Mathematics learner route · Vol 1715
Find the exact idea before you answer
When an ordered list has an odd count, removing one value below the median and one above it leaves the same central observation—provided the median itself is not removed.
This guide addresses a precise Primary 6 problem: learners recalculate mechanically without seeing the symmetry that preserves the middle position. The goal is not a slogan. It is a dependable method that survives new wording, altered numbers and unfamiliar diagrams.
Read the mechanism first, work through the cases with the tempting error visible, complete the independent practice without looking at the explanations, and return later for delayed transfer. The sequence moves from recognition to independent control.
Reading routes
1. The reader problem
The median depends on ordered position. In an odd list, the central value has the same number of observations on both sides.
If one smallest and one largest observation are removed, one item disappears from each side. The central observation stays in the list and still has equal counts around it.
This is not a rule about removing any two observations. Both removals must lie on opposite sides of the median. Removing two from the same side shifts the centre.
Repeated values do not invalidate the rule. What matters is observation position, not whether values are distinct.
The new count remains odd because odd minus two is odd. The median position number decreases by one, but the value occupying the centre can remain the same.
Recognising this symmetry saves time and strengthens proof skills. The answer should state the conditions, not merely announce that the median is unchanged.
2. The mechanism
Let the original count be 2k + 1. The median is at position k + 1, with k observations below its position and k above.
Remove one observation from the lower side and one from the upper side. The new count is 2k − 1, with k − 1 observations on each side of the same central observation.
The new median position is ((2k − 1) + 1) ÷ 2 = k. The old central value shifts from position k + 1 to position k because one earlier observation was removed.
The value remains, even though its position number changes. This distinction between position and value is essential.
If an endpoint equals the median value because of repetitions, removing it can still preserve the median. Position analysis is safer than saying “remove a value smaller and larger” when ties exist.
For a frequency table, subtract one from the lowest occupied category and one from the highest occupied category, then use cumulative frequency to confirm the centre.
The reverse operation also works: adding one observation at or below the lower side and one at or above the upper side can preserve the median when the original centre remains central.
Always check the exception: if the list has only one observation, there are no two endpoints to remove; if removals include the unique central observation, the rule does not apply.
In an odd ordered list, removing one observation from below the central position and one from above it preserves the median value because equal numbers remain on both sides.
3. Worked cases and transfer audits
WORKED CASE 1 OF 10
Five values
Set-up. 2, 4, 7, 10, 13 loses 2 and 13.
Tempting error. The median is recalculated from the removed numbers’ average.
Disciplined method. Keep the ordered centre and remove one item from each side.
Answer and check. The new list 4, 7, 10 still has median 7. One value remains on each side.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Five values, the decisive move was: Keep the ordered centre and remove one item from each side. The original check was: One value remains on each side. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “2, 4, 7, 10, 13 loses 2 and 13.” under Given in shortened form, place the reasoning under Process, and write only “The new list 4, 7, 10 still has median 7.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—The median is recalculated from the removed numbers’ average.—into an explicit repair rather than a vague warning.
WORKED CASE 2 OF 10
Seven values
Set-up. 1, 3, 5, 8, 11, 14, 20 loses 1 and 20.
Tempting error. The median is assumed to move because the count changes.
Disciplined method. Track the original central value 8.
Answer and check. 3, 5, 8, 11, 14 has median 8. Its position changes from fourth to third, not its value.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Seven values, the decisive move was: Track the original central value 8. The original check was: Its position changes from fourth to third, not its value. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “1, 3, 5, 8, 11, 14, 20 loses 1 and 20.” under Given in shortened form, place the reasoning under Process, and write only “3, 5, 8, 11, 14 has median 8.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—The median is assumed to move because the count changes.—into an explicit repair rather than a vague warning.
WORKED CASE 3 OF 10
Repeated values
Set-up. 2, 2, 4, 4, 4, 9, 12 loses one 2 and 12.
Tempting error. Duplicate 4s are collapsed.
Disciplined method. Keep each observation and count positions.
Answer and check. 2, 4, 4, 4, 9 has median 4. Two observations remain on each side of the third position.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Repeated values, the decisive move was: Keep each observation and count positions. The original check was: Two observations remain on each side of the third position. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “2, 2, 4, 4, 4, 9, 12 loses one 2 and 12.” under Given in shortened form, place the reasoning under Process, and write only “2, 4, 4, 4, 9 has median 4.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—Duplicate 4s are collapsed.—into an explicit repair rather than a vague warning.
