problem solving with heuristics p4

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Primary 4 Problem Solving Heuristic

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• Enhance students’ conception in Mathematics through various techniques and methods of solving.
• Inspire students to investigate Mathematical concepts and acquire analytical thinking skills to further understand and appreciate the concepts.

Topics covered are subjected to change depending on the ability of students.

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Understand Heuristic Math Concepts and What They Are Used For

Heuristics are problem-solving methods which are crucial in solving upper primary Math problem sums. In this book, you will learn how to apply each of the heuristics.  

Learn How to Explain to Your Child in a Way that He will Understand 

If your child has problems understanding the way you teach, it is probably due to he was taught different ways in schools. In this eBook, you will learn the heuristics which schools use so that you can learn the same methods to teach your child too. 

Suitable From P4 to P6

Most students struggle with upper primary problem sums becasuse they are not familiar with the heuristics. Use this book to understand when and how to apply each heuristic. 

Help Your Child to Achieve a Break-Through in Math! 

If you are able to learn the heuristics in this eBook well, and teach them to your child, you will see improvements in your child's results in a short time!

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• Primary 4 Heuristic Concepts

SUPPOSITION CONCEPT

DIRECT SUPPOSITION

SUPPOSITION WITH PENALTY

Problem sums with external transfer.

EXTERNAL TRANSFER, GIVEN DIFFERENCE

EXTERNAL TRANSFER, UNCHANGED DIFFERENCE

EXTERNAL TRANSFER, GIVEN UNITS

Problem sums with internal transfer.

INTERNAL TRANSFER, GIVEN UNITS

INTERNAL TRANSFER, TOTAL UNCHANGED

APPROXIMATION & ESTIMATION

• An even number is 500 when rounded off to the nearest hundred. What is the greatest possible value of the even number?
• Which is the best estimate for 55 x 42?

MULTIPLICATIONS AND GROUPING

TOTAL CONCEPT, REGROUPING

COMPARISON MODEL PROBLEM SUMS

STACKING MODEL

COMPARISON OF QUANTITY AND UNITS

Shortage and surplus problem sums.

LISTING FOR HIGHEST COMMON FACTORS

Peter wants to lay his floor with square carpet tiles. The rectangular shaped floor measures 160 cm by 90 cm.

a) Find the largest possible length of the side of each carpet tile.

b) Find the number of tiles that are needed to cover the floor.

SHORTAGE AND SURPLUS, LISTING METHOD WITH NO FIXED GROUPS

SHORTAGE AND SURPLUS, UNITS METHOD

SHORTAGE AND SURPLUS, LISTING METHOD FIXED NUMBER OF GROUPS

Geometry and angles.

CONSTRUCTING PARALLEL LINES

IDENTIFYING PERPENDICULAR LINES

CONSTRUCTING ANGLES

Common fractions problem sums.

FRACTIONS AS PART OF REMAINDER WITH CHANGING DENOMINATOR

FRACTIONS AS PART OF REMAINDER WITH REDRAWING REMAINDER

TOTAL CONCEPT, DIRECT APPLICATION

EXTERNAL TRANSFER WITH 1 GROUP UNCHANGED

EXTERNAL TRANSFER WITH 1 GROUP UNCHANGED.

INTERNAL PROPORTION TRANSFER

Common decimals problem sums.

DECIMALS, ESTIMATION AND APPROXIMATION

a) A ribbon is 2.33 m when rounded off to 2 decimal places. What is the smallest possible length of the ribbon?

b) Yenni’s mass is 47.6 kg when rounded off to the nearest tenth. What is her largest possible mass? Leave your answer in 2 decimal places.

EXTERNAL TRANSFER

INTERNAL TRANSFERS

BEFORE CHANGE AFTER, FIND CHANGE.

DECIMALS AND GROUPING

Area and perimeter.

AREA AND PERIMETER IGNORE REMAINDER

AREA AND PERIMETER – COMPARISON BY UNITS

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5 Most Commonly Tested Math Problem Sums for Primary 4

Primary 4 mathematics introduces students to a range of problem sums, challenging their analytical and mathematical reasoning skills. These P4 math problem sums often encompass various concepts and problem-solving strategies, each designed to enhance students' mathematical proficiency. Understanding the type of problem sum and mastering the appropriate solving techniques are crucial for success in primary mathematics. Let's explore the most commonly tested problem sums for Primary 4 and delve into effective solving methods.

1. Comparison Model

A comparison bar model displays known and unknown quantities using solid bars, organised vertically, one below the other. This arrangement facilitates easy observation of differences in bar lengths, aiding students in solving the questions effectively.

Example Question:

Leon has a total of 2380 stamps. He has 566 more local stamps than foreign stamps. How many local stamps does he have? 

2 units = 2380 - 566                                

          = 1814

1 unit   = 1814 ÷ 2

         = 907

907 + 566 = 1473

Leon has 1473 local stamps

2. Assumption Method

The assumption method, is also known as the supposition method.

The method is used when: 

There are 2 types of items and each item has a value.

E.g. A car (item 1) has 4 wheels (value), while a bicycle (item 2) has 2 wheels (value).

Total value is given but exact number of each item is unknown.

