GCSE Maths Practice: integers-and-directed-numbers

Higher-Tier BIDMAS with Nested Negatives

At advanced GCSE level, questions often involve multiple brackets, negative signs, and mixed operations. The goal is not only to calculate correctly but to manage the hierarchy of operations without losing track of signs. Expressions such as 18 − [−7 × 2 − (−4)] require accurate sequencing of steps to avoid sign errors.

Step-by-Step Reasoning

1. Start with the innermost brackets. Simplify any expression inside them before addressing outer operations. For instance, if the bracket contains −7 × 2 − (−4), you must first handle both multiplications before subtraction.
2. Apply multiplication before subtraction. Multiplying a negative by a positive yields a negative. When you encounter −(−4), remember that subtracting a negative changes direction and becomes +4.
3. Combine results carefully. Each bracket outcome affects the next layer. It helps to rewrite intermediate results after every stage so signs stay visible.

Common Pitfalls

• Sign inversion confusion: Students often forget that a minus before a bracket reverses every sign inside when the bracket is expanded.
• Ignoring nested order: Working left-to-right without regard for bracket hierarchy leads to inconsistent answers.
• Dropping parentheses around negatives: Always keep negatives in brackets when substituting results into further calculations.

Worked Examples (Different from the Question)

• Example 1: 15 − [−6 × (−3)] → Inner multiply: (−6×−3)=18 → then subtract: 15−18=−3.
• Example 2: 10 − {−4 × [2 − (−5)]} → Inside: 2−(−5)=7 → multiply: −4×7=−28 → subtract: 10−(−28)=38.
• Example 3: −8 − [−2 × (−3 + 5)] → Inside: (−3+5)=2 → multiply: −2×2=−4 → subtract: −8−(−4)=−4.

Visualising the Operations

Picture the number line. Negative multiplication means reflecting direction: moving left rather than right. Subtracting a negative then reverses direction again. Visualising movement helps confirm the final sign of a result before you even compute the value.

Exam Strategy

In higher-mark reasoning questions, the examiner wants to see method discipline. Write each intermediate result clearly; show brackets for negatives; and highlight operation order with arrows or underlines. This clarity prevents lost marks for minor sign slips.

Real-World Parallels

Nested operations mirror complex systems: combining losses and gains in finance, temperature shifts, or electric charge differences. A double negative often indicates recovery or reversal—a concept common across mathematics, science, and coding.

FAQs

Q1: Why does subtracting a negative become addition?
A: Because the negative sign before a negative flips its direction, effectively increasing the value.

Q2: How can I confirm I used BIDMAS correctly?
A: Re-insert brackets showing the order: compute from inner to outer, then compare sign consistency with a quick estimate.

Q3: What’s the most common trap with double negatives?

A: Forgetting to apply both sign reversals—one from multiplication, another from subtraction—often flips the result incorrectly.

Study Tip

Train yourself to scan for bracket levels first. Write small superscripts (¹,²,³) near brackets to mark operation order. This simple habit drastically reduces sign confusion and helps you tackle long expressions confidently in your GCSE exams and beyond.