WORKED CASE 4 OF 10
Two removals same side
Set-up. 2, 4, 7, 10, 13 loses 2 and 4.
Tempting error. Any two removals are said to preserve the median.
Disciplined method. Notice both removed observations were below the old median.
Answer and check. 7, 10, 13 has median 10, so the median changes. The symmetry condition failed.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Two removals same side, the decisive move was: Notice both removed observations were below the old median. The original check was: The symmetry condition failed. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “2, 4, 7, 10, 13 loses 2 and 4.” under Given in shortened form, place the reasoning under Process, and write only “7, 10, 13 has median 10, so the median changes.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—Any two removals are said to preserve the median.—into an explicit repair rather than a vague warning.
WORKED CASE 5 OF 10
Removing the centre
Set-up. 2, 4, 7, 10, 13 loses 7 and 13.
Tempting error. One low-looking and one high value are assumed enough.
Disciplined method. Check whether the old central observation survives.
Answer and check. 2, 4, 10 has median 4. Removing the median itself invalidates the rule.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Removing the centre, the decisive move was: Check whether the old central observation survives. The original check was: Removing the median itself invalidates the rule. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “2, 4, 7, 10, 13 loses 7 and 13.” under Given in shortened form, place the reasoning under Process, and write only “2, 4, 10 has median 4.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—One low-looking and one high value are assumed enough.—into an explicit repair rather than a vague warning.
WORKED CASE 6 OF 10
Negative values
Set-up. −9, −4, 0, 3, 8 loses −9 and 8.
Tempting error. Negative signs are thought to reverse the idea.
Disciplined method. Use ordered positions.
Answer and check. −4, 0, 3 has median 0. One observation remains on each side.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Negative values, the decisive move was: Use ordered positions. The original check was: One observation remains on each side. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “−9, −4, 0, 3, 8 loses −9 and 8.” under Given in shortened form, place the reasoning under Process, and write only “−4, 0, 3 has median 0.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—Negative signs are thought to reverse the idea.—into an explicit repair rather than a vague warning.
WORKED CASE 7 OF 10
Decimal data
Set-up. 1.2, 1.5, 1.8, 2.4, 3.1 loses 1.2 and 3.1.
Tempting error. Decimals trigger unnecessary averaging.
Disciplined method. Preserve the central observation.
Answer and check. Median remains 1.8. The list stays odd and ordered.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Decimal data, the decisive move was: Preserve the central observation. The original check was: The list stays odd and ordered. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “1.2, 1.5, 1.8, 2.4, 3.1 loses 1.2 and 3.1.” under Given in shortened form, place the reasoning under Process, and write only “Median remains 1.8.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—Decimals trigger unnecessary averaging.—into an explicit repair rather than a vague warning.
WORKED CASE 8 OF 10
Frequency table
Set-up. Values 1, 2, 3, 4, 5 have frequencies 1, 2, 3, 2, 1; one 1 and one 5 are removed.
Tempting error. Category labels are averaged.
Disciplined method. Update endpoint frequencies and locate the central observation.
Answer and check. The median remains 3. The distribution stays balanced around the centre.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Frequency table, the decisive move was: Update endpoint frequencies and locate the central observation. The original check was: The distribution stays balanced around the centre. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “Values 1, 2, 3, 4, 5 have frequencies 1, 2, 3, 2, 1; one 1 and one 5 are removed.” under Given in shortened form, place the reasoning under Process, and write only “The median remains 3.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—Category labels are averaged.—into an explicit repair rather than a vague warning.
WORKED CASE 9 OF 10
Unknown median
Set-up. An odd ordered list has median m; the smallest and largest observations are removed.
Tempting error. A numerical answer is demanded.
Disciplined method. Use the theorem symbolically.
Answer and check. The new median remains m. The conditions are sufficient without knowing m.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Unknown median, the decisive move was: Use the theorem symbolically. The original check was: The conditions are sufficient without knowing m. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “An odd ordered list has median m; the smallest and largest observations are removed.” under Given in shortened form, place the reasoning under Process, and write only “The new median remains m.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—A numerical answer is demanded.—into an explicit repair rather than a vague warning.
WORKED CASE 10 OF 10
Large list
Set-up. A 101-value ordered list loses its smallest and largest values.
Tempting error. All 99 remaining values are rewritten.
Disciplined method. Use count symmetry.
