We introduce the unit by rolling dice and investigating the numbers that come up.

1. Begin the session by showing the students the large die and asking them which number they think will come up if you roll it.
   What number do you think I will roll?
   Why do you think that?
   Roll the die and see whether students' predictions were correct. Repeat a couple of times.
2. What are the possible numbers that I can roll?
   List these on the board and tell the students that this list of all the possible outcomes is called the sample space.
3. What if I rolled the die twenty times. What do you think will happen? Why?
   List these predictions on the board or on chart paper.

4. With the class, roll the die twenty times keeping track with a tally chart. Summarise onto a class chart.

   1	2	3	4	5	6
   lll	llll	l	lll	llll l	lll

5. Give pairs of students a die and ask them to work together to roll it 20 times. As they finish, ask them to record their results on the class chart.

   Pairs	1	2	3	4	5	6
   Mr Tihi	3	4	1	3	6	3
   Ben & Tane	2	5	3	2	4	4

6. Discuss the results with the class. Look back at their earlier predictions.
   Why are all our results different?
   If you rolled the die another twenty times what do you think would happen? Why?

7. Now let's add our results together.
   What do you think that we will find?
   Use a calculator to sum down each of the columns

   
   Number rolled

   Pairs	1	2	3	4	5	6
   Mr Tihi	3	4	1	3	6	3
   Ben & Tane	2	5	3	2	4	4
   Jay & Sarah	5	3	3	2	5	2
   Class totals 240 rolls	45	36	42	31	39	47

   At this level it is likely that the students may attribute uneven results to "special luck". It is through continued experience that students come to appreciate the mathematics of probability.

Exploring

Over the next 3 days the students play a number of probability games. Copymasters for each game are attached to this unit.  They are encouraged to think about the sample space and to make predictions as they play the games. They will also begin to think about whether the games are fair.

Tell the students that they are going to play a number of games in pairs over the next 3 days and there are some general things they need to do with each game:

• as they play each game they are to write down the possible outcomes (the sample space). They are also to write a prediction about what they think will happen in the game
• play the game, recording the results
• compare what happens with their prediction.

Note: At this level do not expect the students to make mathematically sound predictions or systematically identify all possible outcomes. It is likely that they will make incomplete lists of possible outcomes. In future work, as they have similar experiences, their thinking will become more systematic and mathematically sound. The main aim of this unit is to start the students thinking about possible outcomes and notions of fairness.
Probability games to select from:

Bunny hop (Copymaster 1)
Sample space = {Heads, Tails}
As there is an equal chance of getting a head or a tail you would expect the bunny to be close to 0 after a number of turns.

Doubles (Copymaster 2)
Sample space

There are 6 ways of getting a double or 6 out of 36.

It is unlikely that the students will be this systematic about identifying the sample space. However they should identify that you are more likely to get non-doubles.

Pūkeko racing (Copymaster 3)

Sample space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
The table shows the ways of getting each sum. 7 (6 ways) is the most likely followed by 6 and 8 (5 ways).

Odds or evens (Copymaster 4)

Sample space = {1 3, 5 (odd) and 2, 4, 6 (even)}
For this game the probability of each player winning is equal.

Sums (Copymaster 5)
From the table for Pūkeko racing you can see that there are 24 ways of rolling a 5, 6, 7, 8 or 9 which is a probability of 24 out of 36 or 2/3. The probability of rolling a 2, 3, 4, 10, 11, or 12 out of 35 or 1/3.

Up or down (Copymaster 6)
Sample space = {heads, tails}
This is very similar to the bunny hop game. You expect the climber to be close to 0 after a number of turns.

At the end of each session have a class sharing time to discuss a couple of the games.

• Tell us about one of the games you played today
• What were the possible outcomes?
• What did you think would happen?
• What happened when you played the game?
• Did anyone else play the same game?
• Did you get the same results?
• Do you think that the game was fair? Why? Why not?

Reflecting

On the final day of the unit ask the students to invent their own games using either coins or dice. Share the games with others in the class.

Which number came up the most? Why do you think it did?
If we rolled the die again do you think we would get the same results? Why?

Session 1

In this introductory session we discuss the theoretical probabilities of drawing cubes from a bag containing coloured cubes.