E.g. There are 56 wheels in total. There are 16 cars and bicycles together. 

There were 57 vans and motorcycles on the road. There were 138 wheels in total. What was the total number of vans present?

Tip: If the question asks us to solve for vans, we assume the opposite (motorcycles).

Assuming all motorcycles,

57 × 2 = 114

138 – 114 = 24

24 ÷ 2 = 12

There were 12 vans.

3. Grouping Concept

Grouping concept problem sums involves grouping items together to make 1 group. 

Then, find the number of such groups. 

Look out for the proportion of one item to the other.

Take note that the items may have different values per unit.

Find the total value of 1 group first.

For some questions, you may need to find the lowest common multiple of 2 numbers.

A shop owner sold an equal number of cakes and drinks. What quantity of cakes was sold, given that each cake was priced at $4 and each drink at $3, resulting in a total collection of $84 by the shop owner?

Find one group.

Group 1 cake and 1 drink.

$4 + $3 = $7

Find the number of groups.

$84 ÷ $7 = 12

1 group will have 1 cake and 1 drink.

12 groups will have 12 cakes and 12 drinks.

Hence, there were 12 cakes sold.

4. Stacking Model

Stacking models involve two or more items with multiple quantities of each item. These problems often require careful organization and calculation to determine the quantities or amounts involved. For example, “Kaylee bought 2 cups, 3 plates and 5 spoons.”

At a cinema, an adult ticket costs $6 more than a child ticket. Mr. Tan and his family bought 2 adult tickets and 4 children’s tickets for $54. How much does an adult ticket cost?

Extra  = $6 × 2

         = $12

6 units = $54 – $12

         = $42

1 unit  = $42 ÷ 6

         = $7

Adult ticket = $7 + $6

               = $13

The adult ticket cost $13.

5. Equal Stage

Equal stage problem sums involve scenarios where two people or items have equal quantities at the beginning or the end. Drawing models and understanding keywords are essential for effectively solving the problem sums.

Here are some tips for equal stage problem sums: 

Draw models of equal length when the two people/items have the same quantity.

Keywords like “gave away” means you should cut out from the model.

Keywords like “bought more” means you should add to the model.

For 2 equal fractions, we make the numerator the same.

In a carnival, 23 of the children tickets sold was equal to 1112 of the adult tickets sold. There were 108 more children tickets than adult tickets sold. How many children tickets were sold?

Children → 23 = 2233                   

Adults → 1112 = 2224

Children = 33 units

Adults = 24 units

33 units – 24 units = 9 units

9 units = 108

1 unit  = 108 ÷ 9

        = 12

33 units = 12 × 33

        = 396

U nderstanding and mastering the various types of problem sum encountered in Primary 4 Mathematics is essential for students' success. With effective problem-solving strategies and practice, students can tackle math word problems confidently and excel in their mathematical journey. This problem sums guide for parents provides invaluable insights into helping students navigate through mathematical challenges, ensuring a solid foundation for future learning.

At AGrader , primary students are equipped with efficient techniques to solve 90% of challenging math problems swiftly. Our experienced tutors employ heuristics to teach problem-solving skills, enabling children to tackle even the most perplexing problem sums with ease.

Targeted at students from Primary 1 to 6, our tuition programme follows the latest syllabus outlined by the Ministry of Education in Singapore. The Primary Math Tuition Programme  objectives include ensuring students develop a strong grasp of core mathematical concepts, applying these concepts through critical thinking and heuristics in problem-solving, and fostering confidence for school examinations.

Moreover, AGrader Learning Centre  lessons are tailored to accommodate students of different proficiency levels. Emphasizing conceptual clarity and application-driven learning, students acquire the essential skills and knowledge essential for success in their mathematics assessments. With convenient accessibility across 19 locations throughout the island, parents have the flexibility to select the nearest centre, enabling effortless engagement in the programme.

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Tackle math word problems with greater confidence: The parent’s guide to heuristics

What are math heuristics.

Heuristics are simple rules or mental shortcuts that help us understand a problem or arrive at a decision quickly.

If you’ve followed a “rule of thumb”, you’ve used a heuristic.

If I’m sending an important e-mail, I will always review it twice, at different timings during the day, to minimise the possibility of making an embarrassing mistake!

Likewise, math heuristics are proven tactics we can use to solve problems effectively — by being more strategic, systematic, and conscientious.

Let’s say you had to tighten a screw. You could go through the screwdrivers in your toolbox one by one, or you could approximate the size of the screw and narrow down your selection to the last 3 screwdrivers. Which method gets the job done faster?

How do math heuristics help my child with problem-solving?

As your child approaches upper primary (P5, P6), they’ll find that word problems become more complex — the approaches and solutions are less obvious.

Hence, knowing common math heuristics will give them the tools they need to tackle challenging problem sums that come up in their homework or exams.

Not only can it dramatically increase your chances of solving any Math problem and help you get started, it can also guide you along your thinking processes to reduce the effort and time needed when problem solving.

To prepare your child for PSLE questions, it helps to know a variety of ways to tackle any problem that comes their way. PSLE questions are complex, so mastering heuristics is like having every tool you could ever need in your toolbox!

