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.

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 and problem-solving skills to solve complex math questions.
As new concepts and strategies are being taught, and homework turns from arithmetic exercises to using multiple ways to solve a math word problem, you may feel unsure or have no idea how to help your child.

In this series, the curriculum team at Seriously Addictive Mathematics (S.A.M) shares expert tips on math heuristics and how to use them to solve math word problems.
What are Math Heuristics?

Heuristics – a word that baffles many primary school students and their parents.

To define it simply, math heuristics are strategies that students can use to solve complex word problems.

Word problems can be solved in several ways using different heuristics, while some word problems are solved using a combination of heuristics.

To solve word problems efficiently, students must be familiar with both the problem-solving methods (heuristics) and the problem-solving process.
How many Heuristics are there?

In Singapore Math, there are 12 heuristics in the primary math syllabus that can be grouped into four main categories:

  • To give a representation: Draw a diagram/model, draw a table, make a systematic list
  • To make a calculated guess: Look for pattern(s), guess and check, make suppositions
  • To go through the process: Act it out, work backwards, use before-after concept
  • To change the problem: Restate the problem in another way, simplify the problem, solve part of the problem
  • What is the Problem-Solving Process?
    Another important point to note is that Singapore Math adopts Polya’s four-step problem-solving process:

    1. Understand the problem: What to find? What is known and unknown?
    2. Devise a plan: Choose the most suitable heuristic
    3. Carry out the plan: Solve the problem
    4. Look back: Check the answer

    So, how can we use heuristics to solve math word problems? Here are examples of word problems with solutions provided by the curriculum team at S.A.M.


    Heuristic: Act it out

    Word Problem (Grade 1):

    Alan, Ben and Carol are in the school’s Art Club. Their teacher, Mr Tan, wants two of them to join a contest. How many ways can Mr Tan choose two pupils?

    Alan and Ben, Alan and Carol, Ben and Carol.

    Mr Tan can choose two pupils in 3 ways.

    Heuristic: Draw a diagram/model

    Word Problem (Grade 3):

    The smaller of two numbers is 1217. The greater number is 859 more than the smaller number.
    (a) What is the greater number?
    (b) What is the sum of the numbers?

    The greater number is 1217 + 859 = 2076.

    The sum of the numbers is 2076 + 1217 = 3293.

    Heuristic: Look for pattern(s)

    Word Problem (Grade 1):

    Draw the shape that comes next.

    Label all shapes, the pattern is in repeating blocks of A, B, C.

    Heuristic: Make a systematic list

    Word Problem (Grade 2):

    A shop sells apples in bags of 3. It sells lemons in bags of 4. Paul buys some bags of apples and lemons. He buys the same number of each fruit. He buys more than 20 and fewer than 30 pieces of each fruit. How many apples does Paul buy?
    Paul buys 24 apples.

    Heuristic: Guess and check

    Word Problem (Grade 5):

    Vijay is presented with the equations below. Insert one pair of brackets in each equation to make it true.
    4 × 11 + 18 ÷ 3 + 6 = 46
    The equation is 4 x 11 + 18 ÷ (3 + 6) = 46.

    Heuristic: Restate the problem in another way

    Word Problem (Grade 2):

    There are some identical pens and erasers. 2 pens and 3 erasers are 45 centimetres long altogether. 6 erasers and 2 pens are 60 centimetres long altogether. What is the length of 3 erasers?

    If we subtract the total length of 2 pens and 6 erasers from that of 2 pens and 3 erasers, we get the length of 3 erasers.
    60 cm – 45 cm = 15 cm

    The length of 3 erasers is 15 cm.

    Heuristic: Solve part of the problem

    Word Problem (Grade 6):

    The diagram below shows 4 shaded triangles in Triangle ABC. All the triangles in the diagram are equilateral triangles. If the area of Triangle ABC is 64 cm2, find the total area of the shaded triangles.
    Triangle ABC is made up of 16 small equilateral triangles.
    7 out of 16 small equilateral triangles are shaded.
    7/16 × 64 = 28

    The total area of the shaded triangles is 28 cm2.

    Heuristic: Simplify the problem

    Word Problem (Grade 6):

    The shaded figure below shows a semicircle and two quarter circles. Find the area of the shaded figure. (Take pi = 3.14)
    30 × 15 = 450

    The area of the shaded figure is 450 cm2.

    Heuristic: Work backwards

    Word Problem (Grade 3):

    Darren had some stickers in his collection. He bought 20 more stickers and gave 33 stickers to his sister. He had 46 stickers left. How many stickers did Darren have in his collection at first?

