MATHS300 Lesson 7

Lesson Plan

Summary:

This lesson addresses the hugely important current social issue of the development of Uranium mines and the problems of radioactive waste. Radioactive waste involves the concept of a half-life and exponential decay functions. All radioactive material is described in terms of its half-life.

Arising from this community concern the students 'pretend' to be uranium atoms and model the decay process. A computer simulation then provides an investigative tool to explore the underlying concepts of 'half-life' and exponential decay. Students discover just how long some of this material can stay in the environment.

The mathematical aspects of this lesson should be seen as supporting a joint study of the topic with other subject disciplines to bring out the full range issues.

Resources required:

• Stimulus Sheet A
• Stimulus Sheet B
• Many dice - the more the better, but at least one for each group of four students
• A large size dice for the teacher if possible
• Software ... MAC 8.6 PPC+ or MAC 68K or WIN 95+

Content Outcomes\Links To Curriculum Documents

• Number patterns
• Probability and concepts of chance
• Interpretation of graphs
• Exponential functions and exponential decay
• Mathematics in a social context
• Working Mathematically

Lesson Stages

1. Stimulating interest
2. Simulating radioactivity
3. Defining half-life
4. Experimenting with decay
5. Summarising patterns
6. Discussion

Software: Contributions To Learning

Kinaesthetic involvement of the students as atoms in a decaying radioactive substance is central to this lesson. It helps students build their understanding of the exponential rate of decay. However, there is not enough real time in class to act out more than one or two decay situations. The software allows us to extend the investigation. It can carry out a decay experiment 'in the blink of an eye' and it pauses poignantly to allow students to take in the moment when half the material has decayed. Students have the opportunity to choose both the number of atoms in the 'starting mass' and the decay rate. Time and again the program reinforces the concepts of radioactive half life and exponential functions; and, perhaps more importantly, repeatedly illustrates the dangerous nature of radioactive material.

It is strongly advised that students use the demonstration option first, before the higher speed decay options, so they can believe the computer is indeed modelling the same activity as they act out.

Lesson Notes

Your Photo Opportunity: Maths300 is frequently updated with contributions which help others 'see' the lesson. You are invited to send your electronic photos of this lesson to Doug.Williams@curriculum.edu.au for possible inclusion. We will need written permission from the parent of any child who could be identified.

1. Stimulating Interest

There are several possibilities to getting started. Any of the following can be used, perhaps in any order. All are designed to initiate interest and to see the topic as relevant and worth studying.

• What do students already know?
• Word association.
• Stimulus newspaper articles.
• Internet search.

Whichever of the stimulus items I use, the intent is to highlight that mathematics is involved in understanding why so many people are opposed to the nuclear industry and the problem of disposing of nuclear waste.

What do students already know? Write key words/phrases on the board such as: Jabiluka Maralinga Lucas Heights Nuclear Waste Dump sites What do all of these have in common? I want to discover students' existing knowledge and understanding and listen carefully as they tell me all they know about these issues. At the right moment I ask: Did you know there is mathematics involved in the way radioactive material decays?

Word Association Game I'm going to say some words and I want you to say the first word that comes into your head in response: ... Radioactive material ... Jabiluka ... If everyone writes down a couple of response words and these are collected on the board, it becomes a list for discussion (and an ESL - English as a Second Language - opportunity to build a vocabulary list.) The key word I am hoping to obtain from this word association is URANIUM. Some of the words I often get are : Uranium, Bombs, Cancer, Plutonium, Chernobyl, radioactivity, contamination. Most of the response words do tend to be 'negative'. Nobody ever seems to say 'cheap electricity'.

Stimulus Articles I have a collection from newspapers and magazines about incidents and other issues connected to the nuclear industry. I copy these and ask students to read them and then engage students in a brief discussion about the situations, and their understanding of them. For teachers who have not yet developed such a library for themselves, paraphrased articles are included as Stimulus Sheet A and Stimulus Sheet B.

An Internet Search An Internet search is a wonderful way to alert students to how many other people in the world are also concerned about this issue. Using key search words like Maralinga and Jabiluka, unearths a mountain of sites, many with much valuable and stimulating information. Also accessing the newspaper sites such as AAP or Reuters, and then searching for items about 'Nuclear accidents' or 'Radioactive waste' will show the many situations and incidents that are continually occurring across the world.

2. Simulating Radioactivity The description opposite, while simplified, is basically scientifically accurate. Give one dice to each student. The teacher also has one dice (preferably a large size for class visibility). Use the description to background the activity then set up the physical simulation. Ask all the students to stand up. You are all Uranium Atoms - you are all potentially dangerous, but it's only at that moment when you decay and fire off your 'poisonous ray' that you actually could cause harm. Let's assume the chances of decaying are 1 in 6 per year.

I want you to imagine you are all Uranium atoms and that you are all out there in the paddock. Why are these atoms dangerous to people? The reason is that every now and again, one of the atoms spontaneously (at random) 'decays'. At the instant of decay it fires off a poisonous ray (an alpha or beta particle, or a Gamma ray). If this ray hits someone, that's what does the damage. So before firing off the ray, the atom is not hurting anyone, it is only potentially dangerous. After firing off the ray, the atom is now 'safe' and has been turned into something else, like lead, and is no longer radioactive. It really is a random process. Scientists do not know what triggers any particular atom to decay, but they do know at the end of, say, a year, what fraction of the atoms have decayed.

