Exploring How Meanings are Made in Mathematics Minnie Loh Maryknoll Convent School Topic: Introduction to Probability [Form 3]

3 Elements of Mathematical Language John Polias (2007) stated that teachers can make meaning in Mathematics through three elements. They are (1)Linguistic (2) Visual (3) Symbolic

(1)Linguistic The linguistic includes the language of the mathematical text and the language used by teachers and students in dealing with mathematical knowledge. (2)Visual Visual representation includes diagrams, graphs and pictures. They are used to make meaning in Mathematics. (3)Symbolic Makes mathematical meaning through symbols and the arrangement of those symbols.

Topic (field): F.3 Mathematics – Introduction to Probability Main objectives of the unit of work: (i)Explore the meaning of probability (ii)Calculate the theoretical probability by listing the sample space and counting (iii)Investigate geometric probability (iv)Compare experimental and theoretical probability (v)Find the expected number of occurrences of an event (vi)Find the expected value of an event

Lesson 1 & 2 (i)Explore the meaning of probability –Concept of the likelihood of an event happening. –A number can indicate more clearly the likelihood or chance for an event to happen. –Introduce the terms ‘probability’, ‘sample space’, ‘impossible event’ and ‘certain event’ –Concept of equally likely outcomes through examples.

Lesson 3 & 4 (ii)Calculate the theoretical probability by listing the sample space and counting - Introduce the term ‘theoretical probability’ - - Tree diagram & tabulation method to list all the possible outcomes

Lesson 5 (iii)Investigate geometric probability - Introduce the term ‘geometric probability’ - Calculate the probability by referring to the areas or length of the figures involved.

Lesson 6 & 7 (iv) Compare experimental and theoretical probability - Introduce the term ‘experimental probability’ - Perform an experiment by throwing a fair dice

Lesson 8 (v) Find the expected number of occurrences of an event. - Introduce the ‘expected number of occurrences of an event’. - Calculate the expected number of occurrences of an event through examples.

Lesson 9 & 10 (vi) Find the expected value of an event - Introduce ‘expected value of a trial’ - Introduce the calculation of expected value of an event through examples.

Remarks about the teaching strategy: Introduce the Mathematics definitions to students by unpacking nominal groups.

Pattern for definitions: Word being defined + relating process + class or generalization [Nominal group] [embedded clause or phrase] Example: A ratio is an ordered comparison of quantities of the same kind. This definition pattern can help students to see how definitions are constructed through unpacking the nominal group.

Examples used in this topic: Example: The number used to indicate the likelihood of an event happening is called its probability. Probability is the number used to indicate the likelihood of an event happening. Example: The collection of all possible outcomes is called the sample space. The sample space is the collection of all possible outcomes.

Examples used in this topic: (cont ’ ) Example: If the probability of an event happening in a single trial is p, then we expect after n trials this event will occur np times. Suppose p is probability of an event happening in a single trial. The expected number of occurrences of an event after n trials is np times.

Advantages of Unpacking Nominal Group Catch students’ attention by identifying the word being defined first. Build up students’ understanding by showing the class that the word belongs to. Finally, qualify elements are used to elaborate the term to strengthen students’ understanding.

In fact, the technique of unpacking nominal groups in mathematics is not just for understanding definitions. This can also help students to identify the nominal groups in the math problems, so that students are clear about the meaning being made in the question.

Example: There are 120 apples in box. Given that the probability that an apple is rotten is 0.08, find the expected number of rotten apples in 100 boxes of apples is probability that an apple is rotten. The expected number of rotten apples in a box of 120 apples is 120  0.08 = 9.6.  The expected number of rotten apples in 100 boxes of apples = 9.6  100 = 960 Recall: Suppose p is probability of an event happening in a single trial. The expected number of occurrences of an event after n trials is np times.

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