Answer and check. The original 51st value becomes the new 50th and remains the median. 49 values lie on each side.
Transfer audit. Imagine that one visible detail in this case changes while the governing relationship stays the same. Do not repeat the old answer automatically. First name the detail that carried the reasoning, then ask whether the changed version preserves it. In Large list, the decisive move was: Use count symmetry. The original check was: 49 values lie on each side. A new version is valid only when it still satisfies that check. This audit separates a method from a memorised surface pattern.
Precision drill. Write three lines headed Given, Process and Allowed conclusion. Put “A 101-value ordered list loses its smallest and largest values.” under Given in shortened form, place the reasoning under Process, and write only “The original 51st value becomes the new 50th and remains the median.” under Allowed conclusion. Cross out any extra claim that lacks a returning arrow to the given evidence. Then construct a near-miss answer containing exactly one wrong word, circle that word and replace it. This turns the common error—All 99 remaining values are rewritten.—into an explicit repair rather than a vague warning.
4. Common misconceptions and targeted repair
1. Any two removals work
Location is ignored. The removals must be on opposite sides of the centre.
Targeted repair. Count remaining observations on both sides.
2. Same position number
The old median position is reused. The position number decreases by one.
Targeted repair. Calculate both positions.
3. Duplicates disappear
Equal observations are merged. Frequency creates multiple positions.
Targeted repair. Expand a small list.
4. Mean reasoning
Removed endpoints are averaged. Median is positional.
Targeted repair. Ask whether totals matter.
5. The rule needs positive values
Negative or decimal data are treated differently. Ordering works for all comparable numerical values.
Targeted repair. Use a mixed-sign example.
6. Removing the median is harmless
Only final odd count is checked. The original central observation must remain.
Targeted repair. Mark it before deleting anything.
5. Graded independent practice
Cover the explanations. A choice without a reason is not yet a dependable method. Show the condition, operation or observation that makes your answer legal.
Practice 1: Simple
Question. Remove endpoints from 3, 6, 9, 12, 15. New median?
Explained answer. 9.
Why it matters. The central value remains with one observation each side.
Practice 2: Nine values
Question. What happens when the smallest and largest of nine ordered values are removed?
Explained answer. The fifth value becomes the fourth of seven and remains the median.
Why it matters. One earlier and one later observation disappear.
Practice 3: Same-side test
Question. Remove the two smallest from 1, 4, 6, 9, 12.
Explained answer. New list 6, 9, 12 has median 9, not 6.
Why it matters. Both removals came from one side.
Practice 4: Repeated centre
Question. 1, 2, 2, 2, 7 loses 1 and 7.
Explained answer. 2, 2, 2 has median 2.
Why it matters. Duplicates still occupy positions.
Practice 5: Position
Question. An 11-value list loses endpoints. What is the new median position?
Explained answer. Fifth position in the 9-value list.
Why it matters. (9 + 1) ÷ 2 = 5.
Practice 6: Old position
Question. Which old observation is that?
Explained answer. The original sixth observation.
Why it matters. Removing the first shifts it one place earlier.
Practice 7: Decimals
Question. 0.2, 0.4, 0.9, 1.3, 2.0 loses endpoints.
Explained answer. Median remains 0.9.
Why it matters. Number type does not affect order symmetry.
Practice 8: Symbolic
Question. Median is 18 in a 21-value ordered list. Endpoints removed. New median?
Explained answer. 18.
Why it matters. The central observation survives.
Practice 9: Counterexample
Question. Give removals that change 2, 5, 8, 10, 14.
Explained answer. Remove 2 and 5; median becomes 10.
Why it matters. The example shows the opposite-side condition is necessary.
Practice 10: Frequency
Question. Why update frequencies before locating the median?
Explained answer. Because each removal changes cumulative positions.
Why it matters. Category labels alone do not show observation count.
Practice 11: Add pair
Question. Can adding one small and one large value preserve the median?
Explained answer. Yes, if the old centre remains central with one new observation on each side.
Why it matters. The same symmetry works in reverse.
Practice 12: Proof
Question. Explain with counts.
Explained answer. k observations on each side become k−1 on each side.
Why it matters. A general count proves the rule.
6. Delayed transfer
Leave these until another study session. The changed surface details reveal whether you can rebuild the mechanism without immediate cues.
Transfer 1: Trimmed data
A teacher removes one extreme at each end before reporting the median. What changes?
