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.
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This volume solves one precise learner problem: the learner recalculates a transformed data set item by item, assumes every statistic must change, or confuses the unchanged range with an unchanged mean, maximum or minimum. 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 the greatest value minus the least value. It measures the distance between the two extremes, not the location of the whole set on the number line.
Suppose the values are 3, 7, 8 and 12. Their range is 12−3=9. Add 5 to every value and the new set is 8, 12, 13 and 17.
The new maximum is 17 and the new minimum is 8. The range is 17−8=9, exactly as before.
Both extremes moved five units in the same direction. Their positions changed, but the distance between them did not.
Algebra makes the invariant visible. If the old maximum is M and the old minimum is m, adding c to every value gives maximum M+c and minimum m+c.
The new range is (M+c)−(m+c)=M−m because +c and −c cancel. The argument works for positive, zero or negative c.
This does not mean the data set is unchanged. Every value, the total, the mean, the maximum and the minimum may move, even while the range remains fixed.
The condition “every value” is essential. If only one value changes, especially an extreme, the range may increase, decrease or stay the same depending on the new order.
Subtracting the same number from every value is the same transformation with a negative c, so the range also stays unchanged.
The safe routine is identify extremes, verify a uniform translation, compare the two movements and subtract symbolically before doing unnecessary arithmetic.
Core decision: When the same number is added to or subtracted from every data value, the maximum and minimum shift by the same amount. Their difference, and therefore the range, stays unchanged. The conclusion fails if the transformation is not applied uniformly.
Range is a translation-invariant measure of spread. A common additive shift changes absolute position but preserves every pairwise difference, including the largest-minus-smallest distance that defines the range.
Mechanism: why the rule works
1. Start from the definition
Write range = maximum − minimum before transforming anything.
This blocks the common error of treating range as the number of values, the number of gaps or the maximum alone.
Diagnostic check. Which two values determine the range?
2. Verify uniform addition
Check that exactly the same constant is added to every data value.
If different amounts are added, order and spacing can change, so the invariant no longer follows.
Diagnostic check. Is there one common c for all entries?
3. Track the extremes
Adding one constant preserves order: the old maximum remains a maximum after the shift, and the old minimum remains a minimum.
Every comparison difference stays the same because both values receive c.
Diagnostic check. Which original values become the new extremes?
4. Subtract with brackets
Write (maximum+c)−(minimum+c) and remove brackets carefully.
The second +c is being subtracted, so it becomes −c and cancels the first +c.
Diagnostic check. Have you distributed the subtraction sign correctly?
5. Separate position from spread
Maximum, minimum and mean describe location; range describes one aspect of spread.
A data set can move along the number line without stretching or shrinking.
Diagnostic check. What changed: position, spread or both?
6. Extend to subtraction
Subtracting c from every value is adding −c. Both extremes move left by c and retain the same distance.
The same proof covers temperature offsets, score adjustments and translated coordinates when the data role is legitimate.
Diagnostic check. Can the change be represented as one common additive constant?
7. Test the boundary condition
If only some values change, recompute the extremes rather than invoking the rule.
An interior value may move without changing range, while a changed maximum or minimum can alter it.
Diagnostic check. Was every value transformed, or only selected values?
8. Contrast multiplication
Multiplying every value by a positive factor k multiplies the range by k; it does not usually preserve it.
Addition translates a set, while multiplication stretches or compresses distances.
Diagnostic check. Is the operation additive or multiplicative?
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. Find the range of 3, 7, 8, 12, then add 5 to every value.
Reasoning. Old range is 12−3=9. New extremes are 17 and 8.
Answer. New range is 17−8=9.
Independent check. The maximum and minimum each rose by 5.
Contrast-and-transfer pass. Keep the governing principle from “Positive shift”, but change one surface detail in the task: Find the range of 3, 7, 8, 12, then add 5 to every value. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—New range is 17−8=9.—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 values
Task. The data are −4, 1 and 6. Add 10 to every value.
Reasoning. Old range is 6−(−4)=10. New data are 6, 11 and 16.
Answer. New range is 16−6=10.
Independent check. Crossing zero does not alter the proof.
Contrast-and-transfer pass. Keep the governing principle from “Negative values”, but change one surface detail in the task: The data are −4, 1 and 6. Add 10 to every value. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—New range is 16−6=10.—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: Uniform subtraction
Task. Subtract 7 from every value in 9, 14, 20.
Reasoning. Old range is 20−9=11. New extremes are 13 and 2.
Answer. New range remains 13−2=11.
Independent check. Subtraction is a uniform translation.
Contrast-and-transfer pass. Keep the governing principle from “Uniform subtraction”, but change one surface detail in the task: Subtract 7 from every value in 9, 14, 20. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—New range remains 13−2=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 4: Unknown constant
Task. A data set has range 18. The same unknown number is added to every value.
Reasoning. Call the shift c. New range is (M+c)−(m+c)=M−m.