We teach the heuristics that your child will learn in school in a way that is fun and easy to understand — just check out some of our videos below if you’re curious! We have many more of such videos in our system.

Why do parents need to know about math heuristics too?

Practicle’s math content covers all heuristics that are tested by MOE, but we recommend that parents have foundational knowledge to supervise their children, especially when preparing for exams.

Want to know how your child is performing? We’ll send you reports about how well they’ve mastered their math concepts and heuristics.

You’ll no longer need to rely on teachers for guidance and feedback all the time, or spend exorbitant amounts of money on assessment books.

What heuristics does MOE test for primary school mathematics?

According to the Singapore Mathematics framework developed by the Curriculum Planning and Development Division (CPDD) team at the Ministry of Education Singapore (MOE), the types of heuristics in Mathematics that can be applied to primary school math problems can be grouped as follow:

1. Visualise a problem 2. Make a calculated guess 3. Walk through the process 4. Simplify the problem 5. Consider special cases

Hence, it is crucial for your child to learn how to use them.

Here’s an example of a heuristic that your child will learn from P4-P6:

We have more of such math videos on our YouTube channel , and if you’d like to try to out our questions and receive question-specific video explanations, how about signing up for a free trial ? No minimum commitment, cancel anytime.

How does Practicle teach math heuristics?

As a team of former teachers and game developers, we put a great deal of thought into making the learning experience in Practicle engaging for your kids using a 2-pronged approach:

1. It needs to be educationally sound and aligned with the school’s syllabus 2. It needs to stir interest and encourage kids to learn more, making it truly effective

#2 is where many math learning solutions fall short, especially traditional methods like tuition and assessment books. Students usually tune out quickly.

Practicle makes learning math heuristics fun, while ensuring your kids learn the proper skills needed to do well in their tests and exams.

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Try our learning platform free with a 7 day trial and see if your child likes it.

Practicle is an online gamified Singapore Math adaptive learning platform that helps primary school children master Math through understanding and fun.

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Heuristic Approach to Problem-solving: Examples

Related Topics: More Lessons for Singapore Math Math Worksheets

Videos, worksheets, solutions, and activities to help students learn how to use the heuristic approach to solve word problems in Singapore Math.

Use A Picture / Diagram / Model Example: The total cost of 2 similar bags, 3 wallets and 4 belts is $1188. A bag cost thrice as much as a wallet and a wallet costs twice as much as a belt. How much will Ted have to pay for a bag, a wallet and a belt?

Heuristic Approach to problem-solving Example: 7/10 of the boys who participated in a marathon race were Chinese. The rest of the boys were made up of Eurasians and Malays in the ratio 5:7 respectively. There were 756 more Chinese than Malay boys. Find the total number of boys who participated in the marathon race.

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Math Heuristics: Draw a Table, Make Suppositions and Use Before-After Concept

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Heuristics, in the context of problem-solving, are a set of strategies to help students solve mathematical problems.

Although problem-solving is by and large the process of working towards a goal to which a solution may not be immediately present, it is important that problem solvers (or students) are not only aware of what they are doing and why they are doing it, but also have the ability to self-regulate these thinking processes.

This is where the role of metacognition in problem-solving comes in. Metacognition is the awareness of how your mind works and the ability to control your thinking process.

When students understand and recognise how they learn and are given opportunities to monitor and regulate their thinking during problem-solving, not only do they improve their metacognitive skills but they may also be more successful in solving the problem.

Singapore Math Heuristics

In part one of our Math Heuristics series , we gave an overview of the 12 heuristics in Singapore Primary Math syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to solve various math word problems using them.

In part two of the Math Heuristics series , we expanded on the heuristics – Act It Out, Draw A Diagram and Look For Patterns, and also demonstrated how to apply the Polya’s 4-step problem-solving process in sample word problems.

In the third part of the Math Heuristics series , we focused on the heuristics: Make a Systematic List, Guess and Check and Restate the Problem in Another Way.

In the fourth part of the Math Heuristics series , we looked at three heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards.

In the final part of the series, we will explore the remaining heuristics: Draw a Table , Make Suppositions and Use Before-After Concept .

Heuristic: Draw a Table

Word Problem (Primary 4):

Mary cycles to the park every 6 days. John cycles to the same park every 8 days. Mary and John cycle to the park on 3 June, Tuesday. On which date and day will they cycle to the park together on the same day?

1. Understand: What to find: date and day Mary and John next cycle to the park on the same day. What is known: Mary cycles to the park every 6 days. John cycles to the park every 8 days. Mary and John cycle to the park on 3 June, Tuesday.

2. Choose: Draw a table

3. Solve: Let’s compare the days Mary and John cycle to the park. Mary cycle to the park on days in multiples of 6. John cycle to the park on days in multiples of 8.

Mary and John will cycle to the park on the same day every 24 days. 3 June + 24 days is 27 June 24 days = 3 weeks and 3 days Tuesday + 3 days is Friday

Mary and John will next cycle to the park on 27 June, Friday.

4. Check: Is 24 a multiple of 6? Yes Is 24 a multiple of 8? Yes Is 27 June Friday 24 days after 3 Jun Tuesday? Yes

Try solving the following word problem using Polya’s 4-step process.

