In this unit students play the game "Top Drop" which has them predicting the outcome of dropping a plastic drink bottle top. The outcome of each drop is recorded and used to develop an understanding of relative frequency and probability and to increase the likelihood of winning.

- use relative frequency to predict events
- develop and evaluate strategies to win based on the relative frequency
- explain why probability is not an accurate predictor of events

This unit focuses on understanding the probability of events when all the outcomes are not equally likely and using this to help predict outcomes. The relationship between relative frequency and probability, the accuracy and exactness of numerical values given for probability and variation are key concepts explored in this unit. Events with outcomes not equally likely are different to the ones students are often introduced to first when looking at probability, i.e. rolling a dice, tossing a coin or spinning a spinner. With these all the outcomes are known and the probability of each outcome can be worked out because they are equally likely, e.g. the probability of rolling a 6 with a normal dice is one sixth or the probability of a spinner pointing to blue when three quarters of the spinner is red and one quarter blue is 25%.

Working out how often a plastic bottle top will land facing "up" cannot be worked out in the same way. The questions, "Will it land facing "up" about the same number of times it lands facing "down"?" and "Will it ever land on its "side"?" need to be answered. Simply describing the probability as a fraction with the number of possible outcomes as the denominator does not work with these events. Before the probability of an outcome is assigned, much experimenting is needed to "see" what happens. In this case, the top needs to be dropped many times. Keeping a record of the number of times the top lands facing "up", "down" and "side" is very important. The number of times the top lands facing a particular way out of the total number of times the top was dropped is called its relative frequency. This relative frequency of an outcome is the probability of the outcome. For example, if the top is dropped 500 times and 400 times it landed facing "down" then the probability is likely to be close to 0.8 or 80%. In other words it is likely to land facing "down" in the proportion of 8 out of 10 times when it is dropped many times. The more times dropped the closer to this proportion.

Once students understand relative frequency, then understanding what this means in relationship to predicting outcomes, needs to be explored. Understanding that a relative frequency, i.e. probability, gives some predictability when looking at a large number of outcomes, however its value in predicting the next outcome is not always helpful. Understanding variation is part of understanding probability, i.e. even though the probability of the top landing "down" is 0.8 or 8 out of 10 times, it could land "up" 10 times in a row. This variation also applies to outcomes with equal chances like tossing a coin or rolling a dice. Variation is realising that you could get tails 10 times in a row even though the probability suggest that 5 tails and 5 heads is more likely. Understanding that probability based on relative frequency is at best an indicator or approximation of what might happen with variations occurring. Knowing that probabilities are not precise or an exact way of predicting outcomes is central to understanding probability at this level.

large sheets of paper

plastic bottle tops, one for each student plastic tops from 600ml to 2.25L plastic fizzy drink bottles are best

paper and pencil

probability, relative frequency, outcomes, likelihood, exactness, variation, indicator, approximation, randomness, predictability

### Teaching Sequence

#### Session One

This session introduces the "Top Drop" game to the students.

*The "Top Drop" Game*

This is a game for pairs of students. The players take turns predicting then dropping their own plastic drink bottle top. If the student correctly predicts how the top will end up, they earn one point. The first player to earn 10 points wins.

*Equipment needed*

- one plastic top
- paper and pencil

*How to play*

- The first player predicts which way the top will land ‘down’, ‘up’ or ‘side’.
'down' ‘up’ ‘side’

- Once the player has predicted and told the other player they drop the top. The top must be dropped from a height of approximately 30 cm onto a flat surface. The top must be held on its ‘side’ when it is dropped. The way the top ends once stopped moving is the way it lands, i.e. a top that drops, rolls on its ‘side’ then stops ‘down’ is recorded as ‘down’.
- If the player’s prediction is correct, they are awarded one point. If the player is incorrect, they are awarded no points.
- Each player must keep track of the outcome of each drop, plus the points awarded.
- The first player to be awarded 10 points wins the game.
- The game is to be repeated as many times as possible within the allocated time.

Organize the class into pairs, give out the tops and have each student draw up a recording sheet. Allocate a set amount of time to play the games, e.g. 25 minutes, and get the students to play as many times as they can.

