What are the Chances?

Mathematics / Science

Level 5Level 6Level 7

What is this sequence about?

This learning sequence aims to develop student understanding of 'chance' and 'probability'. These concepts provide rich opportunities to investigate and apply a range of mathematical concepts. In this sequence, students will have opportunities to connect ideas of fractions, decimal and percentages and apply their knowledge of data and statistics to solve probability related problems.

The sequence aims to develop students’ understanding of key concepts of probability such as randomness, sample space, independence of events, law of large numbers and using probability to make decisions and answer questions in real world situations.

 Big understandings Probability is concerned with how often an event, that may seem random, can be expected to occur. Probability can help us answer questions or make complex decisions when there is an element of chance involved. Real-world situations can be represented using probability.

The sequence has been written by teachers for teachers. It has been designed to provide students with rich, engaging learning experiences that address the Victorian Curriculum. The sequence consists of four flexible stages, including suggested learning intentions.

Overview of stages

• 1. Probability of an Event

Suggested Learning Intentions

• To describe the probability of an event using words and numbers
• To understand the concepts of randomness and fairness and their impact on determining likelihood of events

Explore
• 3. Sample Space and Independence of Events

Suggested Learning Intentions

• To analyse probability situations and predict the likelihood of events (independence of events)
• To list and compare all possible outcomes for an experiment or chance situation (sample space)
• To understand the difference between short-term variability and long-term stability (law of large numbers)

Explore
• 2. Theoretical and Experimental Probability

Suggested Learning Intentions

• To describe probability in terms of what happened and what might be likely to happen in future events
• To compare the results of small and large trials to determine which provides a more accurate estimate of probability

Explore
• 4. Real World Probability

Suggested Learning Intentions

• To analyse data to make predictions, answer questions and make decisions about real-world issues

Explore
• Prior knowledge

Before you commence this sequence, it is suggested students are familiar with basic probability concepts such as language of impossible, likely and certain (VCMSP175). It is expected students have also had some experience considering independence of events (VCMSP176; VCMSP177). It is expected students can also count by quarters, halves and thirds and represent these on a number line (VCMNA158). It is also expected students have started to make connections between fraction and decimal notation (VCMNA159) and percentages (VCMNA217)

You can find support for building students’ understanding of these concepts in the following sections of the Mathematics Curriculum Companion. The Teaching Context and Teaching Ideas related to content descriptions VCMSP175; VCMSP176; VCMSP177 may be particularly useful.

Teaching strategies

The Mathematics Curriculum Companion provides teachers with content knowledge, suggested teaching and learning ideas as well as links to other resources. Resources are organised by Mathematics strands and sub-strands and incorporate the proficiencies: Understanding, Fluency, Problem Solving and Reasoning. The Companion is an additional resource that you could refer to when you are planning how you might use the sequence in your school.

The sequence highlights opportunities to apply the High Impact Teaching Strategies (HITS), which are a component of the Victorian Teaching and Learning Model

The sequence highlights the use of a variety of scaffolding practices to help support students in the learning process, in particular:

• Orienting (used specifically in Probability of event)
• Guiding (used specifically in Theoretical and experimental probability)
• Modelling (used specifically in Sample space and independence of events and Simulations).

Vocabulary

Students should be able to understand and use the following concepts and terms by the end of the learning sequence:

 Probability Variation Chance Trials Likelihood Frequency Outcome Sample space Randomness Tables Fairness Tallies Prediction Ratio Expected probability Systematic Observed probability Simulations Relative frequency

You can find definitions of some of these terms in the Literacy Teaching Toolkit and the Mathematics Glossary from the Victorian Curriculum and Assessment Authority.

The Literacy in Mathematics section of the Literacy Teaching Toolkit provides several teaching strategies with worked examples demonstrating how teachers can use literacy to support student understanding of mathematical language. A further set of strategies demonstrate how teachers can develop students' literacy skills to support their mathematical problem solving.