Janice wanted to distribute stickers equally among some children. If each child received 8 stickers, she would have 3 stickers left. If each child received 11 stickers, she would need another 9 stickers. How many children were there?

Answer: Janice had 35 stickers. There were 4 children. See the solution in part one of our Singapore Math Heuristics series .

Heuristic: Make Suppositions

Word Problem (Primary 5):

Mary took a Science test. She answered all 30 questions and obtained 42 marks. For each correct answer, 3 marks were awarded. For each wrong answer, 1 mark was deducted. How many questions did she answer correctly?

1. Understand: What to find: Number of questions Mary answered correctly. What is known: Mary answered 30 questions. She obtained 42 marks. 3 marks were awarded for each correct answer. 1 mark was deducted for each wrong answer.

2. Choose: Make suppositions

3. Solve: Suppose that Mary answered all 30 questions correctly. 30 x 3 = 90 Mary obtained 92 supposed total marks.

90 – 42 = 48 The supposed total marks are 48 more than the actual total marks.

3 + 1 = 4 By replacing 1 correct answer with 1 wrong answer, 4 marks are deducted from the total marks.

48 ÷ 4 = 12 Mary answered 12 questions wrongly.

30 – 12 = 18 She answered 18 questions correctly.

4. Check: What are the total marks awarded for 18 correct answers? 18 x 3 = 54 What are the total marks deducted for 12 wrong answers? 12 x 1 = 12 What are the total marks obtained by Mary? 54 – 12 = 42

Farmer James has some ducks, horses and cows on his farm. He has 30 ducks and cows altogether. The total number of legs the ducks and cows have is 82. The total number of legs the horses have is 28. How many ducks and how many cows are there on the farm?

Answer: There are 19 ducks and 11 cows on the farm. See the solution in part one of our Singapore Math Heuristics series .

Heuristic: Use Before-After Concept

The ratio of Joy’s age to her uncle’s age is 1 : 4 now. In 21 years’ time, the ratio of Joy’s age to her uncle’s age will be 3 : 5. How old is Joy’s uncle now?

1. Understand What to find: Joy’s uncle age now. What is known: The ratio of age now is 1 : 4. The ratio of age 21 years later is 3 : 5.

2. Choose: Use before-after concept

(9 – 2) units = 7 units = 21 years 1 unit = 21 ÷ 7 = 3 years 8 units = 3 x 8 = 24 Joy’s uncle is 24 years old now.

4. Check: How old is Joy now? 3 x 2 = 6 years old Ratio of Joy’s age to her uncle’s age now = 6 : 24 = 1 : 4 How old is Joy 21 years later? 3 x 9 = 27 years old How old is Joy’s uncle 21 years later? 3 x 15 = 45 Ratio of Joy’s age to her uncle’s age 21 years later = 27 : 45 = 3 : 5

Ray and Sam each brought some money for shopping. The ratio of the amount of Ray’s money to the amount of Sam’s money was 3 : 4. After each of them bought a laptop for $1250, the ratio of the amount of Ray’s money to the amount of Sam’s money became 1 : 3. How much money did Sam bring for shopping?

Answer: Sam brought $2000 for shopping. See the solution in part one of our Singapore Math Heuristics series .

This is the final part to S.A.M Math Heuristics series for expert tips on math heuristics.

Read the rest of the “S.A.M Math Heuristics” five-part series below :

Part 1: What Are Heuristics? Part 2: Math Heuristics: Act It Out, Draw A Diagram, Look For Patterns Part 3: Math Heuristics: Make A Systematic List, Guess And Check, And Restate The Problem In Another Way Part 4: Math Heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards

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The curriculum is complemented by S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

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Heuristics, in the context of problem-solving, are a set of strategies to help students solve mathematical problems.

Although problem-solving is by and large the process of working towards a goal to which a solution may not be immediately present, it is important that problem solvers (or students) are not only aware of what they are doing and why they are doing it, but also have the ability to self-regulate these processes.

This is where the role of metacognition in problem-solving comes in. Metacognition is the awareness of how your mind works and the ability to control your thinking process.

When students understand and recognise how they learn and are given opportunities to monitor and regulate their thinking during problem-solving, not only do they improve their metacognitive skills but they may also be more successful in solving the problem.

Singapore Math Heuristics

In  part one of our Math Heuristics series , we gave an overview of the 12 heuristics in Singapore Primary Math syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to solve various math word problems using them.

In  part two of the Math Heuristics series , we expanded on the heursitics – Act It Out, Draw A Diagram and Look For Patterns, and also demonstrated how to apply the Polya’s 4-step problem-solving process in sample word problems.

In  part three of the Math Heuristics series , we focused on the heuristics: Make a Systematic List, Guess and Check, and Restate the Problem in Another Way.

In  part four of the Math Heuristics series , we looked at three heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards.

In the final part of the series, we will explore the remaining heuristics:  Draw a Table ,  Make Suppositions  and  Use Before-After Concept .

_______________________________________________________________________________________________________

Heuristic: Draw a Table

Word Problem (Grade 4) :

Mary cycles to the park every 6 days. John cycles to the same park every 8 days. Mary and John cycle to the park on 3 June, Tuesday. On which date and day will they cycle to the park together on the same day?