As the games are being played, the teacher is to move around the students, getting them to talk about anything they notice, e.g. the top lands ‘down’ most of the time. The students are to be encouraged to add detail to the things they notice, e.g. it is ‘down’ 4 out of 5 drops, rather than "most of the time".

At the conclusion of the game playing time, assemble the class and record on a large piece of paper the total number of times the top landed ‘up’, ‘down’ and ‘side’ from each pair of players.

Ask the students to describe any ideas or methods they used to predict how the top would land. Develop a name for each idea or method and record it on a second large piece of paper. Accept and record all methods or ideas without judgement. Later in the unit the ideas and methods will be looked at again, with misunderstandings exposed as the ideas and methods are thought about and tested.

For homework, encourage the class to keep thinking about methods to predict what will be next. Invite them to play this game at home and ask their family how they would predict what the top would do.

#### Session Two

Place the two large sheets of paper from Session One so all the students can see them. Explain that during this session you want them to test, think about and decide what is the best way to predict what the top will do.

Remind them that they are trying to predict what will happen when the same top is dropped in the same way, from the same height onto the same surface. It is not about trying to trick each other by adding something to the top or changing it so it does what you want.

Before the students start playing the game get them to discuss and select a method for predicting what will happen. Add any new methods to the large sheet of paper from Session One. It is best to have several students trying the same method. Once the methods have been allocated have the students predict how successful they think the method will be. Get them to write down a value from a scale from 1 to 10, with 1 being a hopeless method not successful at all and 10 being an excellent method correct almost every time, to indicate how good they think the method will be.

Example of methods:

- Always select ‘down’.
- If ‘down’, select ‘up’ next time. If ‘up’ select ‘down’ next time. Always select the opposite to the last outcome. Never select ‘side’.
- If two outcomes are the same in a row, select the other one (not ‘side’) e.g. ‘down’, ‘down’ then select ‘up’.
- Drop it with left hand, select ‘up’; drop it with the right hand, select ‘down’. Change hands each turn.
- Every 4th one choose ‘up’, every 10th one choose ‘side’ otherwise choose ‘down’.
- Select ‘down’, then ‘up’, then ‘side’, ‘down’, ‘up’, ‘side’, ‘down’, ‘up’, ‘side’, etc.
- Watch the clock, if the second hand is exactly at the top when you first look select ‘side’, if it is between the top and half past select ‘down’, if it is between half past and the top select ‘up’.
- Before you start, write down what you think the outcomes will be in order, then select according to your list no matter what the previous outcome.

Play the games in pairs as many times as possible for a set amount of time e.g. 20 minutes. Students need to record the outcome of each drop, ‘up’, ‘down’, or ‘side’ and the player’s points.

At the end of the playing time discuss how effective the methods were. This discussion needs to focus on using numbers to work out if a method is more successful than another. Students may support methods they have suggested but by looking at its success rate an unbiased decisions can be made.

Possible questions to ask during this discussion:

- Who thinks the method they tested worked well? Explain why.
- Who won more games today than in Session One? Who won less?
- What part does luck play in winning games?
- Can you change the amount of luck a person has playing the game?
- Is the method you tested today work better than random guessing?
- Did any method work well in one game? Does this make it a good method?
- Is there likely to be a method that is always correct?
- What do you think would be a good method, correct 10 out of 20 times, 15 out of 20 times?

As part of this discussion have the students look at the large sheet of paper containing the number of times each top landed ‘up’, ‘down’ or ‘side’ from Session One and ask the question;

Does the number of times ‘up’, ‘down’ or ‘side’ occurred in Session One, help when predicting?

Add up the total of ‘up’, ‘down’, or ‘side’ on the sheet. Approximate when adding up so the total numbers are tidy numbers e.g. 350 instead of 347. Ask the students whether they think the totals from Session One will be the same, similar or different to the totals from today’s games. On another large sheet of paper write down the total numbers of ‘up’, ‘down’, and ‘side’ from today’s games. Add up the total numbers on this chart, approximating and comparing with the totals from Session One.

At this point introduce the term ‘relative frequency’ and explain its relationship to probability.

