# How Likely is That?

Mathematics

Level 6Level 7Level 8

This learning sequence encourages students to develop their understanding of statistics and probability. Students design and conduct surveys, they organise, analyse, and present data and use it to make generalisations and draw conclusions about a population. They use mean, median and mode to summarise and analyse data.

Throughout this sequence students develop statistical terminology and statistical literacy; they develop an ability to read and interpret graphs and tables and make judgements about the reasonableness of arguments based on data.

Students conduct chance experiments and describe the likelihood of each outcome using fractions, decimals, and percentages. They investigate and compare theoretical and experimental probability.

 Big understandings Whilst we cannot predict many events with certainty, we can use probability to describe how likely they are to happen. We use statistics to investigate and draw conclusions about phenomena.

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 4 flexible stages, including suggested learning intentions.

There is a strong focus within this sequence on supporting students to develop the four mathematical proficiencies set out in the curriculum: Understanding, Fluency, Problem Solving and Reasoning, as well as their capacity for critical and creative thinking.

Overview of stages

• 1. What Does our Village Look Like?

Suggested Learning Intentions

• To understand the difference between a population and a sample
• To design and conduct a survey
• To gather and evaluate categorical and numerical data and represent it using fractions, decimals, and percentages

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• 3. Four Balls

Suggested Learning Intentions

• To predict the likelihood of an outcome
• To understand that some games involve random processes and that previous results have no impact upon future outcomes
• To explain the difference between dependent and independent variables

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• 2. And the Average is...

Suggested Learning Intentions

• To describe, interpret and compare data using mode, median, mean and range
• To explain the difference between a population and a sample

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• 4. Take a Chance

Suggested Learning Intentions

• To understand that a ‘game of chance’ involves chance rather than skill to determine the outcome
• To list the outcomes of chance experiments
• To define and compare experimental and theoretical probability

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• Prior knowledge

Before you commence this sequence, it is recommended that students are familiar with:

• common terminology and vocabulary associated with statistics and probability
• using scaled instruments to measure and compare length, mass, capacity, and temperature
• using metric units of length, mass and capacity to measure, order and compare objects.

Students would also benefit from a basic understanding of two-dimensional and three-dimensional shapes.

You can find support for building students understanding of these concepts in the Mathematics Curriculum Companion. The following resources may also be useful:

• A short interactive resource where experimental probability is compared with theoretical probability (alternatively search: 'experimental probability spinner')
• This clip from ClickView explores the basics of probability using a deck of cards and coins. Students explore sample space, favourable outcomes, possible outcomes, theoretical probability, experimental probability and how to express probability using fractions, decimals, and percentages. Sign into ClickView using your department credentials.

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 for 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

Vocabulary

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

 Chance Long-term stability Probability Trials Experimental probability Sample space Outcome Mean Prediction Median Theoretical probability Mode Relative frequency Discrete data Fair Continuous data Unfair Range Variation Frequency Short-term variability Population

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 this 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 connections

Contents

Level 6

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

 Content description Stage Mathematics Describe probabilities using fractions, decimals and percentages (VCMSP232) And the Average is … Four Balls Take a Chance Conduct chance experiments with both small and large numbers of trials using appropriate digital technologies (VCMSP233) Four Balls Compare observed frequencies across experiments with expected frequencies (VCMSP234) Four Balls Take a Chance Construct, interpret and compare a range of data displays, including side-by-side column graphs for two categorical variables (VCMSP235) What Does our Village Look Like? And the Average is … Pose and refine questions to collect categorical or numerical data by observation or survey (VCMSP237) What Does our Village Look Like? Make connections between equivalent fractions, decimals and percentages (VCMNA217) What Does our Village Look Like? And the Average is… Take a Chance

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

• Students list and communicate probabilities of events using simple ratios, fractions, decimals and percentage
• Students compare observed and expected frequencies of events.

Level 7

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. The following tables show the relevant content descriptions for Level 7:

 Content description Stage Mathematics Calculate mean, median, mode and range for sets of data. Interpret these statistics in the context of data (VCMSP270) What Does our Village Look Like? And the Average is… Describe and interpret data displays using median, mean and range (VCMSP271) What Does our Village Look Like? And the Average is… Assign probabilities to the outcomes of events and determine probabilities for events (VCMSP267) Four Balls Construct and compare a range of data displays including stem-and-leaf plots and dot plots (VCMSP269) Take a Chance Connect fractions, decimals and percentages and carry out simple conversions (VCMNA247) And the Average is…. Multiply and divide fractions and decimals using efficient written strategies and digital technologies (VCMNA244) What Does our Village Look Like? Take a Chance

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

• Students identify or calculate mean, mode, median and range for data sets
• Students describe the relationship between the median and mean in data displays.

Level 8

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. It is primarily designed for Level 7, but also addresses the following content descriptions from Level 8:

 Content description Stage Mathematics Identify complementary events and use the sum of probabilities to solve problems (VCMSP294) Four Balls Represent events in two-way tables and Venn diagrams and solve related problems (VCMSP296) Four Balls Take a Chance Distinguish between a population and a sample and investigate techniques for collecting data, including census, sampling and observation (VCMSP297) What Does our Village Look Like? And the Average is… Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes (VCMSP298) Four Balls Explore the variation of means and proportions of random samples drawn from the same population (VCMSP299) And the Average is …. Investigate the effect of individual data values including outliers, on the range, mean and median (VCMSP300) What Does our Village Look Like? And the Average is… Four Balls

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum in Mathematics Level 8:

• Students choose appropriate language to describe events and experiments.

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 progressions:

 Learning Progression Level 6 Level 7 Level 8 Understanding chance Calculating probabilities Calculating probabilities Calculating probabilities Interpreting and representing data Interpreting data scales Shape of data displays Graphical representation of data Interpreting fractions Using fractions

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