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 recalculates a shifted data set mechanically or assumes that a larger maximum must create a larger range, overlooking that the minimum rises by the same amount. The goal is durable control, not a remembered slogan. A Primary 6 learner should be able to identify the trigger, explain the mechanism, perform the decision under time pressure and reject an attractive answer that exceeds the evidence.
- Start with the exact mistake and the evidence boundary.
- Build the mechanism before memorising a shortcut.
- Study worked contrasts and name why the near-miss fails.
- Repair misconceptions with one targeted check each.
- Attempt graded independent practice before reading the explanations.
- Return after a delay and prove that the method transfers.
- Use the parent and tutor notes after the learner has attempted the work.
The problem beneath the mistake
The range of a data set is maximum minus minimum. If the same number is added to every value, both endpoints move by the same amount, so their difference does not change.
For the set 4, 7, 10, the range is 10 − 4 = 6. Adding 5 to every value gives 9, 12, 15, whose range is 15 − 9 = 6.
The entire set has shifted right on a number line. Its location changes, but its width stays fixed. Range measures that width, not the size of the maximum by itself.
The constant may be positive, zero or negative. Adding −3 is equivalent to subtracting 3 from every value; both endpoints shift left together and the range remains unchanged.
The condition “every value” is essential. If 5 is added only to the maximum, the upper endpoint moves while the lower endpoint does not, so the range usually changes.
For repeated values, the rule still works. Multiplicity can matter for other statistics, but the range depends only on the smallest and largest values.
The mean, median and mode also shift by the added constant under suitable definitions, but that is a separate statement. The present proof for range uses endpoint subtraction.
Multiplication behaves differently. Multiplying every value by a positive factor multiplies the range by that factor; multiplying by a negative factor also reverses order and scales the range by the factor’s absolute value.
In word problems, a fixed adjustment such as adding a calibration offset to every reading changes the reported level but not the spread between highest and lowest readings.
The safe routine is identify the endpoints, verify the same constant reaches every value, cancel the common shift in max minus min, then check whether any condition changes the order.
Core decision: If every value x is replaced by x + c, then the new range is (maximum + c) − (minimum + c) = maximum − minimum. The common constant cancels, so the range is invariant.
Range is a difference measure. Uniform translation changes absolute position but preserves all pairwise differences. Understanding this structure lets the learner reason without rewriting a long list while still knowing exactly when the shortcut is legal.
Mechanism: why the rule works
1. Return to the definition
Write range = maximum − minimum before touching the data.
A remembered claim becomes reliable when it is tied to the statistic’s definition.
Diagnostic check. Which two values alone control the range?
2. Model a uniform shift
Represent every old value x by x + c. Then the old maximum becomes maximum + c and the old minimum becomes minimum + c.
Adding the same constant preserves order: if a < b, then a + c < b + c.
Diagnostic check. Did every observation receive exactly the same c?
3. Cancel the common constant
Compute (M + c) − (m + c) = M + c − m − c = M − m.
The two appearances of c have opposite signs after subtraction. This is algebraic cancellation, not a lucky pattern.
Diagnostic check. Can you show the cancellation in one line?
4. Use a number-line picture
Imagine moving every point the same distance and direction. The leftmost and rightmost points move together.
Translation changes position but not distance. The visual model supports the algebra.
Diagnostic check. Has the interval changed width, or only location?
5. Check the universal condition
If one observation receives a different adjustment, the transformation is not uniform and the proof no longer applies.
A new maximum or minimum may appear, or only one endpoint may move.
Diagnostic check. Which value, if any, was treated differently?
6. Separate translation from scaling
Adding c translates. Multiplying by k scales distances; for k < 0 it also reverses order.
The words add and multiply trigger different structures even when both operations affect every value.
Diagnostic check. Is the transformation x + c or kx?
7. Use inverse checking
Subtract c from the shifted endpoints. If the original maximum and minimum return, the translation has been applied consistently.
A second check can also calculate both ranges directly for a small sample.
Diagnostic check. Does reversing the adjustment restore the original set?
Worked examples: make the decision visible
Pause before each answer. State the controlling evidence in one line, then complete the operation. The explanations deliberately contrast a valid route with a plausible but invalid one, because transfer grows when the learner can say why a tempting route does not apply.
Worked example 1: Positive shift
Task. The values are 3, 8, 11, 15. Add 7 to every value and find the new range.
Reasoning. Original range is 15 − 3 = 12. New endpoints are 22 and 10, so 22 − 10 = 12.
Answer. The new range is 12.
Independent check. Both endpoints rose by 7.
