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G1, G2 and G3 are secondary school Mathematics subject levels under Singapore’s Full Subject-Based Banding. Across suitable tasks in these courses, the words “show that” change the job of the answer. The required result may already be printed, but the student still needs to establish it from the information and valid mathematical relationships supplied.
The quick reasoning is that a target tells you where the argument should arrive; it does not automatically provide a fact you may assume along the way. A useful solution identifies what is given, chooses an appropriate relationship and shows the essential steps leading to the result. Copying the target or writing a plausible final number does not explain why it follows.
This guide is for readers who understand a calculation when they see it but feel uncertain about what to write when the answer is already visible. It distinguishes showing, verifying, solving and explaining through original examples. Select those appropriate to the student’s taught material; it is not a claim that every example appears at every level or in every assessment.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
Sections 16–19
Read exactly what must be established. It could be a value, a formula, an equality, an approximation or a conclusion in context. Then identify the supplied facts that can support it. The written working should connect those facts to the destination without quietly treating the unproved conclusion as an extra starting condition.
Working backwards can help you plan a route. A direct substitution can be the right way to verify a stated candidate. A chain of reversible equivalences can also be valid when used correctly. The issue is not a ban on a particular direction of thought; it is whether the finished argument actually establishes the requested claim.
For the broader subject structure, use G1, G2 and G3 Mathematics explained. Our earlier answer-key guide explains why matching a printed answer does not by itself demonstrate independent understanding. This companion focuses on questions where the result is deliberately supplied as the target.
A given fact is information the task authorises you to use. A target is the statement you need to establish. In a rectangle question, the side lengths may be given in terms of x while a perimeter formula is the target. Using the target formula as though it were supplied data would bypass the central reasoning.
Label the two roles briefly. “Given: length x + 4, width x. Show: perimeter 4x + 8.” That small separation can prevent a student from copying every expression into one undifferentiated list. The task is to explain how the supplied dimensions lead to the perimeter relationship.
This distinction matters beyond formal proof. In a cost problem, a displayed total might be the result to establish rather than an amount available for forming an equation. In a graph problem, a stated intersection could be a candidate to verify. Read the instruction before deciding what authority each number has.
Ask the learner, “Would you know this value if the show-that sentence were covered?” If not, it may be part of the target rather than the initial data. This is a useful reading prompt, not a complete logical test for every advanced question. Use it to make the roles visible during early learning.
The final argument should be understandable to someone who has not already accepted the target. That reader needs a reason to believe the conclusion. A chain of valid steps provides the reason. Merely arriving at familiar symbols on the last line does not explain whether the conclusion follows from the actual givens.
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For a rectangle with length x + 4 and width x, the perimeter relationship is two lengths plus two widths. Write P = 2(x + 4) + 2x. Expanding gives 2x + 8 + 2x, and collecting like terms gives P = 4x + 8. The target follows from the rectangle’s stated dimensions.
The formula is not selected because it resembles the printed answer. It is justified by the shape and the quantity requested. A student who begins with area, x(x + 4), may perform valid expansion but address the wrong measurement. Showing a result still requires identifying the correct mathematical relationship.
Explain essential choices in a few words. “Perimeter is the sum of the four sides” makes the first line clear. There is no need for a long paragraph about rectangles if the relationship is familiar and the working shows it. The explanation should support the calculation rather than bury it under unnecessary prose.
Check that the variable definitions remain unchanged. If x is the width in the question, do not silently use x for the length in the solution. A consistent symbol meaning helps the reader follow each expression. An algebraically tidy line can still misrepresent the original situation if its letters have changed roles.
After reaching the target, connect it back to the statement: “Hence P = 4x + 8.” The word hence is useful only because the preceding work establishes the result. It is not a substitute for the missing relationship. Students should learn to make the conclusion follow, not merely add a concluding word.
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Suppose the task asks the learner to show that the perimeter is 4x + 8, and they write only P = 4x + 8. The conclusion has been repeated but not established. The marker or reader cannot see why the dimensions imply it, whether the perimeter formula was understood or whether the expression was copied.
Adding a check at one convenient value may still be insufficient for a general formula. If x = 2 gives perimeter sixteen in both expressions, that confirms agreement in that case. It does not alone establish the formula for every allowed x. General algebraic reasoning is needed to justify the stated general relationship.
This does not mean examples are useless. They can help the learner understand the claim or catch an incorrect formula. If a proposed perimeter formula fails at x = 2, it cannot be correct universally. A counterexample can disprove a general claim, while a few matching examples do not prove it for all cases.