1. Put 4 coloured cubes in a bag, two red, one blue, and one yellow.
2. Show the students what cubes are in the bag.
3. Ask the students to identify what will happen if a cube is selected at random from the bag:
   What colour cube will be selected?
   What colours of cube could be selected?
   What colour is most likely? Why?
   Some students are likely to try to tell you what colour cube will come out. It is important for them to realise that they can not be certain what colour will be selected. If one is chosen at random it could be any of the three colours. Red is more likely than blue or yellow because there are more red cubes than blue or yellow cubes in the bag.
4. Now ask students:
   What is the chance of selecting a red/blue/yellow cube?
   If students have difficulty identifying simple probabilities such as these, some work with coins and dice is probably required before continuing with this unit.
   
   The important thing for students to understand here is how to work out theoretical probabilities. They will probably know that if there are 4 cubes and half of them are red the probability of selecting a red one is 1/2, but it is useful for them to know that this can be stated more generally:
   
   In other words, the probability of something happening is the number of ways it could happen, divided by the number of things that could happen (as long as all events are equally likely). So the probability of rolling an even number with a normal die is 3 (the number of even numbers) divided by 6 (the number of numbers) which equals 3/6 or 1/2.
    
5. As a class draw a graph of the probabilities of selecting each colour of cube. Label the graph "Theoretical robability of selecting coloured cubes", and discuss what the word ‘theoretical’ means.
   
6. Discuss what this means in terms of selecting cubes. You are twice as likely to select a red cube as a blue or yellow cube.
7. If there is extra time students could be challenged to suggest what cubes would need to be put into a bag to make it match a given graph of theoretical probabilities.

Session 2

In this session we discuss theoretical probability in the context of experimental estimates of probability.

1. Reintroduce the bag of cubes and graph of probabilities from Session 1.
2. Discuss what will happen if you select one cube at random. Students should now be able to explain accurately what could occur, and how likely each of the possibilities are.

3. Ask what will happen if you select four cubes at random, replacing each after recording its colour. Students are likely to suggest that there will be 2 red cubes, 1 blue cube and 1 yellow cube selected. While this is possible, it is not the only possible outcome, in fact the probability of exactly 2 red, 1 blue, and one yellow cube is less than 20%!

4. Ask four students to each come up, select a cube, record its colour and replace it in the bag. Graph the results on the board and discuss the results.
5. Repeat several times.
6. Ask students to discuss in small groups why the results do not match with what they had calculated in the previous session.
7. Share ideas as a class. Hopefully some students will recognise that theoretical probability is only a way of predicting what may occur, it does not tell you what will occur. Ensure that all students understand this.

Session 3

In this session students will investigate some long run probability experiments, observing how a larger sample size affects approximations of theoretical probability.

1. Reintroduce the bag of cubes (2 red, 1 blue, 1 yellow) and the graph of theoretical probabilities.
2. Ask students to predict what will happen if you select 20 cubes at random, replacing them as in the previous session.
3. Record students’ predictions to refer back to.
4. Discuss why students made their predictions. Discourage students who have inaccurate probability concepts – estimates should be around 10 red, and 5 each blue and yellow, but not necessarily exactly those numbers.
5. Divide the class into five groups.
6. Get each group to make a bag with 2 red, 1 blue and 1 yellow counter.
7. Ask each group to select four cubes at random, replacing each after recording its colour.
8. Combine the results as a class, in a table. How close were the student’s predictions?
9. Calculate the proportion of each colour and graph the result.
10. Compare the graph to the one from Session 1 and to those from Session 2.
    Is this graph more accurate?
    Why or why not?
    How could we get a more accurate approximation? (Hopefully someone will suggest more samples)
11. Ask students to make a prediction of the results of selecting 50 cubes at random. Record on the board.
12. Get each group to select ten cubes and analyse the resulting class sample of 50.
13. Repeat for twenty cubes per group to give a sample of 100.
14. Discuss and compare the graphs.

Session 4

In today’s session students choose a long run frequency investigation to carry out in groups.

1. Briefly recap the previous three sessions’ work so that students remember what they have done thus far in the unit.

2. Explain that students are going to carry out similar investigations in groups to compare long run experiments with theoretical probabilities.