1. Understand: What to find: date and day Mary and John next cycle to the park on the same day. What is known: Mary cycles to the park every 6 days. John cycles to the park every 8 days. Mary and John cycle to the park on 3 June, Tuesday.

2. Choose: Draw a table

3. Solve: Let’s compare the days Mary and John cycle to the park. Mary cycle to the park on days in multiples of 6. John cycle to the park on days in multiples of 8.

Try solving the following word problem using Polya’s 4-step process.

Heuristic: Make Suppositions

Word Problem (Grade 5) :

Mary took a Science test. She answered all 30 questions and obtained 42 marks. For each correct answer, 3 marks were awarded. For each wrong answer, 1 mark was deducted. How many questions did she answer correctly?

1. Understand: What to find: Number of questions Mary answered correctly. What is known: Mary answered 30 questions. She obtained 42 marks. 3 marks were awarded for each correct answer. 1 mark was deducted for each wrong answer.

2. Choose: Make suppositions

3. Solve: Suppose that Mary answered all 30 questions correctly. 30 x 3 = 90 Mary obtained 92 supposed total marks.

90 – 42 = 48 The supposed total marks are 48 more than the actual total marks.

3 + 1 = 4 By replacing 1 correct answer with 1 wrong answer, 4 marks are deducted from the total marks.

48 ÷ 4 = 12 Mary answered 12 questions wrongly.

30 – 12 = 18 She answered 18 questions correctly.

4. Check: What are the total marks awarded for 18 correct answers? 18 x 3 = 54 What are the total marks deducted for 12 wrong answers? 12 x 1 = 12 What are the total marks obtained by Mary? 54 – 12 = 42

Farmer James has some ducks, horses and cows on his farm. He has 30 ducks and cows altogether. The total number of legs the ducks and cows have is 82. The total number of legs the horses have is 28. How many ducks and how many cows are there on the farm?

Heuristic: Use Before-After Concept

The ratio of Joy’s age to her uncle’s age is 1 : 4 now. In 21 years’ time, the ratio of Joy’s age to her uncle’s age will be 3 : 5. How old is Joy’s uncle now?

1. Understand: What to find: Joy’s uncle age now. What is known: The ratio of age now is 1 : 4. The ratio of age 21 years later is 3 : 5.

2. Choose: Use before-after concept

This is the final part to S.A.M Math Heuristics series. Read  part one ,  part two ,  part three  and  part four  here.

Established in 2010, Seriously Addictive Mathematics (S.A.M) is the world’s largest Singapore Math enrichment program for children aged four to 12. The award-winning S.A.M program is based on the global top-ranking Singapore Math curriculum with a focus on developing problem solving and thinking skills.

The curriculum is complemented with S.A.M’s two-pillared approach of Classroom Engagement and Worksheet Reinforcement, with an individual learning plan tailored to each child at their own skill level and pace, because no two children learn alike.

Singapore Math Heuristics: Solve Part of the Problem, Simplify the Problem and Work Backwards

Problem-solving in mathematics helps children develop reasoning and communication skills that are transferrable and important life skills. Reasoning is required on three levels when children solve word problems. First, they use reasoning to recognise what information is provided or missing. Then, they use reasoning to figure out what information they need to find. Finally, they […]

Singapore Math Heuristics: Make A Systematic List, Guess And Check, Restate The Problem In Another Way

The skills children pick up in math are indispensable; they can be applied to other academic subjects and to solve real-world problems in their daily lives and future work. The Singapore Math curriculum focuses on problem solving. Through problem solving, children develop thinking skills such as creative thinking and critical thinking. When children analyse math […]

Singapore Math Heuristics: Act It Out, Draw A Diagram, Look For Patterns

In part one of our Singapore Math Heuristics series, we gave an overview of the 12 heuristics in Singapore Primary Math syllabus, with tips from the curriculum team at Seriously Addictive Mathematics (S.A.M) on how to solve various math word problems using them. To recap, heuristics are methods or strategies students can use to solve complex […]

What are Singapore Math Heuristics?

Ever tried to help your child with primary math homework and got stumped? Today’s math questions can be challenging – even for adults. Math education is changing. While many parents spent time memorising procedures and formulas, today’s students are expected to not only understand and master the concepts, but also to have strong thinking skills […]

Mastering Heuristics Series: Unit Transfer Method In Solving Challenging Upper Primary Mathematical Problems

Submitted by See Wai Yip

30th May 2010 Sunday, 2 pm to 5 pm $200 nett, inclusive of a complimentary Unit Transfer Method book

The Unit Transfer Method parents workshop is designed for parents who wish to take a more hands-on approach in their children’s Maths education. Parents will learn the very same problem-solving concepts and methodologies that their children need to master in order to score in Maths.

We all know that times have changed, but who would have imagined that the foundational mathematical problem solving skills we were brought up on mean practically nothing for our next generation?

The mathematics questions found in primary schools these days can leave even the most academically inclined parents stumped! Regardless whether you’ve scored A* in your own PSLE maths paper decades ago, we’re sure you’ve had difficulty trying to guide your child with his/her homework using your old-school algebraic equations. Even the relatively newer model approach, involving drawing rectangular boxes/blocks, doesn’t always work.