Keep the distinction between discovering and establishing clear. Trying x = 2 may make the formula plausible and suggest the route. The final solution can then derive it from the side lengths. An exploratory numerical test and the completed algebraic argument have different jobs, both of which can support learning.
If the student repeatedly copies the result, ask for the first valid line before the target. “What relationship can you write from the dimensions?” is a smaller and more useful prompt than asking for the whole proof at once. Once that line is understood, help the learner connect the next steps independently.
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If a question asks you to verify that x = 5 satisfies 3x + 4 = 19, direct substitution is appropriate. Calculate 3(5) + 4 = 19 and compare with the right side. The task is to check the stated candidate, so using it as the input is exactly what verification requires.
If the question instead asks you to solve 3x + 4 = 19, substitution of five confirms a candidate but does not show how it was obtained. A suitable solution subtracts four from both sides and divides by three to get x = 5. The required job differs even though the final value is the same.
Some show-that questions ask for a candidate to be established, and direct checking may be sufficient for the particular claim. Others require derivation, a specified method or a broader conclusion. Read the complete wording. Do not teach that substituting a supplied value is always wrong or always enough.
Completeness is another issue. Verifying one root of a quadratic establishes that it is a root. It does not by itself show that no other root exists. If all solutions are requested, the argument needs to address the whole set. The scope of the claim determines what evidence is sufficient.
When uncertain about assessment expectations, ask the teacher about the particular task. An article can explain the mathematical distinction but should not invent universal marking rules for every use of show, verify or solve. The student needs both valid reasoning and attention to the method and form actually requested.
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A target can help identify useful intermediate steps. Seeing 4x + 8 may suggest two pairs of sides or a factor of four. Working backwards on a spare line can reveal a route from the given dimensions. That is a legitimate problem-solving activity when it is treated as planning rather than automatic proof.
The finished argument should make clear why the starting facts imply the target. A straightforward forward derivation is often easiest to communicate at school level. Start with the justified relationship and proceed through valid transformations. This avoids a page that appears to assume the conclusion and then manipulate it until something familiar emerges.
There are valid arguments written through reversible equivalences. The important condition is that each step preserves the relevant truth in both directions and that domain restrictions are handled. Students should not use a symbol such as a two-way implication casually if they do not know why the transformations are reversible.
For a simple example, 2x + 6 = 16 is equivalent to 2x = 10 and then x = 5 over the usual real-number setting. Each inverse operation is valid. By contrast, some operations can lose or introduce possibilities. The learner must understand what their steps establish, rather than assuming that any algebraic movement can be reversed.
Keep early teaching practical. Use backwards exploration to find ideas, then ask the student to write a short clear route from the givens. More advanced proof structures can be learned when appropriate. The central habit is honest implication: the written solution must establish the required claim under its conditions.
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An equals sign states that the expressions on either side have the same value under the relevant conditions. It does not mean “then I did this.” A chain such as 3 × 4 = 12 + 5 = 17 incorrectly claims that twelve equals seventeen. The sequence of actions has been written as false equalities.
Write 3 × 4 + 5 = 12 + 5 = 17, or use separate labelled steps. Each equality is then true. This matters particularly in show-that work because the visible chain is the argument. If its statements are false, a correct final number does not make the reasoning valid.
For the rectangle, P = 2(x + 4) + 2x = 2x + 8 + 2x = 4x + 8 is a valid chain. Every expression represents the same perimeter. The arrangement communicates the derivation efficiently without requiring a separate explanatory sentence for every arithmetic simplification.
When solving equations, separate successive equations if that is clearer. From 3x + 4 = 19, obtain 3x = 15 and then x = 5. Writing 3x + 4 = 19 = 15 = 5 would conflate different statements. The student should understand why the latter notation misrepresents the work.
Correct this habit through meaning rather than decoration. Ask what one equals sign on the page claims and whether that statement is true. Then rewrite it accurately. A few focused examples can repair the notation more effectively than repeatedly marking a red line under a chain without explaining the problem.
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Consider the claim that the sum of three consecutive integers is divisible by three. Checking 4 + 5 + 6 = 15 gives one example. It does not establish every possible starting integer. To show the general relationship, let the integers be n, n + 1 and n + 2.
Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). Since n is an integer, n + 1 is an integer. The sum is therefore three times an integer, which establishes divisibility by three. The algebra describes every permitted starting value at once.
The domain is part of the explanation. If n were not restricted to integers, the divisibility statement would not have the same meaning. Defining n carefully makes the final conclusion valid. This is a useful example of why variable definitions and conditions matter, even when the manipulation itself is short.
