# Real World, Real Numbers: Ratio and Proportions

Mathematics

Level 6Level 7Level 8

Proportional reasoning is important as it forms the basis of student learning in other mathematical domains such as measurement and geometry and financial mathematics (Siemon et al., 2018). It is a skill we often rely on daily when comparing the relationship of proportions.

This learning sequence aims to develop student understanding of the relationships between ratio and proportion in a range of everyday situations. Students will extend their understanding of fraction equivalence, fraction quantities and percentages to make sense of the multiplicative relationships that underpin proportional reasoning. By investigating proportional situations in real world contexts, students will have opportunities to simplify and compare ratios, divide quantities into ratios, and explore simple rates with a focus on the proficiency strands of problem solving and reasoning.

 Big understandings Ratios are a way to represent a proportion and are closely related to fractions. Ratios express relationships between two quantities or measures. Ratios and proportions involve multiplicative rather than additive comparisons.

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.

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. Building Understanding of Ratios

Suggested Learning Intentions

• To recognise the relationships between quantities or measures in ratios

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• 3. Comparing Ratios

Suggested Learning Intentions

• To apply our knowledge of equivalence to compare ratios

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• 2. Finding the Unknown in Ratios

Suggested Learning Intentions

• To build fluency in applying our understanding of ratios to solve problems

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• 4. Finding the Better Deal

Suggested Learning Intentions

• To evaluate different rates and prove that one is better than the other

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• Prior knowledge

Before you commence this sequence, it is suggested that you ensure your students are familiar with calculating equivalent fractions, simplifying fractions and comparing fractions. Students should also have some familiarity with making connections between fractions, decimals and percentages.

You can find support for building students’ understanding of these concepts in the following sections of the Mathematics Curriculum Companion:

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

Vocabulary

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

 Ratio Fraction Proportion Rate Proportional Per Additive Comparison Multiplicative Relationship Part Unit Equivalence Covariation

You can find definitions of some of these terms in the Glossary for the Mathematics Curriculum.

It is recommended that the explicit teaching of vocabulary occur throughout this learning sequence. 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 connections

Contents

Level 6

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

 Content description Stage Mathematics Compare fractions with related denominators and locate and represent them on a number line (VCMNA211) Comparing Ratios Finding the Better Deal Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies (VCMNA213) Finding the Unknown in Ratios Critical and Creative Thinking Consider the importance of giving reasons and evidence and how the strength of these can be evaluated (VCCCTR025) Comparing Ratios Finding the Better Deal

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

• Students locate fractions and integers on a number line and connect fractions, decimals and percentages as different representations of the same number
• Students calculate simple fractions of a quantity.

Level 7

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

 Content description Stage Mathematics Compare fractions using equivalence. Locate and represent positive and negative fractions and mixed numbers on a number line (VCMNA242) Comparing Ratios Finding the Better Deal Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (VCMNA243) Comparing Ratios Multiply and divide fractions and decimals using efficient written strategies and digital technologies (VCMNA244) Comparing Ratios Finding the Better Deal Express one quantity as a fraction of another, with and without the use of digital technologies (VCMNA245) Finding the Unknown in Ratios Recognise and solve problems involving simple ratios (VCMNA249) Building Understanding of Ratios Finding the Unknown in Ratios Comparing Ratios Investigate and calculate 'best buys', with and without digital technologies (VCMNA250) Finding the Better Deal Critical and Creative Thinking Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Building Understanding of Ratios Finding the Unknown in Ratios Comparing Ratios Finding the Better Deal Examine a range of learning strategies and how to select strategies that best meet the requirements of a task (VCCCTM041) Building Understanding of Ratios Finding the Unknown in Ratios Comparing Ratios Finding the Better Deal Consider how problems can be segmented into discrete stages, new knowledge synthesised during problem-solving and criteria used to assess emerging ideas and proposals (VCCCTM042) Finding the Unknown in Ratios Comparing Ratios Finding the Better Deal

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
• Students express fractions in their simplest form.

Level 8

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

 Content description Stage Mathematics Solve a range of problems involving rates and ratios, including distance-time problems for travel at a constant speed, with and without digital technologies (VCMNA277) Finding the Unknown in ratios Comparing Ratios Finding the Better Deal Critical and Creative Thinking Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Building Understanding of Ratios Finding the Unknown in Ratios Comparing Ratios Finding the Better Deal Examine a range of learning strategies and how to select strategies that best meet the requirements of a task (VCCCTM041) Building Understanding of Ratios Finding the Unknown in Ratios Comparing Ratios Finding the Better Deal Consider how problems can be segmented into discrete stages, new knowledge synthesised during problem-solving and criteria used to assess emerging ideas and proposals (VCCCTM042) Finding the Unknown in Ratios Comparing Ratios Finding the Better Deal

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

• Students estimate answers and solve everyday problems involving rates, ratios and percentages, with and without the use of digital technology.

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 Interpreting fractions Using Fractions N/A N/A Comparing units N/A Building ratios Ratios and Rates

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