Contrast-and-transfer pass. Keep the governing principle from “Positive shift”, but change one surface detail in the task: The values are 3, 8, 11, 15. Add 7 to every value and find the new range. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The new range is 12.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 2: Negative shift
Task. The readings are −2, 4, 9. Add −5 to each reading.
Reasoning. Original range is 9 − (−2) = 11. New endpoints are 4 and −7; 4 − (−7) = 11.
Answer. The range remains 11.
Independent check. Subtraction of a negative minimum is handled carefully.
Contrast-and-transfer pass. Keep the governing principle from “Negative shift”, but change one surface detail in the task: The readings are −2, 4, 9. Add −5 to each reading. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The range remains 11.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 3: Decimal shift
Task. A set runs from 1.6 to 5.9. Add 0.4 to every value.
Reasoning. The new endpoints are 2.0 and 6.3. Their difference is still 4.3.
Answer. The range remains 4.3.
Independent check. The decimal shift cancels exactly.
Contrast-and-transfer pass. Keep the governing principle from “Decimal shift”, but change one surface detail in the task: A set runs from 1.6 to 5.9. Add 0.4 to every value. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The range remains 4.3.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 4: Long list shortcut
Task. One hundred readings have minimum 18 and maximum 47. Every reading is increased by 6.
Reasoning. Only the endpoints are needed. New range = 53 − 24 = 29, equal to 47 − 18.
Answer. The range is 29.
Independent check. There is no need to rewrite all one hundred values.
Contrast-and-transfer pass. Keep the governing principle from “Long list shortcut”, but change one surface detail in the task: One hundred readings have minimum 18 and maximum 47. Every reading is increased by 6. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The range is 29.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 5: One value changed
Task. The set 2, 6, 9 has 5 added only to 9.
Reasoning. The new set is 2, 6, 14. The minimum stays 2 while the maximum rises.
Answer. The range changes from 7 to 12.
Independent check. The every-value condition is false.
Contrast-and-transfer pass. Keep the governing principle from “One value changed”, but change one surface detail in the task: The set 2, 6, 9 has 5 added only to 9. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The range changes from 7 to 12.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 6: Uniform subtraction
Task. Subtract 12 from every score in a data set whose range is 35.
Reasoning. Subtracting 12 means adding −12 uniformly.
Answer. The range remains 35.
Independent check. The rule includes negative constants.
Contrast-and-transfer pass. Keep the governing principle from “Uniform subtraction”, but change one surface detail in the task: Subtract 12 from every score in a data set whose range is 35. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The range remains 35.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 7: Calibration offset
Task. A thermometer reports every reading 1.5°C too low, so 1.5°C is added to every recorded value.
Reasoning. The correction changes every level equally. Highest and lowest both rise by 1.5°C.
Answer. The corrected range equals the original range.
Independent check. This does not prove the thermometer was otherwise precise.
Contrast-and-transfer pass. Keep the governing principle from “Calibration offset”, but change one surface detail in the task: A thermometer reports every reading 1.5°C too low, so 1.5°C is added to every recorded value. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The corrected range equals the original range.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 8: Unknown endpoints
Task. After 4 is added to every value, the new minimum is 13 and maximum is 29. Find the original range.
Reasoning. The original endpoints were 9 and 25, but their difference can be found directly as 29 − 13.
Answer. The original range is 16.
Independent check. Translation preserves the range in both directions.
Contrast-and-transfer pass. Keep the governing principle from “Unknown endpoints”, but change one surface detail in the task: After 4 is added to every value, the new minimum is 13 and maximum is 29. Find the original range. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The original range is 16.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 9: Compare multiplication
Task. The set 2, 5, 8 is doubled.
Reasoning. The old range is 6; the new set 4, 10, 16 has range 12.
Answer. The range doubles, so the addition rule does not apply.
Independent check. Scaling changes distances.
Contrast-and-transfer pass. Keep the governing principle from “Compare multiplication”, but change one surface detail in the task: The set 2, 5, 8 is doubled. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The range doubles, so the addition rule does not apply.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 10: Proof completion
Task. Complete: (M + c) − (m + c) = __.
Reasoning. Expand the subtraction: M + c − m − c. The c terms cancel.
Answer. The result is M − m.
Independent check. The proof states why every uniform shift works.
Contrast-and-transfer pass. Keep the governing principle from “Proof completion”, but change one surface detail in the task: Complete: (M + c) − (m + c) = __. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The result is M − m.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
The method lab: change one condition at a time
Place five counters on a drawn number line. Slide every counter four spaces right and measure the distance between the extreme counters before and after. Repeat by sliding left seven spaces. The physical movement makes translation visible.
