Secondary 4 “show that” questions test more than getting the stated answer. The result is already visible. The task is to build a valid route from the given information to that result without assuming what you are trying to prove.
Students often feel confused because the answer is printed in the question. That changes the purpose of the task: the examiner is looking at the reasoning chain.
At eduKatePunggol, Secondary 4 Mathematics is taught in focused groups of up to three students in 1.5-hour lessons near Punggol MRT. This guide supports the wider Secondary 4 Mathematics year plan.
The printed result is a destination, not a starting assumption
If a question says “show that x = 4”, do not begin by writing x = 4 and manipulating it until the original condition appears. That risks circular reasoning.
Begin from information already established or explicitly given, and move toward x = 4.
Worked example 1: algebraic show-that
Given 3x + 5 = 17, show that x = 4.
Start from the given equation:
3x + 5 = 17
3x = 12
x = 4.
The result appears at the end of a valid chain.
Worked example 2: identity verification
Show that (a + b)² − (a − b)² = 4ab.
Start from the left-hand side:
(a² + 2ab + b²) − (a² − 2ab + b²)
= a² + 2ab + b² − a² + 2ab − b²
= 4ab.
That is a direct proof from one side to the stated expression.
Do not use the conclusion before it has been earned
A common weak response starts with the required result and works backwards. Backward thinking can help privately to discover a route, but the final written proof should normally be presented forward from valid known information.
This keeps the argument logically transparent.
Worked example 3: geometry reasoning chain
Suppose AB is a diameter of a circle and C lies on the circle. Show that triangle ACB is right-angled.
By the angle-in-a-semicircle theorem:
∠ACB = 90°.
Therefore triangle ACB is right-angled at C.
The reason is part of the proof. The picture alone is not enough.
Use earlier sub-part results only when allowed
If part (a) establishes a result and part (b) builds on it, that earlier result can often be used as a new known fact.
But do not assume the final result of the current part. Track which statements have already been justified.
A “show that” answer still needs enough working
Because the destination is printed, a bare final line is usually uninformative. Keep the transformations, substitutions or theorem reasons that make the result follow.
This is also why written working matters even for students who can do some steps mentally.
Worked example 4: verify a stated numerical result
Suppose a right triangle has legs 9 and 12. Show that its hypotenuse is 15.
By Pythagoras:
c² = 9² + 12² = 81 + 144 = 225.
Since c is a positive length:
c = 15.
The conclusion follows from the theorem and the positive-length condition.
How we diagnose show-that mistakes
Circular-reasoning error: the result is assumed at the beginning.
Missing-reason error: a geometric equality or algebraic move is written without justification.
Gap error: several non-obvious steps are skipped.
Direction error: the student discovers the route backwards but submits only the backwards chain.
Dependency error: an earlier result is used before it has been established.
Why the three-student format helps
In a group of up to three students, one learner can propose the starting fact, another can challenge a logical gap and another can test whether the final result was assumed too early. This makes mathematical reasoning visible.
What a 90-minute lesson could look like
An illustrative lesson could use ten minutes distinguishing given facts from targets, twenty minutes on algebraic show-that questions, twenty minutes on geometry reasoning, twenty minutes on proof repair and twenty minutes for independent written arguments, checking and review.
Repair, stabilisation and extension
Repair: use short proofs with one or two explicit steps.
Stabilisation: mix algebraic identities, equations and geometry theorems.
Extension: use multi-step questions where several earlier facts must be assembled before the printed conclusion becomes available.
Try a short independent check
- Given 5x − 3 = 22, show that x = 5.
- Show that (p + q)(p − q) = p² − q².
- If AB is a diameter and C is on the circle, state why ∠ACB = 90°.
Answers: subtract 3? No—add 3 to get 5x = 25, hence x = 5; expand the product; angle in a semicircle.
What progress should look like
- the target is recognised without being assumed;
- the proof starts from valid given information;
- non-obvious steps are justified;
- geometry reasons are named;
- earlier results are used only after establishment;
- the written chain reaches the required result cleanly.
Punggol class details and consultation inputs
eduKatePunggol Secondary Mathematics tutorials run for 1.5 hours in groups of up to three students near Punggol MRT. Confirm current availability, fees and meeting arrangements directly.
Bring the student’s subject level, examination year and recent “show that”, “prove” or geometry-reason questions with the original working preserved.
Frequently asked questions
Can I work backwards to figure out the proof?
Yes, as private problem-solving. But rewrite the final argument so it proceeds from established facts toward the required result.
Why show working when the answer is already printed?
Because the reasoning is the point of the question.
Use the target as a destination
Return to the Secondary 4 Mathematics year plan. For geometry theorem chains, use the circle-properties guide.
Start from what is known, justify each important step and arrive at the printed result without assuming it. Families can WhatsApp eduKatePunggol with recent work to discuss a suitable next step.
