Working With Negative Numbers Quizzes
Visual overview of Working With Negative Numbers.
Introduction
Negative numbers represent values less than zero. They are used to describe losses, temperatures below freezing, elevations below sea level, or directions opposite to a reference point. Understanding how to calculate with negative numbers is essential for GCSE Maths, forming the basis of algebra, coordinates, and real-life problem solving.
Example: \( -3^\circ\text{C} \) means 3 degrees below zero; a bank balance of \( -£20 \) means you owe £20.
Core Concepts
Number Line and Position
Negative numbers lie to the left of zero on a number line. As you move left, numbers decrease; as you move right, they increase.
- \(-5 < -2 < 0 < 3 < 6\)
Adding Negative Numbers
Think of addition as moving along the number line.
- Adding a positive moves right.
- Adding a negative moves left.
Examples
- \(5 + (-3) = 2\)
- \(-4 + (-2) = -6\)
- \(-3 + 7 = 4\)
Subtracting Negative Numbers
Subtracting a negative is the same as adding a positive — two negatives make a positive.
- \(6 - (-2) = 6 + 2 = 8\)
- \(-3 - (-5) = -3 + 5 = 2\)
Multiplying and Dividing Negatives
Use the sign rules:
- (+) × (+) = (+)
- (–) × (–) = (+)
- (+) × (–) = (–)
- (–) × (+) = (–)
Examples
- \(3 \times (-4) = -12\)
- \(-5 \times 6 = -30\)
- \(-7 \times -2 = 14\)
- \(-12 \div 3 = -4\)
- \(-15 \div -5 = 3\)
Order of Operations (BIDMAS) with Negatives
Follow BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) carefully with negatives.
Example
- \(3 - (-2) \times 4 = 3 - (-8) = 3 + 8 = 11\)
- \((-5)^2 = 25\) but \(-5^2 = -25\)
Comparing and Ordering Negatives
For negative numbers, a smaller absolute value means the number is greater.
- \(-2 > -5\) because it is closer to zero.
Absolute Value
The absolute value of a number is its distance from zero on the number line, ignoring the sign.
- \(|-7| = 7\)
- \(|3| = 3\)
Worked Examples
Example 1 (Foundation): Adding Negatives
\(-8 + 3 = -5\)
Move 3 right from -8 → -5.
Example 2 (Foundation): Subtracting Negatives
\(4 - (-6) = 4 + 6 = 10\)
Example 3 (Higher): Multiplying Negatives
- \(-9 \times -2 = 18\)
- \(-9 \times 2 = -18\)
Example 4 (Higher): Mixed Operations
Calculate \( -2^2 + (-3)^2 \)
- \(-2^2 = -4\) (only 2 is squared)
- \((-3)^2 = 9\)
- Total \( = -4 + 9 = 5\)
Example 5 (Higher): Real-Life Context
Temperature rises from \(-6^\circ\text{C}\) to \(3^\circ\text{C}\).
- Change \( = 3 - (-6) = 9^\circ\text{C}\) increase.
Common Mistakes
- Forgetting that subtracting a negative becomes addition.
- Confusing \((-5)^2\) with \(-5^2\).
- Multiplying negatives incorrectly (sign errors).
- Assuming larger negative numbers are “bigger.”
- Dropping negative signs in multi-step calculations.
Applications
- Temperature changes and weather graphs
- Financial profit/loss and debt problems
- Coordinates and gradients in geometry
- Physics: forces and velocity in opposite directions
- Algebraic simplification with negative terms
Strategies & Tips
- Mark negatives clearly with brackets when substituting into formulas.
- Remember “minus minus = plus.”
- Use patterns: \( -1 \times 7 = -7\); \( -1 \times -7 = 7\).
- Practise order-of-operations with negatives until automatic.
- Draw a quick number line when unsure.
Summary / Call-to-Action
Negative numbers describe values below zero and follow consistent rules for addition, subtraction, multiplication, and division. Mastering these rules prevents sign errors in algebra, coordinates, and real-world problems.
- Practise combining positives and negatives in all operations.
- Use brackets to clarify sign changes.
- Check results on a number line when learning.
Confidence with negatives builds accuracy and fluency across GCSE Maths.