Planning tool
Year levels
Strands
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 twoway 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 twoway 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.
Students will:
 know and be able to use the language of probability accurately
 be able to draw tree diagrams, Venn diagrams and twoway tables from given data
 be able to interpret Venn diagrams and twoway tables, and calculate probabilities from a variety of contexts in relation to all possibilities of two events.
Some students may:
 believe that when there are two possibilities, they must be equally likely (which may be because so many classroom and textbook examples use coin tossing).
 be able to read and interpret a twoway table or Venn diagram but have difficulty constructing them.
Students whose first language is a language or dialect other than English may need particular support in this topic.
The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.
Learning intention
 We are learning the language of probability in the contexts of two events.
 We are learning all possible outcomes of two events.
Why am I learning this?
Information is all around us. We use information to find patterns and make predictions to make our lives easier. We can do this by analysing scenarios, events and data. Probability is useful to us because we can look at history and present events to try to understand the future. We make comparisons on whether two events are equal, bigger or smaller, unequal or whether they are related or independent of one another.
What to do
 Go to this link.
 First, see what you know! Do the background intro quiz. You should know a bit on this topic by now. To learn more, go to the next section.
 Watch the video and then consider the scenarios that appear in the following worksheet.
 Now test yourself. Begin the quiz 'Comparing probabilities'.
 Now go and find a friend and redo the quiz. Stop and discuss the scenarios. Did that help?
 You should know more now and that means you’ve got this when you get to school.
Success criteria
 I understand that there are many possible outcomes for two events.
 I have learnt that there are different contexts requiring different thought processes when thinking about two events in probability and in the world around us.
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Teaching strategies
A collection of evidencebased 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.

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 threephased 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.

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 problemsolving activities.
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Twoway 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.

Representing probabilities: medical testing
In this lesson students apply their understanding of probability in a medical context. They use twoway 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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Twoway tables
Use this resource to introduce the language of ‘and’ and ‘or’ and relate this to probabilities.
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