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The prompt
The prompt
Mathematical inquiry processes: Experiment; reason and extend to other cases. Conceptual field of inquiry: Sample spaces; probability of independent events; Pascal's (Yang Hui's) triangle.
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 being both half the number of coins) might lead students to speculate that the prompt is true. In fact, it is false.
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.
Question, notice, and wonder
Question, notice, and wonder
A mixed attainment year 8 class started a unit on probability with the coins prompt. The students responded to the prompt in the following ways:
What does the 'P' stand for?
Why are the head and tail in brackets?
The probability of landing on a head is 50% no matter what it landed on before.
There is an equal chance of landing on a head and a tail.
Why does it sat two heads and two tails because the coins will land randomly?
The number of heads and tails in the prompt are equal.
The number of heads and the number of tails are each half the number of coins.
P(1 head and 1 tail) when you flip two coins is a half because it is half for a head and half for a tail.
We are wondering if the two probabilities are equal.
Extending the inquiry
Extending the inquiry
The inquiry could develop into finding the probabilities of outcomes with more coins, leading to a generalisation of the probability of flipping n heads with 2n coins?
Updated July 2025
Lines of inquiry
Lines of inquiry
Structured inquiry
Structured inquiry
The structured inquiry explores the difference between experimental and theoretical probability. It involves students in:
Carrying out practical experiments with coins (and also using an online random coin flip generator);
Drawing sample space diagrams; and
Calculating theoretical probabilities for different numbers of coins.
The teacher might draw out the pattern in the number of permutations for each outcome (see the picture), which leads into lines of inquiry related to Pascal's (Yang Hui's) triangle.
Lines of inquiry using Pascal's triangle
Lines of inquiry using Pascal's triangle
After the structured inquiry, the slides contain different lines of inquiry with Pascal's triangle.
The initial tasks explore patterns in the triangle formed by multiples of different numbers, the sum of each row, and sequences in the diagonals.
In more lines of inquiry, students use the terms in Pascal's triangle to work out probabilities. For example, from the the sixth row (1 6 15 20 15 6 1) whose sum is 64, the second number gives the probability of flipping five heads and one tail - that is, 6 out of 64. After practising the procedure, students verify some intriguing statements from which they form generalisations:
P(2 heads) when you flip three coins = P( 2 heads) when you flip four coins
2 x P(1 head) when you flip seven coins = P( 2 heads) when you flip eight coins
Question-driven inquiry
Question-driven inquiry
The picture shows the questions and observations from a year 8 mixed attainment class. Students ask about the notation and grapple with the meaning of the prompt.
They make assertions about the theoretical probability of getting two heads and two tails from four coins and wonder what would happen with six 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.
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.
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.
Structured inquiry
Structured inquiry
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. The question about the outcome with three coins opens up another line of inquiry in which students explore probabilities for different numbers of coins.
Amy decided to develop the students' understanding of probability trees by running a structured inquiry. Students requiring support were offered a probability trees inquiry sheet.
A curious discovery
A curious discovery
In an Inquiry Maths workshop at the University of Birmingham, two trainee teachers made a curious discovery. John Wood and Ben Crossley who were studying for a Postgraduate Diploma in Secondary Education noticed that:
P(2 heads) from 3 coins = P(2 heads) from 4 coins
Their observation can be generalised, as John and Ben went on to show using Pascal's triangle (see the picture of their notes).
P(1 head) from 1 coin = P(1 head) from 2 coins
P(3 heads) from 5 coins = P(3 heads) from 6 coins
P(n heads) from (2n - 1) coins = P(n heads) from 2n coins
The workshop was organised by Tom Francome, the course leader.
December 2019
An alternative prompt
An alternative prompt
The coins prompt was inspired by a different prompt devised by Nicola Stokes (a Head of Mathematics in East Sussex, UK):
P(2 heads and 2 tails) when you flip 4 coins = 2 x P(1 head and 1 tail) when you flip 2 coins
Nicola tried out the prompt with her year 9 class. The students responded by asking the questions and making the statements shown in the picture.
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.
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