3. Ask students to brainstorm the kind of events that they might do an investigation of. The key requirements are: defined theoretical probabilities; and measurable probabilities for the experiment. If students can not think of ideas, suggest a few to get them started:
   Probability of suits for a deck of cards.
   Probabilities of numbers on dice.
   Probabilities of different results for tossing two coins.

4. Explain to students that they are to produce a report of their findings which will include:

   • An introduction explaining what they are investigating.
   • A graph of the theoretical probability for their event.
   • Trials of 5, 20, 50 and 100 samples, with graphs of the experimental probabilities for each.
   • A summary statement explaining what they have found out.

5. Students work in small groups.

Session 5

In the final session of the week we complete our investigations and share the results with the class.

1. Students to be given time as required to complete their investigations.

2. Students share their results with the class.

3. Discuss as a class:
   What is the difference between theoretical probability and experimental estimates of probability?
   What is theoretical probability?
   What are experimental estimates of probability?

4. Make a poster with class definitions of the two terms which can be displayed to remind students. Possible definitions are:
   Theoretical probability is what you would guess would happen, and can be worked out by dividing the ways a thing can happen by the possible things that can happen.
   Experimental estimates of probability are the probabilities you find when you do an experiment, and get closer to the theoretical probability the more samples you do.

The Problem

When you toss 2 coins at once, will they usually land with the same side up or different sides up?

Teaching Sequence

1. Introduce the problem as a game to be played with pairs.
   Players take turns, one tossing the coins while the other guesses whether the coins will land with the same side up or different sides up. Players record the results of each guess as same or different.
2. As the game is being played get the students to observe what is happening to the totals. Ask the students to toss the coins 20 times and write a statement on their results. You may need to model how to record the results from the coin toss (e.g. in a frequency table). They should do this again after 50 tosses.
   What can you say about the totals?
   Does one way of landing seem to come up more often than the other?
   Is it better to guess same or different?
3. Share findings from the game.
   Why are there different totals? (develops the notion of chance)
4. Pose the question: What are the different ways the coins could land?
5. Let the pairs find all the possible outcomes of tossing the 2 coins. Ask that they record their work in a way that would convince others that they had found all the possible outcomes.
6. Share strategies for recording outcomes.

Extension

Repeat the game with 3 coins.

Solution

When the game is played there will be variation in the results that helps develop intuitive understandings of chance. As more trials are made the results will begin to approach ½ , although it may take at 50 for this to happen.

There are 4 different outcomes when 2 coins are tossed:

• HH TT TH HT

This means that it is equally likely that they land with the same sides up as they will land with different sides up.

Solution to the Extension

(8 outcomes)

HHH HHT HTT HTH TTT TTH THH THT

Here it is less likely that all the sides will be the same.

Session 1

Today we make predictions about the cubes that are hidden in a bag. We find out that even when we can’t peek in the bag we can still make a good prediction about what is in it. Think of cubes in the bag like all the people in New Zealand or fish in our seas. We cannot know exactly about all of them but we can use statistics to get an idea.

1. Put four cubes in a paper bag (3 red and 1 blue) without students seeing. Provide a context such as red are eels and blue are mudfish.
   Here is a bag with four cubes. The cubes are either red or blue and we’re going to try to find out  how many of each colour there are selecting cubes one at a time.
2. Shake the bag and ask a student to select one cube to show the class. Record the colour on the board and get the student to put the cube back in the bag.
   This is called sampling with replacement. Replacement means we put it back – like a fish.
   (Note: Each time a student takes a cube it must be returned before the next student draws a cube. Otherwise, the probabilities will change.)
3. Ask another student to select a cube.
   What colour have you got?
   If it is the same colour as the one previously drawn ask: 
   Do you think that it is the same cube? Why or why not?
   If it is a different colour ask:
   Does that mean that half the cubes are red and half are blue?
   The important idea is that students acknowledge that such conclusions are speculative. There is no certainty except that at least one red (same colour drawn), or at least one red and one blue (different colours drawn) are in the bag.
4. Ask a third student to draw a cube but this time get them to predict what the cube might be.
   Why did you guess that? How certain are you?
5. Add the third cube colour to the data.
   Has that changed your mind about what is in the bag? Why? Why not?
6. Ask a fourth student to draw a cube.
7. Look at the result of the four draws.
   Do you think that we have seen all the cubes?
   Do we know what the colours of the four cubes are? Why or why not?
   Would we find out more if we had more turns?
8. Let another four students select a cube one at a time with replacement. Add the colour to the data on the board. Before selecting each time, ask the student to predict the colour of the cube.
   Record your best prediction about the colours of the four cubes in the bag.
9. Ask students to discuss their predictions with a buddy and to justify their thinking. Look for acknowledgement of certainty and uncertainty. For example, if all the draws were red, we cannot say for certain that there is no blue cube. If one or more of each colour have been drawn, we can be certain that there is at least one cube of each colour in the bag.

