If you ask a Year 6 student, “What is 3 – (-5)?”, they will likely recite a rule they learned in class: “Two minuses make a plus, so it’s 8.”
But if you ask them why, the room often goes silent.
In the high-pressure environment of the 11+ exams, rote memorisation is fragile. When questions get complex—involving brackets, multiplication, and division simultaneously—students who rely on tricks often panic. Students who understand the concept thrive.
At Singapore Maths Academy, we believe in the Concrete-Pictorial-Abstract (CPA) approach. That is why we have developed our new Interactive Directed Numbers Tool to replace memorisation with visualisation.
The Problem with “Tricks”
Directed numbers (positive and negative integers) are one of the biggest stumbling blocks in Key Stage 2 and 3. The rules seem arbitrary:
- Why does subtracting a negative number result in a bigger number?
- Why is a negative times a negative a positive?
Without a visual model, these are just abstract rules. With Algebra Tiles, they become physical actions.
Visualising the Impossible: How to “See” the Maths
Our new tool uses red and green tiles to represent negative and positive numbers. The most powerful concept students learn here is the Zero Pair.
The Rule of Zero: A positive tile (+1) and a negative tile (-1) cancel each other out to make Zero.
This simple fact unlocks the answers to the hardest directed number questions.
Challenge 1: Subtracting a Negative
Question: 3 – (-5) = ?
The Confusion: “I have 3. How can I take away negative 5? I don’t have any negatives!”
The Conceptual Solution:
- Start: Imagine you have 3 Green tiles (+3).
- The Problem: You need to remove 5 Red tiles (-5), but there aren’t any on the board.
- The Fix: Add 5 Zero Pairs (+5 and -5). Adding zero doesn’t change the value; you still have a total of 3.
- The Action: Now you do have 5 Red tiles to remove. Drag them to the trash.
- The Result: You are left with your original 3 Green tiles, plus the 5 Green tiles from the Zero Pairs.
3 + 5 = 8
Try this yourself using the “Visualise” button on our Directed Numbers App.
Challenge 2: Multiplying Negatives
Question: -2 × -3 = ?
The Confusion: “How can I have negative groups?”
The Conceptual Solution: We read this equation not as “Minus 2 times minus 3,” but as “Remove 2 groups of -3.”
- Start with a blank board (Zero).
- To remove something, you first need to have it. Add enough Zero Pairs to the board.
- Identify “2 groups of -3” (which is 6 Red tiles).
- Remove those red tiles.
- What is left behind? The 6 Green tiles (+6) from the Zero pairs.
Answer: +6
By seeing that “taking away debt” leaves you with more money, the concept clicks instantly.
Applying this to the 11+: Typical Exam Questions
In 11+ and Common Entrance exams, directed numbers rarely appear as simple sums. They are hidden inside word problems or algebra. Here is how conceptual understanding helps.
Example 1: The Temperature Trap
Question: At midnight, the temperature is -4°C. By midday, it has risen by 7°C. By 10pm, it drops by 5°C. What is the new temperature?
Visual Approach: Instead of juggling signs in their head, the student visualises the tiles.
- Start with 4 Red tiles (-4).
- “Rise by 7”: Add 7 Green tiles (+7).
- 4 Reds and 4 Greens annihilate each other (Zero Pairs).
- Left with 3 Green tiles (+3).
- “Drop by 5”: Add 5 Red tiles (-5).
- 3 Greens and 3 Reds annihilate.
- Left with 2 Red tiles.
Answer: -2°C
Example 2: The Algebra Substitute
Question: If a = -2 and b = 3, what is the value of 4a – 2b?
Visual Approach:
- 4a: This means “4 groups of -2”.
- Visualise: 4 sets of 2 Red tiles = 8 Red tiles (-8).
- – 2b: This means “Subtract 2 groups of 3”.
- We have -8 (Red). We need to take away +6 (Green).
- We don’t have Greens! Add 6 Zero Pairs.
- Take away the 6 Greens.
- We are left with the original 8 Reds plus the 6 Reds from the zero pairs.
- Total: 14 Red tiles.
Answer: -14
Try It Yourself
We have made this digital manipulative available for free to help parents and students bridge the gap between primary maths and complex algebra.
Whether you are struggling with -6 ÷ -2 or just want to see why the rules work, jump into Questions Mode and use the “Watch Explanation” feature to see the maths come to life.