Worked example · PSE / Well-being · Year 9
DCF: Data and computational thinking → Problem-solving and modelling
A secure Year 9 response, annotated to show which success criterion each part meets. It is also your answer key: every number below has been checked. The key idea is the multiplier (fraction who forward it × people each one sends it to). Above 1 the rumour grows, exactly 1 it stays level, below 1 it dies out. Model A passes the whole school (1,000) in round 6, at 20:30, and passes the population of Wales (3.1 million) in round 14, at 22:30, only 3.5 hours after the first post. A correction with a multiplier of 1.5 beats a Model B rumour (multiplier 1) by round 5 but never catches a Model C rumour (multiplier 2). Accept decimals rounded sensibly (for example 9.48 or 9.5) and spreadsheet formulas with or without capital letters.
| 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 |
| 3 | 45 | 27 | 33 | 40 |
| 4 | 60 | 81 | 34 | 121 |
| 5 | 75 | 243 | 35 | 364 |
| 6 | 90 | 729 | 36 | 1,093 |
| 7 | 105 | 2,187 | 37 | 3,280 |
| 8 | 120 | 6,561 | 38 | 9,841 |
| 9 | 135 | 19,683 | 39 | 29,524 |
| 10 | 150 | 59,049 | 310 | 88,573 |
| 11 | 165 | 177,147 | 311 | 265,720 |
| 12 | 180 | 531,441 | 312 | 797,161 |
| 13 | 195 | 1,594,323 | 313 | 2,391,484 |
| 14 | 210 | 4,782,969 | 314 | 7,174,453 |
(a) New people in round n = 3n. Round 0 is 30 because 30 = 1: nobody has forwarded it yet, so there is just the 1 person who posted it. Each round after that multiplies by 3 once more.
(b) Round 10: 310 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 59,049 new people. This matches the table.
(c) After round 5 the total is 364, which is below 1,000. After round 6 it is 1,093, which is above 1,000. So the total first goes above 1,000 in round 6. 6 × 15 = 90 minutes after 19:00, which is 20:30. In an hour and a half, more people have seen it than there are pupils in the whole school.
(d) After round 13 the total is 2,391,484, which is below 3.1 million. After round 14 it is 7,174,453, which is above 3.1 million. So it is round 14. 14 × 15 = 210 minutes = 210 ÷ 60 = 3.5 hours after 19:00, which is 22:30.
Every row is the row above × 3, the totals add each new round on, and the powers of 3 match the new people. The pupil uses the rule 3n to check the table, and answers (c) and (d) by quoting the totals either side of the target and turning rounds into a time. Evidences criteria 1 and 3.
| 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 |
| Model C: half forward it to 4 | ½ × 4 | 2 | Grows: it doubles every round |
| Model D: 1 in 4 forward it to 2 | ¼ × 2 | 0.5 | Dies out: it halves every round |
| 1 in 10 forward it to a group chat of 20 | 1/10 × 20 = 20 ÷ 10 | 2 | Grows: it doubles every round, like Model C |
| 1 in 5 forward it to 5 | ⅕ × 5 | 1 | Stays level, like Model B |
| Round | Model B | Model C | Model D | |||
|---|---|---|---|---|---|---|
| New | Total | New | Total | New | Total | |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 3 | 4 | 3 | 4 | 3 | 4 |
| 2 | 3 | 7 | 6 | 10 | 1.5 | 5.5 |
| 3 | 3 | 10 | 12 | 22 | 0.75 | 6.25 |
| 4 | 3 | 13 | 24 | 46 | 0.375 | 6.625 |
| 5 | 3 | 16 | 48 | 94 | 0.1875 | 6.8125 |
| 6 | 3 | 19 | 96 | 190 | 0.09375 | 6.90625 |
(a) When the multiplier is above 1, the rumour grows, and faster and faster. When it is exactly 1, the rumour stays level: the same number of new people every round, so the total only goes up steadily. When it is below 1, the rumour dies out: fewer new people each round, and the total levels off (Model D never gets past 7).
(b) Model C new people carry on 192, 384, 768. Totals: round 7 is 382, round 8 is 766, round 9 is 766 + 768 = 1,534. So the total first goes above 1,000 in round 9. 9 × 15 = 135 minutes = 2 hours 15 minutes after 19:00, which is 21:15.
(c) 1 + 3n > 1,000, so 3n > 999, so n > 333. Round 333 gives exactly 1 + 999 = 1,000, which is not above 1,000, so the answer is round 334 (total 1,003). 334 × 15 = 5,010 minutes. 5,010 ÷ 60 = 83.5 hours, about 3.5 days. By then nearly everyone would have lost interest.
Each multiplier is worked out as a fraction × a whole number and linked to what happens. The pupil notices that a big group chat with only 1 in 10 forwarding is as dangerous as Model C. In (c) the pupil solves an inequality and spots that exactly 1,000 is not “above” 1,000. Evidences criteria 2 and 3.
(a) Model A is a curve that gets steeper and steeper: flat at first, then shooting up (a J shape). Model B is a flat straight line at 3 after round 1, because every round has the same number of new people. Model C is a gentle curve that is starting to bend upwards. B and C look almost flat because the scale has to go up to 800 to fit Model A, and 1 square is 50 people. Model C’s biggest value, 96, is less than 2 squares high, and Model B’s 3 people is a tiny fraction of a square. Model C is still doubling: if the graph carried on, it would also shoot up, just later.
