A fair retest keeps the skill the same and changes everything else: the numbers, the wording and, later, the shape of the question. If you can still do it, the skill has transferred. If you cannot, you learned one question, not one method.
This page is for students who understand examples but cannot repeat the method alone. It gives you a way to test yourself honestly, without needing someone else to write questions.
What can you change in a question?
Think of three dials. Turn one at a time at first, then two together.
- Numbers. Use different values, including one that is less tidy.
- Context. Move from a shop price to a population, a speed or a mass.
- Shape. Ask the same thing backwards, or add one extra step.
A retest that only turns the first dial is the gentlest. A retest that turns the third is the real check.
Worked example: reverse percentages
The original question you got wrong:
After a 15% increase, a pair of shoes costs RM69. Find the original price.
Correct method: the new price is 115% of the original, so original × 1.15 = 69, and original = 69 ÷ 1.15 = RM60. Check: 60 × 1.15 = 69.
The mistake that happened: the student found 15% of 69 (RM10.35) and subtracted it, giving RM58.65. This treats 69 as the original, but 15% was taken of the original, not of 69.
Now build the retest ladder:
| Level | Question | Dial turned | Answer |
|---|---|---|---|
| 1 | After a 20% increase, a bag costs RM84. Find the original price. | Numbers | 84 ÷ 1.2 = RM70 |
| 2 | A town’s population grew by 8% to 5,400. What was it before? | Numbers and context | 5,400 ÷ 1.08 = 5,000 |
| 3 | In a sale, a jacket is reduced by 12% to RM66. Find the original price. | Shape (decrease) | 66 ÷ 0.88 = RM75 |
Check each: 70 × 1.2 = 84; 5,000 × 1.08 = 5,400; 75 × 0.88 = 66.
Level 3 is the useful one. A student who memorised “divide by 1.15” will divide by 1.12 or add 12%, because the word “reduced” changes the multiplier. If you get level 3 right without looking at notes, you have moved past the original numbers.
A simple schedule
- Day 0: correct the original question and log the cause.
- Day 2 or 3: try level 1 and level 2, closed notes.
- Day 7: try level 3 and one mixed question where you are not told which skill it is.
Write the date and result beside each attempt. Two correct answers on different shapes is stronger evidence than five on the same one. For a structured mixed set, use the original mixed-practice builder.
The mistake to avoid
The trap is changing only the digits and calling it a new question. If the question says “increase by 15%” and your retest says “increase by 25%”, you still know which button to press.
Another trap is making the retest easier without noticing. Rewriting 69 as 60 and 15% as 10% makes the question trivial. Keep the difficulty level, or raise it a little.
Check yourself
1. After a 25% increase, a phone costs RM150. Find the original price.
Show answer
150 ÷ 1.25 = RM120. Check: 120 × 1.25 = 150.
2. A shirt in a sale is 20% cheaper at RM48. What was the price before the sale?
Show answer
The sale price is 80% of the original: 48 ÷ 0.8 = RM60. Check: 60 × 0.8 = 48.
3. In a retest you answer correctly, but you noticed you recognised the question. What should you do next?
Show answer
Write a new question that turns a different dial, for example a decrease instead of an increase, or a two-step question. Treat the recognised retest as only weak evidence.
Where this leads next
If your retests expose a repeated cause, go back to using a mistake log to spot a repeated pattern. When you are ready for questions that do not announce their method, moving from following an example to choosing a method is the next step.
Self-made retests work well for many students.
A one-to-one teacher becomes worth considering when you cannot tell whether a retest was fair, or when you keep passing retests and still lose marks on new papers. In online one-to-one Mathematics tuition, a teacher can set questions that turn one dial at a time and explain why you slipped.
You can see how it works in a paid one-hour trial. Other routes are under student learning routes.