Answer. The new range is 18 without knowing c.
Independent check. The invariant removes unnecessary information.
Contrast-and-transfer pass. Keep the governing principle from “Unknown constant”, but change one surface detail in the task: A data set has range 18. The same unknown number is added to every value. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The new range is 18 without knowing c.—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: Mean contrast
Task. The set 2, 5, 8 receives +4.
Reasoning. The range remains 8−2=6, but the mean rises from 5 to 9.
Answer. Range unchanged does not mean mean unchanged.
Independent check. Different statistics respond differently to the same transformation.
Contrast-and-transfer pass. Keep the governing principle from “Mean contrast”, but change one surface detail in the task: The set 2, 5, 8 receives +4. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Range unchanged does not mean mean unchanged.—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: One value only
Task. Change 2, 5, 8 by adding 4 only to the maximum.
Reasoning. The new set is 2, 5, 12. This is not a uniform shift.
Answer. Range changes from 6 to 10.
Independent check. The phrase every value is the controlling condition.
Contrast-and-transfer pass. Keep the governing principle from “One value only”, but change one surface detail in the task: Change 2, 5, 8 by adding 4 only to the maximum. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Range changes from 6 to 10.—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: Interior change
Task. Change only the middle value in 2, 5, 8 from 5 to 7.
Reasoning. The extremes remain 2 and 8.
Answer. Range stays 6 in this case, but not because of the uniform-shift rule.
Independent check. A different reason can produce the same numerical outcome.
Contrast-and-transfer pass. Keep the governing principle from “Interior change”, but change one surface detail in the task: Change only the middle value in 2, 5, 8 from 5 to 7. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Range stays 6 in this case, but not because of the uniform-shift rule.—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: Common bonus
Task. The values 11, 16, 19 and 23 each receive a bonus of 4.
Reasoning. The old range is 23−11=12. The new extremes are 27 and 15 because every value receives the same addition.
Answer. The new range is 27−15=12.
Independent check. The bonus changes each level but preserves every pairwise difference.
Contrast-and-transfer pass. Keep the governing principle from “Common bonus”, but change one surface detail in the task: The values 11, 16, 19 and 23 each receive a bonus of 4. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The new range is 27−15=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 9: Coordinate translation
Task. Points have vertical coordinates −2, 3 and 9. The whole figure moves up 6 units.
Reasoning. The vertical coordinates become 4, 9 and 15. The greatest-minus-least vertical separation remains 11.
Answer. The coordinate range stays 15−4=11.
Independent check. A translation moves position without stretching the figure.
Contrast-and-transfer pass. Keep the governing principle from “Coordinate translation”, but change one surface detail in the task: Points have vertical coordinates −2, 3 and 9. The whole figure moves up 6 units. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The coordinate range stays 15−4=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 10: Multiplicative contrast
Task. Double every value in 2, 5, 8.
Reasoning. Old range is 6. New extremes are 4 and 16.
Answer. New range is 12, twice the old range.
Independent check. Multiplication changes distances by the scale factor.
Contrast-and-transfer pass. Keep the governing principle from “Multiplicative contrast”, but change one surface detail in the task: Double every value in 2, 5, 8. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—New range is 12, twice the old 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.
The method lab: change one condition at a time
Place five points on a number line, translate all of them four units right and compare every pairwise gap. Explain why the maximum-to-minimum gap is preserved.
Create three transformations: uniform addition, non-uniform addition and multiplication. Predict the range effect before calculating.
Use symbols M, m and c to rebuild the cancellation proof from memory, paying attention to the subtraction bracket.
Design two non-uniform changes that nevertheless leave range unchanged. Explain why they are examples of a different mechanism rather than proof of the uniform-shift rule.
Common misconceptions and targeted repairs
Misconception 1: Every statistic stays unchanged
What the learner may think. The learner treats one invariant as a claim about the whole data set.
Why it fails. Maximum, minimum, total and mean change under a common addition.
Targeted repair. Name exactly which statistic is invariant.
Misconception 2: Range is the maximum
What the learner may think. The learner reports the highest value after shifting.
Why it fails. Range is a difference between two extremes.
Targeted repair. Write maximum minus minimum first.
Misconception 3: Adding increases range
What the learner may think. The learner assumes larger values must create larger spread.
Why it fails. Common addition moves all values without changing gaps.
Targeted repair. Draw a translated number line.
Misconception 4: The constant must be positive
What the learner may think. The learner distrusts subtraction or negative shifts.
Why it fails. The algebra works for any real additive constant.
Targeted repair. Represent subtraction as addition of a negative.
Misconception 5: Changing one value uses the same rule
What the learner may think. The learner ignores the word every.
Why it fails. Uniformity is the condition that makes cancellation valid.
Targeted repair. Mark which entries received the change.
Misconception 6: Multiplication is another shift
What the learner may think. The learner expects doubling to preserve range.