Select this example only if its language and algebra fit the student’s taught work. The broader lesson can be demonstrated more simply through a perimeter formula or a repeated grouping arrangement. Do not introduce unfamiliar proof terminology merely to make an ordinary explanation look more advanced.
Ask the learner which line connects the expression to the conclusion. Reaching 3n + 3 is not yet the full explanation of divisibility. Writing 3(n + 1) exposes the factor of three, and identifying the integer factor completes the reason. A target often becomes clear when the expression is put into a useful form.
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A question may ask you to show that a quantity is 2.50 to two decimal places. A calculation giving 2.496 supports that rounded result. Writing only 2.50 does not reveal whether the underlying value was close enough or whether the required rounding was performed correctly. Show sufficient precision before the final approximation.
For an original example, 17.472 litres shared equally among seven containers gives 2.496 litres each. Rounded to two decimal places, that is 2.50 litres. The trailing zero communicates the requested decimal-place form. It does not mean the unrounded value equals 2.50 exactly.
Use the approximation sign appropriately when displaying the transition. The exact quotient in this example is 2.496, while its two-decimal-place form is approximately 2.50. A chain that writes 2.496 = 2.50 as an exact equality is misleading. The target’s stated accuracy tells the reader how the final value is being represented.
Do not round so early that the evidence for the requested result disappears or changes. Retain an adequate intermediate value and apply the final instruction. If the task specifies a particular degree of accuracy, follow it. A longer decimal does not automatically improve an answer when a particular presentation is required.
Our guide to exact values and graph estimates explains the wider distinction. Here, the point is the argument connecting a calculated value to a printed rounded target. The student should show why the target is justified, not merely imitate its number of digits.
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A geometry show-that question often needs more than a numerical answer. If two angles are claimed equal, identify the property that establishes their equality in the given configuration. A diagram’s appearance is not automatically evidence. Use stated parallel lines, equal sides or other valid facts and deductions.
For example, if a triangle has two stated equal sides, the opposite base angles are equal under the relevant theorem. If the diagram merely looks symmetrical without that information, the same conclusion may not be justified. The reason must connect to the actual givens rather than a shape guessed from a sketch.
If a right angle is needed for a method, establish it from the supplied information or use the given marking. Applying Pythagoras’ theorem because a triangle looks nearly right-angled is not enough. A correct numerical target reached through an unsupported condition still leaves the central argument incomplete.
Add known and derived facts to the diagram as they become available. Keep track of what is given and what has been established. This helps prevent a student from using a later target as an earlier reason. The diagram becomes a record of justified information rather than a source of visual guesses.
The appropriate amount of written explanation depends on the task. Use concise mathematical reasons where needed and show the relationships clearly. A list of theorem names unconnected to the actual angles is not stronger than a short relevant argument. The reader needs to know which property applies to which part of the figure.
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An original exercise says that twelve visitors include eight adults and the rest children. Adult tickets cost four dollars and child tickets two dollars. Show that the total ticket payment is forty dollars. Assume these are the only charges and the stated prices apply to every visitor in the group.
First find the child count: twelve minus eight equals four. The adult payment is eight times four, or thirty-two dollars. The child payment is four times two, or eight dollars. Adding gives forty dollars. The result follows from the group composition and ticket prices, rather than being used to invent an unsupported count.
A compact solution can show all essential steps: children = 12 − 8 = 4; total payment = 8 × $4 + 4 × $2 = $40. Each line has a clear job. The first identifies a missing quantity from the givens, while the second prices the two groups and combines them.
Suppose a student instead starts with forty dollars and subtracts thirty-two to obtain eight, then divides by two to get four children. That may help check consistency, but if presented alone it assumes the target payment while trying to establish it. The given total visitor count already supplies the independent route to the child count.
A reverse consistency check can be useful after the forward derivation. Four children plus eight adults recovers twelve visitors, and the payments total forty. The distinction is not that reverse thinking is forbidden. It is that the evidence establishing the target should not depend solely on accepting that same target as a starting fact.
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SECTION 12 OF 19
11. A Supplied Result Can Support Later Parts Once Established or Authorised
Multi-part questions sometimes supply a result in one part so later work can continue. The instructions may explicitly allow its use even if a student cannot complete the demonstration. Read those instructions. A later calculation can have a valid starting point because the task authorises it, while the earlier show-that argument remains unfinished.
Do not confuse this with pretending the earlier part was proved. Label what you are using and continue according to the question. This can help preserve productive work on later parts. Assessment credit depends on the actual instructions and scheme, so ask the teacher about specific cases rather than assuming a universal rule.
