Presentation: Modelling How Fast a Rumour Spreads

Slide 1 of 7 · Settling starter · PSE / Well-being · Year 9

3 minutes

Modelling how fast a rumour spreads: 1 message, 1 Friday evening

You have a worksheet and, between 2, a stimulus sheet about an invented school, Ysgol Glan Tywi, with 1,000 pupils. At 19:00 on a Friday an untrue message about a pupil called Tomos is posted to 3 friends. Today you build a mathematical model of how fast it could spread, change the rules to see what makes a rumour explode, stay level or die out, and test whether a correction can ever catch up.

While the register is called: write your name, class and date on your worksheet, then read What happened and How we will model it on the stimulus sheet.

Starter question: In Model A everyone sends the rumour to 3 new people and each round takes 15 minutes. How many new people see it in round 2, and what time is it then?

DCF: Data and computational thinking → Problem-solving and modelling No computers needed · calculator, ruler and pencil · the school and Tomos are invented

Supply teacher script (no subject knowledge needed)

Hand out: one worksheet per pupil, one stimulus sheet (One message, one Friday evening: a rumour at Ysgol Glan Tywi) between 2, a calculator, a ruler and a pencil each; coloured pencils help with the graph but are optional. No computers are needed. The school, the pupils and the numbers about the school are invented; the only real figure is the population of Wales, 3,107,500. Safeguarding: if a pupil says a rumour is being spread about them or a friend, do not discuss it in front of the class. Say “come and see me at the end”, take their name, and pass it to the class teacher or the safeguarding lead the same day.

Say aloud: “Name, class and date at the top. While I take the register, read What happened and How we will model it on the stimulus sheet. Then: in Model A, everyone who has just seen it sends it to 3 new people, and each round is 15 minutes. Round 1 is 3 people at 19:15. How many new people in round 2, and what time is it?”

Starter answer: 9 new people (3 × 3), at 19:30 (2 rounds × 15 minutes). The total who have seen it is then 1 + 3 + 9 = 13.

Slide 2 of 7 · Today’s brief and key words · PSE / Well-being · Year 9

3 minutes

Your brief: build the model, change one rule at a time, judge what it leaves out

You will complete the Model A table and find when the rumour passes 1,000 and 3.1 million, work out the multiplier for Models B, C and D, plot 3 models on a graph, test whether Nia’s correction catches up, name the model’s assumptions and write the spreadsheet formulas that would run it, then use the multiplier to explain how to stop a rumour.

Model

A simplified version of something real, built from a few clear rules, so you can work out what might happen. Useful even though it is not perfect.

Multiplier

How many times bigger each round is than the round before: the fraction who forward it × the number of people each one sends it to.

Power of 3

3ⁿ means 3 multiplied by itself n times. 3² = 9, 3³ = 27, and 3⁰ = 1.

DCF: Data and computational thinking → Problem-solving and modelling LNF: Numeracy → Number → Relationships within the number system

Slide 3 of 7 · Step 1: Model A · PSE / Well-being · Year 9

11 minutes

Step 1: Model A, everyone forwards it to 3 people

Complete the table to round 14: each “new people” is the one above × 3, and each total is the total above + the new people. Rounds 0 to 2 are on the board. Then answer (a) to (d): the rule as a power of 3, round 10 from the rule, and when the total first passes 1,000 (the whole school) and 3.1 million (the population of Wales).

Round	Minutes after 19:00	New people this round	As a power of 3	Total who have seen it so far
0	0	1	3⁰	1
1	15	3	3¹	4
2	30	9	3²	13

Step 1 on your worksheet · 11 minutes Quote the 2 totals either side of 1,000 and of 3.1 million

Supply teacher script (no subject knowledge needed)

Say aloud: “Round 2 is on the board: 3 times 3 is 9 new people, 3 squared, total 13. Copy it, then carry on to round 14 with your calculator. Then (a) to (d). For (c) and (d) write down the total just below the target and the total just above it, then turn the round number into a time: 15 minutes a round.”

