PSLE Mathematics Even Spacing: Count Gaps, Not Marks

PSLE Learner’s Guide · Vol 0366 · Patterns, measurement and inclusive counting

Find the exact decision this guide helps you make

This guide helps a Primary 6 learner solve evenly spaced arrangement questions by distinguishing marked positions from intervals, deciding whether the arrangement is open or closed and attaching the spacing length to the gaps. The short rule is identify whether the arrangement is open or closed, count the intervals that carry length, then multiply or divide using those gaps. You will learn the mechanism, see worked examples, repair common misconceptions, practise in stages and use a compact examination routine.

This is an examination-performance guide, not a replacement for full topic teaching. Use the PSLE Learning Guide for the wider sequence and the Mathematics Learning Hub for deeper subject instruction.

For complete teaching, worked practice and topic progression beyond this performance decision, continue through the Mathematics Learning Hub. This volume keeps a narrower job: it helps the learner recognise, execute and check one examination decision without competing with the wider topic owner. Use the complete PSLE Learner’s Guide volume directory as the permanent series route. For a nearby performance routine that is already published, continue to Vol 0362 on constraining a missing digit before trying values.

A careful note about the current examination year

This article is written for the 2026 PSLE preparation context. SEAB’s official 2026 formats page lists English Language as subject 0001, revised Mathematics as 0008 and revised Science as 0009. Format details can change, so confirm any paper-specific timing, component or administrative instruction with the current official SEAB 2026 source and your school. The reasoning routine taught here is deliberately broader: it strengthens how you read evidence, make a decision and check an answer.

Why this mistake happens

A row of posts, trees or marks contains objects and the spaces between them. Distance is built from the gaps, not from the objects themselves. In an open row with both ends occupied, one more object than gap appears. In a closed loop, the final object connects back to the first, so the number of gaps equals the number of objects. Some questions remove an end object, share a corner or specify a mark only at one end, so a memorised “minus one” is unsafe. The diagram and boundary condition must decide the count.

The mind likes a fast label. A familiar word, operation, result or rule triggers an answer before the conditions have been checked. That shortcut sometimes works in easy practice, so it becomes a habit. In a demanding question, however, a nearby distractor preserves the surface pattern while changing one important condition. The learner then performs a correct-looking move on the wrong object. The repair is to slow only the decision point: identify what is being linked or measured, preserve its labels, test the conditions, and then continue at normal speed.

Think of the method as a chain with four locks. Object: what exact idea or quantity is under discussion? Relationship: how are the relevant parts connected? Condition: what must be true for the move to be valid? Claim: what is the strongest conclusion the evidence actually supports? A weak answer often skips from object to claim. A strong answer makes the two middle locks visible.

The one-line operating rule

Identify whether the arrangement is open or closed, count the intervals that carry length, then multiply or divide using those gaps. Say it before the first practice set. During timed work, shorten it to four cue words you can recall without stopping.

The step-by-step method

Step 1: Sketch the first few marked positions and draw a short bar for every gap between neighbouring positions

Sketch the first few marked positions and draw a short bar for every gap between neighbouring positions. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

At this stage, do not decorate the answer with extra facts. Ask one diagnostic question: “What evidence would make this step wrong?” If you can name a possible failure, check it now. If the failure is absent, continue. This negative check prevents confidence from replacing proof.

Step 2: Decide whether the arrangement is an open line, a closed loop or a line with one or both ends unmarked

Decide whether the arrangement is an open line, a closed loop or a line with one or both ends unmarked. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Step 3: Count objects and intervals separately before inserting any measurement

Count objects and intervals separately before inserting any measurement. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Step 4: Attach the spacing length to one interval, not to one post, tree, lamp or mark

Attach the spacing length to one interval, not to one post, tree, lamp or mark. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Step 5: Use total distance equals number of intervals times spacing, reversing the relationship when required

Use total distance equals number of intervals times spacing, reversing the relationship when required. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Step 6: Check the boundary by testing a tiny case of two or three positions and by rebuilding the final count

Check the boundary by testing a tiny case of two or three positions and by rebuilding the final count. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Misconceptions to diagnose and repair

Misconception 1

Tempting move: Multiplying the number of posts by the spacing in an open row with both ends occupied. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Repair: return to the line, number, setup or statement that supplies the condition. Mark it with a short label. Then rebuild only the affected link. Keep all earlier work that remains valid. The purpose of checking is targeted correction, not anxious repetition.

