Coin throws
0
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about the chance of heads or tails.
Task administration:
This task can be completed with pencil and paper or online (with SOME auto marking).
Copyright:
Image source: MS Clipart
Levels:
4, 5
Curriculum info:
Key Competencies:
Keywords:
Description of task:
Students identify the most likely combination of throws for a fair coin, and explain their reasons.
Curriculum Links:
This resource can be used to help to identify students' understanding of ordering the likelihoods for outcomes involving chance.
Key competencies
This resource involves justifying a conclusion using written communications, which relate to the Key Competencies: Thinking and Using language, symbols and text.
For more information see https://nzcurriculum.tki.org.nz/Keycompetencies
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.Answers/responses:
Y10 (04/2016)  
a) 
"A total of 2 heads and 2 tails." [Option 3]
Explanation  Any 1 of:
No student in the sample came up with a completely satisfactory explanation. None mentioned the different ways that 2 heads and 2 tails could occur.

moderate
very difficult

Based on a sample of 123 Year 10 students.
Teaching and learning:
The key concept in this resource is that there are more combinations of outcomes which have 2 heads and 2 tails than there are for any other outcomes. There is only one way of getting 4 heads (HHHH), four ways that 3 heads and 1 tail can occur, and six ways 2 heads and 2 tails can occur. However, each individual outcome is equally likely (e.g., TTTT is as likely as TTHH or HHTT).
See Throwing a coin which explores a very similar situation.
For more information about probability, look at the Probability concept map
Diagnostic and formative information:
Many of the following misconceptions suggest that the student is not yet fully grasping some central aspects of probability.
Incorrect or incomplete responses  Likely misconception 
"A total of 2 heads and 2 tails" [Option 3]
with an explanation similar to these:

Ignores all combination of events (Representative approach)
Sees 2 Heads and 2 Tails as the best representation of all possible throws of a coin four times.

"A total of 2 heads and 2 tails" [Option 3]
with an explanation similar to these:

Confuses frequency with probability (Representative approach)
Thinks a probability of ½ (2/4) means that a half of the throws should be Heads. The student believes that 2 out of the 4 throws should be Heads, and that this is therefore most representative of what should happen.

"They are all equally likely" [Option 4]
with an explanation similar to these:

Believes nothing can be predicted with probability (Outcome approach) These students believe that you cannot tell anything about probability. They would think that 2 Heads and 2 Tails in total is just as likely as 4 Heads and 0 Tails. 
"They are all equally likely" [Option 4]
with an explanation similar to this:

Probability can be influenced (Lack of independence)
Students believe that external factors such as how hard the coin is flipped, whether a wind is blowing, etc. will influence how the coin lands. These students generally still get part a) correct. 
Based on a sample of 123 Year 10 students.
Next steps:
Further discussions with students may help. These can be linked with practical experiences involving probability.
Ignores all combinations of events (Representative approach)
This seems to conflict with the idea that the most likely total number of Heads is 2.
Activity 1
Get these students to work with a simpler situation involving two coins of different denominations (10c and 20c for example).
This seems to conflict with the idea that the most likely total number of Heads is 2.
Activity 1
Get these students to work with a simpler situation involving two coins of different denominations (10c and 20c for example).
 Get them to list the different possible combinations. If they only list 3, get them to separate out "10c Heads and 20c Tails" as different from "10c Tails and 20c Heads", i.e. there are two ways to get 1 head and one tail (HT and TH). One way to get all combinations is to draw a tree diagrams.
 Get them to throw the coins a large number of times, recording their results. They should observe that each of the four options is approximately as common, i.e. they are equally likely. However a total of 1 Head and 1 Tail are twice as likely because there are two ways of getting it (HT and TH).
Confuses frequency with probability (Representative approach)
Students need to distinguish between the probability of one throw being Heads (^{1}/_{2}) and the frequency of Heads, which could be as low as 0 or as high as 4.
Students need to distinguish between the probability of one throw being Heads (^{1}/_{2}) and the frequency of Heads, which could be as low as 0 or as high as 4.
Students may need to use the steps in Activity 1 to distinguish between the most likely number of Heads and Tails (which has several ways of occurring), and the probability of any particular sequence.
Believes nothing can be predicted with probability (Outcome approach)
Students need many experiences of random events, i.e., events driven by probability. Students who answer "It’s all just a matter of chance" may need to be asked to explain their reasoning. One effective approach is to use the Predict, Observe, Explain (POE) approach. They could be asked "What will the sum of 2 dice be?" and "Is each sum equally likely?" If students are using the outcome approach, they may well predict that each sum between 2 and 12 is equally likely "because it is all just chance". Doing an experiment and graphing their results will quickly dispel this misconception. If students do a tally chart or a dot plot of their results they will see that the data starts to form a shape (distribution) which becomes closer to a triangle the more times the dice are thrown. This shows that chance itself follows its own set of rules, but these are based upon the laws of probability.
Students need many experiences of random events, i.e., events driven by probability. Students who answer "It’s all just a matter of chance" may need to be asked to explain their reasoning. One effective approach is to use the Predict, Observe, Explain (POE) approach. They could be asked "What will the sum of 2 dice be?" and "Is each sum equally likely?" If students are using the outcome approach, they may well predict that each sum between 2 and 12 is equally likely "because it is all just chance". Doing an experiment and graphing their results will quickly dispel this misconception. If students do a tally chart or a dot plot of their results they will see that the data starts to form a shape (distribution) which becomes closer to a triangle the more times the dice are thrown. This shows that chance itself follows its own set of rules, but these are based upon the laws of probability.
Probability can be influenced (Lack of independence)
Students could explore if they can flip a coin in a way which could lead to them predicting accurately what the outcome will be. We generally assume that throwing a coin or rolling a dice is "random", but is it? This could lead to a rich class discussion.
Further discussions with students on these and other misconceptions may help. These can be linked with practical experiences involving probability.
Figure it out
 Left to Chance, (Statistics L34 page 21) gives an example of throwing a coin several times. While the outcome "Lose" is most probable, each of the eight paths through the network is equally likely.
 Canterbury Colours, (Statistics L34 page 19) gives an almost equivalent activity to throwing a coin two times. It is slightly different because a sweet is not put back into the bag after it is selected (so it is called sampling without replacement, which is not covered until Level 8 of the curriculum).