Series route: This guide belongs to the complete PSLE Learner’s Guide volume directory. The original PSLE Learning Guide supplies the wider English, Mathematics and Science route, while PSLE Mathematics provide verified subject-specific continuation.
Find your route through this guide
This volume solves one precise learner problem: the learner calculates every consecutive-number product from scratch and misses a simple bound that can reject impossible answers or compare expressions quickly. The aim is not to memorise a slogan. It is to understand the mechanism well enough to recognise the idea when the wording, numbers, diagram or context changes. A Primary 6 learner should be able to explain the decision, perform it under time pressure, and detect a tempting answer that only looks familiar.
- Understand the mechanism before memorising a form.
- Study the worked examples and compare near-misses.
- Repair common misconceptions with targeted tests.
- Complete graded independent practice before reading the answers.
- Use delayed transfer and examination control to make the skill durable.
- Use the parent and tutor notes only after the learner has attempted the work.
The problem beneath the mistake
The product12×13 is larger than12² because one factor has increased from12 to13. It is smaller than13² because its first factor is still12 rather than13. Therefore144<156<169.
This three-part relationship creates a bracket around the product before exact calculation. A proposed answer outside the bracket can be rejected immediately.
Positivity matters. For positive factors, increasing a factor while holding the other positive increases the product. With zero or negative numbers, order arguments require extra care.
The lower difference is easy: n(n+1)−n²=n. The consecutive product is exactly n above the smaller square.
The upper difference is also easy: (n+1)²−n(n+1)=n+1. The product is exactly n+1 below the larger square.
For20×21, the product is20 above400 and21 below441, so it is420. The bounds and differences agree from both sides.
The method supports estimation. If an answer key gives3,590 for59×60, the product must be between59²=3,481 and60²=3,600. The proposed value fits the broad bound, but the exact difference test shows the product should be3,540.
A bound is not an exact solution unless the offset is used. It narrows possibilities and supplies a powerful independent check.
The structure also connects to rectangular area. An n-by-(n+1) rectangle has more area than an n-by-n square but less area than an (n+1)-by-(n+1) square.
Two consecutive whole numbers have opposite parity, so their product is even. That is a different property from the square bounds; both can be used as separate checks.
The relationship can be reversed to locate a number. If a value lies exactly n above n² and n+1 below the next square, it is the product n(n+1).
The reliable routine is identify consecutive positive factors, write the neighbouring squares, compute one offset, find the product, then use the other offset and parity as independent checks.
Core decision: For any positive whole number n, n² is less than n(n+1), which is less than (n+1)², because multiplying n by a factor greater than n raises the product, while replacing n with the larger factor n+1 raises it again.
A useful rule has three parts: a trigger, an operation and a check. The trigger tells you when the rule applies. The operation tells you what to do. The check tells you whether the result still matches the meaning of the question. Learners who remember only the operation often apply it in the wrong place; learners who remember only an example often freeze when the surface details change.
Mechanism: why the rule works
1. Verify consecutiveness
Check that the larger factor is exactly one more than the smaller.
Diagnostic question: Can the factors be written n and n+1? Reliable response: Do not use the rule for a wider gap.
2. Confirm positivity
Ensure the comparison uses positive whole numbers.
Diagnostic question: Will increasing a factor increase the product? Reliable response: Keep this Primary 6 rule inside its stated domain.
3. Build the lower bound
Compare n×n with n×(n+1).
Diagnostic question: Which factor increased? Reliable response: Write n²<n(n+1).
4. Build the upper bound
Compare n×(n+1) with (n+1)×(n+1).
Diagnostic question: Which remaining factor is still smaller? Reliable response: Write n(n+1)<(n+1)².
5. Use an exact offset
Add n to n² or subtract n+1 from (n+1)².
Diagnostic question: Which nearby square is easier? Reliable response: Choose one route and calculate.
6. Check independently
Use the opposite square, parity or direct multiplication.
Diagnostic question: Does a different relationship reach the same value? Reliable response: Reject answers that only fit one weak estimate.
Worked examples: make the decision visible
Do not race through these examples. Before reading each explanation, name the evidence that matters and the information that does not. Then compare the correct route with the tempting route. That comparison is where the transferable learning sits.
Worked example 1: Small pair
Task. Find7×8 using square bounds.
Reasoning. The product lies between49 and64 and is7 above49. The controlling idea is the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 49+7=56.
