In Singapore Math, a heuristic refers to a problem-solving strategy or approach that students can use to tackle complex mathematical problems, especially when the solution is not immediately obvious.
Instead of following a fixed method, students use flexible approaches like model drawing, logical reasoning, or working backwards. These strategies help students break down complex math questions into more manageable parts, allowing them to systematically work through the problem to find a solution.
Importance of Heuristics in Singapore Math
- Promotes Deeper Understanding: Heuristics encourage students to understand the underlying principles of a problem rather than just memorising formulas or procedures.
- Enhances Problem-Solving Skills: By using various strategies, students become more flexible and creative in their approach to solving different types of problems.
- Encourages Independent Thinking: Heuristics empower students to tackle problems on their own by giving them a toolkit of strategies to apply in various situations.
- Builds Confidence: As students become more proficient in using heuristics, they gain confidence in their ability to solve complex problems, which can lead to improved performance in math.
- Supports the CPA Approach: Heuristics align well with the Concrete-Pictorial-Abstract (CPA) approach in Singapore Math, as they often involve moving from concrete examples to abstract reasoning, reinforcing the concepts learned. This approach is also used at Bluetree Math and we call it the 3E approach for solving math problems.
Our Unique Heuristic Method: 3E Approach
At Bluetree, the 3E approach—Explore, Explain, Extend—guides teaching, closely aligning with the CPA (Concrete-Pictorial-Abstract) method in math education:
- Explore (Concrete): Students use hands-on manipulatives to explore and understand mathematical concepts.
- Explain (Pictorial): Visual aids like bar models are used to break down and explain these concepts.
- Extend (Abstract): Students apply their understanding to solve abstract problems and connect concepts to real-life scenarios.
This approach ensures that students build a solid, step-by-step understanding of math, moving from concrete experiences to abstract reasoning.
Overall, heuristics are crucial in Singapore Math as they equip students with essential problem-solving methods that not only help them excel in mathematics but also develop critical thinking skills applicable beyond the classroom.
8 Types & Examples of Heuristics
Some examples of the commonly used heuristics in Singapore Math are:
- Model Drawing: This involves using bar models to visually represent the relationships between different parts of a problem, making it easier to solve.
- Table Drawing: This method organises information in a structured format, facilitating analysis and enhancing clarity in the problem-solving process.
- Work Backwards: This strategy is used when the final outcome is known, and students need to determine the starting point.
- Making a Systematic List: This heuristic helps in organising information and exploring different possibilities to ensure all options are considered.
- Guess and Check: Students make educated guesses and then adjust their guesses based on feedback until they find the correct solution.
- Look for Patterns: Identifying patterns in numbers or operations can simplify complex problems and lead to a solution.
- Make a Supposition / Assumption Method: The student makes assumptions and forms a hypothetical scenario to explore different outcomes to guide problem-solving.
- Before-After Concept: This concept compares quantities before and after a change, illustrating how values transform as an effect of the mathematical problem.
1. Model Drawing
Bar modelling helps students visualise mathematical problems, making abstract concepts more concrete and understandable. This method is particularly effective in primary math, where visual learning plays a crucial role in comprehension.
Here are different bar modelling techniques commonly used:
A) Part-Whole Model
Description: This model represents a whole quantity divided into parts.
Use: It is used to illustrate addition and subtraction problems for Lower Primary.
Example: Sarah has 8 apples, and she gives 3 to John. How many apples does she have left?
8 – 3 = 5 (Ans)
Sarah has 5 apples left.
B) Comparison Model
Description: This model compares two quantities to show the difference between them.
Use: It is useful for comparing numbers in addition, subtraction, multiplication and division for Lower Primary.
Example: John has 5 more marbles than Tim. Tim has 7 marbles. How many marbles does John have?
7 + 5 = 12 (Ans)
John has 12 marbles.
2. Table Drawing
Tic Tac Toe Table (Start-Change-End / Before-Change-After) Table
Description: The Start-Change-End Tic-Tac-Toe table is a variation of the classic tic-tac-toe game but adapted for educational purposes. It consists of a 3×3 grid, where each cell represents a different aspect of a process or task. Since the information are displayed in a table form with proper headers, it makes it clearer for student to understand their work.
- Start: The initial situation or starting point.
- Change: The transformations or modifications applied during the process.
- End: The final outcome or result.
Use: It is useful problem sums questions that involves Fractions, Ratio and Percentage usually for Upper Primary.
Example: PSLE 2018 Question: Annie had a total of 285 red and blue balloons. She used 45 red balloons and 40% of the blue balloons. After that, the ratio of the number of red balloons to blue balloons Annie has was 1:3.
- What fraction of her blue balloons did Annie use? (Give your answers in the simplest form)
- How many balloons did Annie have in the end?
Tip: Always convert percentages to fractions. 40% = \(\frac{40}{100} = \frac{2}{5}\)
\(\frac{Blue Used}{Total}\) = \(\frac{2}{5}\) (Ans) A
Annie used \(\frac{2}{5}\) of her blue balloons.
1u + 45 + 5u = 285
6u → 285 – 45 = 240
6u → 240
1u → 240 ÷ 6 = 40
4u → 40 x 4 = 160 (Ans) B
Annie has 160 balloons in the end.
3. Work Backwards
Description: For working backward questions, start from the final result and reverse the operations or steps to reach the initial information.
Use: You will usually see key words like “at first” in the question. You will start from the end and you work your way to find the amount at first. All the signs will be reversed. (+ becomes – and – becomes +)
Example: Henry finished reading a book in a month. He read \(\frac{1}{5}\) of the book during the first week and 88 pages during the second week. He read 7 more pages than \(\frac{5}{9}\) of the remaining pages during the third week and completed the rest of the 89 pages in the last week. How many pages were there in the book?