Misconception 2

Tempting move: Subtracting one automatically even when the arrangement is a closed loop. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Misconception 3

Tempting move: Counting a corner twice when two sides share the same marked position. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Misconception 4

Tempting move: Assuming both endpoints are marked when the wording specifies only interior positions. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Misconception 5

Tempting move: Reporting the number of gaps when the question asks for objects, or the reverse. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Eight worked cases

Worked case 1: Open row, both ends marked

Scenario. Six trees are planted in a straight row 4 m apart. Find the distance from the first to the last.

The tempting error. Using six spaces because there are six trees. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Six marked positions create five intervals in an open row. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. Distance = 5 × 4 = 20 m. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Check. Re-read the command word, substitute the conclusion into the original situation and ask whether every noun, number and comparison still has the same referent or unit. If the answer is stronger than the evidence, narrow it.

Delayed transfer. Draw only three trees and observe the two gaps. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 2: Find number of posts

Scenario. A 30 m straight fence has posts at both ends, 5 m apart. How many posts?

The tempting error. Dividing to get six and stopping. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. 30÷5 gives six intervals. Both ends are marked, so there is one more post. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. There are 7 posts. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Rebuild the row with positions at 0, 5, …, 30. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 3: Closed loop

Scenario. Eight lamps are equally spaced around a circular path, with 3 m between neighbours. Find the perimeter.

The tempting error. Subtracting one because of the open-row rule. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. A closed loop has a gap from the last lamp back to the first, so eight lamps create eight intervals. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. Perimeter = 8 × 3 = 24 m. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Mark the closing gap explicitly. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 4: One end unmarked

Scenario. Markers are placed every 2 m along a 10 m path, starting 2 m from the entrance and including the far end.

The tempting error. Adding one for the entrance even though no marker is there. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. The positions are 2, 4, 6, 8 and 10 m. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. There are 5 markers and five 2 m intervals from the entrance. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. List the coordinates instead of relying on a formula. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 5: Both ends unmarked

Scenario. Trees are planted every 5 m inside a 30 m stretch, but not at either endpoint.

The tempting error. Using seven positions from the both-ends rule. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Possible interior positions are 5, 10, 15, 20 and 25 m. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. There are 5 trees. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Separate six total intervals from interior marked positions. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 6: Shared corner

Scenario. Posts run along two perpendicular 12 m sides, 3 m apart, with posts at all endpoints.

The tempting error. Counting the corner once for each side. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Each side has five post positions, but the shared corner is one physical post. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. Total posts = 5+5−1=9. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Label the shared corner before combining counts. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 7: Missing post does not remove distance

Scenario. Eleven equally spaced posts span 50 m. One interior post is removed. What is the first-to-last distance?

The tempting error. Reducing the distance because only ten posts remain. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. The original endpoints and interval width still define the span; removing an interior marker joins two adjacent gaps but does not move endpoints. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. The span remains 50 m. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. State what changed: marker count, not endpoint distance. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 8: Reverse spacing

Scenario. Nine trees form an open row from first to last over 32 m. Find the spacing.

The tempting error. Dividing by nine trees. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Nine trees create eight equal intervals. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. Spacing = 32÷8=4 m. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Multiply eight gaps by 4 m to rebuild 32 m. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Practice progression: from recognition to transfer

Round 1: mark only

Use six short items. Do not answer them. Mark the object, reference, unit, interval, condition or rule that controls the decision. The goal is accurate noticing without the burden of full writing.