(b) 2,187 is far above the top of the grid, which stops at 800. It would need a grid almost 3 times as tall, and round 8 (6,561) would need one 8 times as tall. This kind of growth (multiplying by the same number each round, called exponential growth) starts slowly and then becomes so fast that each jump is bigger than everything before it put together, so it quickly runs off any graph.
The points are plotted accurately against the scale of 1 square = 50 people, each model has its own mark, the lines are joined with a ruler and the key is complete. The pupil explains the shape of each line and why the scale hides Model C. Evidences criterion 4.
| Round | Correction: new people | Correction: total so far | Model B rumour: total so far | Model C rumour: new people |
|---|---|---|---|---|
| 3 | 4 | 4 | 10 | 12 |
| 4 | 6 | 10 | 13 | 24 |
| 5 | 9 | 19 | 16 | 48 |
| 6 | 13.5 | 32.5 | 19 | 96 |
| 7 | 20.25 | 52.75 | 22 | 192 |
(a) Round 4: correction 10, rumour 13, so the rumour is still ahead. Round 5: correction 19, rumour 16. So in round 5 (20:15) more people have seen the correction than the rumour.
(b) Round 3: 12 ÷ 4 = 3 times as many. Round 7: 192 ÷ 20.25 = 9.48... = about 9.5 times as many. The gap is getting bigger, not smaller.
(c) No. Each round the rumour’s new people are multiplied by 2 but the correction’s are only multiplied by 1.5. So every round the rumour pulls further ahead: 2 ÷ 1.5 = 1.33..., so the rumour gets about 1.33 times further ahead each round (3, then 4, then 5.33, then 7.11, then 9.48 times as many). The correction started later and smaller, and its multiplier is smaller, so it can never catch up. To catch up it would need a bigger multiplier than the rumour, or the rumour’s multiplier would have to fall below the correction’s.
Correction values are the row above × 1.5, kept as decimals, and the totals are correct. The pupil compares with division, not subtraction, and explains the result using the two multipliers. Evidences criteria 3 and 4.
| Assumption the model makes | What really happens | Real spread faster or slower than the model? |
|---|---|---|
| Everyone who sees it forwards it. | Many people ignore it, do not believe it, or decide it is unkind to pass on. | Slower |
| Each person always reaches 3 new people who have not seen it. | Friends share friends, so the same people get it again and again. In a school of 1,000 it soon runs out of new people. | Slower |
| Each person sends it to only 3 people, and only once. | One post in a big group chat or on a public account can reach 20 or 200 people at once, and screenshots get passed on. | Faster |
Also acceptable: every round takes exactly 15 minutes (real people check their phones at different times, and some are asleep, so it is slower overnight but can be faster when everyone is online); nobody stops it (real apps remove reported posts, which slows it down).
(a) Make the multiplier fall as time goes on, because fewer people are interested in an old rumour and more of the people they send it to have already seen it. Or stop the total at 1,000, because there are only 1,000 pupils in the school.
(b)
| What the formula must do | Formula |
|---|---|
| Cell A4: the round number, 1 more than the row above | =A3+1 |
| Cell B4: Model A new people, 3 times the row above | =B3*3 |
| Cell C4: Model A total, the total above plus the new people in this row | =C3+B4 |
| Cell D4: Model C new people | =D3*2 |
Then fill each formula down the column. Better still, type the multiplier into one cell, for example F1, and use =B3*$F$1, so changing one number changes the whole model.
(c) A spreadsheet can fill down hundreds of rounds in a second with no calculation mistakes, and it can change the multiplier in one cell and update every row and the graph at once, so you can test lots of “what if” rules quickly. It can also draw the graph for you, with a scale that fits.
Each assumption is stated clearly, compared with real life and judged as faster or slower. The improvement comes from an assumption, and the formulas use cell references, * for × and the cell above, so they work when filled down. Evidences criterion 5.
| Action | Which part of the multiplier it lowers, or how it helps |
|---|---|
| Do not forward it, not even “as a joke” or to ask “is this true?”. | Lowers the fraction who forward it. If enough people stop, the fraction drops until the multiplier is below 1 and the rumour dies out. |
| Never post it into a big group chat or share a screenshot. | Lowers the number of people each one sends it to. A group chat of 20 turns a small fraction into a big multiplier. |
| Report the message in the app and tell a trusted adult, such as a form tutor or head of year. | The post can be removed, so nobody else can forward it, and an adult can stop it at its source and support Tomos. |
Also acceptable: send Nia’s correction to the people you know saw the rumour (it raises the correction’s multiplier); tell the sender privately that it is not true and ask them to delete it; check before you believe or share anything about a real person.
Tomos could get help from Meic (080880 23456, free and confidential, in Welsh or English, 8am to midnight every day) or Childline (0800 1111). He could also talk to a parent or carer, his form tutor or his head of year.
Each action is linked to one part of the multiplier, or to removing the message altogether, so the pupil is using the model to explain real choices, not just listing rules. Two real sources of help are named correctly. Evidences criterion 6.