    Label the changes as C1 and C2.
    Before C2: 46 + 33 = 79
    At first: 79 – 20 = 59

    Darren has 59 stickers in his collection at first.

    Heuristic: Draw a table

    Word Problem (Grade 4):

    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?
    Janice had 35 stickers. There were 4 children.

    Heuristic: Make suppositions

    Word Problem (Grade 4):

    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?

    Suppose that James has 30 ducks.
    30 × 2 = 60
    30 ducks have 60 legs altogether.
    82 – 60 = 22
    The total number of legs is 22 less than the actual total number.
    A cow has 2 more legs than a duck.
    22 ÷ 2 = 11
    James has 11 cows.
    30 – 11 = 19

    There are 19 ducks and 11 cows on the farm.

    Heuristic: Use before-after concept

    Word Problem (Grade 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?
    (8 – 3) = 5 units = $1250
    8 units = $1250 ÷ 5 × 8 = $250 × 8 = $2000

    Sam brought $2000 for shopping.



    At a time when the world is becoming only more dependent on ever-evolving technology and given that computer science continues to be among the most lucrative professions in the world, it is perhaps not an understatement that learning coding is no less than a life skill today. Kids, grownups — we could all do with learning coding and what’s more, it doesn’t have to be that difficult either. The best way to start then is to begin with the simplest and work your way up. And while adults sure can learn coding, starting at an early age is highly recommended.

    At a time when the world is becoming only more dependent on ever-evolving technology and given that computer science continues to be among the most lucrative professions in the world, it is perhaps not an understatement that learning coding is no less than a life skill today. Kids, grownups — we could all do with learning coding and what’s more, it doesn’t have to be that difficult either. The best way to start then is to begin with the simplest and work your way up. And while adults sure can learn coding, starting at an early age is highly recommended.

    In an increasingly competitive world, programming basics can help one have an edge over others. The question then is: which programming language must one start with? Rather which coding language is an ideal start for kids? There are several programming languages out there, so in this blog we bring you the top programming languages for kids:

    1. Scratch

    Scratch is a visual programming language and online community targeted mainly at children, by coding with ‘blocks’ in the editor. Users of Scratch can create online projects using a block-like interface. Developed by the MIT Media Lab, Scratch is thus an event-driven, block-based programming language that has been translated into 70+ languages, and is used in most parts of the world. It is used as an introductory language because creating interesting programs using Scratch is fairly easy, and skills learned in Scratch can be applied to other basic programming languages such as Python and Java. This is probably why it is popular in after-school centers, schools and colleges. Moreover, it has a very interactive online community where people share their artwork and games with each other; community statistics on Scratch’s official website state that more than 35 million projects have been shared as of October 2018. It’s an ideal choice given that kids can create animations, interactive stories, art or music using Scratch.

    2. Python

    Python is a programming language that is very similar to normal speech. One doesn’t have to add many comments to the code because Python code — if well written — can just do the job without extra comments. If you want your kids to learn coding, Python is a great starting point as it will offer your kid a basic understanding of how programming generally works. With Python, students can develop programming ideas and then convert these ideas into instructions that the machine can interpret. It’s an easy coding language to learn mainly because several common functionalities that programmers need are already built into this programming language.

    In fact, Python provides constructs that enable clear programming on both small and large scales. Also, it features a dynamic type system and automatic memory management. It supports multiple programming paradigms, including object-oriented, imperative, functional and procedural, and has a large and comprehensive standard library; plus Python interpreters are available for many operating systems. CPython, the reference implementation of Python, is an open source software and has a community-based development model, just as most of Python’s other implementations. Using Python, a lot can be achieved by simply researching and using the core Python libraries.

    3. JavaScript

    If your kid is interested in web development and design and wishes to learn the basics of programming, then JavaScript is just the programming language for your kid. An object-oriented and procedural programming language (which means its actions are carried out on the user’s computer) that’s used for front-end or client-facing applications, JavaScript is indeed a popular programming language. It is, in fact, native on all web browsers and commonly used to create complex interactive web applications. Along with HTML and CSS, JavaScript is one of the three core technologies of the World Wide Web. What’s more, these days JavaScript engines are also being used to run software such as including web servers, databases, desktop widgets and non-web programs such as word processors and PDF software. Learning this language typically gives quick tangible results, in other words, the student actually sees something on the web page and how it is being put together.

