Mistakes are useful
Everyone makes mistakes in maths. Finding and fixing them, called error analysis, is one of the best ways to understand a topic deeply.
Types of errors
| Type | Example | Fix |
|---|---|---|
| Computational error | 7 × 8 = 54 | Recalculate; know facts |
| Sign error | −3 − 5 = −2 | Use a number line; rewrite as adding a negative |
| Conceptual error | 1/2 + 1/3 = 2/5 | Relearn the method (common denominators) |
| Copying error | writing 47 as 74 | Re-read the problem |
| Place value error | 0.3 × 0.2 = 0.6 | Count decimal places |
| Order of operations | 3 + 4 × 2 = 14 | Use BEDMAS |
| Units error | adding 2 m + 50 cm = 52 | Convert first |
Common misconceptions
A misconception is an idea that feels right but isn’t.
| Misconception | Reality |
|---|---|
| Multiplying always makes bigger | 6 × 0.5 = 3 |
| Longer decimals are bigger | 0.25 < 0.3 |
| (a + b)² = a² + b² | (a + b)² = a² + 2ab + b² |
| −x is always negative | if x = −4, then −x = 4 |
| Dividing always makes smaller | 6 ÷ 0.5 = 12 |
How to find an error
- Check the answer against an estimate.
- Substitute the answer back into the original problem.
- Re-do each step and compare.
- Look at the first step where things go wrong.
- Name the type of error and fix it.
Example 1: solving an equation
A student wrote:
- 3x + 5 = 20
- 3x = 25
- x = 25/3
Error: In step 2, 5 was added instead of subtracted. Fix: 3x = 15, x = 5. Check: 3(5) + 5 = 20 ✓.
Example 2: distributing
- −2(x − 4) = −2x − 8 ✗
Error: sign error. −2 × −4 = +8. Correct: −2x + 8.
Example 3: percentage
"A $50 item is 20% off, then 20% is added back. It’s $50 again."
Error: conceptual. $50 × 0.8 = $40; $40 × 1.2 = $48.
Explaining errors clearly
When you explain an error:
- Say where it happens (which step).
- Say what went wrong.
- Say why it’s wrong.
- Show the correct work.
Worked examples
Example 1: Find the error: 2³ = 6.
2³ means 2 × 2 × 2 = 8, not 2 × 3.
Example 2: Find the error: 4.5 + 2.35 = 2.80.
The decimal points weren’t lined up. Correct: 6.85.
Example 3: Find the error: the mean of 2, 4, 9 is 15.
The sum (15) wasn’t divided by 3. Mean = 5.
Common mistakes in error analysis
- Saying only "it’s wrong" without explaining why.
- Fixing the answer without fixing the method.
- Not checking the corrected answer.
Try it yourself
Find and fix each error.
- 5 − (−3) = 2
- 1/4 × 8 = 1/32
- Area of a circle with r = 3: 6π
- 2x + 3x = 5x²
Answers: 1) Subtracting a negative means adding: 8. 2) 8/4 = 2. 3) That’s the circumference formula (2πr). Area is πr² = 9π. 4) Like terms add coefficients only: 5x.
Quick recap
- Error analysis builds understanding.
- Know the types: computational error, sign error, conceptual error.
- Check answers by estimating and substituting.