Presentation: Modelling Radioactive Half-Life

Slide 1 of 7 · Settling starter · Science · Year 11

3 minutes

Modelling radioactive half-life: 2 models, 1 constant

You have a worksheet with a table of random digits printed on it. Nobody can say when 1 particular unstable nucleus will decay, but a large number of them decay in a completely predictable way. Today you show that with a simulation of 20 nuclei, a mathematical model of 600 dice, a graph, and 3 real isotopes, including caesium-137 on the hills of north Wales.

While the register is called: write your name, class and date on your worksheet, get your scientific calculator and ruler out, then read What you are doing and the rules for Part 1. Find the row with the number of your birth month (January = row 1, December = row 12).

Starter question: In Part 1, which digits mean a nucleus decays, and what chance of decaying does that give each nucleus in each round?

DCF: Data and computational thinking → Problem-solving and modelling No computers, dice or radioactive sources needed

Supply teacher script (no subject knowledge needed)

Hand out: one worksheet per pupil. Every pupil needs a scientific calculator and a ruler. No computers, dice or radioactive sources are needed: the random events come from the table of digits printed on the worksheet. If you happen to be in a computer room, the 600-dice results are also in the file half-life-dice-simulation.csv, but the same numbers are printed in Part 2.

Say aloud: “Name, class and date at the top. Calculator and ruler out. While I take the register, read What you are doing and the rules for Part 1. Find the row of the digit table that matches your birth month. Then tell me: which digits mean decay?”

Starter answer: a digit of 0 or 1 means the nucleus decays; 2 to 9 means it survives. That is a 1 in 5 chance of decaying each round, so on average 0.8 of the nuclei survive each round. Neighbours start on different rows, so they get different results. That is deliberate.

Slide 2 of 7 · Today’s brief and key words · Science · Year 11

2 minutes

Your brief: simulate it, model it, graph it, then use it on real isotopes

You will run a 20-nucleus simulation for 6 rounds and compare it with the model 20 × 0.8ⁿ, complete the model column for 600 dice and plot both on 1 graph, read off the half-life twice and check it by calculation, then use the half-life formula on iodine-131, technetium-99m and caesium-137, and evaluate where each model matches real decay and where it breaks down.

Simulation

A rule applied to random numbers. Here: cross out a nucleus if its digit is 0 or 1, or remove a die if it shows a 6. Each run gives a different result.

Mathematical model

A formula that gives the average result: multiply by 0.8 each round for the nuclei, or by 5/6 each throw for the dice. It never wobbles.

Half-life

The time it takes for the number of undecayed nuclei to halve. It is the same whatever number you start from, so 1 number describes the whole curve.

DCF: Data and computational thinking → Problem-solving and modelling Keep the unrounded value in your calculator between steps

Slide 3 of 7 · Part 1: your own simulation of 20 nuclei · Science · Year 11

15 minutes

Part 1: run your 20-nucleus simulation for 6 rounds, then compare it with the model

Start on the row of your birth month. Round 1 uses that row, round 2 the next row down, and so on for 6 rounds. In each round, look only at the digits under nuclei you have not crossed out: 0 or 1 means it decays, so cross its number out. Record each round. Then fill the model column (× 0.8 each round, 1 decimal place) and explain why your numbers differ from it.

Round	Row used	Nuclei that decayed this round	Nuclei left	Model: 20 × 0.8ⁿ (1 d.p.)
0	·	·	20	20.0
1 (example: born in May, row 5)	5	2, 7, 11, 15	16	16.0

Part 1 on your worksheet · 12 minutes to simulate, 3 to compare Only check nuclei that are still undecayed

Supply teacher script (no subject knowledge needed)

Say aloud: “The board shows round 1 for someone born in May: row 5 has 0 or 1 under nuclei 2, 7, 11 and 15, so those 4 decay and 16 are left. Your row is your birth month. Do 6 rounds, moving down a row each time, and only look at nuclei you have not crossed out yet. Twelve minutes. Then fill in the model column: 20, times 0.8, times 0.8 again, to 1 decimal place, and write why your numbers are not the same as the model.”

Quick check, nuclei left after rounds 1 to 6 by start row: row 1: 14, 12, 6, 4, 2, 1. Row 2: 18, 12, 9, 7, 5, 3. Row 3: 14, 9, 7, 5, 3, 2. Row 4: 14, 11, 9, 7, 5, 4. Row 5: 16, 12, 10, 8, 6, 5. Row 6: 15, 13, 10, 8, 6, 5. Row 7: 16, 13, 10, 7, 6, 4. Row 8: 16, 13, 10, 9, 4, 4. Row 9: 17, 14, 13, 7, 6, 4. Row 10: 17, 16, 9, 8, 6, 6. Row 11: 19, 11, 10, 7, 6, 6. Row 12: 12, 10, 7, 6, 6, 5. Model column: 16.0, 12.8, 10.2, 8.2, 6.6, 5.2.

Walk the room: the usual mistake is checking a nucleus that was already crossed out, or forgetting to move down a row. Why results differ from the model: the model gives the average over a huge number of runs and can give fractions of a nucleus; with only 20 nuclei, 1 extra 0 or 1 in a row changes the answer by 5% of the whole sample. Neighbours differ because the rule is the same but the chance events are different: someone born in January ends with 1 nucleus, someone born in October or November with 6.

Slide 4 of 7 · Part 2: scaling up to 600 dice · Science · Year 11

15 minutes

Part 2: fill in the 600-dice model column, draw the graph, read the half-life

Each die is a nucleus; a 6 means it decayed. A 6 comes up 1 time in 6, so 5/6 survive each throw on average. Multiply the previous unrounded value by 5/6 each time and round to the nearest whole number. Plot the simulation as points (do not join them) and draw the model as a smooth curve. Then answer (a), (b) and (c).