Resolution. The median stays if the original count is odd and the central observation remains.
Transfer 2: Tied endpoints
The smallest value equals another observation. Does removal still work?
Resolution. Yes; remove one observation at the low end and track positions.
Transfer 3: Three pairs
Remove three observations from each end.
Resolution. The median remains while the original centre survives and equal numbers are removed from both sides.
Transfer 4: Even original count
Does the same statement automatically hold for an even list?
Resolution. No. The median is an average of two central values, and endpoint removal must be checked separately.
Transfer 5: Graphical dot plot
How can the rule be seen on a dot plot?
Resolution. Cross off one outer dot on each side; the middle dot remains balanced.
7. Examination control
Mark the median observation before deleting anything.
- Order the data.
- Confirm the original count is odd.
- Mark the central observation.
- Identify each removal’s side.
- Recount the remaining values.
- Verify equal counts around the marked value.
A one-sentence proof using equal counts is stronger than a repeated full calculation.
In unfamiliar tables, update frequencies but keep category values separate from cumulative positions.
If a question removes more than one pair, apply the symmetry repeatedly until the centre would be touched.
8. Deepening laboratory
Use this laboratory to test the rule rather than admire it. Change one condition at a time, predict the effect before calculating or rereading, and compare the prediction with the evidence. If a prediction fails, identify whether the definition, the relationship or the data was misunderstood.
A. Algebra proof
Let n = 2k + 1 and derive the new position.
Model analysis. After two removals, n′ = 2k − 1 and the centre is position k; the old centre shifts there.
B. Minimal counterexample
Find the smallest odd list showing same-side removal failure.
Model analysis. A five-value list is sufficient because removing two from one side leaves three values with a shifted centre.
C. Ties
Create a list where the removed endpoint equals the median value.
Model analysis. Position reasoning still decides whether the median value remains; equality of values can make several observations share the median number.
D. Repeated trimming
Trim a 15-value list symmetrically until five remain.
Model analysis. Five pairs can be removed while preserving the original eighth value as centre.
E. Reverse construction
Start with five values and add symmetric outer observations.
Model analysis. The old third value remains central in the seven-value list.
F. Representation
Show the theorem with numbered cards.
Model analysis. Pair cards from the ends; removing a complete outer pair leaves the unpaired centre.
9. Parent and tutor guidance
Use numbered cards and physically remove paired endpoints so position shifts are visible.
Ask for a counterexample whenever the learner says “removing two values keeps the median.”
Reward condition statements: odd count, opposite sides and centre retained.
10. Frequently asked questions
Does the median position stay the same?
No. Its position number decreases by one, but the value can stay the same.
Do values need to be distinct?
No. Observations, including duplicates, occupy positions.
Can more than one pair be removed?
Yes, while removals stay balanced and the centre survives.
What about even lists?
Check separately; the median uses two middle values.
Why is this useful?
It saves calculation and reveals how positional statistics work.
Synthesis workshop: compare cases without losing the rule
Synthesis is harder than solving one familiar example because two cases may share vocabulary while requiring different conclusions. Work through the comparisons below by naming the invariant condition first, then the changed detail. A correct answer must explain why the rule survives or why it stops.
Comparison 1: Five values and Negative values
Case A begins with 2, 4, 7, 10, 13 loses 2 and 13. Case B begins with −9, −4, 0, 3, 8 loses −9 and 8. Do not merge their visible details. In A, the disciplined move is: Keep the ordered centre and remove one item from each side. In B, the disciplined move is: Use ordered positions. The shared skill is to identify the exact relationship before interpreting the result. The difference is that each case supplies a different trigger, data arrangement or evidential boundary. State that difference in one sentence before giving either answer.
Now hide the original answers and rebuild them. For A, the permitted conclusion is: The new list 4, 7, 10 still has median 7. For B, it is: −4, 0, 3 has median 0. Check A with “One value remains on each side.” and B with “One observation remains on each side.” If an explanation passes only one check, it cannot be copied into the other case. Finally, write a third example that preserves A’s governing condition but borrows B’s surface setting. This controlled remix is a strong test of transfer because it separates the rule from the nouns that first carried it.