10. Before we look in the bag, discuss all the possible combinations for the colours of the four cubes.
    Record these combinations on the board. Students might forget the four of one colour, zero of the other colour possible combinations.

    Possible combinations for 4 cubes
    Red	Blue
    0	4
    1	3
    2	2
    3	1
    4	0

11. Ask the students to decide which combination they think is most likely.
12. Look inside the bag and check the cubes. Discuss how reliable their prediction was.
    How could we have improved the prediction before checking?
    Students might suggest that more selections might have improved the reliability. Larger samples tend to be more representative than smaller samples
13. Put the cubes back in the bag and ask:
    I am going to draw a cube. Which colour do you think it will be?Why?
    Can we be sure that I will get that one? Why?
    How could we record your chance of success?
    If students choose red their chance of success is 3 out of 4 or 3/4.
    If they choose blue their chance of success is 1 out of 4 or 1/4.
    Though there is a greater chance that the cube drawn will be red, there is still a 1/4 chance that it might be blue.

Exploring

Over the next three days we work in pairs to make our own bags of cubes. We swap them with our friends to see if they can guess "What’s in the bag?"

Session 2 

1. Invite each pair of students to choose 10 cubes. There should be two colours available. Ask them to put 10 cubes in their bag using any combination of the two colours they want.
2. Swap bags with another pair of students. Students can create a context for their bag. E.g. rugby balls and soccer balls, tui and fantail. Each pair must predict how many cubes there are of each colour in the bag by taking turns drawing cubes from the bag, one at a time with replacement. Remind them to put the cube back in the bag after each draw. Tell students that they have 5 minutes to make as many draws as they can. It is important that they record their results. The results can be recorded on a chart or digitally.
3. Ask the students to make a prediction about the colours of the ten cubes in the bag. How will you use the data to make the best prediction you can?
4. Gather the class and share predictions. Do the students:
   • acknowledge that their predictions are uncertain?
   • relate fractions to their predictions, e.g. 20 out of 30 trial cubes were red, that’s 2/3 so 6 or 7 of the ten cubes might be red?
   • provide a range of what events might occur, e.g. 5-7 red and 3-5 blue?

Ask the students to 'think aloud' so other students can build on their understanding.

Session 3

1. In this session students predict events from complete knowledge of the set of cubes. In doing so, they consider the likelihood of the colour of the cube selected next.
2. Show the students a paper bag with 24 cubes in it of varying colours, e.g. 10 yellow, 8 blue, 3 red, 2 green, 1 white. Tip the contents of the bag onto the mat or tabletop. Ask some students to sort the cubes by colour. Ask the students to provide a context for this activity that they can relate to. 
   I want you to create a data display of the colours. You are free to use whatever display you want. What display might you use? (Students might suggest bar graphs, pie charts, frequency tables, pictographs, etc.)
   You can make as many trips as you need up here to view the data but you must create your display back at your desk. Give your students adequate time to create their displays.

3. Explain the rules of the game to your students.
   I am putting all the cubes back into the paper bag. Nothing in the set of cubes has been changed.
   With each turn, one person from the class will take out a cube, tell you what colour it is, show you the cube then put it aside. It will not be put back into the bag.
   Before the cube is taken out you need to make a prediction about what colour it will be. Record your predictions like this:

   Round	Prediction	Correct/Incorrect	Points
   1	Green	I	-1
   2	Yellow	C	3
   3	Red	I	-1
   …

