PRIMARY 5 MATHEMATICS LEARNING GUIDE · BATCH 10 · GUIDE 40
Guess, Check, Improve is not random guessing. A useful trial is chosen inside sensible bounds, checked against the problem conditions, and then adjusted in a direction that reduces the error. Each trial should create information for the next one.
This guide develops intelligent trial, upper and lower bounds, systematic tables, monotonic change, step-size refinement, feasibility checks and stopping rules. It connects with systematic listing, assumption method and non-routine problem solving without duplicating them.
Return to the Primary 5 Mathematics Learning Hub. Related guides: Non-Routine Problems, Heuristics & Strategy Choice · Systematic Listing, Case Organisation, Tables & Invariant Thinking · Assumption Method, Fixed Totals & Equalisation.
1. A good first guess lies inside known bounds
If a number of tickets must be between 20 and 40, guessing 100 provides no useful test. Start inside the feasible interval.
2. Bounds reduce the search space
If 15 items each cost either $6 or $10, total cost must lie between $90 and $150. A target of $200 is impossible before any trial begins.
3. Guessing becomes intelligent when direction is predictable
If increasing the number of $10 tickets always raises total revenue, then a revenue that is too low means the next guess should use more $10 tickets.
The check tells you which direction to move.
4. Use a table to preserve trials
| $10 tickets | $6 tickets | Total revenue |
|---|---|---|
| 5 | 10 | $110 |
| 8 | 7 | $122 |
| 10 | 5 | $130 |
A table prevents repeated guesses and makes the change pattern visible.
5. Each replacement can create a constant change
Replacing one $6 ticket with one $10 ticket raises revenue by $4. Once that pattern is noticed, continued guessing may no longer be necessary.
The trial process should stop when a stronger relation emerges.
6. Start near the middle when no better information exists
If a whole-number answer lies between 0 and 40, a midpoint guess around 20 can divide the search space. The check then tells you whether to move up or down.
7. Large first steps, smaller later steps
If the target is far away, adjust substantially. As the trial approaches the target, reduce the step size.
This is refinement rather than random wandering.
8. Example: mixed ticket prices
Twenty tickets cost $8 or $12 and total $208.
Try 10 premium: revenue = 10×12 + 10×8 = 200. Too low.
Each extra premium adds $4. Need $8 more, so add 2 premiums.
Answer: 12 premium tickets.
9. Example: animal legs
There are 15 animals, chickens or goats, with 42 legs.
Try 5 goats: 5×4 + 10×2 = 40 legs. Too low.
Each extra goat replacing a chicken adds 2 legs. One more goat gives 42.
Answer: 6 goats.
10. Guess-check can reveal the assumption method
Once the constant change per replacement is recognised, the problem can often be solved directly with the assumption method.
Guess-check is therefore sometimes a discovery route to a more efficient structure.
11. Not every trial problem has constant change
If the relationship is nonlinear or the conditions change at thresholds, each adjustment may not change the output by a constant amount.
In such cases, keep using controlled trials but do not assume a constant step prematurely.
12. Use monotonicity when available
If more of one variable always increases the target quantity, the search has one direction. This property is called monotonic change.
You do not need the formal term to use the idea: too low → increase; too high → decrease.
13. Feasibility checks come before refinement
If a trial violates a whole-number, capacity or minimum condition, discard it even if the numerical total looks close.
A valid answer must satisfy all constraints, not only one equation.
14. Guessing with whole-object constraints
A bus problem may require a whole number of buses and capacity no greater than 45 passengers per bus. A trial that produces 6.4 buses is not itself a valid operational solution.
15. Guessing can solve digit and number puzzles
If a two-digit number has digits summing to 11 and the tens digit is 3 more than the ones digit, test feasible digit pairs systematically rather than randomly.
Pairs summing to 11: (2,9), (3,8), (4,7), (5,6), (6,5), (7,4), (8,3), (9,2). Only (7,4) has tens digit 3 greater. Number = 74.
16. Guessing can solve distribution problems
If an unknown number of groups receive either 8 or 10 items and one condition leaves a surplus, trials can be organised by group count. But once the excess/shortage gap structure becomes visible, switch methods.
17. Guessing can solve geometric integer cases
If a rectangle has integer side lengths, fixed perimeter and another area condition, list feasible factor or perimeter pairs systematically. Bounds and parity can eliminate many cases before testing.
18. Stop when the answer is uniquely determined
Do not continue producing cases after one case satisfies every condition and the structure guarantees uniqueness.
A stopping rule keeps the method efficient.
19. Prove that no other case works
If several values could still satisfy the conditions, the search is incomplete. Use bounds, monotonicity, exhaustive listing or a derived equation to prove uniqueness.
20. Guess-check-improve versus systematic listing
Systematic listing aims to cover all cases in an organised way. Guess-check-improve aims to converge quickly on a solution by using feedback from each trial.
They overlap, but their search strategies differ.
21. Guess-check-improve versus assumption method
Assumption method begins from an extreme baseline and uses a constant replacement difference directly. Guess-check-improve may discover that same structure through a few trials.
22. Error map
| Error | Cause | Repair question |
|---|---|---|
| Random repeated guesses | No directional rule | What did the last check tell you to change? |
| Guess outside feasible range | Bounds ignored | What is the minimum and maximum possible? |
| Keeps guessing after constant pattern appears | Scaffold not released | Can the change now be calculated directly? |
| Accepts closest trial | Exact constraints ignored | Does the trial satisfy every condition exactly? |
23. Practice laboratory
- 20 tickets cost $8 or $12 and total $208. Find $12 tickets using intelligent trial.
- 15 chickens/goats have 42 legs. Find goats.
- 15 items cost $6 or $10. Can total be $200? Use bounds.
- Find the two-digit number whose digits sum to 11 and tens digit is 3 more than ones digit.
- A whole-number answer lies from 1 to 60. A trial of 30 gives a result too high and increasing the guess always increases the result. Which direction should the next trial move?
24. Answers
1. Try 10 premium → $200; need $8 more; $4 per replacement → 12 premium.
2. Try 5 goats → 40 legs; one more goat adds 2 → 6 goats.
3. Maximum 15×10 = $150, so impossible.
4. 74.
5. Move below 30.
25. Full intelligent-trial problem
A theatre sold 50 tickets. Adult tickets cost $18 and student tickets $12. Revenue was $720. Find the number of adult tickets.
Try 20 adults: 20×18 + 30×12 = 360 + 360 = 720.
The first sensible trial happens to satisfy the target exactly, so 20 adult tickets.
Check uniqueness: replacing one student with one adult changes revenue by $6. Any different adult count would therefore change revenue away from $720.
26. Final checkpoint
A strong Primary 5 learner can define feasible bounds, choose informative trials, use each check to set the direction of the next move, reduce step size near the target, switch to a stronger method when a constant pattern appears, and prove that the final case satisfies all constraints uniquely.
Return to the Primary 5 Mathematics Learning Hub.