Perhaps the alternative Heuristics approach will put those seemingly impossible-to-solve problem sums in a “less intimidating” light. Heuristics simply refers to rules, processes or methods that the student can discover.

Mr Sunny Tan, Principle Trainer at mathsHeuristics™ covers six heuristics concepts in his first book titled, Unit Transfer Method (Part 1 of 4 in his Mastering Heuristics Series of guidebooks for tackling Upper Primary Mathematics). They are namely – Before and After, Excess and Shortage, Repeated Identity, Equal, Two Variables, and Ratio and Proportion. The Ministry of Education in Singapore has incorporated 11 Problem-Solving Heuristics into all primary-level mathematical syllabuses.

An NIE-trained ex-teacher specialising in Mathematics, with more than 10 years’ teaching experience, Sunny observed that schools do not actually teach a standard method nowadays. Parents who depend on tuition agencies or private tutors to improve their kid’s performance often overlook the fact that most tutors were also raised on the same Mathematics syllabus as themselves, thus may not be effective in coaching their kids.

With the aim of closing this gap, Sunny established the mathsHeuristics™ Programme as well as penned the Mastering Heuristics Series guidebook. His own academic research focuses on mathematical problem-solving, problem-posing and thinking skills.

Since its launch in January 2009, the book has received good press coverage such as in The Straits Times, and ParentsWorld and Wawa magazines; parents and academics too have raved about its simple, logical and powerful technique. The handbook boldly challenges you to solve a sample problem sum in five minutes – and it really can be done once you grasp the Unit Transfer concept, which emphases the use of ratio as a way to analyse the problem sums. Equipped with this concept, almost 90% of the challenging problems involving the six above-mentioned PSLE topics can be solved.

Dr Lua, a overseas Singaporean parent for the past eight years, was concerned that his son, Anthony, could not adapt to the method of solving problem sums taught in our local primary schools when Anthony joined in Primary 3. Anthony was not schooled in Singapore since pre-school to Primary 2 – he had undergone the British and Japanese education systems, which do not focus on examinations and grade achievements. “The experiential and applied Maths training acquired by my child in overseas schools would not allow him to cope with the aggressive and rigorous solving of Maths problems set by in the Singapore curriculum. In addition, my child lacks the visualisation skill needed in using and applying model heuristics to solve Maths problems,” said Dr Lua.

Being an academic himself, Dr Lua held many research discussions with Sunny to experiment how Anthony coped with using the Unit Transfer Method to understand and solve problem sums. Anthony now has a non-visualisation heuristics and tool to help him solve the higher-order Maths problem sums. In turn, from Anthony’s experience, Sunny was able to refine and re-invent his method further.

The main idea is to identify which variable in the problem sum remains unchanged, interpret the statement and convert everything to units for a solution. The method further teaches you to convert decimals and percentages to fractions first since decimals and percentages may be confusing to children.

Although students should also be trained in other problem-solving methods, many who have learnt Sunny’s Unit Transfer Method feel that this particular method has been effective in solving many of the difficult problem sum questions. Requiring visualisation, the Unit Transfer Method explains all sums in simple tabulation format and complements the widely taught model approach, which uses diagrams. This helps the student to systematically organise information in the problem sums.

The Mastering Heuristics Series gives two perspectives:

• Concepts in action – to show how easy and efficient Mathematical Heuristics are in solving challenging mathematical questions. 
• Concepts being applied – to systematically show every step involved in applying each Mathematical Heuristics concept.

The Unit Transfer Method guidebook is a useful standalone resource for hands-on parents who want to be able to guide their children in their maths homework. The easy-to-follow step-by-step instructions make the Mastering Heuristics Series an ultimate practical guide for parents. Needless to say, it also serves as a companion for students enrolled in the mathsHeuristics™ Programme, while keeping their parents abreast with what they are learning.

Purchase online at or self collect at the mathsHeuristics ™ centre

Interested parents can also sign up for workshops, such as the three-hour Unit Transfer Method parents workshop conducted by mathsHeuristics™ at Goldhill Centre (Thomson). Upcoming Workshop: 30 May (Sunday), 2-5pm. See more information .

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Some Helpful Problem-Solving Heuristics

A  heuristic  is a thinking strategy, something that can be used to tease out further information about a problem and thus help you figure out what to do when you don’t know what to do. Here are 25 heuristics that can be useful in solving problems. They help you monitor your thought processes, to step back and watch yourself at work, and thus keep your cool in a challenging situation.

