Welcome to the Records Room, where seized numbers get filed so their shape shows. Three lessons, six cases. Green drawer = the Three Knocks nudge. Purple = a smaller case to warm up on. Gold = the full solution, once you have given it a real go.
Filing the evidence
The pouch ledger (how evidence gets filed)
The agency confiscated 21 coin pouches and weighed them in grams. The evidence book, in the order they were logged:
23, 41, 17, 35, 28, 44, 31, 19, 27, 38, 22, 45, 30, 26, 33, 12, 29, 36, 24, 40, 31
A list like that hides everything. A stem-and-leaf plot files it: the stem is the drawer label (the tens), each leaf is one value filed in that drawer (its ones digit). Nothing gets averaged away and nothing gets binned into bars. Every pouch keeps its identity, which means the leaf count always equals n. That one fact is your self-check for the whole unit.
The two-pass build
- Rough copy. Write the stems 1 to 4, then file each value's leaf in raw order, ticking values off the list as you go. Rough means unsorted, and that is fine.
- Fair copy. Rewrite the plot with each drawer's leaves in order, smallest first. Leaves in order, always.
- Count the leaves. 3 + 7 + 7 + 4 = 21 = n. Only now do you answer anything.
Sorted, the evidence reads: 12, 17, 19, 22, 23, 24, 26, 27, 28, 29, 30, 31, 31, 33, 35, 36, 38, 40, 41, 44, 45. Note the repeat 31: two pouches, two leaves. The plot is the dataset, not a set.
Reading the ledger
- Median by position, never by eyeballing. n = 21, so the median sits at position (21 + 1) ÷ 2 = 11th. Count leaves cumulatively: 3 through stem 1, 10 through stem 2, so the 11th value is the first leaf of stem 3. Median = 30 g. (Fallback method: cross values off from both ends until one is left.)
- Range = biggest VALUE minus smallest VALUE. 45 − 12 = 33 g.
- Shape, for later. One hump through the 20s and 30s, roughly symmetric, no stragglers. File that thought: Lesson 3 comes back for it.
The overstuffed drawer (split stems, demo only)
15 trainee goblins paced out the distance from the lamp post to the hideout. Paces, as logged: 50, 43, 57, 41, 51, 46, 53, 44, 58, 47, 51, 43, 55, 52, 48.
Sorted: 41, 43, 43, 44, 46, 47, 48, 50, 51, 51, 52, 53, 55, 57, 58. Plain stems give exactly two drawers, both overstuffed, and zero shape. So split each drawer into a low half (leaves 0 to 4) and a high half (leaves 5 to 9):
Leaf count: 4 + 3 + 5 + 3 = 15 = n ✓. Median = 8th value = 50 paces; range = 58 − 41 = 17 paces. Now a shape appears where the plain plot showed a wall.
Reading only, no building. The first move on any plot is always the same: count the leaves.
👀 Stuck? Knock three times
Knock, knock, knock: What am I doing? Why? Is it helping?
The number of leaves IS the number of cases. Count them before anything else. Then (n + 1) ÷ 2 tells you which position the median lives at, and you count to that position leaf by leaf.
🔍 Show me a smaller case
Warm up on a tiny ledger:
Values: 12, 15, 20, 23, 27. That is n = 5 leaves. Median position (5 + 1) ÷ 2 = 3rd, so median = 20. Range = 27 − 12 = 15. Now the real ledger, exactly the same moves.
✅ Show the full solution
Decode. The key says 1 | 3 means 13 clues, so stems are tens and each leaf is the ones digit of one closed case. Reading the drawers out gives: 6, 8, 11, 13, 14, 14, 17, 19, 20, 22, 25, 25, 28, 31, 34.
- Count the leaves first: 2 + 6 + 5 + 2 = n = 15 cases.
- Smallest = first leaf of the top drawer: 0 | 6 = 6 clues. Largest = last leaf of the bottom drawer: 3 | 4 = 34 clues.
- Median position = (15 + 1) ÷ 2 = 8th. Cumulative count: 2 through stem 0, then 2 + 6 = 8 through stem 1. The 8th value is the last leaf of stem 1: median = 19 clues.
- Range = 34 − 6 = 28 clues.
- 20 clues or more = everything in stems 2 and 3: 5 + 2 = 7 cases.
- What does one leaf stand for, in your own words?
- How do you find n from a plot when there is no list in sight?
- Which leaf almost tricked you?
Two passes: a rough plot in the order given, then the fair copy with every drawer sorted. Tick each value off as you file it.
👀 Stuck? Knock three times
Knock, knock, knock: What am I doing? Why? Is it helping?