Throw n	Dice remaining (simulation)	Sixes removed this throw	Model: 600 × (5/6)ⁿ
0	600	·	600
1	515	85	500 (600 × 5/6)

Part 2 on your worksheet · 10 minutes for the column and graph, 5 for the half-life Points for data, a smooth curve for the model, title and key

Supply teacher script (no subject knowledge needed)

Say aloud: “Throw 1 is on the board: 600 times 5/6 is 500. Keep going, multiplying the unrounded value in your calculator by 5/6 each time, and round only when you write it down. Then draw the graph: simulation as points, model as a smooth curve. Ten minutes. Then answer (a), (b) and (c): read the half-life off your graph twice and check it with (5/6) to the power 3 and to the power 4.”

Quick check, model column throws 2 to 12: 417, 347, 289, 241, 201, 167, 140, 116, 97, 81, 67. (a) From 600 to 300 takes about 3.8 throws on the model curve (the simulation hits exactly 300 at throw 4); from 300 to 150 takes about another 3.8 throws, reaching 150 at about throw 7.6. The time to halve is the same whatever number you start from: that is what a half-life is. (b) (5/6)³ = 0.579 and (5/6)⁴ = 0.482, so the half-life is between 3 and 4 throws and closer to 4, because 0.482 is only 0.018 from 0.5 while 0.579 is 0.079 away. (c) At throw 1 the model expects 100 sixes and the simulation gave 85, 15% fewer; at throw 12 the model expects about 14.5 sixes and the simulation gave 9, 38% fewer. Random variation does not shrink as fast as the sample does, so with fewer dice it becomes a bigger share of the result, which is why the points scatter more near the end of the graph.

Slide 5 of 7 · Parts 3 and 4: real isotopes, then evaluate the models · Science · Year 11

10 minutes

Part 3 & Part 4: use the model on real isotopes, then judge it

Part 3 (7 min): for a half-life T, the fraction left after time t is (½)^{t ÷ T}. On a calculator: 0.5, the power key, then (t ÷ T). Answer (a) iodine-131, (b) technetium-99m and (c) caesium-137, with units. Part 4 (3 min): 2 ways the dice simulation is like real decay, 2 ways it is different, and why the smooth model fits a real sample well but your 20 nuclei poorly.

Isotope	Half-life	Worked model
(a) Iodine-131, 800 MBq on arrival	8 days	24 days ÷ 8 days = 3 half-lives: 800 → 400 → 200 → 100 MBq. One more half-life gives 50 MBq, so 32 days.

Parts 3 and 4 on your worksheet · 10 minutes Show the number of half-lives, and give every answer a unit

Supply teacher script (no subject knowledge needed)

Say aloud: “Iodine-131 is worked on the board. Copy the method: divide the time by the half-life to get the number of half-lives, then halve that many times. Do (b) and (c), then Part 4. Ten minutes.” MBq is a unit of activity, megabecquerels; pupils just carry the unit through.

Quick check, Part 3: (b) 24 hours ÷ 6 hours = 4 half-lives, so (½)⁴ = 1/16 is left, or 6.25%. A half-life of hours is long enough to make, inject and scan the tracer while it is active, and short enough that almost none is left a day later; seconds would be gone before the scan, years would stay in the patient long after. (c) (½)^{26 ÷ 30} = 0.5^0.867 = 0.548, about 55% of the 1986 caesium-137 was still there in 2012. So decay alone removed less than half of it, and cannot be the whole reason the controls were lifted: over 26 years the caesium became more tightly held in the soil and less was taken up by the grass the sheep ate. The half-life model predicts how much isotope is left; it says nothing about how it moves through soil, plants and animals.

Quick check, Part 4: like real decay: it is random, so you cannot predict which die goes next, only how many on average; every die has the same chance every throw however long it has survived, just as a nucleus does not age, which is why the half-life is constant. Unlike real decay: real decay is continuous, not in separate throws, and the chance per second is fixed by physics, not set at 1 in 6; a real sample has something like a billion billion nuclei, not 600, and a decayed nucleus becomes a different nucleus rather than being taken away. The smooth model fits a real sample almost perfectly because the random ups and downs are too small to see; it fits 20 nuclei poorly because each nucleus is 5% of the sample.

Slide 6 of 7 · Self and partner check · Science · Year 11

2 minutes

Swap with your partner and check against the success criteria

Tick each success criterion your partner’s work meets. Then correct 1 thing: a missing unit in Part 3, or joined-up points on the graph.

I ran a rule-based simulation accurately and recorded the result of every round.
I built a mathematical model by repeated multiplication and compared it with simulated data, explaining the differences using chance and sample size.
I plotted data as points and the model as a curve on 1 graph, and read off a half-life that stays the same each time the number halves.
I used the half-life model to make correct predictions for real isotopes, with units.
I judged where each model matches real decay and where it breaks down, including what the model cannot explain in the Chernobyl case.

2 minutes · partner check Finished early? By 2026, 40 years on: what fraction of the caesium-137 was left?

Slide 7 of 7 · Plenary review · Science · Year 11

3 minutes

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

Everyone says how many nuclei they had left after round 6 and their birth month. Notice how different the numbers are from the same rule. Then check these answers.

Question	Correct answer
What is the half-life of the 600-dice model, in throws?	About 3.8 throws, between 3 and 4 and closer to 4. The same from 600 to 300 as from 300 to 150.
Iodine-131, half-life 8 days: activity after 24 days from 800 MBq?	100 MBq (3 half-lives). 50 MBq after 32 days.
What fraction of the 1986 caesium-137 was still there in 2012?	About 55%, so decay alone does not explain why the controls could be lifted.

Check your name and class are on your worksheet Worksheets in · calculators and rulers back