Modelling how fast a rumour spreads

What you are doing

An untrue message about a pupil called Tomos is posted in a group chat on a Friday evening. You will build a mathematical model of how fast it could spread, then change the rules to see what makes a rumour explode, stay level or die out. You will find out whether a correction can ever catch up, judge what your model gets wrong, and use it to explain what each person can do to stop a rumour.

The key idea is the multiplier: how many times bigger each round is than the round before.

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

In Model A, everyone who has just seen the rumour sends it to 3 new people. Each round takes 15 minutes. Complete the table. Use your calculator: each “new people” number is the one above it × 3. “Total so far” is the total above it + the new people. Round 0 and round 1 are done for you.

(a) Write the rule for Model A. New people in round n = ......... Why is round 0 equal to 3⁰?

(b) Use your rule, not the table, to work out the number of new people in round 10. Show your working, then check it against the table.

(c) In which round does the total first go above 1,000, the whole of Ysgol Glan Tywi? What time is it then? Show the two totals either side of 1,000.

(d) In which round does the total first go above 3.1 million, the population of Wales? How many hours after 19:00 is that? Show your working.

Step 2: change the rule, change the multiplier

The multiplier tells you how many new people there are for every person in the round before:

multiplier = fraction of people who forward it × number of people each one sends it to

Use the rules on the stimulus sheet. Model A is done for you.

Now complete the table for Models B, C and D. In every model, round 0 is 1 person and round 1 is 3 people. After that, multiply each round’s new people by that model’s multiplier.

(a) Complete the rule. When the multiplier is above 1, the rumour ............... . When it is exactly 1, the rumour ............... . When it is below 1, the rumour ............... .

(b) Model C doubles every round. Carry on the table in your head or on a calculator: in which round does the Model C total first go above 1,000? What time is it then?

(c) In Model B, the total goes up by 3 every round, so the total after round n is 1 + 3n. Work out the first round when the total goes above 1,000, and how many hours that would take.

Step 3: draw the graph

Plot the new people in each round, from round 0 to round 6, for Models A, B and C. The axes are drawn and numbered for you. Use a different mark or colour for each model (for example crosses for A, squares for B, circles for C), join each model’s points with a ruler, and fill in the key.

(a) Describe the shape of each line. Why do Models B and C look almost flat on this graph, even though Model C is growing?

(b) Model A’s round 7 would be 2,187 new people. Why can it not be plotted on this grid, and what does that tell you about this kind of growth?

Step 4: can the correction catch up?

Nia’s correction starts in round 3 with 4 people and has a multiplier of 1.5. Complete the table. Round 3 is done for you. Keep decimals: they are averages.

(a) If the rumour follows Model B, in which round have more people seen the correction than the rumour? Quote both totals.

(b) If the rumour follows Model C, how many times more new people see the rumour than the correction in round 3? In round 7? Show your division.

(c) Will the correction ever catch up with a Model C rumour? Explain using the two multipliers, 2 and 1.5.

Step 5: judge the model

Every model makes assumptions: things it treats as true to keep the maths simple. Find 3 assumptions in Model A. For each one, say what really happens and whether the model makes the spread look faster or slower than real life.

(a) Suggest one way to improve the model so that it is more realistic.

(b) A spreadsheet could run this model for you. Here is how the first rows would look. Write the formulas a spreadsheet would need.

(c) Name 2 things a spreadsheet would let you do with this model that are slow on paper.

Step 6: lowering the multiplier, on your own

A rumour stops spreading when its multiplier drops below 1. Write 3 things a pupil could do when they receive the message about Tomos. For each one, say which part of the multiplier it lowers (the fraction who forward it, or the number of people each one sends it to), or how else it helps.

Name 2 places Tomos could go for help.

Step by step

1. 5 min Read the stimulus sheet: what happened, how we will model it and the rules for each model.
2. 12 min Step 1: complete the Model A table, write its rule and find when it passes 1,000 and 3.1 million.
3. 10 min Step 2: work out the multipliers and complete the table for Models B, C and D.
4. 8 min Step 3: plot Models A, B and C on the grid.
5. 8 min Step 4: test whether the correction can catch up.
6. 7 min Step 5: find the model’s assumptions and write the spreadsheet formulas.
7. 5 min Step 6: 3 actions that lower the multiplier, and where to get help, on your own.

If you finish early

• In Model A, the total after round n is (3ⁿ⁺¹ − 1) ÷ 2. Check this rule for round 6 and round 14 using your table.
• In Model D the rumour dies out. If you kept going for ever, what is the largest number of people who could ever see it? Use your table to make a sensible guess.
• Ysgol Glan Tywi only has 1,000 pupils, so in real life the rumour cannot keep tripling there. Sketch what you think the “total so far” graph would really look like over the evening, and explain the shape.