Rough plot first, in the order given. Fair copy second, leaves sorted. Tick each raw value as you file it, then count the leaves: if you do not count 20, an informant went missing, and no question gets answered until they are found.
🔍 Show me a smaller case
Warm up: file 14, 8, 21, 17, 9, 12.
Sorted: 8, 9, 12, 14, 17, 21.
n = 6, an even count, so the median is the average of the 3rd and 4th values: (12 + 14) ÷ 2 = 13. Range = 21 − 8 = 13. Same moves on the real case.
✅ Show the full solution
Decode. Stems are tens (1 to 4), leaves are ones, unit is seconds. Sorted fair copy: 12, 14, 16, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 38, 41, 44.
- Leaf count: 4 + 8 + 6 + 2 = 20 = n. Nothing went missing, so now we may answer.
- Even n, so the median averages the 10th and 11th values. Cumulative: 4 through stem 1, 12 through stem 2, so the 10th and 11th are the 6th and 7th leaves of stem 2: 27 and 28.
- Median = (27 + 28) ÷ 2 = 27.5 seconds.
- Range = 44 − 12 = 32 seconds.
- What is your check that no evidence went missing?
- Why can a median be a number nobody actually recorded?
- What would you tell a rookie about the fair copy?
Spine to spine
Two shifts, one cabinet
One spine, two stories. When two datasets share the same stems, a back-to-back plot lets you compare them at a glance. The day shift and the night shift each cracked the daily cipher 12 times last month. Minutes taken, in the order logged:
Day: 22, 35, 28, 41, 19, 33, 27, 38, 24, 30, 26, 45
Night: 31, 24, 39, 46, 28, 35, 42, 33, 37, 29, 44, 36
Sorted Day: 19, 22, 24, 26, 27, 28, 30, 33, 35, 38, 41, 45. Sorted Night: 24, 28, 29, 31, 33, 35, 36, 37, 39, 42, 44, 46. Both shifts logged exactly 12 solves, and every paired ledger in this unit keeps the group sizes equal, so centres and spreads compare straight across.
The left side files OUTWARD
The left side is a mirror image in layout only. On Day's stem-2 row the printed digits are 8 7 6 4 2, and the 2 hugging the spine is the SMALLEST value, 22. Put your finger on the spine and read outward: 22, 24, 26, 27, 28, growing as you go, exactly like the right side. Each side also has its own key direction: Day 8 | 2 means 28, but 2 | 4 Night means 24.
Each side answers on its own
- Counts: Day 1 + 5 + 4 + 2 = 12 ✓. Night 3 + 6 + 3 = 12 ✓.
- Day median: n = 12, so average the 6th and 7th values. Cumulative: 1 through stem 1, 6 through stem 2, so the 6th is the last leaf of stem 2 (28) and the 7th is the first leaf of stem 3 (30): (28 + 30) ÷ 2 = 29 minutes. Day range = 45 − 19 = 26.
- Night median: cumulative 3 through stem 2, so the 6th and 7th are the 3rd and 4th leaves of stem 3: 35 and 36. (35 + 36) ÷ 2 = 35.5 minutes. Night range = 46 − 24 = 22.
One comparison sentence, modelled once (full reports are Lesson 3's job): "A typical day-shift solve is about six and a half minutes faster (median 29 against 35.5)."
Reading only. Left side: start at the spine and read outward. Keep Grub's numbers and Snag's numbers in separate labelled workings.
👀 Stuck? Knock three times
Knock, knock, knock: What am I doing? Why? Is it helping?
Left side rule: start at the spine and read outward, the values grow. Then treat each stall as its own single plot: count its leaves, then position its median with (n + 1) ÷ 2.
🔍 Show me a smaller case
Warm up on a tiny mirror ledger. Trace each left leaf with your finger, spine first:
Left values, reading outward from the spine: 11, 13, 25. Median = 13, range = 25 − 11 = 14. Right values: 12, 14, 20. Median = 14, range = 20 − 12 = 8. Same moves on the stalls.
✅ Show the full solution
Decode. Two stalls share one spine. Grub files leftward (smallest at the spine), Snag files rightward. Read out, in separate workings: Grub 8, 12, 15, 17, 21, 23, 24, 26, 30, 32, 35. Snag 14, 18, 22, 25, 26, 27, 29, 31, 33, 36, 41.
- Leaf counts: Grub 1 + 3 + 4 + 3 = 11, Snag 2 + 5 + 3 + 1 = 11. Equal groups, fair fight.
- Median position for both: (11 + 1) ÷ 2 = 6th.
