child to score one mark (out of four to five marks). That one mark can make a difference between passing and failing, or one mark away from the top grade of 85 marks.
By now, you would have realised I keep emphasising on UNDERSTANDING.
Why?
Let’s imagine you enter a furniture store and ask for a table from the sales staff.
The sales staff gives you a lampshade instead. Would you still hand over money to buy the lampshade?
No! You want to buy a table, not a lampshade.
Similarly, examiners will only hand over marks to your children when they understand what the word problems are about.
Your children still get the mark for understanding even if they make calculation errors.
Your action step for this chapter is to reread Chapter 2 for the five common tools that are used to solve word problems regardless of difficulties. Remember,
UNDERSTANDING = AWARDING OF MARKS
Chapter 4: How hard should you push your child?
Are you pushing your child to be a doctor, banker, or some prestigious job so that you can brag about your child’s achievements to your friends? Or you harbour hopes that one day, they will take over your family business?
If so, your children will never buy in to your explanation why it is important to chase after the ‘A’ in math exams.
‘Push’ is a strong indicator that your children are not convinced by what you want them to achieve. Stress occurs in you and your children. Friction surfaces and both of you get into quarrels and cold wars. In some cases, the wars last a lifetime.
What if I tell you there is a way to get them to pull (attract) themselves towards the ‘A’ in math?
I used to teach a student who wants to become a professional footballer. He attends soccer training with almost 100% attendance, and goes to the National Football Academy with his boots and gear in the hope that one day, he will represent the national football team as the best defender.
The mother doesn’t need to push him. He pulls himself towards his goal.
More importantly, he is allowed to make mistakes while training and his coaches will give feedback to him on his strengths and areas of improvements during and after the training. More importantly, he is allowed to make mistakes and learn without being judged.
What pulls children to do what they do? They feel safe to fail and are given feedback and time to learn.
My action step is simple for you.
Mistakes + Feedback + Time to learn from mistakes = ENDLESS SUCCESS IN ANYTHING THEY DO.
You will no longer need to push your children anymore by then. You can go for a swim while they pull themselves to excellence.
EPIC BONUS: Cheat sheets to help your child score 90 marks or more for Primary 3 math
By this time, you might be worried if you can remember the main points of this guide.
Well, use these cheat sheets to help you and your child score 90 marks confidently in math exams (and take a swim)!
Here are not one, two but FOUR cheat sheets for your children to score 90 marks or more for Primary 3 math!
Cheat sheet 1: What are the mustknow topics and concepts for Primary 3 math?
Cheat sheet 2: How to solve word problems and nonroutine problems?
Cheat sheet 3: How to score for word problems?
Cheat sheet 4: How hard should you push your child?
Mistakes + Feedback + Time to learn from mistakes = ENDLESS SUCCESS IN ANYTHING THEY DO.
Just one more thing…
Thank you for reading the Ultimate Guide to Math Tutoring for Primary 3 parents. I spent weeks writing this guide so that you can help your children score an A in Primary 3 math exams with a wide smile on their faces.
If you are interested to read and use more of the ‘secrets’ found in this Ultimate Guide to help your child score an A in Math exams with maximal confidence and interest, sign up for my weekly newsletter in the link below!
Not even a doodle.
Next, you see your 10 year old daughter busy doodling away at her word problems with cats, fishes, mermaids and all kinds of animals…but not the solutions to her word problems.
Then, your son ‘writes something’.
It turns out to be a doodle of a dinosaur.
A frown soon surfaces from your face.
Then, you yelled at them to work harder but they only paused what they were doing…and picking up where they left off.
You looked at their textbooks but get overwhelmed like them. Your textbook in the past had lots of worked examples for you to refer and relate to in your homework but their way of learning is to understand concepts deeply with 1 or 2 worked examples and figure out the rest of the details themselves.
It all seems in vain for you and doomed for your children.
Well, not quite.
There is a simple solution to the feelings your children are experiencing. In fact, you can also try out this solution yourself whenever you are stuck at solving a problem.
It is called…giving yourself a break.
How does giving yourself a break, a nonmath activity, help your children move from stuck to solving their word problems! Isn’t this supposed to be a blog post about math concepts that help your children solve math problems in 3 minutes?!
This is what happens when you give yourself a break.
Your mind goes from feeling like it’s going to explode into bits and pieces to a relaxed and calm state, like you were taking a vacation at the seaside.
With a relaxed and calm mind, you see through the ‘fog’ that surrounds the solutions to the problem, and you feel like you were released from the chains of your previous thoughts.