Tip: Whenever you see a remaining, drop the model down and slice the remaining units into 9 units.
Find the LCM of 3 and 9 = 9
4p → 7 + 89 = 96
1p → 96 4 = 24
9p → 24 x 9 = 216
12u → 216 + 88 = 304
3u → 304 4 = 76
15u → 76 x 5 = 380 (Ans)
There are 380 pages in the book.
4. Making A Systematic List
Description: Systematic listing is a problem-solving strategy where students list all possible outcomes, combinations, or arrangements in an organized manner. This approach ensures that no possibilities are overlooked and helps in counting, probability, or combinatorics problems.
Use: Systematic listing is particularly useful in problems that require finding all possible combinations, arrangements, or solutions. It helps students approach problems methodically, reducing errors and ensuring that every option is considered.
Example: How many different two-digit numbers can be formed using the digits 2, 4, and 6, without repeating any digit?
Systematic Listing Example:
Tip: Always list it out systematically, starting from the first digit. (blublu talking)
First digit: 2
Possible numbers: 24, 26
First digit: 4
Possible numbers: 42, 46
First digit: 6
Possible numbers: 62, 64
Complete List of Numbers:
24, 26, 42, 46, 62, 64
There are 6 different 2-digit numbers that can be formed.
5. Guess and Check
Guess and Check Table
Description: A Guess and Check table is a systematic method used to find a solution by making an educated/calculated guess, checking if it satisfies the conditions of the problem, and refining guesses as needed. It helps students organize their guesses, track results, and adjust their approach based on feedback.
Use: The Guess and Check method is useful in problems where the solution isn’t immediately obvious and can be approached by trial and error. It’s especially helpful in word problems, such as finding numbers that satisfy certain conditions or determining quantities that add up to a specific total.
Example: There are 12 ducks and cows. They have a total of 38 legs. How many cows are there?
Marking Criteria for guess and check:
- 6 columns for table
- Minimum 3 guesses (even if you got the answer on the 1st / 2nd attempt, do a third one. Teacher wants to see that students understand the relationship between the numbers)
Tips for guess and check:
Always guess the halfway point, eg 12 ducks and cows = 6 cows and 6 ducks each
There are 7 cows.
6. Look for Patterns
Description: There are many kinds of pattern questions from Primary 1 all the way to Primary 6. Pattern questions can range from simple ones like identifying the elements of the pattern, such as numbers, shapes, colors, or objects to complex one which involves numbers (square numbers, cube numbers, triangular numbers and more)
Use: For complex pattern questions, students need to be able to deduce if special numbers are used. They have to try to determine the relationship that governs the pattern. This might involve addition, subtraction , multiplication, division and repetition.
Example: Janet uses circles and triangles to form figures that follows a pattern as shown below.
a) The table shows the number of triangles and circles for the first 4 figures. Complete the table for figure 5.
| Figure Number | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Number of Triangles | 2 | 6 | 12 | 20 | 5 + (5×5) = 30 |
| Number of Circles | 6 | 10 | 14 | 18 | 18 + 4 = 22 |
| Total Number of Triangles and Circles | 8 | 16 | 26 | 38 | 22 + 30 = 52 |
Number of circles → Constant addition of +4.
18 + 4 = 22 (Ans)
Number of circles → 4n + 2
Number of triangles → n + (n x n)
5 + ( 5 x 5 ) = 30 (Ans)
Total Shapes → 22 + 30 = 52 (Ans)
b) A figure in the pattern has 240 triangles. What is the figure number?
N + (n x n ) = 240
Simple guess and check
Try: n = 10
10 + (10 x 10) = 110 (x)
15 + (15 x 15) = 15 + 225 = 240 (✓)
The figure number is Figure 15 (Ans)
c) What is the total number of triangles and circles in figure 100?
Number of circles in figure 100 → 4 x 100 + 2 = 402
Number of triangles in figure 100 → 100 x (100 x 100) = 10100
Total triangles and circles → 10100 + 402 = 10502 (Ans)
There are 10502 triangles and circles in figure 100.
7. Make a Supposition / Assumption Method
Description: Supposition / Assumption involves making assumption based on given information to explore possible outcomes or solve a problem.
Use: The Supposition / Assumption method are usually taught to the P4 / P5 after Guess and Check is taught. It is a faster method to derive the solutions as compared to guess and check where students have to use trial and error to guess the right numbers.
Example: There are 12 ducks and cows. They have a total of 38 legs. How many cows are there?
Assume the opposite (All are ducks)
To: Total → 12 x 2 = 24 legs
Be: Big gap → 38 – 24 = 14 legs
So: Small gap → 4 – 2 = 2 legs
Opp → (Opposite of what you assumed) Number of cows → Big gap Small gap → 14 2 = 7 (Ans)
There are 7 cows.
8. Before-After Concept
Description: This model shows a “before and after” internal transfer concept. It is used to illustrate how quantities change due to mathematical operations such as addition, subtraction, or transfers between two quantities.
Use: It helps in understanding and visualising the portion that was given away / added in. Found in P3 and P4 questions.
Example: Mary had 20 more stamps than John at first. John gave 12 of his stamps to Mary. Mary now has 3 times as many stamps as John. How many stamps did John have at first?
2 units → 12 + 20 + 12 = 42
1 unit → 42
2 = 21
21 + 12 = 33 (Ans)
John has 33 stamps at first.
Want to know how your child can build a solid foundation in Math Heuristics?
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- P5 Exam Preparatory Course (EPC)
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- PSLE Intensive Course (PIC)
- Headstart and holiday programmes (M.A.P.)
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