Create a table with columns for original minimum, original maximum, constant, new minimum, new maximum and both ranges. Include positive, negative and decimal constants. The equal-range pattern should be explained algebraically, not merely noticed.
Now break the condition deliberately: adjust only the largest value, only an interior value, and different values by different amounts. Record when the range changes and when it happens to stay the same. An accidental unchanged result is not the uniform-shift theorem.
Compare three transformations on the same set: add 5, multiply by 5, and add 5 only to the maximum. Explain location, order and width for each transformation.
Common misconceptions and targeted repairs
Misconception 1: Larger maximum means larger range
What the learner may think. The learner sees the shifted maximum and stops.
Why it fails. The minimum also rises by the same amount.
Targeted repair. Compute maximum minus minimum after both endpoints move.
Misconception 2: The constant is added to the range
What the learner may think. The learner writes new range = old range + c.
Why it fails. Range is a difference, and the same c cancels.
Targeted repair. Substitute into (M+c)−(m+c).
Misconception 3: Only positive constants work
What the learner may think. The learner thinks subtraction must reduce the range.
Why it fails. Uniform leftward translation preserves distance too.
Targeted repair. Treat subtract c as add −c.
Misconception 4: Every operation behaves alike
What the learner may think. The learner extends the result to multiplication.
Why it fails. Scaling changes distances even when applied uniformly.
Targeted repair. Classify the transformation before using a rule.
Misconception 5: One endpoint is enough
What the learner may think. The learner updates only the maximum.
Why it fails. Range requires both endpoints.
Targeted repair. Write the new minimum and maximum on separate lines.
Misconception 6: A chance result proves the theorem
What the learner may think. Different adjustments happen to leave the same range once.
Why it fails. The theorem requires a uniform constant; coincidence does not establish the condition.
Targeted repair. Check the transformation rule, not only the final number.
Graded independent practice with explained answers
Attempt each item without looking ahead. Write the evidence or relationship before the final response. Early items check recognition; middle items require explanation; later items change the surface form so that copying a worked example is insufficient.
Practice 1: Definition
Question. What is the range of a data set?
Explained answer. Maximum minus minimum.
Why this earns trust. The statistic is stated before transformation.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Maximum minus minimum.” from memory, but to recreate a justified answer from the new evidence.
Practice 2: Simple shift
Question. The range is 14. Add 3 to every value. What is the new range?
Explained answer. 14.
Why this earns trust. A uniform translation preserves the difference.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “14.” from memory, but to recreate a justified answer from the new evidence.
Practice 3: Subtraction
Question. The range is 22. Subtract 9 from every value.
Explained answer. The range remains 22.
Why this earns trust. This is a uniform addition of −9.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “The range remains 22.” from memory, but to recreate a justified answer from the new evidence.
Practice 4: Endpoints
Question. Minimum 5, maximum 19; add 8 to all values. Find new endpoints and range.
Explained answer. New minimum 13, new maximum 27, range 14.
Why this earns trust. Both endpoints move and the difference stays 14.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “New minimum 13, new maximum 27, range 14.” from memory, but to recreate a justified answer from the new evidence.
Practice 5: Negative data
Question. Minimum −7, maximum 6; add 10 to all.
Explained answer. New endpoints 3 and 16; range 13.
Why this earns trust. Order is preserved through the shift.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “New endpoints 3 and 16; range 13.” from memory, but to recreate a justified answer from the new evidence.
Practice 6: Decimal
Question. Range 2.75; add 0.6 to every observation.
Explained answer. The range remains 2.75.
Why this earns trust. The constant cancels regardless of decimal form.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “The range remains 2.75.” from memory, but to recreate a justified answer from the new evidence.
Practice 7: Condition
Question. Five is added only to the maximum. Is invariance guaranteed?
Explained answer. No. The transformation is not uniform.
Why this earns trust. The proof’s essential condition fails.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No. The transformation is not uniform.” from memory, but to recreate a justified answer from the new evidence.
Practice 8: Interior change
Question. An interior value changes but endpoints do not. Must the range change?
Explained answer. No. If the minimum and maximum stay fixed, the range stays fixed.
Why this earns trust. This is a separate endpoint fact, not uniform-shift reasoning.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “No. If the minimum and maximum stay fixed, the range stays fixed.” from memory, but to recreate a justified answer from the new evidence.
Practice 9: Scaling
Question. All values are multiplied by 3. What happens to the range?
Explained answer. It is multiplied by 3 for a positive factor.