Quick check, new people then total: round 3: 27, 40 · 4: 81, 121 · 5: 243, 364 · 6: 729, 1,093 · 7: 2,187, 3,280 · 8: 6,561, 9,841 · 9: 19,683, 29,524 · 10: 59,049, 88,573 · 11: 177,147, 265,720 · 12: 531,441, 797,161 · 13: 1,594,323, 2,391,484 · 14: 4,782,969, 7,174,453. (a) new people in round n = 3ⁿ; round 0 is 3⁰ = 1 because nobody has forwarded it yet, just the 1 person who posted. (b) 3¹⁰ = 59,049, matches the table. (c) after round 5 the total is 364 (below 1,000), after round 6 it is 1,093 (above), so round 6, 6 × 15 = 90 minutes, 20:30. (d) after round 13 it is 2,391,484 (below 3.1 million), after round 14 it is 7,174,453 (above), so round 14, 14 × 15 = 210 minutes = 3.5 hours, 22:30.

Slide 4 of 7 · Steps 2 and 3: multipliers, then the graph · PSE / Well-being · Year 9

15 minutes

Step 2 & Step 3: change the rule, change the multiplier, then plot it

Step 2 (9 min): work out each multiplier as fraction × whole number, say whether it grows, stays level or dies out, then complete the table for Models B, C and D to round 6 and answer (a) to (c). Model A and Model B are on the board. Step 3 (6 min): plot new people for Models A, B and C, rounds 0 to 6, a different mark for each, joined with a ruler, key filled in, then answer (a) and (b).

Rule	Working	Multiplier	Grows, stays level or dies out?
Model A: everyone forwards it to 3	1 × 3	3	Grows: it triples every round
Model B: 1 in 3 forward it to 3	⅓ × 3	1	Stays level: 3 new people every round

Steps 2 and 3 on your worksheet · 15 minutes Decimals are fine: the models use averages · 1 square on the grid = 50 people

Supply teacher script (no subject knowledge needed)

Say aloud: “Model B is on the board: a third times 3 is 1, so 3 new people every round, it stays level. Copy it, do the other 4 rules, then the B, C, D table: every model starts 1 then 3, after that multiply by that model’s multiplier. Decimals are fine. Then the graph: crosses for A, squares for B, circles for C, ruler to join, fill in the key.”

Quick check, multipliers: Model C ½ × 4 = 2, grows, doubles · Model D ¼ × 2 = 0.5, dies out, halves · 1 in 10 to a chat of 20: 20 ÷ 10 = 2, doubles like C · 1 in 5 to 5: 1, stays level like B. Table, new then total: Model B rounds 2 to 6: 3 each round, totals 7, 10, 13, 16, 19. Model C: 6, 10 · 12, 22 · 24, 46 · 48, 94 · 96, 190. Model D: 1.5, 5.5 · 0.75, 6.25 · 0.375, 6.625 · 0.1875, 6.8125 · 0.09375, 6.90625. (a) above 1 grows faster and faster; exactly 1 stays level, total goes up steadily; below 1 dies out, total levels off (D never passes 7). (b) Model C new people carry on 192, 384, 768; totals round 7 is 382, round 8 is 766, round 9 is 1,534, so round 9, 135 minutes, 21:15. (c) 1 + 3n > 1,000 so n > 333; round 333 is exactly 1,000, not above, so round 334 (total 1,003); 334 × 15 = 5,010 minutes = 83.5 hours, about 3.5 days.

Quick check, graph: Model A points 1, 3, 9, 27, 81, 243, 729 (a J-shaped curve, steeper and steeper). Model B a flat line at 3 after round 1. Model C 1, 3, 6, 12, 24, 48, 96, a gentle curve starting to bend up. (a) B and C look flat because the scale goes to 800 to fit A and 1 square is 50 people; C’s 96 is under 2 squares high, but C is still doubling and would shoot up later. (b) 2,187 is far above 800, almost 3 grids tall, and round 8 (6,561) would need 8; growth that multiplies by the same number each round (exponential growth) starts slowly then runs off any graph.

Slide 5 of 7 · Steps 4, 5 and 6: the correction, the model’s limits, your actions · PSE / Well-being · Year 9

13 minutes

Step 4, Step 5 & Step 6: can the correction catch up, what the model gets wrong, and what you can do

Step 4 (5 min): Nia’s correction starts in round 3 with 4 people, multiplier 1.5. Complete the table (round 4 is on the board) and answer (a) to (c). Step 5 (4 min): 3 assumptions Model A makes, 1 improvement, the 4 spreadsheet formulas, 2 things a spreadsheet does faster. Step 6 (4 min, on your own): 3 actions that lower the multiplier, and 2 places Tomos could get help.