Independent check. It is also8 below64. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 2: Twenty and twenty-one
Task. Find20×21.
Reasoning. Use20²+20. The controlling idea is the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 400+20=420.
Independent check. Also441−21=420. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 3: Near one hundred
Task. Find99×100.
Reasoning. Use100²−100 or99²+99. The controlling idea is the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 10,000−100=9,900.
Independent check. The result is even and between9,801 and10,000. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 4: Reject high answer
Task. Could13×14 equal200?
Reasoning. The product must be below14²=196. The controlling idea is the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. No;200 is impossible.
Independent check. The exact product is182. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 5: Reject low answer
Task. Could25×26 equal620?
Reasoning. The product must exceed25²=625. The controlling idea is the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. No;620 is impossible.
Independent check. The exact product is650. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 6: Answer fits bound but wrong
Task. Could59×60 equal3,590?
Reasoning. It lies between the squares, so the broad bound alone cannot reject it. Use the exact offset. The controlling idea is the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. No;59²+59=3,540.
Independent check. Also3,600−60=3,540. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 7: Non-consecutive warning
Task. Can12×15 use the consecutive offset rule?
Reasoning. The factors differ by3, not1. The controlling idea is the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. No; use another method.
Independent check. The product may still lie between some squares, but not with these offsets. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Worked example 8: Area model
Task. Compare a9-by-10 rectangle with9-by-9 and10-by-10 squares.
Reasoning. Adding one strip of9 squares creates the rectangle; another strip of10 would create the larger square. The controlling idea is the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed. This prevents a learner from choosing an answer merely because one word, number or observation resembles a remembered example.
Answer. 81<90<100.
Independent check. The area strips show offsets9 and10. A good check approaches the conclusion from a different direction. Repeating the same steps more slowly may repeat the same hidden assumption.
Deepening lab: compare cases, conditions and limits
A method becomes independent when you can say where it applies, where it stops and which change would alter the conclusion. The cases below deliberately vary one structural feature at a time. For each case, pause before the analysis and predict whether the core rule remains available. Then name the exact condition that controls the decision. This is harder than repeating a worked solution, but it is also closer to what an unfamiliar examination item demands.
Deepening case 1: 1×2
Case. The product lies between1 and4.
Analysis. It is1 above the smaller square. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Calculate. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 2: 2×3
Case. The product lies between4 and9.
Analysis. It is2 above4. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Check parity. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 3: 9×10
Case. Use10²−10.
Analysis. The upper square is especially easy. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. State the value. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 4: 10×11
Case. Use10²+10.
Analysis. The lower square is especially easy. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Compare method choice. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 5: 24×25
Case. Use25²−25.
Analysis. A known square makes the upper route efficient. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Calculate carefully. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 6: 25×26
Case. Use25²+25.
Analysis. The lower route is equally direct. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Explain the symmetry. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 7: 49×50
Case. Use50²−50.
Analysis. The product is just below2,500. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Estimate first. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 8: 50×51
Case. Use50²+50.
Analysis. The product is just above2,500. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Compare with49×50. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 9: 99×100
Case. The upper factor is a friendly base.
Analysis. Subtract100 from10,000. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Do not write9,990. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 10: 100×101
Case. Add100 to10,000.
Analysis. The lower square is friendly. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. State place value. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 11: Lower offset
Case. The product exceeds n² by n.
Analysis. The added strip has n unit squares. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Draw it. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 12: Upper offset
Case. The larger square exceeds the product by n+1.
Analysis. The missing strip has n+1 squares. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Draw the missing strip beside the rectangle. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 13: Evenness
Case. One of two consecutive whole numbers is even.
Analysis. Their product is even. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Use as a separate check. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 14: Bound only
Case. An answer between the squares can still be wrong.
Analysis. Bounds narrow; offsets determine. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Give a counterexample. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 15: Wrong gap
Case. Factors8 and10 are not consecutive.
Analysis. The exact offsets8 and9 do not apply. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Choose a new strategy. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 16: Reverse location
Case. A number is31 above31².
Analysis. It equals31×32. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Verify from the upper square. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 17: Mental comparison
Case. Which is larger:30×31 or29×30?
Analysis. Both connect around30²; one is30 above and the other30 below. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Find the difference60. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 18: Rectangle growth
Case. Moving from n² to n(n+1) adds one column of height n.