4. If your prediction is correct, write C and give yourself 3 points. If your prediction is incorrect give yourself minus 1 point.
   There will be 24 rounds. At the end of the game the +5 and -1 scores will be combined. The player with the highest score wins.
5. Before the game starts, remind your students about the display they created.
   How might your display be useful?
6. Play the full 24 rounds of the game. As the number of cubes in the bag reduces, the students’ chances of correctly predicting increases to the point that on the last draw they should be certain of the outcome. Students can play this game with a buddy to provide scaffolding if required.
   Watch for the following behaviour from your students:
   • Do they keep track of the colours of remaining cubes, using their data display?
   • Do their predictions match the likelihoods, as expected from the colour frequencies of the remaining cubes?
   • Do they successfully cope with the plus and minus nature of getting their score?
7. At the end of the game interview the winner or winners about the secret to their success. Focus on the way they tracked the number of cubes of each colour left in the bag and how they used the frequencies of colours to make predictions.
8. Pose various scenarios of cubes remaining in the bag and invite students to make predictions about the next cube. For example:
   In the bag are 6 yellow, 4 blue, and 2 red cubes. What coloured cube do you think will be drawn next?
9. Calculate the probabilities for each colour. For example, there are 12 cubes all together so the chance of yellow equals 6/12 or ½, 4/12 or 1/3 of the cubes are blue, and 2/12 or 1/6 of the cubes are red. Yellow has the greatest chance of being selected.
10. Trial selecting the next cube in each scenario. Students should come to realise that a colour may have the best chance but still not be selected. A high probability does not guarantee certainty.

Session 4

In this session students explore the impact of ‘tag and release’ methods of sampling. Such methods are common to biological research in which animals need to be returned to their habitat. The lake in this session could be named as a local lake to relate to students' experiences. 

1. Pose the following problem:
   Suppose you are a scientist. The fish in the lake are precious so you want to return each fish after you have tagged it. How can you get an accurate picture of:
   • The total number of fish in the lake?
   • The fraction of all the fish that are each species (type of fish)?
     Types of freshwater native fish.
2. Invite the students to offer ideas. Question b is like the previous inquiries. Taking fish out of the lake one at a time, noting the species, and returning the fish will allow a reasonable prediction of species as more and more data are gathered.
   What is the advantage of tagging the fish?
   Students will know if a tagged fish is caught that it is already represented in the data.
3. Produce a bag of cubes (fish), e.g. 8 red, 6 yellow, 4 blue, 2 green.
   I have a lake full of fish here. You are the scientist and I am your fish catching assistant. Our aim is to find out the fraction of each species and get an estimate of the total number of fish.
   Take some time to think about how you will record the data.
4. Take ten fish from the bag, one at a time, tagging the cube with a sticker before returning it to the bag. Watch how students record the data. Discuss the methods they are choosing.
   You might need to suggest using a tally chart and to indicate with an asterisk if the cube is tagged (caught before).
5. After the sample of ten cubes is complete, ask students to predict the fraction of each species and the total number of cubes in the bag. Note that the total number will be impossible to estimate if no tagged fish are caught. Check:
   Do their fractions match the distribution of cube colours in the sample?
   Do they acknowledge that the predictions are very uncertain, given the small sample size?
6. Carry out another sample of 20 cubes, tagging and replacing. At the end of the sample ask your students to predict the fractions and total number.
   Is it better to think about each sample as separate or treat the combined results as one big sample? 
   Pay particular attention to the issue of the total number of cubes.
   For example, suppose five of the 20 cubes selected in the last sample are tagged.
   What can this tell us about the total number of cubes in the bag?
   About one quarter of all the cubes in the bag are tagged. 
   How might we predict the total number of fish from that?
7. Draw a pie chart like this to support your students:
    
   If the ten cubes we tagged at first make up one quarter of all the cubes in the bag, how many cubes might be in the bag?
   Students might realise that four quarters make the whole population so 4 x 10 = 40 is the best prediction they can make.
   How certain are you? What could we do to improve the reliability?
   Students might say that tagging as many fish as possible would improve the reliability of the prediction. In fact, if we keep tagging fish we might reach a stage when every fish we catch is tagged.
   Why is 'tag until every fish is tagged' not feasible in real life? (Population may be very large, the more animals that you tag the harder it is to find an animal that isn’t tagged.)
8. Check the actual contents of the bag. Students might realise that the estimate of total number of cubes (population) is much more reliable than the example above (It should be).
9. Ask students to create a bag for someone else to sample. Limit the colours to a maximum of four but allow students to use any number of cubes between 20 and 50. Provide each student with marking materials like stickers, tape or blu tac.
10. Students exchange bags and carry out an initial sample of ten cubes, tagging and replacing. They then take a sample of 20 cubes, tagging and replacing.
    Tell them to record the data in an organised way and justify the predictions of fractions and total number.
    How do they deal with fish that get selected but are already tagged? Do they tag them again? Does double or triple tagging improve the data? How? Monitor for any students who may need further support and provide as required. 
11. Finish the lesson with a video about tag and release methods used by scientists. Many videos are available online.