Record the result as one of four codes: N for noticing error, M for method error, I for interpretation error, and C for checking error. Practise the weakest code with a fresh item. Do not repeat an entire worksheet when only one decision needs repair.

Round 2: explain the choice

Solve four items and add one sentence beginning “This is valid because…”. If the sentence merely repeats the answer, the condition is still hidden.

Round 3: repair a wrong answer

Take four plausible wrong answers. Identify the earliest faulty link, preserve the correct work and rewrite only what must change. This develops error diagnosis rather than answer copying.

Round 4: mixed questions

Mix this job with two older skills. Remove headings that reveal the topic. Decide which routine applies before solving. Selection is part of performance.

Round 5: timed set

Use a small timed set with a fixed check budget. Mark a cue at the decision point, finish the answer, then perform one positive check and one rejection check.

Round 6: delayed transfer

Return after two or three days with different surface details. Before seeing notes, write the one-line rule from memory and solve. Compare the new error pattern with the earlier one.

A compact practice log

Question source and date
Decision cue you marked
Your first answer in one line
Earliest wrong link, if any
Corrected rule in twelve words or fewer
Date for delayed transfer
Whether the new item was solved without notes

A useful log is small enough to maintain. It should help you predict the next mistake, not become a scrapbook of completed pages.

Checking that tests meaning, not neatness

A check should have a target. First run a positive test: identify the evidence, calculation or condition that supports your answer. Then run a rejection test: name the closest tempting alternative and the exact reason it fails. This pair is more powerful than rereading the same line while hoping to feel certain.

Use a reverse check where possible. Reconstruct the original amount, replace a connector with its expanded meaning, map a conclusion back to the data, or apply the repaired rule to the apparent exception. Reverse checking exposes lost labels and unsupported leaps because the answer has to return to its source.

Stop when the check has answered a real question. Repeatedly changing a sound answer because it feels unfamiliar is not checking. If the object, relationship, condition and claim all remain consistent, keep the answer and move on.

Examination control

When time is tight, use the cue ORCC: Object, Relationship, Condition, Claim. Write only the smallest useful mark—an arrow, a unit, a time interval, a boundary or a qualifying word. The mark should prevent the known error without slowing the rest of the paper.

Read the command word and identify the decision.
Mark the object or quantity that must keep its identity.
State the relationship or operation in plain language.
Check the condition that makes it valid.
Give the claim at the strength supported by evidence.
Use one positive and one rejection check, then stop.

If you become stuck, do not produce a longer answer immediately. Shorten the task: write what is known, what must be decided and what condition separates the two best options. A precise partial structure often unlocks the next step and also makes any partial credit more defensible.

A seven-day delayed-transfer plan

Day 1: learn the one-line rule and complete two worked cases with notes.
Day 2: solve four short items without notes; classify errors by N, M, I or C.
Day 3: mix two new items with two older PSLE skills and hide the topic labels.
Day 4: explain one case aloud to a parent, tutor or imaginary learner in under two minutes.
Day 5: complete a timed mini-set and use the ORCC marks only at decision points.
Day 6: rest this skill or practise a different subject so memory has to consolidate.
Day 7: attempt one unfamiliar transfer item, then write the smallest rule revision needed.

Success on Day 7 matters more than fluency immediately after reading. Delayed transfer shows whether you can retrieve the method, recognise its job under new surface details and control it without a model beside you.

Guidance for parents and tutors

Ask for the decision, not a performance speech. Useful prompts are: “What exactly are you deciding?”, “Which word, unit or condition controls it?”, “What is the closest wrong option?” and “How can you check without starting again?” These questions keep ownership with the learner while making thinking visible.