• Ask somebody else  how to do the problem. This strategy is probably the most used world-wide, though it is not one we encourage our students to use, at least not initially.
• Guess and try  (guess, check, and revise). Your first guess might be right! But incorrect guesses can often suggest a direction toward a solution. (N.B. A spreadsheet is a powerful aid in guessing and trying. Set up the relationships and plug in a number to see if you get what you want. If you don’t, it is easy to try another number. And another.)
• Restate the problem  using words that make sense to you. One way to do this is to explain the problem to someone else. Often this is all it takes for the light to dawn.
• Organize information  into a table or chart. Having it laid out clearly in front of you frees up your mind for thinking. And perhaps you can use the organized data to generate more information.
• Draw a picture  of the problem. Translate problem information into pictures, diagrams, sketches, glyphs, arrows, or some other kind of representation.
• Make a model  of the problem. The model might be a physical or mental model, perhaps using a computer. You might vary the problem information to see whether and how the model may be affected.
• Look for patterns , any kind of patterns: number patterns, verbal patterns, spatial/visual patterns, patterns in time, patterns in sound. (Some people define mathematics as the science of patterns.)
• Act out the problem , if it is stated in a narrative form. Acting it out can have the same effect as drawing a picture. What’s more, acting out the problem might disclose incorrect assumptions you are making.
• Invent notation . Name things in the problem (known or unknown) using words or symbols, including relationships between problem components.
• Write equations . An equation is simply the same thing named two different ways.
• Check all possibilities  in a systematic way. A table or chart may help you to be systematic.
• Work backwards  from the end condition to the beginning condition. Working backwards is particularly helpful when letting a variable (letter) represent an unknown.
• Identify subgoals  in the problem. Break up the problem into a sequence of smaller problems (“If I knew this, then I could get that”).
• Simplify the problem . Use easier or smaller numbers, or look at extreme cases (e.g., use the minimum or maximum value of one of the varying quantities).
• Restate the problem again . After working on the problem for a time, back off a bit and put it into your own words in still a different way, since now you know more about it.
• Change your point of view . Use your imagination to change the way you are looking at the problem. Turn it upside down, or pull it inside out.
• Check for hidden assumptions  you may be making (you might be making the problem harder than it really is). These assumptions are often found by changing the given numbers or conditions and looking to see what happens.
• Identify needed and given information clearly . You may not need to find everything you think you need to find, for instance.
• Make up your own technique . It is your mind, after all; use mental actions that make sense to you. The key is to do something that engages you with the problem.
• Try combinations of the above heuristics .

These heuristics can be readily pointed out to students as they engage problems in the classroom. However, real-world problems are often confronted many times over or on increasingly complex levels. For those kinds of problems, George Polya, the father of modern problem-solving heuristics, identified a fifth class (E) of looking-back heuristics. We include these here for completeness, but also with the teaching caveat that solutions often improve and insights grow deeper after the initial pressure to produce a solution has been resolved. Subsequent considerations of a problem situation are invariably deeper than the first attempt.

• Check your solution . Substitute your answer or results back into the problem. Are all of the conditions satisfied?
• Find another solution . There may be more than one answer. Make sure you have them all.
• Solve the problem a different way . Your first solution will seldom be the best solution. Now that the pressure is off, you may readily find other ways to solve the problem.
• Solve a related problem . Steve Brown and Marion Walter in their book,  The Art of Problem Posing , suggest the “What if not?” technique. What if the train goes at a different speed? What if there are 8 children, instead of 9? What if . . .? Fascinating discoveries can be made in this way, leading to:
• Generalize the solution . Can you glean from your solution how it can be made to fit a whole class of related situations? Can you prove your result?

Heuristic Problem Solving: A comprehensive guide with 5 Examples

What are heuristics, advantages of using heuristic problem solving, disadvantages of using heuristic problem solving, heuristic problem solving examples, frequently asked questions.

• Speed: Heuristics are designed to find solutions quickly, saving time in problem solving tasks. Rather than spending a lot of time analyzing every possible solution, heuristics help to narrow down the options and focus on the most promising ones.
• Flexibility: Heuristics are not rigid, step-by-step procedures. They allow for flexibility and creativity in problem solving, leading to innovative solutions. They encourage thinking outside the box and can generate unexpected and valuable ideas.
• Simplicity: Heuristics are often easy to understand and apply, making them accessible to anyone regardless of their expertise or background. They don’t require specialized knowledge or training, which means they can be used in various contexts and by different people.
• Cost-effective: Because heuristics are simple and efficient, they can save time, money, and effort in finding solutions. They also don’t require expensive software or equipment, making them a cost-effective approach to problem solving.
• Real-world applicability: Heuristics are often based on practical experience and knowledge, making them relevant to real-world situations. They can help solve complex, messy, or ill-defined problems where other problem solving methods may not be practical.
• Potential for errors: Heuristic problem solving relies on generalizations and assumptions, which may lead to errors or incorrect conclusions. This is especially true if the heuristic is not based on a solid understanding of the problem or the underlying principles.
• Limited scope: Heuristic problem solving may only consider a limited number of potential solutions and may not identify the most optimal or effective solution.
• Lack of creativity: Heuristic problem solving may rely on pre-existing solutions or approaches, limiting creativity and innovation in problem-solving.
• Over-reliance: Heuristic problem solving may lead to over-reliance on a specific approach or heuristic, which can be problematic if the heuristic is flawed or ineffective.
• Lack of transparency: Heuristic problem solving may not be transparent or explainable, as the decision-making process may not be explicitly articulated or understood.
• Trial and error: This heuristic involves trying different solutions to a problem and learning from mistakes until a successful solution is found. A software developer encountering a bug in their code may try other solutions and test each one until they find the one that solves the issue.
• Working backward: This heuristic involves starting at the goal and then figuring out what steps are needed to reach that goal. For example, a project manager may begin by setting a project deadline and then work backward to determine the necessary steps and deadlines for each team member to ensure the project is completed on time.
• Breaking a problem into smaller parts: This heuristic involves breaking down a complex problem into smaller, more manageable pieces that can be tackled individually. For example, an HR manager tasked with implementing a new employee benefits program may break the project into smaller parts, such as researching options, getting quotes from vendors, and communicating the unique benefits to employees.
• Using analogies: This heuristic involves finding similarities between a current problem and a similar problem that has been solved before and using the solution to the previous issue to help solve the current one. For example, a salesperson struggling to close a deal may use an analogy to a successful sales pitch they made to help guide their approach to the current pitch.
• Simplifying the problem: This heuristic involves simplifying a complex problem by ignoring details that are not necessary for solving it. This allows the problem solver to focus on the most critical aspects of the problem. For example, a customer service representative dealing with a complex issue may simplify it by breaking it down into smaller components and addressing them individually rather than simultaneously trying to solve the entire problem.