Assessment

Opportunities for formative and summative assessment are identified at different stages of the learning sequence. Look for the 'Assessment Opportunity' icon.

You may want to develop a rubric to assess students’ progress. A range of Formative Assessment resources are available from the Victorian Curriculum and Assessment Authority. This includes a Guide to Formative Assessment Rubrics, a series of modules to support you to develop your own formative assessment rubrics, and sample rubrics across six curriculum areas that demonstrate how you can put formative assessment rubrics into practice in the classroom.

In developing a rubric, you may wish to co-construct assessment criteria with your students. Each stage of the sequence provides sample success criteria for students working at Level 7.

The Victorian Curriculum and Assessment Authority has published annotated work samples that provide teachers with examples of student learning achievement at multiple levels for each strand of the Mathematics curriculum.

Victorian Curriculum onnections

Contents

Level 5

This sequence addresses content from the Victorian Curriculum in Mathematics, Science and Critical and Creative Thinking. It is primarily designed for Level 6, but also addresses the following content descriptions from Level 5:

 Content description Stage Mathematics Recognise that probabilities range from 0 to 1 (VCMSP204) Probability of an Event List outcomes of chance experiments involving equally likely outcomes and represent probabilities of those outcomes using fractions (VCMSP203) Probability of an Event Theoretical and Experimental Probability Sample Space and Independence of Events Construct displays, including column graphs, dot plots and tables, appropriate for data type, with and without the use of digital technologies (VCMSP206) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Describe and interpret different data sets in context (VCMSP207) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Science With guidance, pose questions to clarify practical problems or inform a scientific investigation, and predict what the findings of an investigation might be based on previous experiences or general rules (VCSIS082) Theoretical and Experimental Probability Real World Probability With guidance, plan appropriate investigation types to answer questions or solve problems and use equipment, technologies and materials safely, identifying potential risks (VCSIS083) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Decide which variables should be changed, measured and controlled in fair tests and accurately observe, measure and record data (VCSIS084) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Construct and use a range of representations, including tables and graphs, to record, represent and describe observations, patterns or relationships in data (VCSIS085) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Compare data with predictions and use as evidence in developing explanations (VCSIS086) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Critical and Creative Thinking Examine how different kinds of questions can be used to identify and clarify information, ideas and possibilities (VCCCTQ021) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Investigate common reasoning errors including contradiction and inconsistency, and the influence of context (VCCCTR024) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Consider the importance of giving reasons and evidence and how the strength of these can be evaluated (VCCCTR025) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Investigate thinking processes using visual models and language strategies (VCCCTM029) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 5:

• Students list outcomes of chance experiments with equally likely outcomes and assign probabilities as a number from 0 to 1
• Students pose questions to gather data and construct various displays appropriate for the data, with and without the use of digital technology
• Students compare and interpret different data sets.

Level 6

This sequence addresses content from the Victorian Curriculum in Mathematics, Science and Critical and Creative Thinking. It is primarily designed for Level 6, addressing the following content description:

 Content description Stage Mathematics Describe probabilities using fractions, decimals and percentages (VCMSP232) Probability of an Event Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Conduct chance experiments with both small and large numbers of trials using appropriate digital technologies (VCMSP233) Theoretical and Experimental Probability Sample Space and Independence of Events Compare observed frequencies across experiments with expected frequencies (VCMSP234) Theoretical and Experimental Probability Sample Space and Independence of Events Construct, interpret and compare a range of data displays, including side-by-side column graphs for two categorical variables (VCMSP235) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Pose and refine questions to collect categorical or numerical data by observation or survey (VCMSP237) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Design algorithms involving branching and iteration to solve specific classes of mathematical problems (VCMNA221) Sample Space and Independence of Events Real World Probability Make connections between equivalent fractions, decimals and percentages (VCMNA217) Probability of an Event Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Science With guidance, pose questions to clarify practical problems or inform a scientific investigation, and predict what the findings of an investigation might be based on previous experiences or general rules (VCSIS082) Theoretical and Experimental Probability Real World Probability With guidance, plan appropriate investigation types to answer questions or solve problems and use equipment, technologies and materials safely, identifying potential risks (VCSIS083) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Decide which variables should be changed, measured and controlled in fair tests and accurately observe, measure and record data (VCSIS084) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Construct and use a range of representations, including tables and graphs, to record, represent and describe observations, patterns or relationships in data (VCSIS085) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Compare data with predictions and use as evidence in developing explanations (VCSIS086) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Critical and Creative Thinking Examine how different kinds of questions can be used to identify and clarify information, ideas and possibilities (VCCCTQ021) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Investigate common reasoning errors including contradiction and inconsistency, and the influence of context (VCCCTR024) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Consider the importance of giving reasons and evidence and how the strength of these can be evaluated (VCCCTR025) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Investigate thinking processes using visual models and language strategies (VCCCTM029) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 6:

