Colm Sweet (a maths teacher in West Sussex, UK) devised this prompt in 2007 and was the first to use it in a classroom. The prompt has led to highly successful inquiries, combining student-led exploration and teacher instruction. The inquiries have moved from calculations of theoretical probabilities to models that attempt to take account of 'real world' considerations.
As it is not always evident to students that the prompt can be linked to probability, the teacher might guide the orientation phase of the inquiry more than is the case for other inquiries on the website. For example, the teacher could restrict questions and comments to those that relate to picking balls from the triangle or to the arrangement of the 'racked' balls. Thus, questions and comments have included the probability of picking certain colours (or of not picking a certain colour) and the probability of picking a given sequence of two or three balls. They have also involved the probability of racking the balls in the way shown or racking the reds and blues in rows of two and three respectively. In the early phases of the inquiry, the teacher is advised to establish that each outcome - that is, the result of picking one of the six balls - is equally likely.
The prompt has exposed students' misconceptions of probability. Some examples of statements that have generated class discussion or led to requests for teacher instruction are:
- There are 27 possible permutations of blue, yellow and red balls if you select three balls with no replacement (not taking account of what balls are left in the triangle).
- The probability of choosing blue, red and yellow in that order is 6/15.(attempting, incorrectly, to add the fractions instead of multiplying them).
- The probability of picking three balls without any blues is 3/19 (believing that all outcomes are equally likely).
- The probability of a combination of red, blue and yellow is 1/20 (confusing 'combination' with 'permutation').
In classroom inquiry to date, the prompt has been extended in two ways:
- Teachers have introduced students to the meaning of the nCr and nPr buttons on a scientific calculator; and
- Different constraints have been applied to the prompt to increase the complexity of the inquiry. For example, students co-constructed a context (under the teacher's guidance) in which the outcomes of selecting different balls are not equally likely. They proposed that the selection is made by 'standing' at the bottom right-hand corner. In that case, it was surmised that the probability of picking the first red ball is twice as high as the probability of picking the two balls in the second row (yellow and blue), which, in turn, are twice as likely to be picked as the three remaining balls. The probability calculations that followed took into account the new context.
These documents show the questions and comments that arose in the initial phase of two inquiries. The teachers collated the responses for students to choose a problem to continue the inquiry.
Promethean flipchart download
Smartboard notebook download