Session 5

In this final session, pose the following problem to your students (Copymaster 1). Use their response to consolidate learning and to assess achievement of your students on the probability outcomes.

Look for the following:

Question One: Does the next sample of ten reflect the previous sample but also show variation from that sample? Students should show that the two samples will vary.

Question Two: Can students represent the part-whole relationships as fractions?

Question Three: Do students acknowledge that a green jelly bean could still be in the bag, but it has not been selected in the two samples? A sophisticated response will state something about the likelihood of a green being in the bag.

Question Four: Does the sample of five reflect proportions similar to those in the collected samples, with some variation?

After the students complete the task independently gather the class to discuss the ideas above. Students might reflect on how they might change their answers following the discussion.

Session 1

Today we make predictions about the cubes that are hidden in a bag. We find out that even when we can’t peek in the bag we can still make a good prediction about what is in it. Think of cubes in the bag like all the people in New Zealand or fish in our seas. We cannot know exactly about all of them but we can use statistics to get an idea.

1. Put four cubes in a paper bag (3 red and 1 blue) without students seeing. Provide a context such as red are eels and blue are mudfish.
   Here is a bag with four cubes. The cubes are either red or blue and we’re going to try to find out  how many of each colour there are selecting cubes one at a time.
2. Shake the bag and ask a student to select one cube to show the class. Record the colour on the board and get the student to put the cube back in the bag.
   This is called sampling with replacement. Replacement means we put it back – like a fish.
   (Note: Each time a student takes a cube it must be returned before the next student draws a cube. Otherwise, the probabilities will change.)
3. Ask another student to select a cube.
   What colour have you got?
   If it is the same colour as the one previously drawn ask: 
   Do you think that it is the same cube? Why or why not?
   If it is a different colour ask:
   Does that mean that half the cubes are red and half are blue?
   The important idea is that students acknowledge that such conclusions are speculative. There is no certainty except that at least one red (same colour drawn), or at least one red and one blue (different colours drawn) are in the bag.
4. Ask a third student to draw a cube but this time get them to predict what the cube might be.
   Why did you guess that? How certain are you?
5. Add the third cube colour to the data.
   Has that changed your mind about what is in the bag? Why? Why not?
6. Ask a fourth student to draw a cube.
7. Look at the result of the four draws.
   Do you think that we have seen all the cubes?
   Do we know what the colours of the four cubes are? Why or why not?
   Would we find out more if we had more turns?
8. Let another four students select a cube one at a time with replacement. Add the colour to the data on the board. Before selecting each time, ask the student to predict the colour of the cube.
   Record your best prediction about the colours of the four cubes in the bag.
9. Ask students to discuss their predictions with a buddy and to justify their thinking. Look for acknowledgement of certainty and uncertainty. For example, if all the draws were red, we cannot say for certain that there is no blue cube. If one or more of each colour have been drawn, we can be certain that there is at least one cube of each colour in the bag.

10. Before we look in the bag, discuss all the possible combinations for the colours of the four cubes.
    Record these combinations on the board. Students might forget the four of one colour, zero of the other colour possible combinations.

    Possible combinations for 4 cubes
    Red	Blue
    0	4
    1	3
    2	2
    3	1
    4	0

11. Ask the students to decide which combination they think is most likely.
12. Look inside the bag and check the cubes. Discuss how reliable their prediction was.
    How could we have improved the prediction before checking?
    Students might suggest that more selections might have improved the reliability. Larger samples tend to be more representative than smaller samples
13. Put the cubes back in the bag and ask:
    I am going to draw a cube. Which colour do you think it will be?Why?
    Can we be sure that I will get that one? Why?
    How could we record your chance of success?
    If students choose red their chance of success is 3 out of 4 or 3/4.
    If they choose blue their chance of success is 1 out of 4 or 1/4.
    Though there is a greater chance that the cube drawn will be red, there is still a 1/4 chance that it might be blue.