If the result was established in an earlier part, refer back clearly. “Using the perimeter expression from part a” connects the later substitution to a justified relationship. This is efficient communication. There is no need to repeat a full derivation when the previous part already supplies the required result and the task permits its use.
If a later answer disagrees with the key, inspect the inherited result and the new operation separately. A student may have a correct formula but substitute the wrong value. Another may carry an earlier error consistently. The review should identify the point where the actual task stopped being represented correctly.
This teaches an important organisational habit. A long problem is a chain of established relationships and new decisions. Knowing which results are available at each stage helps the student plan. It also makes the final working easier to follow than an unlabelled collection of expressions whose status is never explained.
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Verifying one candidate establishes that it works, not necessarily that it is the only candidate. Checking one example supports that example, not every possible case. Reading one graph window shows the displayed region, not automatically the whole real domain. The conclusion should match the scope of the evidence.
For a simple linear equation, valid solving operations can establish the unique solution under the usual conditions. For a quadratic, more than one solution may need consideration. The learner should not treat “I found a value that works” as a complete substitute for “I found all values requested.” Read what the claim actually covers.
A counterexample has a different role. If the task asks whether every three consecutive integers have an odd sum, one even-sum example disproves that universal claim. The sum 1 + 2 + 3 = 6 is enough to show it is not always odd. The appropriate evidence depends on whether a claim is being established or rejected.
These distinctions can be taught through short contrasts. Ask what each calculation proves and what remains unproved. The student does not need to attach advanced logical vocabulary to every answer. They need to avoid turning a limited check into an unsupported general conclusion.
A strong final sentence is precise. “This verifies that five satisfies the equation” or “This expression is a multiple of three for integer n” tells the reader exactly what follows. A broad statement such as “therefore it works for everything” may go beyond the argument and should be replaced with the actual justified conclusion.
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| Stage | Question to answer | What the working should show |
|---|---|---|
| Identify | What is given and what is the target? | Consistent quantities and definitions |
| Select | Which relationship applies? | A justified formula, property or comparison |
| Develop | How does the result follow? | Valid operations and essential intermediate steps |
| Check | Are conditions and requested form satisfied? | Domain, units, completeness and accuracy |
| Conclude | What has been established? | The target linked to the preceding argument |
This structure is a planning aid, not a required five-paragraph answer. A short problem may fit into two lines. The stages describe the decisions behind those lines. Use as much writing as the task needs to make the reasoning clear, and no more than is useful for that purpose.
During revision, ask which stage caused the difficulty. Some students cannot choose the first relationship. Others choose it correctly but lose a sign. Others perform the algebra but omit the final interpretation. The same printed target can hide different learning needs, so feedback should identify the first failed decision.
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These original tasks are informal learning checks. Select examples suitable for the student’s current course. Before doing the full work, ask the learner to name an appropriate starting relationship and the type of conclusion needed. This makes the reasoning goal visible before calculation begins.
One: a rectangle has width x and length x + 3. Show that its perimeter is 4x + 6. Use P = 2x + 2(x + 3), then expand and collect. Two: verify that x = 4 satisfies 2x + 1 = 9. Substitute four and show that the left side equals nine.
Three: show that n + (n + 1) is odd for integer n. The sum is 2n + 1, one more than an even integer. Four: a quantity is calculated as 2.496 and must be shown as 2.50 to two decimal places. State the unrounded value and apply the rounding instruction; do not claim exact equality between the two forms.
Five: an exercise asks whether every multiple of four is a multiple of eight. Four is a counterexample, so the universal claim is false. Six: a triangle looks isosceles but no equal sides or equivalent information are supplied. Appearance alone does not establish equal base angles. Identify the missing condition rather than inventing it.
Seven: a worked solution checks a proposed identity at x = 1 and announces that it holds for all real x. That check is insufficient for the general claim. Use valid algebra or the appropriate taught argument. Eight: a question asks for all roots, but a student verifies one supplied root only. The candidate is checked, but completeness still needs attention.
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Keep the original attempt and identify the first unsupported step. Was the target assumed? Was a theorem applied without its condition? Was one numerical example treated as a general proof? Was the correct quantity never formed? These failures need different explanations, even when the final answer matches the printed result.
For a missing start, ask the student to write only the relationship justified by the givens. For an invalid transformation, use a smaller algebraic example. For an overgeneralisation, compare an example check with a general expression. Repair the precise gap before requiring another long written demonstration.
