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.
 “I find that kids tend to have problems with word problems. It would be useful if you can share how they can score for word problems.”
 “What are the mustknow topics, concepts for primary math?”
 “Most times, when I see the question in my mind, I will be thinking how to solve it and what should I start with.”
 “If every chapter in the primary math textbooks has a summary guide or an outline, it will be wonderful. I haven’t seen any textbook that has summary guides or outlines.”
This Ultimate Guide addresses all of the above concerns.
How can you benefit from this guide?

When your child understands the key concepts and topics for Primary 3, they can solve 80% of the word problems (also known as Long Answer Questions or LAQ in the math exams) and score almost 100% in MultipleChoice Questions (MCQs) and the openended ShortAnswer Questions (SAQ).
 The higher they score, the wider your smile. The wider your smile, the higher your happiness (and relief) levels.
 The higher they score, the wider your smile. The wider your smile, the higher your happiness (and relief) levels.

Knowing how to solve common word problems correctly boosts your child’s confidence and preparation going into the exams.
 The more confidence they have, the higher their chances of scoring 90% or more.
 The higher their chances, the more you feel at ease and return home to peace.
 The more confidence they have, the higher their chances of scoring 90% or more.

The cheat sheets serve as quick revision guides for your child to use and prepare for math exams easily.
 You can sip a cup of tea, read a book or watch Netflix while your child uses the cheat sheets to revise.
 You can sip a cup of tea, read a book or watch Netflix while your child uses the cheat sheets to revise.
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.
 “Do you know he looks forward to solving word problems now? He would always finish what you have assigned to him weekly and finish it quickly as soon as you go off. He now sits for two hours at a time on his own, trying to solve the problems his teacher gives to them as homework.”
 “Many thanks to you for guiding my son. My husband and I have not doubted your ability to guide him and always thought he was in good hands with your coaching. We appreciate your patience and efforts in guiding him”
 “She is feeling so much more confident with her skills and I’m very happy to see that. She was really upset when I signed her up for tutoring but now, she is motivated and trying hard to learn. I appreciate everything you have done.”
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:
 Pay attention in class to understand the math teacher’s explanation of concepts
 Ask questions when I don’t understand
 Complete the homework
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
 Pay attention in class to understand the math teacher’s explanation of concepts
 Ask questions when I don’t understand
 Complete the homework
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
 Elementary Mathematics – A1 (75 marks and above)
 Additional Mathematics – A2 (70 to 74 marks)
‘A’ levels
 Mathematics Syllabus C – B (60 to 69 marks)
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.
 Graduated with a Master of Education (Mathematics) degree
 Presented a research paper on the teaching and learning of decimals in the International Congress of Mathematical Education (ICME) conference in Seoul, 2012, to a group of 14 teachers who come from as far as South Africa ( I still remember the teacher in Tshirt and kneelength shorts and a bandana from South Africa and the Vietnamese lady who was also presenting her research on decimals)
This is my life journey in learning and teaching of mathematics. It is a journey of:
 Patient to break down complicated math concepts
 Listen to understand how the concepts work
 Ask questions to clarify understanding
 Do the homework and hand it up on time
 Read textbooks and references to deepen understanding
Are you and your child ready to:
 Be Patient?
 Listen?
 Ask?
 Complete the homework?
 Read?
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.
 Basket A has more circles than Basket B.
 Basket B has lesser circles than Basket A.
Learning point:
 “More than” simply means more in value than the other.
 “Lesser than” simply means less in value than the other.
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:
 ‘More than’ does not always mean adding of two numbers all the time.
 ‘More than’ means one number has a higher value than the other.
 Similarly, ‘lesser than’ does not mean subtracting two numbers all the time. It simply means one number has a lower value than the other.
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:
 AD and DC
 DC and CB
 BA and AD
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
 you are standing at point A and
 your friend is standing at Point D.
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,
 AB and DC are of equal distance of AD (or BC) from each other and
 AB and DC don’t meet each other.
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:
 AD and BC
 AB and CD
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.
 PERPENDICULAR lines meet to form a RIGHT ANGLE but PARALLEL lines never meet.