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Working Backwards: Heuristic for Problem Solving

Working Backwards is a non-routine heuristic that all pupils learn in primary schools . Many pupils learn this heuristic as early as when they were in primary two. You can easily find this heuristic being included in one of the topics in assessment books .

Though common, this heuristic is in fact one of the toughest in primary Mathematics syllabus . Problem sums that can be solved using Working Backwards are usually wordy with a sequence of events taking place which make them complex. You need endurance and clarity of mind to follow through a series of events that are unfolding in sequence, not to mention in a backward manner.

Without further ado, let’s delve into one typical upper primary problem sum to find out how we can tackle this category of problem sums using the heuristic of Working Backwards .

Xiaoming and Ali were playing a card game using 96 pokemon cards. In the first game, Ali lost 1 5 of his cards to Xiaoming. In the second game, Xiaoming lost 1 3 of his cards to Ali. After the second game, both boys had the same number of pokemon cards. How many pokemon cards did Xiaoming have at first?

Study and Understand the Problem

In this problem sum, there are only two variables – Xiaoming and Ali.

Cards are transferred between Xiaoming and Ali in a series of games, but the total number of cards between them is still the same – internal transfer.

Think of a Plan

There are contextual clues of a typical “Working Backwards” problem sum:

• Final information is given on how a situation ends and you need to find the answer in the beginning.
• There is a focus on sequence of events.

Act on the Plan

Reverse the solution steps by working backwards in a systematic manner using a table. A table will help to organise your working in a more orderly manner and track the steps in sequence.

Final number of cards each person had = 96 ÷ 2 = 48

Second game

After Xiaoming lost 1 3 of his cards to Ali, he was left with 2 units as 1 unit was won by Ali (Refer to the numerator and denominator). Thus, 2 units = 48

After Ali lost 1 5 of his cards to Xiaoming, he was left with 4 units as 1 unit was won by Xiaoming (Refer to the numerator and denominator). Thus, 4 units = 24

Reflect on my Answer

Work forward with the answer you have.

XM → 66 A → 30

A → of 30 = 6 30 – 6 = 24 XM → 66 + 6 = 72

XM → of 72 = 24 72 - 24 = 48 (correct!) A → 24 + 24 = 48 (correct!)

More Examples of Problems Sums Involving Working Backwards

Try to pick out the contextual clues that tell you that Working Backwards can be used.

P3 Math question

Alan bought some fish. One day, 6 of his fish died. After that, he bought the same number of fish as those which were still alive. He gave away all his fish equally among 8 friends and each friend had 4 fish. How many fish did Alan have at first?

P6 Math question

A MRT train left Bugis station with some passengers. At Lavender station, no passengers alighted and the number of passengers who boarded the train was  1 4 of the original number of passengers in the train. At Kallang station,  2 5 of the passengers alighted and 51 passengers boarded the train. At Aljunied station, 2 3 of the passengers alighted and 24 passengers boarded the train. At Paya Lebar station, all 122 passengers alighted from the train. How many passengers were there when the train left Bugis Station?

Whenever possible, use a table or draw boxes to help you solve Working Backwards questions in an orderly and systematic manner.

If this article has benefitted you, do support us by giving it a “Like” in OwlSmart Facebook or “Share” it with friends who have children in upper primary levels. With an OwlSmart subscription , you also gain access to more than 10 questions on Working Backwards.

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About the Author

Teacher Zen has over a decade of experience in teaching upper primary Math and Science in local schools. He has a post-graduate diploma in education from NIE and has a wealth of experience in marking PSLE Science and Math papers. When not teaching or working on OwlSmart, he enjoys watching soccer and supports Liverpool football team.

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P4 New Problem-Solving Processes in Mathematics

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New Problem-solving Processes in Mathematics is a series ofPrimary 1–6 Mathematics books that provides comprehensive practices in routine andchallenging word problems.Pupils are exposed to the various essential types of problem-solving heuristics identifiedin the routine practices in each topic. The guided examples that employ the four-stepproblem-solving approach and the scaffolded practice questions in the routine practicesfacilitate structured drilling exercises. Through ample drilling exercises in the routinepractices, pupils will thus be ready to take on challenging word problems in the reviewsets.

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