    4. Hypertext Markup Language (HTML) / Cascading Style Sheets (CSS)

    Learning HTML / CSS is certainly a good way to get introduced to programming as it lends the student a sense of accomplishment on having learnt languages that form an important part of most programming concepts (along with JavaScript). Both are easy to learn and use and are also very useful for kids who wish to pursue web designing. Bear in mind that almost every browser supports the HTML language; it is by default in every computer with Windows OS, so you don’t need to purchase any extra software to work with this programming language. HTML elements are denoted by tags, written using angle brackets. At the same time, it must be pointed out that HTML/ CSS do not really teach the student concepts such as looping or branching and they are thus not ideal for making dynamic pages. With HTML/ CSS, it’s more about learning to make tags for different things on a web page and learning how to perform tasks such as making a Word document colorful. Indeed, HTML can use a wide range of colors, objects and layouts. One of the biggest advantages of learning HTML is that it can embed programs written in JavaScript, which affects the behavior and content of web pages. Knowing CSS is helpful as it helps define the look and layout of content on these pages.

    5. C#

    C# is your best bet if your kid wishes to learn how to make 3D games. Pronounced “see sharp”, C# was developed around the year 2000 by Microsoft. It is a hugely popular programming language that’s used to develop most third-party applications for Windows. It’s certainly an in-demand programming language when it comes to software programming jobs, and is a great starting point for people who have never learnt coding. With a syntax similar to that of Java, C# is easier to learn if you have worked with the former language. C# can be used to make web applications, games, and other programs. It’s ideal for students who are interested in making applications for Windows. Some of its advantages include a strong memory backup, automatic garbage collection and rich class libraries.

    6. C++

    Best for a deeper understanding Pronounced “see plus plus,” this programming language is used to create applications that run locally on machines such as your computer. Despite the time and complexity required to learn C++, this language will provide teens with a very deep understanding of programming. C++ can be used to create systems software, games, and a variety of other programs. Great for: Teens who want a complex understanding of programming principles Students who want to program in the gaming industry Your student’s résumé — jobs expect programmers to have a level of familiarity with C++.

    7. Ruby

    For beginner programmers, Ruby has the most readable syntax. Which means that a large part of the code will be self-explanatory and less time is needed to explain the code to students. This makes Ruby an ideal choice for children who are still new to programming concepts. Also a dynamic, interpreted, reflective, object-oriented, general-purpose programming language, Ruby too has a dynamic type system and automatic memory management. It was originally used to create Twitter and its syntax is broadly speaking similar to that of Python.

    8. Java

    Easily one of the most widely used programming languages in the world — it is said that around 90 percent of Fortune 500 companies use it — learning Java gives one a comfortable advantage over those who don’t. Java can be used to make Android mobile apps, large backend environments and game engines, among other things. Given that it is a statically typed language, Java needs one to specify a variable type. But it is also designed to have minimal implementation dependencies. Allowing application developers to “write once, run anywhere” (WORA), the compiled Java code can run on all platforms that support Java without the need for recompilation. So on one hand, applications created using this programming language offer greater scalability and stability, but at the same time, it can take longer to learn Java when compared to other languages. Students wanting a deeper understanding of coding can turn to Java. So can intermediate coders wishing to consolidate their skill set, kids wanting to learn Java principles in a game format (think Minecraft) and even teens who are preparing to take the Computer Science AP Exam.

    9. Go

    Go, also known as Golang, is a programming language designed by Google engineers Robert Griesemer, Rob Pike, and Ken Thompson. It has been used by leading companies such as Google, YouTube, Apple Dropbox, BBC, IBM and Twitter among many others. Statically typed, compiled and syntactically similar to the programming language C, Go has a simple structure and syntax and is devoid of classes and type inheritance. Since it is based on functions, it is simple and easy to learn. It’s a compiled language, so developers need to be more accurate and attentive and the resulting code is usually neater and safer. Being simple, it is easily maintainable, and its development is faster and cheaper. Go additionally offers memory safety, garbage collection, structural typing, and CSP-style concurrency (it allows multiple processes to run effectively at the same time). The compiler, tools, and source code are all free and open source. Moreover, it can be used for different platforms, be it Windows, Linux or Unix devices. All of these are reasons why learning Go is an exciting idea.

    10. Dart

    Developed by Google, Dart is an object-oriented, class defined programming language. Like Go, it is a garbage-collected language using a C-style syntax that transcompiles optionally into JavaScript. Supporting interfaces, mixins, abstract classes, reified generics, static typing, and a sound type system, Dart is used to build web, server, desktop, and mobile applications. In fact, Flutter, the open-source mobile application development framework created by Google to develop Android and iOS apps, is written in Dart. This language is AOT (Ahead Of Time) compiled to fast, predictable, native code and because it has features that are familiar to users of both static and dynamic languages, it is quite easy to learn. Also, with Dart, it is easier to make smooth animations and transitions that run at 60fps (frames per second).