• Students connect fractions, decimals and percentages as different representations of the same number
• Students specify rules used to generate sequences involving whole numbers, fractions and decimals
• Students interpret and compare a variety of data displays, including displays for two categorical variables
• Students compare observed and expected frequencies of events, including those where outcomes of trials are generated with the use of digital technology
• Students specify, list and communicate probabilities of events using simple ratios, fractions, decimals and percentages.

Level 7

This sequence addresses content from the Victorian Curriculum in Mathematics, Science and Critical and Creative Thinking. It is primarily designed for Level 6, but also addresses the following content descriptions from Level 7:

 Content description Stage Mathematics Connect fractions, decimals and percentages and carry out simple conversions (VCMNA247) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Recognise and solve problems involving simple ratios (VCMNA249) Probability of an Event Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Construct sample spaces for single-step experiments with equally likely outcomes (VCMSP266) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Assign probabilities to the outcomes of events and determine probabilities for events (VCMSP267) Probability of an Event Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Science In fair tests, measure and control variables, and select equipment to collect data with accuracy appropriate to the task (VCSIS109) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Construct and use a range of representations including graphs, keys and models to record and summarise data from students’ own investigations and secondary sources, and to represent and analyse patterns and relationships (VCSIS110) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Reflect on the method used to investigate a question or solve a problem, including evaluating the quality of the data collected, and identify improvements to the method (VCSIS112) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Critical and Creative Thinking Consider how to approach and use questions that have different elements, including factual, temporal and conceptual elements (VCCCTQ032) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Examine common reasoning errors including circular arguments and cause and effect fallacies (VCCCTR035) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Investigate when counter examples might be used in expressing a point of view (VCCCTR037) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Theoretical and Experimental Probability Sample Space and Independence of Events Real World Probability

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 7:

• Students solve problems involving all four operations with fractions, decimals, percentages and their equivalences, and express fractions in their simplest form
• Students determine the sample space for simple experiments with equally likely outcomes, and assign probabilities outcomes.

Learning Progressions

The Numeracy Learning Progressions support teachers to develop a comprehensive view of how numeracy develops over time. You can use the Numeracy Learning Progressions to:

• identify the numeracy capability of your students
• plan targeted teaching strategies, especially for students achieving above or below the age-equivalent expected level in the Victorian Curriculum: Mathematics
• provide targeted feedback to students about their learning within and across the Progressions.

The sequence is related to the following Progression(s):

 Learning Progression Level 5 Level 6 Level 7 Understanding chance Probabilities Calculating probabilities Calculating probabilities Interpreting and representing data Collecting and displaying data Interpreting data scales Shape of data displays Operating with decimals Understanding and estimating relative size of decimals Understanding the effects of multiplication and division with decimals Flexible strategies for multiplication and division of decimals Interpreting fractions Fractions as numbers Using fractions n/a Operating with percentages n/a Understanding percentages and relative size Find percentage as part of a whole Comparing units n/a n/a Building ratios

Click on the Learning Progression to access more detailed descriptions of student learning at each level.