Round	Correction: new	Correction: total	Model B rumour: total	Model C rumour: new
3	4	4	10	12
4	6	10	13	24

Steps 4, 5 and 6 on your worksheet · 13 minutes Compare with division, not subtraction · link every action to a part of the multiplier

Supply teacher script (no subject knowledge needed)

Say aloud: “Round 4 is on the board: 4 times 1.5 is 6 new, total 10. Copy it, finish rounds 5 to 7, keep the decimals. Then Step 5: 3 things Model A pretends are true, and the 4 formulas, using the cell above, like =B3*3. Then Step 6 on your own: 3 things a pupil could do when the message about Tomos arrives, and which part of the multiplier each one lowers.”

Quick check, Step 4: round 5: 9, 19, 16, 48 · round 6: 13.5, 32.5, 19, 96 · round 7: 20.25, 52.75, 22, 192. (a) round 5 (20:15): correction 19, Model B rumour 16. (b) round 3: 12 ÷ 4 = 3 times; round 7: 192 ÷ 20.25 = about 9.5 times; the gap is growing. (c) No. The rumour multiplies by 2 each round and the correction only by 1.5, so the rumour pulls about 1.33 times further ahead every round; the correction started later, smaller and with a smaller multiplier, so it can never catch up unless its multiplier becomes bigger than the rumour’s.

Quick check, Step 5: assumptions: everyone forwards it (really many ignore it: slower); each person reaches 3 new people (really friends share friends and a school of 1,000 runs out of new people: slower); each sends to only 3, once (really one post in a big group chat reaches 20 or 200: faster); also every round is exactly 15 minutes, and nobody reports it. (a) let the multiplier fall over time, or cap the total at 1,000. (b) A4 =A3+1, B4 =B3*3, C4 =C3+B4, D4 =D3*2; capitals optional. (c) fill down hundreds of rounds instantly without mistakes; change the multiplier in 1 cell and update everything; draw the graph with a scale that fits. Step 6: do not forward it, even “as a joke” (lowers the fraction who forward); never post it into a big group chat or share a screenshot (lowers the number each one sends it to); report it in the app and tell a trusted adult (removes the post at source). Help: Meic, 080880 23456, and Childline, 0800 1111, or a parent or carer, form tutor or head of year.

Slide 6 of 7 · Self and partner check · PSE / Well-being · Year 9

2 minutes

Swap with your partner and check against the success criteria

Tick each success criterion your partner’s work meets. Then swap back and correct 1 thing: a box with an answer but no working, or a formula without the cell above in it.

I can complete a table of repeated multiplication and write its rule using powers, such as 3ⁿ.
I can work out a multiplier as a fraction × a whole number and say whether a model grows, stays level or dies out.
I can use a model to answer a question, such as when it passes 1,000 or 3.1 million, showing my working.
I can plot and compare models on a graph, and explain why a correction with a smaller multiplier never catches up.
I can name the assumptions a model makes, suggest an improvement and write spreadsheet formulas that would run it.

2 minutes · partner check Finished early? Check the rule: total after round n = (3ⁿ⁺¹ − 1) ÷ 2 for round 6

Slide 7 of 7 · Plenary review · PSE / Well-being · Year 9

3 minutes

Round the room, then quick-check answers before handing in

Everyone names 1 action from Step 6 and the part of the multiplier it lowers, nobody repeats one. Then check these answers.

Question	Correct answer
In Model A, when does the rumour pass the whole school of 1,000?	Round 6, at 20:30: total 364 after round 5, 1,093 after round 6. It passes 3.1 million in round 14, at 22:30.
What is the multiplier when half forward it to 4 people, and what happens?	½ × 4 = 2: it doubles every round. Above 1 grows, exactly 1 stays level, below 1 dies out.
Can a correction with multiplier 1.5 catch a rumour with multiplier 2?	No. The rumour pulls about 1.33 times further ahead every round. Only lowering the rumour’s multiplier below 1 stops it: do not forward, do not post to big chats, report and tell an adult.

Check your name and class are on your worksheet Worksheets in · stimulus sheets, rulers and calculators back to the front