Exploring

Over the next three days we work in pairs to make our own bags of cubes. We swap them with our friends to see if they can guess "What’s in the bag?"

Session 2 

1. Invite each pair of students to choose 10 cubes. There should be two colours available. Ask them to put 10 cubes in their bag using any combination of the two colours they want.
2. Swap bags with another pair of students. Students can create a context for their bag. E.g. rugby balls and soccer balls, tui and fantail. Each pair must predict how many cubes there are of each colour in the bag by taking turns drawing cubes from the bag, one at a time with replacement. Remind them to put the cube back in the bag after each draw. Tell students that they have 5 minutes to make as many draws as they can. It is important that they record their results. The results can be recorded on a chart or digitally.
3. Ask the students to make a prediction about the colours of the ten cubes in the bag. How will you use the data to make the best prediction you can?
4. Gather the class and share predictions. Do the students:
   • acknowledge that their predictions are uncertain?
   • relate fractions to their predictions, e.g. 20 out of 30 trial cubes were red, that’s 2/3 so 6 or 7 of the ten cubes might be red?
   • provide a range of what events might occur, e.g. 5-7 red and 3-5 blue?

Ask the students to 'think aloud' so other students can build on their understanding.

Session 3

1. In this session students predict events from complete knowledge of the set of cubes. In doing so, they consider the likelihood of the colour of the cube selected next.
2. Show the students a paper bag with 24 cubes in it of varying colours, e.g. 10 yellow, 8 blue, 3 red, 2 green, 1 white. Tip the contents of the bag onto the mat or tabletop. Ask some students to sort the cubes by colour. Ask the students to provide a context for this activity that they can relate to. 
   I want you to create a data display of the colours. You are free to use whatever display you want. What display might you use? (Students might suggest bar graphs, pie charts, frequency tables, pictographs, etc.)
   You can make as many trips as you need up here to view the data but you must create your display back at your desk. Give your students adequate time to create their displays.

3. Explain the rules of the game to your students.
   I am putting all the cubes back into the paper bag. Nothing in the set of cubes has been changed.
   With each turn, one person from the class will take out a cube, tell you what colour it is, show you the cube then put it aside. It will not be put back into the bag.
   Before the cube is taken out you need to make a prediction about what colour it will be. Record your predictions like this:

   Round	Prediction	Correct/Incorrect	Points
   1	Green	I	-1
   2	Yellow	C	3
   3	Red	I	-1
   …

4. If your prediction is correct, write C and give yourself 3 points. If your prediction is incorrect give yourself minus 1 point.
   There will be 24 rounds. At the end of the game the +5 and -1 scores will be combined. The player with the highest score wins.
5. Before the game starts, remind your students about the display they created.
   How might your display be useful?
6. Play the full 24 rounds of the game. As the number of cubes in the bag reduces, the students’ chances of correctly predicting increases to the point that on the last draw they should be certain of the outcome. Students can play this game with a buddy to provide scaffolding if required.
   Watch for the following behaviour from your students:
   • Do they keep track of the colours of remaining cubes, using their data display?
   • Do their predictions match the likelihoods, as expected from the colour frequencies of the remaining cubes?
   • Do they successfully cope with the plus and minus nature of getting their score?
7. At the end of the game interview the winner or winners about the secret to their success. Focus on the way they tracked the number of cubes of each colour left in the bag and how they used the frequencies of colours to make predictions.
8. Pose various scenarios of cubes remaining in the bag and invite students to make predictions about the next cube. For example:
   In the bag are 6 yellow, 4 blue, and 2 red cubes. What coloured cube do you think will be drawn next?
9. Calculate the probabilities for each colour. For example, there are 12 cubes all together so the chance of yellow equals 6/12 or ½, 4/12 or 1/3 of the cubes are blue, and 2/12 or 1/6 of the cubes are red. Yellow has the greatest chance of being selected.
10. Trial selecting the next cube in each scenario. Students should come to realise that a colour may have the best chance but still not be selected. A high probability does not guarantee certainty.