Atoms in Year Zero (assuming 20 students) Year Atoms 0 20

Draw up a table on the board which shows the number of atoms (students) in Year 0. Let's see if this is your year to decay. All students roll their dice and generate a number between 1 and 6. The teacher then declares that the teacher's dice is the 'killer dice' that will determine if ...this is your year to decay. The teacher rolls the dice. Suppose the rolled number is a 4... All the 4s - you have decayed - fire off your poisonous ray and then sit down because you are now 'safe' and no longer radioactive. Those of you still standing are still potentially dangerous.

It depends on my class, but one humorous addition I sometimes include is to say, as students realise their number has come up: As you sit down, if it makes you feel any better, you are allowed to (gently) thump the nearest person. This is showing that the poisonous ray has fired off!!.

Year Atoms 0 20 1 17 Assuming 3 atoms have fired off during Year 0 ie: in this case, three students rolled 4. Year Atoms 0 20 1 17 2 13 3 10

Record the numbers of atoms that decay on the class table. Then repeat this for the next couple of years. So after about 3 or 4 years the table might look like the lower one opposite. The teacher is deliberately waiting for this key moment in the simulation when about half the class is sitting down. At this important moment (setting up the situation which defines a half-life), ask students to guess how many years it will be until all the atoms are gone. Half the class is sitting down - half the atoms have decayed and it has taken 3 years. How long do you think it will be until all the atoms have decayed and it is safe to walk across our paddock?

I write the estimates down on a chart because these are very useful for later comparisons. Often many of my students reason that if it has taken 3 years for half the atoms to 'go', then they will all go in 6 or 7 years.

Keep the simulation going until the 'last remaining atoms' have been 'decayed', noting the steps year by year until this moment.

3. Defining Half-Life Compare the actual number of years taken against student estimates. Students are often surprised by how long it takes and the final answer is often greater than anyone guesses. State the half-life concept and emphasise that all radioactive material is described in terms of its half-life. After 3 years half of the atoms decayed, after another 3 years, half of the remaining atoms decay, after another 3 years, half of those still remaining decay, and so on until all are gone. The fact that the students personally and physically participated in the simulation makes this concept much clearer. Introduce a factual example such as the main contaminating material at the Chernobyl disaster in Ukraine was Caesium-30 which has a half-life of 30 years. That means that 30 years after the disaster (which happened in 1986), half of the atoms will still be radioactive. 30 years after that, half of the remainder will still be there, and 30 years after that ... This shows that there will still be contamination there for at least the next 200 years.

Continuing the table from above, with a half-life of about 3 years we would expect: Year Atoms 0 20 1 17 2 13 3 10 ... ... 6 5 ... ... 9 2 ... ... 12 1 ... ... 15 0

4. Experimenting With Decay

This can be done in three stages:

i) Repeating the experiment physically
which allows the ideas to be reinforced.

ii) Using 100 dice

Continue until all 100 are gone - this should take about 20 to 25 'years'.

iii) Using the Computer Simulation

The first option is Demonstration. It is designed to show students that the computer simulates the experiment just like they did physically. The screen shows that 100 atoms were 'rolled', 3 was the 'killer' number so all the threes have been coloured, and the user can now press the space bar to have those 13 'spent' (now safe) atoms removed. Students can continue using the option in Demo mode until they are confident that the computer is simulating decay and half-life as they now understand them.

When they are ready, they can click the Auto radio button in the bottom right of the screen and the software will continue the visual sequence of decay automatically. In this case the half-life has occurred at 5½ years.

The second option is the key to allowing students to be 'independent investigators'. It allows them to choose any number of atoms (up to 10,000) and any probability of decay in one year. The physical simulation assumed a chance of an atom decaying in one year to be 'one in six' since this is so easy to model with a dice. The computer lets students assume the chance is say 1 in 20.

The visual presentation of the decay graph helps those visual learners to 'see' the half-life concept.

Ask students to: Select any number of atoms and any probability. At this stage 'estimate' what the half-life may be. Run the experiment until the computer pauses (after half the atoms are gone). At this point, estimate how long until all the atoms are gone. Run the simulation to the end to check this estimate.

Often my students verbalise observations such as: This stuff hangs around for a long time!!

As students work through various combinations, my hope is that they begin to see the half-life concept evident in each case. This allows an intuitive understanding of an exponential function to develop.

5. Summarising Patterns In this section students summarise findings and attempt to formalise their learning. In their work books, they can record each situation, including a screen capture of the graph. This can then be annotated to highlight how the half-life concept consistently appears in every graph. For example, if the number of atoms = 10,000 and p = 1/20, the half life is 14 years, as shown above. That is, we find from the program that after 14 years there are about 5,000 atoms left. This means that after another 14 years, there should be about 2,500 atoms left. This continues showing that it takes around 210 years for all of the 10,000 atoms to be safe.

6. Discussion

Encourage students to continue to explore, reflect and discuss.

• How much is 10,000 atoms anyway? Does it matter how many atoms there are?
• What is the real chance of decay of particular radioactive material?
• We know the half-life for the Caesium-30 that spewed out of Chernobyl. What chance of decay should we use to get that result?

In their journal students should also include responses under headings such as:

• What I learned in this activity.
• How I felt about this learning - enjoyable? important?

Encourage students to invite the community to join the debate by publicly explaining what they have been learning.

I challenged students to think about how we could inform others of the mathematics of radioactive waste, so that when public debates occur about such issues as the Jabiluka mine site, then this mathematics can be part of the debate. We considered display posters for the community as well as possibly writing a letter to the local newspaper.

MATHS300 is a living site. This lesson will be enriched through further teacher development in classrooms across the world. You are invited to contribute to that process by submitting:

• variations
• extensions
• inspirations
• photos
• student work

Please email material to the address below. If it is included it will be acknowledged. You can review current contributions in the Classroom Contributions folder for this lesson.