 AREA is the total number of squares INSIDE a figure but PERIMETER is the TOTAL LENGTH around a figure.
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
 one black rectangle and one white rectangle
 two equal rectangles form the figure
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 smaller black rectangles and two smaller white rectangles
 Four smaller equal rectangles form the figure
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 black smaller rectangles and three white smaller rectangles
 Six equal smaller rectangles forming the same figure
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
 splitting each part in a figure into smaller and equal units or
 joining smaller and equal units in a figure into a larger part
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.
 Step 1: Understand the problem
 Step 2: Plan the solution
 Step 3: Solve the problem
 Step 4: Check if the solution solves the problem
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:
 Jane and Mary
 180 stamps
Next, read sentence 2 again.
Who has 48 stamps? (Jane)
Sentence 2 now looks like this:
 Jane 48 stamps
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:
 Jane and Mary, 180 stamps
 Jane, 48 stamps
Unknown information:
 Mary, number of stamps
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
 Randomly plug in numbers
 Work backwards
 Draw a diagram
 Make a table
 Buy 180 stamps, ask one of your children to be Jane and the other to be Mary. Then, act out the problem.
 _______________________________
 _______________________________
 _______________________________
 _______________________________
 _______________________________
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.
 Step 1: Understand the problem (Read and understand each sentence)
 Step 2: Plan the solution (Generate ten ideas to solve the problem)
 Step 3: Solve the problem (Test out one of the ideas)
 Step 4: Check if the solution solves the problem (Check your calculations by testing out the calculated numbers)
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.
 How many marbles does Alan have?
 What is the difference in the number of marbles between Hamid and Alan?
(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,
 divide 23 by 3 to see if it has a remainder of 2
 divide 23 by 6 to see if it has a remainder of 5
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:
 there are 15 tarts left
 9 tarts are given to Jim’s sister and
 Jim ate 5 tarts
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
 ‘A’ appears in the first, third, fifth, and ninth letters.
 ‘B’ appears in the second and fourth letters.
Step 2 of 4: Plan it
Aha! You have found three rules for this pattern.
 Rule 1: ‘A’ takes the odd positions
 Rule 2: ‘B’ takes the even positions
 Rule 3: The set ‘A’‘B’ repeats itself
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.
 The ‘A’ are in the odd positions
 The ‘B’ are in the even positions
 ‘B’ always comes after ‘A’
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
 1^{st} is (0+1) position,
 4^{th} is (3+1) position,
 7^{th} is (6+1) position, and
 10^{th} is (9+1) position.
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:
 12 chickens and dogs
 A chicken has 2 legs
 A dog has 4 legs
 32 legs altogether
Unknown:
 Number of dogs
 Number of chickens
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
 a deep understanding
 a basic understanding
 or worst, no understanding
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:
 My Pals are Here!
 Shaping Math
 Targeting Math
 New Syllabus Primary Mathematics
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
 read and understand the word problems,
 plan a strategy,
 solve them and
 check if their solutions are correct.
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,
 your heart starts to beat faster like you are being chased by a tiger
 your neck muscles tightened around you
 you see numbers swimming in front of you.
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.
 GO FORTH AND LET YOUR CHILDREN MAKE MANY MISTAKES IN THEIR MATH HOMEWORK!
 Act on the feedback from your children’s math teachers to help your children learn better and faster
 Give them time to learn from their mistakes and identify what works and what doesn’t
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?
 More than, less than
 Parallel and perpendicular lines
 Area and perimeter
 Equivalent fractions
Cheat sheet 2: How to solve word problems and nonroutine problems?

Solving nonroutine problems
 Step 1: Understand them
 Step 2: Plan your solutions
 Step 3: Solve them
 Step 4: Check if your solutions work
 Step 1: Understand them

Use these five tools to solve common problems quickly and correctly:
 Bar models
 Listing
 Work backwards
 Patterns
 GuessandCheck
 Bar models
Cheat sheet 3: How to score for word problems?
 First, solve the easiest problems
 Then, solve the challenging problems
Cheat sheet 4: How hard should you push your child?
 GO FORTH AND LET YOUR CHILDREN MAKE MANY MISTAKES IN THEIR MATH HOMEWORK!
 Act on the feedback from your children’s math teachers to help your children learn better and faster
 Give them time to learn from their mistakes and identify what works and what doesn’t
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!