GCSE Maths Practice: integers-and-directed-numbers

Understanding Negative Multiplication in Context

Mathematical operations with negative numbers appear in many scientific and real-world situations. Temperature, altitude, and direction-based quantities can all move below zero, creating negative values. This problem uses such a scenario to illustrate how multiplication and addition interact when both positive and negative quantities are involved.

Scenario

Imagine that a weather balloon is recording the temperature in the upper atmosphere. The temperature drops by 4 °C every hour during a 3-hour flight. After this period, a cold front moves through and reduces the temperature by another 6 °C. The question asks for the total change in temperature. Mathematically, the hourly drop can be represented by multiplying −4 (degrees per hour) by 3 (hours). Then, to include the extra drop from the cold front, we add another −6. The full operation follows the order of operations, or BIDMAS, to determine the final outcome.

Step-by-Step Process

1. Identify all operations: Multiplication and addition are present.
2. Apply BIDMAS: Perform multiplication before addition to avoid misinterpretation.
3. Handle negative signs correctly: A negative multiplied by a positive remains negative, because the signs differ.
4. Combine results: Once the multiplication is complete, add the remaining negative amount to find the total change.

Each step mirrors a real-world temperature change sequence where cooling rates combine over time.

Worked Examples (Different from This Question)

• Example 1: The temperature drops 2 °C each hour for 5 hours, then falls another 3 °C. Expression: (−2 × 5) + (−3).
• Example 2: A mountain climber descends 7 m every minute for 4 minutes, then continues 10 m lower. Expression: (−7 × 4) + (−10).
• Example 3: A submarine moves downward at 5 m per minute for 6 minutes and then dives an extra 8 m. Expression: (−5 × 6) + (−8).

In all these examples, multiplication is carried out first because it represents a repeated or continuous change, followed by the addition (or subtraction) of the final adjustment.

Common Mistakes

• Adding before multiplying, which produces an incorrect total.
• Forgetting that multiplying a negative by a positive gives a negative result.
• Misreading (−6) as subtraction instead of a negative number being added.

Real-Life Relevance

Negative operations are crucial in fields such as meteorology, physics, and economics. In meteorology, multiple layers of air cooling or warming combine to form the net temperature change. In physics, negative acceleration and directional velocity changes follow similar mathematical rules. Mastering sign operations ensures that calculations in such topics remain reliable and consistent.

FAQs

Q1: Why is multiplication done before addition?
A: Multiplication shows a repeated rate of change. If you added first, you would incorrectly combine the quantities before calculating the rate’s total effect.

Q2: How can I tell if the final result will be negative?
A: Estimate the direction: if all changes describe cooling or downward movement, the result must remain negative.

Q3: Does adding a negative always mean the same as subtracting?
A: In arithmetic terms, yes—it means moving further in the negative direction. But writing it as + (−6) helps visualise BIDMAS order more clearly.

Study Tip

When solving multi-step problems involving negatives, underline each sign before starting. Always carry out multiplications or divisions first, then handle addition or subtraction. This approach prevents careless sign errors and builds logical accuracy for more complex topics such as algebra, motion, and energy equations later in GCSE and A-level Maths.