Why this earns trust. Scaling is distinguished from translation.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “It is multiplied by 3 for a positive factor.” from memory, but to recreate a justified answer from the new evidence.
Practice 10: Proof
Question. Simplify (M+4)−(m+4).
Explained answer. M−m.
Why this earns trust. The +4 and −4 cancel.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “M−m.” from memory, but to recreate a justified answer from the new evidence.
Practice 11: Reverse
Question. Shifted endpoints are 12 and 31 after adding 7. Find the original range.
Explained answer. 31−12 = 19.
Why this earns trust. The range survives translation in either direction.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “31−12 = 19.” from memory, but to recreate a justified answer from the new evidence.
Practice 12: Routine
Question. State the safe check.
Explained answer. Identify endpoints, verify the same constant reaches every value, cancel it in the difference and distinguish scaling.
Why this earns trust. All conditions are explicit.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Identify endpoints, verify the same constant reaches every value, cancel it in the difference and distinguish scaling.” from memory, but to recreate a justified answer from the new evidence.
A four-session mastery cycle
Session 1: construct. Read the mechanism once, cover it and write a three-line model containing the trigger, the operation and the check. Add one example and one near-miss. Compare only after the model is complete. If the operation is right but the trigger is missing, the knowledge is unsafe because it may be applied in the wrong place.
Session 2: discriminate. Mix six items: two that clearly require this method, two that tempt the method but violate a condition, and two that require another skill. For every item, write “use” or “do not use” before answering. Accuracy in method selection matters more than speed at this stage.
Session 3: explain. Give a complete answer to an imagined classmate. Include the smallest sufficient evidence, the mechanism and the independent check. Remove any sentence that merely repeats the question or adds an unsupported claim. A concise causal explanation is stronger than a longer collection of familiar words.
Session 4: perform. Attempt a timed mixed set without notes. Mark uncertain decisions, but continue. During review, classify each error as trigger, evidence, operation, scope, unit or check. Reattempt one changed version after a delay. Mastery is shown when the learner chooses and adapts the method without needing the original example beside it.
Delayed transfer: prove that the learning lasts
Transfer test 1: Statistics family
After two days, investigate how mean, median, mode and range respond to adding the same constant.
Success criterion. Each claim is justified from a definition, not copied from the range rule.
Transfer test 2: Broken-condition set
Sort transformations into uniform translation, scaling and non-uniform adjustment.
Success criterion. The range rule is applied only to the first group.
Transfer test 3: Word-problem transfer
Solve calibration, elevation and temperature examples that use fixed offsets.
Success criterion. The learner identifies the common shift before calculating.
Transfer test 4: Blank-page proof
Write the algebraic proof and a number-line explanation from memory.
Success criterion. Both show why the constant cancels and name the every-value condition.
Examination control
1. Write range = maximum − minimum. The definition anchors the reasoning.
2. Check “every”. One unequal adjustment breaks the theorem.
3. Move both endpoints. Never update only the maximum.
4. Cancel before calculating. A symbolic line can replace a long rewritten list.
5. Classify the operation. Translation preserves range; scaling generally does not.
Under time pressure, compress the method to endpoints → same constant → cancel → range unchanged. The compressed routine is useful only after the full reasoning is understood. If it produces uncertainty, expand the step instead of guessing.
A compact error log
Classify errors as definition missed, minimum not shifted, constant added to range, negative shift mishandled, universal condition broken or scaling confused with translation.
Retest with a negative constant and a non-uniform near-miss. The learner should explain why one preserves range and the other is not guaranteed to do so.
Notes for parents and tutors
Use a ruler or number line to show the whole interval sliding without stretching.
Ask the learner to prove the result before accepting “because the rule says so”. The cancellation is short enough for Primary 6.
Mix questions where only an interior value changes so the learner distinguishes endpoint reasoning from uniform translation.
When speed matters, praise the decision not to rewrite a long list, provided the condition has been checked aloud.
Frequently asked questions
Does the range always stay unchanged when a number is added?
Only when the same number is added to every value.
What if the number is negative?
The rule still works; the set shifts left uniformly.
Why does the maximum not make the range larger?
The minimum increases by the same amount, so their difference stays fixed.
Do repeated values matter?
Not for range unless they affect which values are smallest and largest.
What happens to the mean?
It increases by the same constant, but that requires its own definition-based explanation.
What if all values are multiplied?
The range is scaled, not generally preserved.
Can an unequal change still leave the range unchanged?
It can happen if endpoints stay fixed, but that is not guaranteed by the uniform-shift theorem.
What is the one-line proof?
(M+c)−(m+c)=M−m.
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 where the learner must choose a method rather than imitate one.