- Grub: cumulative 1 through stem 0, 4 through stem 1, so the 6th is the 2nd leaf of stem 2: median = 23 apples.
- Snag: cumulative 2 through stem 1, so the 6th is the 4th leaf of stem 2: median = 27 apples.
- Ranges: Grub 35 − 8 = 27. Snag 41 − 14 = 27.
- Typical day: Snag sells more, by 27 − 23 = 4 apples.
- Say the left-side rule in your own words.
- Two stalls, same range, different medians: what does each number tell you that the other cannot?
A spicy one: build, compute, then argue. Keep the two trainees' workings in separate labelled columns from the very first tick.
👀 Stuck? Knock three times
Knock, knock, knock: What am I doing? Why? Is it helping?
Build each side as its own single plot first, then marry them on one spine. Fern's leaves file rightward as normal; Nix's file leftward, smallest hugging the spine.
🔍 Show me a smaller case
Warm up: A scored 12, 9, 15, 21 and B scored 14, 18, 11, 23.
Sorted A: 9, 12, 15, 21. Median = (12 + 15) ÷ 2 = 13.5, range = 21 − 9 = 12. Sorted B: 11, 14, 18, 23. Median = (14 + 18) ÷ 2 = 16, range = 23 − 11 = 12.
Now the real rivals, exactly the same build.
✅ Show the full solution
Decode. Sorted Nix: 22, 27, 29, 31, 34, 36, 38, 41, 43, 45. Sorted Fern: 31, 35, 36, 38, 39, 40, 42, 44, 46, 47.
- Counts: Nix 3 + 4 + 3 = 10 ✓. Fern 5 + 5 = 10 ✓.
- n = 10, even, so each median averages the 5th and 6th scores.
- Nix: cumulative 3 through stem 2, so the 5th and 6th are the 2nd and 3rd leaves of stem 3: 34 and 36. Median = (34 + 36) ÷ 2 = 35. Range = 45 − 22 = 23.
- Fern: stem 3 holds 5 leaves, so the 5th is 39 and the 6th is the first leaf of stem 4, which is 40. Median = (39 + 40) ÷ 2 = 39.5. Range = 47 − 31 = 16.
- The sentence: Fern is performing better: she scores higher on a typical day (median 39.5 against 35) AND more consistently (range 16 against 23).
- Which side of the plot was harder to build, and what fixed it?
- What is your even-n median rule in six words or fewer?
The verdict
The acorn alarms (numbers are evidence, sentences are the verdict)
The chief has never once looked at a plot. The chief reads sentences. So this is the so-what lesson: four lines and a verdict. Shape, Centre, Spread, Outliers, then one sentence that answers the actual question in context.
The evidence: 19 emergency acorn alarms last month, minutes until a detective arrived, in alarm order:
12, 15, 9, 14, 18, 11, 16, 13, 44, 10, 17, 12, 15, 8, 13, 16, 11, 14, 19
Sorted: 8, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 19, 44.
Leaf count: 2 + 16 + 0 + 0 + 1 = 19 = n ✓. The empty drawers STAY in the cabinet: stems 2 and 3 hold nothing, and that gap is the evidence that exposes the straggler at 44.
- Centre. Median position (19 + 1) ÷ 2 = 10th. Cumulative: 2 through stem 0, so the 10th is the 8th leaf of stem 1: median = 14 minutes.
- Spread. Range = 44 − 8 = 36 minutes, but quote it honestly: without the 44, the range is 19 − 8 = 11. The range is hostage to outliers, so it travels with a caveat.
- Why detectives trust the median. The 19 times sum to 287, so the mean is 287 ÷ 19 ≈ 15.1 minutes. One sleepy goblin dragged the mean up more than a whole minute. The median never flinched from 14.
The four-line report, modelled
| Line | Frame to keep | The acorn alarms |
|---|---|---|
| Shape | "Most values cluster between ... with ..." | "Most values cluster between 8 and 19, with a gap through the 20s and 30s and one straggler at 44." Formal stamp: positively skewed, because the tail points high. |
| Centre | "The median is ..., so a typical ..." | "The median is 14, so a typical alarm gets a detective in about 14 minutes." |
| Spread | "The range is ... (smallest to largest), but ..." | "The range is 36 (8 to 44), but almost all of that is one value; the pack itself spans only 11." |
| Outliers | "... sits far from everything ..." | "44 sits far from everything; one detective was asleep on the 44-minute night, and it gets its own interview." |
| Verdict | One sentence, in context | "Response is fast and dependable, one bad night aside." |
The plot is given so all your effort goes into the sentences. Every line quotes a number or a plain-words shape word; a report never just gestures at the picture.