    (This blog was originally posted in the Blogs section of the YoungWonks website on January 2, 2019. To read more such blogs and to claim a free trial coding class for your child, please visit: https://www.youngwonks.com/

    Insight Into Autism: The Latest Brain Research


    The Centers for Disease Control recently announced that overall autism prevalence rates in the U.S. have increased, indicating that one in 68 American children are on the autism spectrum (an estimated one in 42 boys and one in 189 girls). Millions of Americans of all ages currently fall within the autism spectrum.

    The Centers for Disease Control recently announced that overall autism prevalence rates in the U.S. have increased, indicating that one in 68 American children are on the autism spectrum (an estimated one in 42 boys and one in 189 girls). Millions of Americans of all ages currently fall within the autism spectrum.

    Most people know something about autism, yet few recognize it as a social cognition disorder. Autism Spectrum Disorder (ASD) impacts communication, social interaction and learning. Unlike a broken leg, for example, a brain disorder such as autism, cannot be seen, except through behavior considered "abnormal" or "different."

    April is National Autism Awareness Month, a good time to put to rest some of the misconceptions about autism and to highlight exciting new brain research that will help more individuals with autism achieve productive life success.

    As the fastest growing developmental disability with an annual growth rate of 10-17 percent, symptoms of autism differ from individual to individual. Some are only mildly affected and perhaps highly intelligent. They may have trouble making friends but function well in most, if not all, academic areas. Those more severely affected may not even be able to engage in meaningful give and take of communication, while still others may exhibit isolating behaviors or uncontrollable emotional outbursts, acting out in frustration.

    No matter where individuals fall on the autism spectrum, those diagnosed often struggle to succeed in a world based on human interdependence. Individuals diagnosed with autism often have impaired social cognition or difficulty observing social rules, participating in social routines, or understanding and expressing emotions. For parents, this can be heartbreaking, as they wonder whether their child will be able to finish school; teens with autism yearn for friendships and romantic relationships, and adults with autism desperately want to be successful in holding down a meaningful job.

    Autism may be compared to dyslexia, where the different types of reading problems in dyslexia are due to different regions of the brain not working properly. In autism, for example, networks in the brain allowing one to understand and express normal emotions may be poorly wired. To believe that individuals with autism do not need or want friends is to suggest that people with dyslexia do not want to read. They want it desperately. Individuals with autism long to belong.

    Fortunately, research at the Center for BrainHealth is providing new insight into autism spectrum disorders and ways to train individuals with the diagnosis reach their fuller potential. We know certain areas of the brain are used to navigate through dynamic social interactions. We can actually see the differences in the brains of people with autism through advanced brain imaging. And we have learned that through short-term, intensive stimulation, we can "re-wire" the brain to some degree, improve the emotional awareness in individuals with high functioning autism and help them connect with others.

    One of our most exciting projects uses a unique virtual reality intervention that maps and tracks facial expressions during interactions. We have hired experts in gaming technology to create a digital environment that simulates a variety of scenarios and social interactions that happen in daily life; the computer program employs dynamic technology as well as visual and auditory feedback to enhance the brain's response.

    article continues after advertisement

    Our scientific team is partnering with researchers at Yale's Child Study Center to test the feasibility of providing the research-based training program to young adults across the country. A person with autism can use the technology to "practice" and hone their skills initiating a conversation with a person they would like to meet, interviewing for a job, or standing up for themselves by confronting a friend or colleague. Practicing social interaction in a safe, non-threatening, gaming environment helps people reduce anxiety and gain the confidence and skills they need to attempt more social interactions in their daily lives.

    Participants are experiencing great results with this intervention. As one young woman put it, "It feels real. I knew it was an alternate reality, but I felt the same emotions I would feel in the actual situation I was practicing." She says she has, "now made real friends, long-lasting friends" and that the program showed her the significance of friendship.

    Autism is a very difficult diagnosis, because it affects areas of the brain governing social interactions and relationships, the very foundation of family and community life. But those with the diagnosis need not feel hopeless or believe isolation is their only future. Research to enhance social cognition in autism are showing great promise, and interventions are bringing new hope for better, more connected lives for these individuals.

    Copyright Sandra Bond Chapman

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