Session 4

In this session students explore the impact of ‘tag and release’ methods of sampling. Such methods are common to biological research in which animals need to be returned to their habitat. The lake in this session could be named as a local lake to relate to students' experiences. 

1. Pose the following problem:
   Suppose you are a scientist. The fish in the lake are precious so you want to return each fish after you have tagged it. How can you get an accurate picture of:
   • The total number of fish in the lake?
   • The fraction of all the fish that are each species (type of fish)?
     Types of freshwater native fish.
2. Invite the students to offer ideas. Question b is like the previous inquiries. Taking fish out of the lake one at a time, noting the species, and returning the fish will allow a reasonable prediction of species as more and more data are gathered.
   What is the advantage of tagging the fish?
   Students will know if a tagged fish is caught that it is already represented in the data.
3. Produce a bag of cubes (fish), e.g. 8 red, 6 yellow, 4 blue, 2 green.
   I have a lake full of fish here. You are the scientist and I am your fish catching assistant. Our aim is to find out the fraction of each species and get an estimate of the total number of fish.
   Take some time to think about how you will record the data.
4. Take ten fish from the bag, one at a time, tagging the cube with a sticker before returning it to the bag. Watch how students record the data. Discuss the methods they are choosing.
   You might need to suggest using a tally chart and to indicate with an asterisk if the cube is tagged (caught before).
5. After the sample of ten cubes is complete, ask students to predict the fraction of each species and the total number of cubes in the bag. Note that the total number will be impossible to estimate if no tagged fish are caught. Check:
   Do their fractions match the distribution of cube colours in the sample?
   Do they acknowledge that the predictions are very uncertain, given the small sample size?
6. Carry out another sample of 20 cubes, tagging and replacing. At the end of the sample ask your students to predict the fractions and total number.
   Is it better to think about each sample as separate or treat the combined results as one big sample? 
   Pay particular attention to the issue of the total number of cubes.
   For example, suppose five of the 20 cubes selected in the last sample are tagged.
   What can this tell us about the total number of cubes in the bag?
   About one quarter of all the cubes in the bag are tagged. 
   How might we predict the total number of fish from that?
7. Draw a pie chart like this to support your students:
    
   If the ten cubes we tagged at first make up one quarter of all the cubes in the bag, how many cubes might be in the bag?
   Students might realise that four quarters make the whole population so 4 x 10 = 40 is the best prediction they can make.
   How certain are you? What could we do to improve the reliability?
   Students might say that tagging as many fish as possible would improve the reliability of the prediction. In fact, if we keep tagging fish we might reach a stage when every fish we catch is tagged.
   Why is 'tag until every fish is tagged' not feasible in real life? (Population may be very large, the more animals that you tag the harder it is to find an animal that isn’t tagged.)
8. Check the actual contents of the bag. Students might realise that the estimate of total number of cubes (population) is much more reliable than the example above (It should be).
9. Ask students to create a bag for someone else to sample. Limit the colours to a maximum of four but allow students to use any number of cubes between 20 and 50. Provide each student with marking materials like stickers, tape or blu tac.
10. Students exchange bags and carry out an initial sample of ten cubes, tagging and replacing. They then take a sample of 20 cubes, tagging and replacing.
    Tell them to record the data in an organised way and justify the predictions of fractions and total number.
    How do they deal with fish that get selected but are already tagged? Do they tag them again? Does double or triple tagging improve the data? How? Monitor for any students who may need further support and provide as required. 
11. Finish the lesson with a video about tag and release methods used by scientists. Many videos are available online.

Session 5

In this final session, pose the following problem to your students (Copymaster 1). Use their response to consolidate learning and to assess achievement of your students on the probability outcomes.

Look for the following:

Question One: Does the next sample of ten reflect the previous sample but also show variation from that sample? Students should show that the two samples will vary.

Question Two: Can students represent the part-whole relationships as fractions?

Question Three: Do students acknowledge that a green jelly bean could still be in the bag, but it has not been selected in the two samples? A sophisticated response will state something about the likelihood of a green being in the bag.

Question Four: Does the sample of five reflect proportions similar to those in the collected samples, with some variation?

After the students complete the task independently gather the class to discuss the ideas above. Students might reflect on how they might change their answers following the discussion.