👀 Stuck? Knock three times
Knock, knock, knock: What am I doing? Why? Is it helping?
Four lines, in order: shape, centre, spread, outliers. Then the verdict answers the actual question: WHO witnessed this crime? Numbers first (n, median, range), sentences second.
🔍 Show me a smaller case
Warm up: report on this mini ledger.
Values: 18, 19, 20, 21, 22, 24, 41. n = 7, median = 4th = 21, range = 41 − 18 = 23. Mini report: "Cluster 18 to 24, gap through the 30s, straggler at 41. Median 21. Range 23, mostly the straggler's doing. 41 gets an interview." Now the witnesses, four full lines.
✅ Show the full solution
Decode. Key: 3 | 4 means 34 years. Count first, then numbers, then the report.
- n = 4 + 6 + 4 + 2 + 1 = 17 witnesses.
- Median position (17 + 1) ÷ 2 = 9th. Cumulative: 4 through stem 2, 10 through stem 3, so the 9th is the 5th leaf of stem 3: median = 37 years.
- Range = 63 − 21 = 42 years.
- Shape: hump in the 30s with a tail stretching to the 60s: positively skewed, because the tail points high.
- Outliers: 63 is the oldest, but the tail climbs steadily (48, 52, 56, then 63), so there is no clear outlier. A defensible report says "a straggler, not an obvious outlier".
- Verdict: "A typical witness was in their late 30s (median 37); ages ran widely from 21 to 63, with more younger witnesses than older."
- What is the difference between a tail and an outlier, in your own words?
- Which of your four lines would the chief quote?
Reading, not building: the whole load is the writing. Swamp's four lines first, then High Street's, then let the two reports argue.
👀 Stuck? Knock three times
Knock, knock, knock: What am I doing? Why? Is it helping?
Write Swamp's four lines first, then High Street's, then let them argue. The verdict must quote a centre AND a spread, or it is an opinion, not a finding.
🔍 Show me a smaller case
Warm up: route A takes 5, 6, 7 minutes (median 6, range 2). Route B takes 2, 6, 10 minutes (median 6, range 8).
Same typical time, wildly different reliability. Which would you bet a job interview on? That question is the whole shape of this case.
✅ Show the full solution
Decode. Sorted Swamp: 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 38, 41. Sorted High Street: 21, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28. Counts: Swamp 7 + 3 + 1 + 1 = 12 ✓, High Street 12 ✓.
- n = 12 for both, so each median averages the 6th and 7th values.
- Swamp: stem 1 holds 7 leaves, so the 6th and 7th values are its 6th and 7th leaves: 18 and 19. Median = (18 + 19) ÷ 2 = 18.5 minutes. Range = 41 − 14 = 27.
- High Street: all twelve values live in stem 2. The 6th is 24 and the 7th is 25. Median = (24 + 25) ÷ 2 = 24.5 minutes. Range = 28 − 21 = 7.
- Swamp report: clusters tightly 14 to 22, then two flood-day stragglers at 38 and 41. Median 18.5. Range 27, almost all of it the stragglers' doing. Outliers: 38 and 41 each get their own interview.
- High Street report: one tight symmetric block, 21 to 28. Median 24.5. Range 7. No outliers, no surprises.
- Verdict: the Swamp is about 6 minutes faster on a typical day, but twice last month it took more than twice as long. If being late twice a month costs you, High Street never runs worse than 28 minutes. Centre says Swamp, spread says High Street, and the CONTEXT decides. Either route is defensible IF your sentence argues from both.
- When centre and spread disagree, what breaks the tie?
- Which single sentence of your report earns its place on the chief's desk?
- What did the outliers change about your verdict?
Optional spice · The middle half
Only if you are cruising. Back to the acorn alarms: the range shouted 36, but almost all of that was one sleepy goblin. Here is a measure that refuses to listen to him.
- Split the 19 sorted values at the median (14). Common school convention: the median itself sits out, and you name the convention whenever you use it.
- Lower nine: 8, 9, 10, 11, 11, 12, 12, 13, 13. Middle = 5th = 11.
- Upper nine: 14, 15, 15, 16, 16, 17, 18, 19, 44. Middle = 5th = 16.
- The middle half of all alarms runs 11 to 16 minutes, a band just 16 − 11 = 5 wide, even while the range shouts 36.
The range listens to the straggler; the middle half ignores it. That is the whole taster. The full machinery (quartiles, boxplots) is Year 10 business, and the Records Room will still be here.
Filed all six? The Records Room stamps your ledger.
Archive them in your case journal, and tell the goblin which ledger you want pulled next.
Australian Curriculum v9 · AC9M9ST03