Mathematical inquiry processes: Experiment; reason and extend to other cases. Conceptual field of inquiry: Sample spaces; probability of independent events.
The prompt is suitable for introducing the concept of probability. It uses mathematical notation in a familiar context of flipping coins. The superficial similarity of the cases (the number of heads and tails both being half the number of coins) might lead students to speculate that the prompt is true. In aiming to set the prompt just above the level of the class, the teacher might expect the students to be able to explain the case of two coins, but four coins would be beyond their powers of reasoning. In the classroom, the prompt has developed into the following lines of inquiry:
Carrying out practical experiments with coins or using software such as Excel to simulate the experiment (experimental probability);
Drawing sample space diagrams;
Calculating theoretical probability;
Using probability tree diagrams;
Understanding the relevance of Pascal's triangle (combinations, patterns and the binomial expansion);
Changing the prompt (more coins, different outcomes); and
Looking for a pattern in the probabilities of the outcomes with 2, 4, 6 ... coins.
Developing inquiry from students' questions
These are the questions and observations from a year 8 mixed attainment class. They show students asking about notation, grappling with the difference between the meanings of 'probability' and 'outcome', suggesting changes to the prompt and making assertions about the theoretical probability of getting two heads and two tails on four coins. The class went on to create sample spaces for two and four coins. Pascal's triangle proved useful in extending the prompt to other even numbers of coins (see below). The class worked out the probabilities as follows:
Students ended the inquiry by studying patterns in Pascal's triangle. The highest attaining students derived the binomial expansion for the first few cases of (x + 1)n.
Amy Flood (a head of mathematics in Tower Hamlets, London) used the prompt with her mixed attainment year 8 class. The students' questions could lead into different lines of inquiry, including sample spaces and experimental and theoretical probability. Another contribution suggests exploring other cases with different numbers of coins. Amy decided to develop the students' understanding of probability trees by running a structured inquiry. Students requiring support were offered the sheet below.
A curious discovery
In an Inquiry Maths workshop at the University of Birmingham (December 2019), two trainee teachers, John Wood and Ben Crossley, made a curious discovery:
P(2 heads) when you flip 3 coins = P(2 heads) when you flip 4 coins
Their observation can be generalised, as John and Ben went on to show using Pascal's triangle (see their notes above).
P(1 head) when you flip 1 coin = P(1 head) when you flip 2 coins
P(3 heads) when you flip 5 coins = P(3 heads) when you flip 6 coins
P(n heads) when you flip (2n - 1) coins = P(n heads) when you flip 2n coins
John and Ben were studying for a Postgraduate Diploma in Secondary Education. The course leader is Tom Francome who you can follow on twitter @TFrancome.
An alternative prompt
P(2 heads and 2 tails) when you flip 4 coins =
2 x P(1 head and 1 tail) when you flip 2 coins
The main prompt was inspired by this alternative, which was devised by Nicola Stokes (a Head of Mathematics in East Sussex, UK). Nicola tried it out with her year 9 class. The students responded to the prompt by asking the questions and making the statements above. Nicola reports that the prompt "worked really nicely and all of the class chose to continue to investigate if the prompt was true as they couldn't immediately see that it wasn't." During the inquiry, the students developed fluency in constructing sample space diagrams and tree diagrams.