"If an experiment is repeated many, many times without changing the experimental conditions, the relative frequency of any particular event will settle down to some value. The probability of the event can be defined as the limiting value of the relative frequency." from ‘Statistics Glossary’ by Valerie J. Easton and John H. McColl, www.stats.gla.ac.uk

Write up the proportion of ‘up’, ‘down’ and ‘side’ in relationship to the total number of drops, i.e. probability, and discuss how similar proportions are likely to occur if the top was dropped many more times.

After looking at these relative frequencies, i.e. probabilities, discuss how they could be helpful when trying to predict what will happen. Have each student predict how many times ‘up’, ‘down’ and ‘side’ will occur if they dropped a top 20 times.

Have each student drop a top 20 times, recording how many ‘up’, ‘down’ and ‘side’, then discuss the accuracy of their predictions. For some the results will match the relative frequencies while other results will vary widely. The possibility of variation sitting alongside carefully worked out probability figures is a very important part of understanding probability. The figures give some predictability although variation can and do occur. The higher the number of trials, the less likely variation will occur.

It is important that students understand that the relative frequencies from Session One and this session are indicators of the likelihood of each outcome, i.e. probability. For example, if ‘down’ was the outcome 148 times out of 200 drops, then ‘down’ will likely occur in a similar ratio or proportion of the time in the future if things are the same. 148 times out of 200 is approximately 75% or ¾ so it is likely to land ‘down’ 3 out of 4 times when dropped many times. The more times the top is dropped the closer to this proportion. Using these figures would suggest the best strategy to win would be to always select ‘down’ because ¾ of the time you would be correct.

#### Session Three

During this session the students are to create a variation of the "Top Drop" game that is more interesting and challenging. They are to modify and experiment with the number of points awarded for each successful prediction.

The game played during the first two sessions awarded one point if the outcome was successfully predicted no matter which outcome was selected: ‘up’, ‘down’ or ‘side’. Once the relative frequency over many experiments was known, this allowed the strategy most likely to win to be selected, i.e. select the outcome with the highest relative frequency each time.

Students are to change the number of points awarded for successful predictions. For example: if ‘side’ is selected and the next drop is ‘side’ then 3 points are awarded, if ‘top’ is selected and successful then 2 points are awarded and if ‘down’ is selected and successful then 1 point is awarded.

Once two students have modified the game individually, i.e. both have decided on the number of points awarded, they are to get together and play the game using each others point system. Students are to change playing partners and play "Top Drop" using their point system as many times as possible.

At the conclusion of the games’ playing time, discuss the merits, or otherwise, of the different point systems.

#### Session Four

During this session, a "Top Drop" tournament is to take place. The student who wins the most games is declared the tournament winner.

Each student is to play every other student in the class once or as many different students as time allows. Before a game starts, both students are to predict the outcome of a drop and then both drop a top. If both students predict correctly or both predict incorrectly, then they are to predict and drop again, continuing until only one is correct. The player who predicts correctly is the player who decides the number of points awarded for each outcome, the number of points for successfully predicting ‘up’, ‘down’ and ‘side’. The number of points awarded for each outcome is decided at the start of each game. The points can be the same as previous games or different for each game. The other player has the first turn.

Dear Parents and Whanau,

This week in maths we have been playing a probability game called "Top Drop". We have been discussing different ideas and methods of predicting what the next drop might be.

Home Activity: Play a game of "Top Drop" with your child ask them what strategies they are using to make a prediction. "Top Drop" is a game for two people. The players take turns predicting then dropping their own plastic drink bottle top. If the player correctly predicts how the top will end up, they earn one point. The first player to earn 10 points wins. You'll need one plastic top, paper and pencil. Here's how you play. The first player predicts which way the top will land ‘down’, ‘up’ or ‘side’. 'down' ‘up’ ‘side’. Once the player has predicted and told the other player they drop the top. The top must be dropped from a height of approximately 30 cm onto a flat surface. The top must be held on its ‘side’ when it is dropped. The way the top ends once stopped moving is the way it lands, i.e. a top that drops, rolls on its ‘side’ then stops ‘down’ is recorded as ‘down’. If the player’s prediction is correct, they are awarded one point. If the player is incorrect, they are awarded no points. Each player must keep track of the outcome of each drop, plus the points awarded. The first player to be awarded 10 points wins the game. The game is to be repeated as many times as possible within the allocated time.

How good are you at predicting how the top will land? What strategies did you use?