In scientific terms, we call this incubation. This is when the mind flips the problem over and over again like flipping pancakes repeatedly until it finds the right solutions.
According to the book ‘How we learn’ by Benedict Carey, distracting ourselves from the problem which we are stuck at lets us step back from the problem to solve to gain different perspectives to solve the problem.
In the same book, a study was conducted to find out whether a few minutes of rest affects people differently when they are given vague hints to solve a problem as compared to those who are not given vague hints.
The results of the study showed those who got the rest solved two times as many problems as compared to the ones who got the same vague hints but not given a break.
In other words, ‘break’ time or ‘rest’ time WORKS.
Let me give you a personal example on 19 February 2018 at 10 a.m. as a case study.
19 February 2018 at 10 a.m.
A mother of a Primary 6 boy asked me, during her son’s lesson with me at her house, to help her solve a Primary 6 challenging exam math problem in a different way. She wanted to find out if I could come up with a simple and easy to understand solution she could explain to her son.
I agreed to help her out. I was quite confident I could come up with one in less than five minutes.
Here is the word problem.
Shop X sells bags of candies at $7 each.
Shop Y sells bags of candies at $4 each.
Shop X sells $296 more candies than Shop Y but sells 25 bags less than Shop Y.
How many bags of candies did Shop X sell?
Attempt 1
This word problem was a nightmare for me.
It took me THIRTY minutes to understand what the problem meant. For a tutor who has over 12 years of teaching experience in primary math, this was one of the rare moments where I felt like I was trying to pry open a durian with my bare hands.
And after thirty minutes, this was my first attempt to solve the problem.
If you looked at my first attempt, I had drawn diagrams to represent the sentences so that I could understand what the question was clearly.
And you can tell from the image above it looked like I was on a long and tedious way to solve the problem.
Attempt 2
I tried walking around the living room where we were having lessons for 5 minutes to solve the problem.
I was still stuck.
Then, a thought came to me.
Number of bags of candies from Shop X = Number of bags of candies from Shop Y.
And this is how it looks.
Then, as I went on. I got stuck.
For the rest of the time, I had no idea what I was talking about while explaining this to my student.
This is a wildgoosechase.
Attempt 3
So, I decided to solve the problem the third time…and the lesson was ending in 10 minutes.
This was not much different from the first attempt. I could feel my brain tightening in my head, like a sponge being squeezed dry of water.
Translated: STUCK
And the lesson ended.
3 failed attempts in 90 minutes.
I requested to the mother to let me come up with a simple and easytounderstand solution while on my way to lunch.
On my way to lunch, I took a bus. I was still thinking about the simple solution in the bus.
Then, while walking in the underpass, the same attempts clogged my mind.
Then, I had to take the MRT train to my lunch venue at Holland Village.
Even in the train, I still could not figured out the simple solution.
I looked back at the set of 1 bag of candy from Shop X and 1 bag of candy from Shop Y.
It was here that I hear a voice within me.
“Have you thought of making both quantities of candies to be the same from Shop X and Shop Y?”
Another thought hit me.
“Why not add 25 bags of candies to Shop Y so that both have the same number of bags of candies for Shop X and Y? Then, we can compare the difference in sales between both shops and divide it by the difference in cost between 1 bag of candy from X and 1 bag of candy from Y.”
This sounds really exciting!
I took out my paper and sketched out the solution with my pen.
And this is the fourth attempt of solving.
Attempt 4
The problem was solved with a simple and elegant solution.
Looking back at the problemsolving experience, I was stuck at solving the word problem for 90 minutes at the student’s house.
Then, in 5 minutes, I came up with a simple solution to the word problem.
I couldn’t figure out why until I realised what caused me to solve it.
I cast aside the problem when I was lost for directions at a bus stop.
What an irony!
—————————
So, here is your action step.
When your child is stuck at solving a word problem for at least 15 minutes and he or she has tried all possible ways to solve it but to no avail,
It is that simple.
Imagine exchanging 90 minutes of tantrum throwing, doodling, disinterest for 5 to 15 minutes of calmness and peace from your children…
And then leaving your short fuses alone after a long and tiring day of work for you.
FEEL THE SHIFT OF YOUR MOOD FROM A FROWN TO A SMILE ON YOUR FACE…and free up time for yourself to go for a swim with your friends…while knowing your child is happily crushing even the most challenging word problems with a 5 to 15 minute break!
If you are interested to read and use these practical and easytouse habits to help your child score an A in Math exams with minimal stress, greater confidence and interest, sign up for my weekly newsletter in the link below!
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