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Maths

Calculating with Negative Numbers

Overview

Students consolidate addition and subtraction with negatives before extending to multiplication and division, developing both procedural fluency and conceptual understanding of directed numbers.

Learning Objective
Students add, subtract, multiply and divide with negative numbers, understand the rules for signs in multiplication and division, and apply these in a range of contexts.

Resources needed

  • Number lines
  • Mini whiteboards

Lesson stages

0 / 7 done
  1. 1 Temperature: it is −3°C and gets 5 degrees colder. New temperature? (−3 − 5 = −8°C). Debt: I owe £10 (−10) and earn £15. New balance? (−10 + 15 = 5). Money lost: bank charges me twice (2 × −£4 = −£8). Use context to reason before rules.
  2. 2 Adding a negative = subtracting: 5 + (−3) = 5 − 3 = 2. Subtracting a negative = adding: 5 − (−3) = 5 + 3 = 8. Key insight: 'subtracting a negative removes debt, which adds value.' Use number lines to verify.
  3. 3 Build from context: 3 × (−4) = losing £4 three times = −12. So positive × negative = negative. Then: (−3) × (−4): losing a debt of £4 three times = gaining £12 = +12. Negative × negative = positive. Summarise the sign rules.
  4. 4 Division mirrors multiplication: 12 ÷ (−3) = −4 (positive ÷ negative = negative). (−12) ÷ (−3) = 4 (negative ÷ negative = positive). Students complete a sign rule table: ++ = ?, +− = ?, −+ = ?, −− = ?
  5. 5 Students practise 12 calculations mixing all four operations with negatives. Include: (−5)² = 25 (negative × negative), −5² = −25 (square only the 5, then negate). Discuss this important distinction.
  6. 6 Calculate: 3 − 4 × (−2) + 1. Apply BIDMAS: multiplication first: 3 − (−8) + 1 = 3 + 8 + 1 = 12. Students work through four such expressions, showing each step.
  7. 7 Complete the multiplication table for integers from −3 to 3. Students fill in all entries and look for patterns: diagonal symmetry, positive/negative regions, zeros in the middle row and column.

Tap a step to mark it as done.

Variations

  • Extend to calculating with negative decimals and fractions
  • Link to substitution: evaluate 3x − x² for x = −2
  • Plot integer coordinates using negative values
More information

Display the four sign rules with visual examples. Use context consistently — temperatures, money, debt — before abstract numbers.

Use a number line throughout. Restrict to addition and subtraction before introducing multiplication. Provide a sign rule card.

Do students apply sign rules consistently in multiplication and division? Do they distinguish between (−5)² = 25 and −5² = −25? Do they use BIDMAS correctly with negative numbers?

Draw number lines by hand. All calculations on mini whiteboards or plain paper.

Students often think −5² = 25 (squaring the negative). Reinforce: the negative sign is outside the square unless it is in brackets.

Activity details
Secondary Medium 40 minutes No prep Solo, Pairs Indoor
Topics
Negative Numbers Arithmetic Number Line Directed Numbers