Analysis. Area gives the algebraic offset. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Label dimensions. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 19: Domain limit
Case. Negative consecutive integers need sign-aware order reasoning.
Analysis. The positive-number statement should not be overgeneralised. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. State the scope. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Deepening case 20: Final audit
Case. Consecutive factors, positivity, two squares, offset, exact value and independent check complete the method.
Analysis. Each catches a different error. Notice that the answer depends on the fixed positive order n less than n+1 and the effect of increasing one positive factor while keeping the other fixed; changing a decorative detail would not be enough to change the reasoning.
Challenge. Audit six products. State the conclusion in one precise sentence, then add one sentence that names the evidence or relationship that makes it valid.
Finish the lab with a blank-page recall. Write the rule, its conditions, one confirming example, one near-miss and one independent check from memory. Compare that page with the article only after finishing. If the rule is correct but the condition is missing, the learning is not yet safe: an overgeneralised rule can produce confident errors. If the condition is correct but no example can be generated, practise construction rather than rereading.
Common misconceptions and targeted repairs
Misconception 1: Between squares gives the exact value
What the learner may think. Knowing the interval finishes the calculation.
Why it fails. Many integers can lie between consecutive squares. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Use the offset n or n+1. Use this short test immediately: How far from the square?
Misconception 2: Any near factors qualify
What the learner may think. Factors with a small gap use the same rule.
Why it fails. The offsets are derived specifically for n and n+1. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Verify a gap of exactly1. Use this short test immediately: Are they consecutive?
Misconception 3: Product is halfway
What the learner may think. n(n+1) lies halfway between n² and(n+1)².
Why it fails. The gaps are n and n+1, differing by1. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Calculate both offsets. Use this short test immediately: Are they equal?
Misconception 4: Add one to the square
What the learner may think. Since factors differ by1, add1 to n².
Why it fails. The added strip contains n units, not1. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Use n²+n. Use this short test immediately: What is the strip area?
Misconception 5: All signs behave the same
What the learner may think. The positive inequality works unchanged for negative numbers.
Why it fails. Multiplication order with negatives needs separate analysis. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Keep the rule in its stated domain. Use this short test immediately: Are both factors positive?
Misconception 6: One check is enough
What the learner may think. An answer inside the interval is correct.
Why it fails. A plausible wrong value can still fit the broad bounds. The error feels efficient because it removes a decision, but it also removes the evidence boundary that makes the answer valid.
Targeted repair. Use offset and parity or direct multiplication. Use this short test immediately: Does the exact gap match?
Graded independent practice
Attempt each item without looking ahead. Write one line that identifies the controlling evidence before writing the final answer. Items 1–3 check recognition, 4–7 require explanation, and the later items change the surface form so that memory alone is insufficient.
Practice 1: Recognition
Question. Can34×35 use the method?
Explained answer. Yes.
Why this answer earns trust. The factors are consecutive positive whole numbers. Transfer variation: Write both square bounds.
Practice 2: Lower route
Question. Find34×35 from34².
Explained answer. 1,156+34=1,190.
Why this answer earns trust. The product is34 above the lower square. Transfer variation: Check from35².
Practice 3: Upper route
Question. Find74×75 from75².
Explained answer. 5,625−75=5,550.
Why this answer earns trust. The product is75 below the upper square. Transfer variation: Check evenness.
Practice 4: Bound
Question. Between which squares does18×19 lie?
Explained answer. Between18²=324 and19²=361.
Why this answer earns trust. The rectangle lies between the two squares. Transfer variation: Find the exact value.
Practice 5: Reject
Question. Can18×19 equal370?
Explained answer. No.
Why this answer earns trust. It must be less than19²=361. Transfer variation: State the exact product.
Practice 6: Plausible wrong answer
Question. Can18×19 equal350?
Explained answer. No; the exact product is324+18=342.
Why this answer earns trust. The bound alone was insufficient. Transfer variation: Check361−19.
Practice 7: Reverse
Question. A number is40 more than40². Identify it.
Explained answer. 40×41=1,640.
Why this answer earns trust. The lower-square offset identifies the consecutive product. Transfer variation: Check against41².
Practice 8: Comparison
Question. Which is larger:40×41 or39×40?
Explained answer. 40×41, by80.
Why this answer earns trust. They are40 above and40 below40² respectively. Transfer variation: Verify exact products.
Practice 9: Rule recall
Question. State the full inequality.