When an error occurs, locate its earliest point. Praise accurate diagnosis even before the final answer is corrected. If the learner repeatedly loses the same label, introduce one stable notation: arrows for reference, units beside remainders, interval brackets for rates, or boxes around claim conditions. Avoid adding several new routines at once.

Build variation gradually. Begin with clean examples, then add one plausible distractor, then mix the skill with older content, and finally delay the next attempt. A learner who succeeds only when the worksheet title names the method has learned recognition, not transfer.

Keep emotional temperature low. A surprising exception or remainder is information, not proof that the learner is “bad at” a subject. The practical question is which link changed. Once that link is named, practise a small set that isolates it and then return to mixed questions.

The independent-reader test

Before you call the skill secure, explain one solution to a reader who cannot see your thoughts. Name the starting information, the decision, the condition that controls it and the reason the rejected alternative fails. If the explanation depends on “you know what I mean”, a label is still missing. Add that label to the answer, then remove any sentence that does not help the reader follow the chain.

Next, cover the worked example and rebuild the method from a fresh prompt. Do not copy the surface details. Recover the sequence of decisions. This separates genuine control from familiarity with one page. A learner who can transfer the decision routine to new names, numbers or apparatus has a more dependable examination skill than a learner who can reproduce only the model answer.

Finally, compress the method without losing its safety condition. Write one sentence of no more than fifteen words, then test that sentence against an awkward case. If the short rule gives the wrong result, restore the missing condition. Good compression keeps the mechanism available under time pressure while the awkward case prevents the cue from becoming a careless shortcut.

A timed diagnosis drill

Set aside twelve minutes and choose three unfamiliar questions that could trigger this error. Spend the first minute identifying the decision in each question without solving it. Mark the object, relationship, condition and permitted claim. This preview is not wasted time: it prevents you from entering a long method before you know what the method must preserve.

Solve the three questions in the next eight minutes. After each answer, write one tiny prediction about what a correct answer should preserve: a reference word, a whole-number boundary, a pathway, a unit, a direction or an observable result. Then compare the finished answer with that prediction. If they disagree, repair the earliest link and keep the later work only when its meaning remains valid.

Use the final three minutes to classify the errors. A noticing error means the controlling detail was never marked. A method error means the right information was present but the wrong operation or reasoning chain was used. An interpretation error means the result was calculated or observed correctly but described at the wrong level. A checking error means the answer contained a detectable contradiction that the final audit missed.

Repeat the drill a week later with different surface details and without looking at the old corrections. Improvement is not merely a higher score. Look for a shorter pause at the decision point, fewer unnecessary restarts, more stable labels and a clearer reason for rejecting the nearest wrong answer. Those are signs that the performance routine is becoming portable.

Frequently asked questions

Why are there fewer gaps than posts in an open row?

The first and last posts act as boundaries, so each new gap after the first post adds one new post.

Why are gaps and objects equal in a loop?

The last object connects back to the first, creating a closing interval.

Should I always subtract one?

No. First decide the boundary and which endpoints are marked.

What if one endpoint has no object?

List or sketch the actual positions; the usual both-ends formula no longer applies.

What carries the length?

The intervals between positions.

How do shared corners affect counting?

Count the corner once even if it belongs to two sides.

What is the fastest check?

Use a tiny two- or three-position model and verify the boundary.

What should I study next?

Use the Mathematics Learning Hub for full teaching on patterns, measurement and problem solving.

Your next route

Use this guide until you can state the rule, solve an unfamiliar case and explain why the closest alternative fails. Then move to a deeper subject route: the English Learning Hub, Mathematics Learning Hub, Science Learning Hub, or the full learning-hubs directory. Return to the PSLE Learning Guide when you want to place this skill inside a broader revision sequence.

Mathematics turns direction into boundary control: the learner decides which objects are positions, which spaces carry length and whether the final interval closes back to the start.

How to Perform in PSLE | Learner’s Guide Vol 0366 | Mathematics: Count Gaps, Not Marked Positions, in Evenly Spaced Arrangements