Why it fails. Multiplication scales gaps instead of translating them.
Targeted repair. Compare additive and multiplicative transformations.
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: Basic range
Question. The values 4, 9 and 15 each receive +6. What is the new range?
Explained answer. The old range is 15−4=11, so the new range is 11.
Why this earns trust. A uniform addition preserves range.
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 old range is 15−4=11, so the new range is 11.” from memory, but to recreate a justified answer from the new evidence.
Practice 2: Subtraction
Question. The values 12, 18 and 25 each receive −5. Find the new range.
Explained answer. 25−12=13, so the new range is 13.
Why this earns trust. Both extremes fall by 5.
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 “25−12=13, so the new range is 13.” from memory, but to recreate a justified answer from the new evidence.
Practice 3: Unknown shift
Question. A set has range 27. Every value receives the same additive adjustment.
Explained answer. The range remains 27.
Why this earns trust. The common constant cancels.
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 27.” from memory, but to recreate a justified answer from the new evidence.
Practice 4: Negative data
Question. For −8, −2 and 3, add 20 to every value.
Explained answer. The new range remains 3−(−8)=11.
Why this earns trust. Sign changes do not affect the invariant.
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 new range remains 3−(−8)=11.” from memory, but to recreate a justified answer from the new evidence.
Practice 5: Condition check
Question. Only the smallest value receives +3. Can the uniform rule be used?
Explained answer. No. Recompute the ordered values and extremes.
Why this earns trust. Not every value received the same change.
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. Recompute the ordered values and extremes.” from memory, but to recreate a justified answer from the new evidence.
Practice 6: Mean comparison
Question. Every score receives +2. What happens to mean and range?
Explained answer. The mean rises by 2, while the range stays unchanged.
Why this earns trust. Location and spread are separated.
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 mean rises by 2, while the range stays unchanged.” from memory, but to recreate a justified answer from the new evidence.
Practice 7: Maximum comparison
Question. Every value receives −9. What happens to the maximum?
Explained answer. The maximum decreases by 9.
Why this earns trust. The maximum moves even though range does not.
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 maximum decreases by 9.” from memory, but to recreate a justified answer from the new evidence.
Practice 8: Multiplication
Question. Every value is tripled. What happens to a range of 7?
Explained answer. For positive factor 3, the range becomes 21.
Why this earns trust. Distances are scaled, not translated.
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 “For positive factor 3, the range becomes 21.” from memory, but to recreate a justified answer from the new evidence.
Practice 9: Algebra
Question. Simplify (M+c)−(m+c).
Explained answer. M−m.
Why this earns trust. The c terms cancel after the subtraction bracket is opened.
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 10: Routine
Question. State the decision sequence.
Explained answer. Define range, verify one common additive shift, track extremes, subtract and compare statistics.
Why this earns trust. The invariant is used only after its condition is checked.
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 “Define range, verify one common additive shift, track extremes, subtract and compare statistics.” 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: Number-line translation
After two days, translate a mixed-sign data set without recalculating every difference.
Success criterion. The learner explains preserved spacing visually and algebraically.
Transfer test 2: Statistic sorting
Classify mean, maximum, minimum and range under a common +c shift.
Success criterion. Each statistic’s change is stated separately.
Transfer test 3: Condition trap
Mix uniform and non-uniform transformations.
Success criterion. The learner refuses the shortcut when every value is not changed equally.
Transfer test 4: Blank-page proof
Write (M+c)−(m+c)=M−m from memory.
Success criterion. The learner explains cancellation rather than quoting a slogan.
Examination control
1. Write the range definition. Maximum minus minimum controls the reasoning.
2. Check “every” and “same”. Uniformity activates the invariant.
3. Track both extremes. They move by equal amounts.
4. Use brackets. Show why the common shift cancels.
5. Separate statistics. Do not claim the whole data set is unchanged.
Under time pressure, compress the method to range = max−min → common +c → (M+c)−(m+c) → M−m. 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
Label errors as range misdefined, uniformity missed, extremes misidentified, subtraction sign mishandled, mean confused with range or multiplication treated as translation.
Retest with one non-uniform change beside one uniform shift.
Notes for parents and tutors
Use a ruler or number line to slide several marked values together.
Ask what moved and what distance stayed fixed.
Contrast range with mean using a small data set.
Keep examples numerical and avoid implying that real school scores are adjusted.
Frequently asked questions
Why does the range stay the same?
The maximum and minimum move by the same amount, so their difference is unchanged.
Does the mean stay the same?
No. It changes by the common added amount.
What about subtraction?
Subtracting the same number from every value also preserves range.
Can the values be negative?
Yes. The algebra and spacing argument still work.
What if only one value changes?
Recompute the extremes; the shortcut does not apply.
What if every value is doubled?
The range doubles for a positive factor of 2.
Does unchanged range mean unchanged data?
No. It means only the maximum-to-minimum distance is unchanged.
What is the fastest proof?
Write (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.