Explained answer. For positive whole n, n²<n(n+1)<(n+1)².
Why this answer earns trust. Positivity and consecutiveness matter. Transfer variation: Recall after two days.
Practice 10: Transfer
Question. Create eight consecutive products around known squares.
Explained answer. Every solution includes bounds, one exact offset and an independent check.
Why this answer earns trust. Construction exposes the pattern. Transfer variation: Include one non-example.
Delayed transfer: prove that the learning lasts
Transfer test 1: Bound-and-offset sort
Sort proposed answers as impossible by bounds, plausible but unverified, or exact by offset. Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. Every decision cites the correct square relationship. If the criterion is not met, return to the smallest missing decision rather than repeating the entire chapter.
Transfer test 2: Delayed area derivation
After one week, rebuild the inequalities with square and rectangle areas. Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. The learner explains both offsets without memorising symbols alone. If the criterion is not met, return to the smallest missing decision rather than repeating the entire chapter.
Transfer test 3: Mental comparison set
Compare adjacent consecutive products without full multiplication. Do this after a delay, without rereading the worked examples. The purpose is to test retrieval and decision-making, not immediate imitation.
Success criterion. Differences are derived from their positions around a common square and then checked exactly. If the criterion is not met, return to the smallest missing decision rather than repeating the entire chapter.
Examination control
1. Read the job before the story. Underline the command and the noun it governs. “Explain the change,” “state the word,” “find the fraction” and “identify the evidence” are different jobs even when the surrounding context is similar.
2. Separate given information from remembered information. Use only the facts supplied for the current task plus secure subject knowledge. Do not let a previous question, a familiar worksheet or an expected pattern silently become new evidence.
3. Write enough working to expose the decision. A marker or teacher should be able to see the relationship that justifies the answer. Extra sentences are not automatically stronger; the strongest working is concise, relevant and checkable.
4. Reserve a different-route check. For language, read the sentence for meaning and grammar. For mathematics, estimate or reverse the operation. For science, check the claim against the stated observation and a conservation or fair-test principle.
5. Stop when the requested job is complete. Unrequested speculation can contradict a correct answer. Add detail only when it explains the mechanism or satisfies another explicit part of the question.
Under time pressure, use a compact four-part margin routine: trigger → evidence → operation → check. Trigger names the feature that activates the method. Evidence identifies the exact words, values or observations allowed. Operation performs the language, mathematical or scientific step. Check asks whether the final statement answers the requested job, preserves units or referents, and avoids a stronger claim than the evidence supports.
A compact error log
After practice, do not record only “careless mistake.” Name the broken decision. Useful labels include trigger missed, evidence imported, relationship reversed, referent lost, unit dropped, operation correct but job wrong, and check repeated the same route. Add one prevention cue beside the label. A short accurate log is more useful than a long emotional account because it tells the learner what to do on the next item.
Review the log after two days and again after a week. Reattempt only one or two representative items first. If the method transfers, move to a mixed set. If it fails, repair the exact link between trigger and operation. This spacing matters because a solution can feel obvious while it is still visible and disappear when the page is closed.
Notes for parents and tutors
Use square-number cards and build an n-by-(n+1) rectangle between two square arrays. The added and missing strips make the offsets visible.
Ask for a bound before an exact product. This habit catches impossible answers early and strengthens number sense without replacing full reasoning.
When the answer lies inside the bounds, do not stop. Require the exact offset from one square and a different check from the other.
The adult’s most useful question is not “Why were you careless?” but “Which decision did the evidence require?” Ask the learner to point to the controlling word, value, relationship or observation. If the learner can explain that link, reduce support. If the learner cannot, model one example and then change the surface details for the next attempt.
Frequently asked questions
What is the main relationship?
For positive whole n, n²<n(n+1)<(n+1)².
How far above the smaller square is the product?
Exactly n.
How far below the larger square is it?
Exactly n+1.
Does the rule work for any nearby factors?
No. The exact offsets require consecutive factors.
Are the products halfway between the squares?
No. The two gaps differ by1.
What is the fastest check?
Use the opposite square and confirm the product is even.
Next routes
Return to the complete volume directory to locate the series, use the original PSLE Learning Guide for the foundation route, and continue through PSLE Mathematics for verified subject guidance. The next useful task is a mixed set in which this idea is sometimes relevant and sometimes irrelevant; that is the point at which a learner must choose, not merely copy.