Comparison 2: Seven values and Decimal data
Case A begins with 1, 3, 5, 8, 11, 14, 20 loses 1 and 20. Case B begins with 1.2, 1.5, 1.8, 2.4, 3.1 loses 1.2 and 3.1. Do not merge their visible details. In A, the disciplined move is: Track the original central value 8. In B, the disciplined move is: Preserve the central observation. The shared skill is to identify the exact relationship before interpreting the result. The difference is that each case supplies a different trigger, data arrangement or evidential boundary. State that difference in one sentence before giving either answer.
Now hide the original answers and rebuild them. For A, the permitted conclusion is: 3, 5, 8, 11, 14 has median 8. For B, it is: Median remains 1.8. Check A with “Its position changes from fourth to third, not its value.” and B with “The list stays odd and ordered.” If an explanation passes only one check, it cannot be copied into the other case. Finally, write a third example that preserves A’s governing condition but borrows B’s surface setting. This controlled remix is a strong test of transfer because it separates the rule from the nouns that first carried it.
Comparison 3: Repeated values and Frequency table
Case A begins with 2, 2, 4, 4, 4, 9, 12 loses one 2 and 12. Case B begins with Values 1, 2, 3, 4, 5 have frequencies 1, 2, 3, 2, 1; one 1 and one 5 are removed. Do not merge their visible details. In A, the disciplined move is: Keep each observation and count positions. In B, the disciplined move is: Update endpoint frequencies and locate the central observation. The shared skill is to identify the exact relationship before interpreting the result. The difference is that each case supplies a different trigger, data arrangement or evidential boundary. State that difference in one sentence before giving either answer.
Now hide the original answers and rebuild them. For A, the permitted conclusion is: 2, 4, 4, 4, 9 has median 4. For B, it is: The median remains 3. Check A with “Two observations remain on each side of the third position.” and B with “The distribution stays balanced around the centre.” If an explanation passes only one check, it cannot be copied into the other case. Finally, write a third example that preserves A’s governing condition but borrows B’s surface setting. This controlled remix is a strong test of transfer because it separates the rule from the nouns that first carried it.
Comparison 4: Two removals same side and Unknown median
Case A begins with 2, 4, 7, 10, 13 loses 2 and 4. Case B begins with An odd ordered list has median m; the smallest and largest observations are removed. Do not merge their visible details. In A, the disciplined move is: Notice both removed observations were below the old median. In B, the disciplined move is: Use the theorem symbolically. The shared skill is to identify the exact relationship before interpreting the result. The difference is that each case supplies a different trigger, data arrangement or evidential boundary. State that difference in one sentence before giving either answer.
Now hide the original answers and rebuild them. For A, the permitted conclusion is: 7, 10, 13 has median 10, so the median changes. For B, it is: The new median remains m. Check A with “The symmetry condition failed.” and B with “The conditions are sufficient without knowing m.” If an explanation passes only one check, it cannot be copied into the other case. Finally, write a third example that preserves A’s governing condition but borrows B’s surface setting. This controlled remix is a strong test of transfer because it separates the rule from the nouns that first carried it.
Comparison 5: Removing the centre and Large list
Case A begins with 2, 4, 7, 10, 13 loses 7 and 13. Case B begins with A 101-value ordered list loses its smallest and largest values. Do not merge their visible details. In A, the disciplined move is: Check whether the old central observation survives. In B, the disciplined move is: Use count symmetry. The shared skill is to identify the exact relationship before interpreting the result. The difference is that each case supplies a different trigger, data arrangement or evidential boundary. State that difference in one sentence before giving either answer.
Now hide the original answers and rebuild them. For A, the permitted conclusion is: 2, 4, 10 has median 4. For B, it is: The original 51st value becomes the new 50th and remains the median. Check A with “Removing the median itself invalidates the rule.” and B with “49 values lie on each side.” If an explanation passes only one check, it cannot be copied into the other case. Finally, write a third example that preserves A’s governing condition but borrows B’s surface setting. This controlled remix is a strong test of transfer because it separates the rule from the nouns that first carried it.
Final self-audit
Before declaring mastery, choose one misconception and design a multiple-choice distractor that expresses it cleanly. Then write the discriminating word, number or observation that defeats the distractor. Next, solve one practice item by a second method or independent check. The purpose is not extra length; it is to make the answer resistant to the exact near-miss that examination options are designed to exploit.
11. Next routes
Return to the complete Learner’s Guide volume directory, revisit the original PSLE Learning Guide foundation, or continue through the Mathematics hub.
The lasting goal is independent control: identify the exact relationship, test a tempting alternative, and write only the conclusion the words, numbers or observations support.