Mathematical inquiry processes: Develop a line of inquiry; make conjectures and reason. Conceptual field of inquiry: Sample spaces; conditional probability.
Colm Sweet, a teacher of mathematics in West Sussex (UK), devised the prompt in 2007 for use in his own 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. Another approach involves the teacher in requiring students to pose questions in the lesson before the inquiry is due to start. This gives the teacher time to restrict the inquiry to particular topics. The selected questions can then be presented to the students in the form of a co-created worksheet (see examples below).
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:
(1) Teachers have introduced students to the meaning of the nCr and nPr buttons on a scientific calculator; and
(2) 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.
Sequencing inquiry from students' questions
Jhahida Miah (a teacher of mathematics at Haverstock School, London) used the prompt with her year 10 class. Although the students had low prior attainment in mathematics, they asked questions about the prompt throughout the inquiry that gave rise to many different lines of inquiry. Jhahida planned a sequence of lessons, basing each lesson on one or more of the students' question.
If we remove or add balls how does this affect the probability of picking different colours? (Joven's question)
The students worked out the probabilities of picking blue, red and yellow when you remove one, two or three balls of each colour.
What is the probability of picking a blue ball and then picking another blue ball?
The students learnt about the distinction between dependent and independent events and worked on the probability of two, three and four events with and without replacement of the balls.
If I pick two balls what are the possible outcomes?
Students used different sample spaces to list outcomes if you remove two, three or four balls. They used two-way tables for two events and lists for more than two events. A pair of students spontaneously devised a probability tree diagram.
Is there another way I can list the outcomes?
Jhahida posed this question in order to introduce tree diagrams to the whole class. She explained the concept of mutually exclusive events before calling on the pair of students from the previous lesson to show their diagrams. Students drew tree diagrams for picking two balls with and without replacement. Towards the end of the lesson, Jhahida introduced another question that Romar had posed during the lesson, How would the tree change if you add 4 more balls?
Can we change the yellow to red? Less yellows, more reds? If we add 4 more balls we could make a bigger triangle?
Jhahida used these questions to suggest making the triangle bigger. In an introductory task students explored the triangular numbers by drawing triangles on isometric dotty paper. They then tried to design triangles with 10, 15, 21, 28 and 36 balls, changing the numbers of blue, red and yellow balls in an attempt to give the same probabilities for picking different colours as they had found for the triangle in the prompt. Their aim was achieved when a student designed a 36-triangle with 18 blue, 12 red and 6 yellow balls. This led the class, guided by the teacher, to speculate that the solution was related to triangular numbers that are also multiples of six. The next one is 66. One student suggested the number of blue balls must be half of 66 and another then proposed 22 red and 11 yellows. The inquiry concluded with the class discussing the ratio of blue to red to yellow balls to achieve the same probabilities (3:2:1).
Lines of inquiry
The picture was posted on twitter by the Mathematics Department of St Clement Danes School (Chorleywood, UK). The questions and observations come from a year 9 class with higher prior attainment. Their teacher, Amanda Kirby, reports on the development of the inquiry:
The class has done 3 or 4 Inquiry Maths lessons before but this time I decided to put them into groups of 4 and rearranged the tables to encourage discussion. They worked really well and it had the best impact of any of the inquiry lessons they have done so far. Their starter activity was to pose questions and make comments, which we then gathered together on the white board. Each group gave at least one idea. They then used the regulatory cards to decide which line of inquiry they wanted to follow. I guided them to stick to probability.
Some groups started with ease by drawing up sample space diagrams; others asked for help. I had printed off copies of the questions from the website and gave them to three groups. I suggested they try to answer some questions and then they could maybe ask their own. One group started a probability tree straight away but asked for some help to decide what happened to the probabilities on the second branches. Through questioning they decided that they would start by replacing the ball (they were looking at taking two balls out and the outcomes) and then they repeated it for conditional probability. A group started by looking at all the different combinations of the six coloured balls in the triangle.
Another group had drawn a sample space diagram for two balls but had got different answers from calculations than the diagram and we discussed why this was. They then discovered that the balls needed numbering as they weren't replacing the second ball yet they had allowed R1R1 to happen! Another group were trying to list all the combinations for taking three balls (replacing them) and had discussions over whether RRB was to be the same as RBR and BRR. At the end of the second lesson of the inquiry, the groups presented their findings to each other. Overall, the inquiry was excellent for providing differentiation, stretch and challenge.
Stretch through inquiry
James Thorpe, a teacher of mathematics at John Taylor High School, Staffordshire (UK), gives his reflections after using the prompt with different classes:
The beauty of the prompt is that it can be used to tackle probability problems from simple outcomes to more complicated combinations. The screen shots show some of the discussion that took place with a year 9 (higher ability) class on what we could try to find out and how.
We tackled these problems over 3 lessons. I used a worksheet that included questions that came from the pupils' ideas. Starting with the triangle, we covered simple probability, combinations, AND, OR and expectation. The nature of the lessons allowed me to stretch them further than I might have done with a more traditional approach and the students had a sense of ownership of the inquiry. We did stretch to consider picking 3 balls. I showed them how to consider the binomial distribution and the N choose R button on the calculator to find the number of calculations, which the students liked.
If you have good ideas on how to further prompt the pupils to come up with interesting questions and a clear vision of the minimum content that needs to be covered, then this is an engaging way to present a probability unit.