After explanation, change the target or data slightly. A student who derived a rectangle perimeter can try a related expression with a different fixed difference in side lengths. Ask them to explain the first line without viewing the old solution. The new attempt tests whether the relationship can be selected independently.
Use delayed practice too. Immediate reproduction may depend on remembering the teacher’s sequence. Returning later asks the learner to reconstruct the argument. Record the kind of hint needed, if any, rather than counting a supported reproduction as identical evidence to an independent derivation.
A useful correction note names the principle. “Start from the supplied dimensions, not the target perimeter” is specific. “Show more working” may be too vague if the learner does not know which relationship is missing. More writing helps only when it adds valid evidence that the original answer lacked.
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Ask, “Which facts are you allowed to use at the start?” and “What are you trying to establish?” These questions help separate givens from targets without supplying the full solution. Allow time for the student to identify the relationship. A brief pause can be productive thinking rather than a sign that an adult must intervene immediately.
When a number is already printed, ask why the working is needed. The student should recognise that the task concerns justification. When a candidate is supplied for verification, ask what substitution would establish. Keeping these purposes distinct prevents a blanket rule that all supplied results must either be ignored or assumed.
In a small group, students can compare two short arguments after attempting their own. One may contain a valid derivation and another a circular assumption. A tutor can identify the exact difference and then require an individual follow-up. Group agreement alone does not show that every learner can construct the reasoning independently.
Our Clementi Secondary 1 Mathematics tutorial guide explains eduKateSG’s three-student teaching approach. Confirm current subject-level and learning fit before arranging support. Good teaching should make the next mathematical decision clearer, rather than merely producing a longer model answer for the learner to copy.
Notice improvements in explanation. The student may now define a variable, identify an essential geometry condition or distinguish a numerical check from a general argument. Test those habits in changed questions. A correct final line is encouraging, but the developing ability to justify it is the particular progress a show-that task is intended to reveal.
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Is it always wrong to work backwards from the printed answer?
No. Backwards thinking can help plan a solution, and valid reversible equivalences can form an argument when used correctly. The finished work must actually establish the requested result under its conditions. At introductory level, a clear forward route from the givens is often the easiest way to communicate that justification.
Can substituting a supplied value be enough?
Yes, when it establishes the claim the task asks you to verify. It may not establish derivation, uniqueness or all solutions if those are required. Read the full wording and method instruction. A candidate check and a complete solution can have different scopes even when they share the same final value.
Why is one numerical example not enough for a general formula?
It confirms only that case. An incorrect formula can match a few values and fail elsewhere. Use a general algebraic relationship or another suitable argument that covers the stated domain. Examples can help discover or test a claim, while a counterexample can disprove a universal claim.
How much working should a student write?
Enough to show the justified relationship and essential valid steps leading to the requested result. The actual question and school guidance matter. More lines are not automatically better. A concise transparent derivation is stronger than a long chain that assumes the target or contains false equalities.
Should a student continue if an earlier show-that part is unfinished?
Read whether the question authorises use of the supplied result in later parts. If so, continue honestly under that instruction without pretending the earlier argument is complete. Assessment treatment depends on the specific task and scheme. Ask the teacher about unclear cases rather than assuming one universal marking rule.
If the stated target appears false, do not force the working
First confirm that the question and target were copied correctly. Then check the definitions and calculations. If a genuine contradiction remains, preserve the evidence and ask the teacher about the particular task. Inventing a sign change or omitting an inconvenient condition to reach the printed result does not create a valid argument. Mathematical justification depends on the statements being true under their conditions, even when a target has been supplied.
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A show-that target is useful because it gives the learner a destination. The work must provide the connection from the authorised information to that destination. Defining quantities, selecting a valid relationship and explaining essential steps makes the conclusion understandable rather than merely recognisable.
Choose one school question with a printed target. Separate the givens from what must be established, then ask for the first justified line. Build the argument from there and check what it actually proves. This small routine helps the learner move from copying a known result to producing a reasoned mathematical explanation.
Continue to How Mathematics Works for related guides. The examples here are original instructional material. Relevant course references are the 2027 G1 Mathematics K110 syllabus, G2 Mathematics K210 syllabus and G3 Mathematics K310 syllabus, checked on 15 October 2026. Follow the applicable examination year and school instructions for formal requirements.
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Continue the G1/G2/G3 Mathematics series
What Is G1 Mathematics | How to Compare Offers When Pack Sizes and Conditions Differ
What Is G2 Mathematics | Why a Fixed Starting Amount Breaks a Proportion Shortcut
What Is G3 Mathematics | How Removing One Value Changes the Mean and Median
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