Expected level of development
Australian Curriculum Mathematics V9: AC9M8P02
Numeracy Progression: Understanding chance: P6
At this level, students are introduced to more complex probability concepts, terminology and visual representations for all combinations of two events. Students learn the language and differences between the connectors: ‘and’, ‘or’ (inclusive or exclusive) and ‘at least’.
It can be difficult for students to understand the importance of the connectors as so often they can be interpreted ambiguously. Use example statements and varying contexts to ensure students fully comprehend and recognise the language of probability before calculating it.
Introduce tree diagrams to allow students to visualise and determine all possible combinations for two events. Use Venn diagrams and two-way tables to illustrate the relationship between two sets of data. Venn diagrams are particularly useful when using the language ‘and’ and ‘or’ to describe sets of items. Use questioning to probe student thinking as they describe what is being represented by these diagrams. For instance, are the events mutually exclusive (cannot occur at the same time) or are the two events inclusive (meaning that the result of the second event is influenced by or related to the first)?
Outcomes of these different events are easily demonstrated with concrete tools or digital simulations. Extend knowledge of probability notation and have students practise using word equations alongside the visual diagrams that represent the context of their practical investigations. For example, in the context of ‘and’:
Pr(A and B) + Pr(A and not B) + Pr(not A and B) + Pr(not A and not B) = 1
Teaching and learning summary:
- Revise previous knowledge of probability.
- Define the terms clearly and use worked examples to highlight the differences between terms and contexts. Encourage students to be very precise in their use of language – in probability ‘and’ and ‘or’ have unique meanings, unlike in common English usage.
- Introduce tree diagrams, Venn diagrams and two-way tables, and discuss how they can be used to compare two data sets and calculate probabilities.
- Introduce and encourage the use of more complex probability notation.
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Teaching strategies
A collection of evidence-based teaching strategies applicable to this topic. Note we have not included an exhaustive list and acknowledge that some strategies such as differentiation apply to all topics. The selected teaching strategies are suggested as particularly relevant, however you may decide to include other strategies as well.
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Explicit teaching
Explicit teaching is about making the learning intentions and success criteria clear, with the teacher using examples and working though problems, setting relevant learning tasks and checking student understanding and providing feedback.
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Mathematics investigation
By giving students meaningful problems to solve they are engaged and can apply their learning, thereby deepening their understanding.
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Concrete, Representational, Abstract (CRA)
The CRA model is a three-phased approach where students move from concrete or virtual manipulatives, to making visual representations and on to using symbolic notation.
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Teaching resources
A range of resources to support you to build your student's understanding of these concepts, their skills and procedures. The resources incorporate a variety of teaching strategies.
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Fan Appreciation Night
This interactive uses video and contextualised problems to guide students through complex ideas in probability. All the materials needed are given, including teacher notes, student workbooks and checks for understanding.
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Sets and Venn diagrams
From the AMSI TIMES project. This resource explains Venn diagrams and sets in some detail. Exercises and references are also provided.
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Introducing Venn diagrams
The first in a series of lessons, Venn diagrams are introduced using various interactive problem-solving activities.
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Two-way tables
Use this resource to introduce the language of ‘and’ and ‘or’ and relate this to probabilities.
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An introduction to tree diagrams
Use this resource to demonstrate how to use a tree diagram as a simple way of representing a sequence of events.
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At least one ...
In this problem, students are introduced to tree diagrams and the concept of mutually exclusive events whose probabilities add up to 1.
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Assessment
By the end of Year 8, students can represent the possible combinations of two events with tables and diagrams, and determine related probabilities to solve practical problems.
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Representing probabilities: medical testing
In this lesson students apply their understanding of probability in a medical context. They use two-way tables and Venn diagrams to explain their thinking and interpret their results. Included in the resource are teacher notes, student handouts and links to interactive apps.
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An introduction to tree diagrams
Use this resource to demonstrate how to use a tree diagram as a simple way of representing a sequence of events.
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Two-way tables
Use this resource to introduce the language of ‘and’ and ‘or’ and relate this to probabilities.
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