# The X Factor: Using Algebra to Solve Problems

Mathematics

Level 6Level 7Level 8

What is this sequence about?

This learning sequence aims to build students’ capability to develop and use algebraic generalisations to solve problems accurately and efficiently. It aims to equip students with the skills to express generalisations in multiple ways. By looking for and noticing patterns, students will be able to describe relationships using variables in spoken, written and symbolic form, and make sense of them in context.

 Big understandings Generalisations allow us to solve problems accurately and efficiently. Generalisations can be expressed in multiple ways, and by looking for and noticing patterns, we can develop generalisations using variables to help describe relationships in spoken, written and symbolic form.

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

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.

• The first two stages provide opportunities for open-ended, contextually based mathematical investigations, through which students can explore concepts of algebra sense, and making meaning of algebra. These stages are dedicated to ‘sense-making’, allowing students to develop the proficiency of Understanding.
• The third and fourth stages are open-ended problem-solving tasks, requiring students to use the proficiencies of Problem Solving and Reasoning, as well as some of the understandings developed throughout the first two stages; specifically, the recognition that generalisations are critically useful to efficient problem solving.

Each task provides opportunities for students to develop Fluency in a range of algebraic concepts and ideas, notably: variable and generalisation; symbolic representation using variables, constants, and operations; substitution; solving equations; equivalent expressions; and graphs of algebraic relationships.

Overview of stages

• 1. Paths in the Park

Suggested Learning Intentions

• To explore, construct and explain generalisations representing a relationship between two variables

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• 3. Snail Trail

Suggested Learning Intentions

• To complete a mathematical investigation involving variables
• To successfully apply a range of problem-solving strategies to a given task

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• 2. Garden Beds

Suggested Learning Intentions

• To explore, construct and explain generalisations representing relationships between two variables
• To understand and justify that there can be different ways of writing a generalisation

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• 4. Chocolate Boxes

Suggested Learning Intentions

• To complete a mathematical investigation involving variables
• To successfully apply a range of problem-solving strategies to a given task

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• Prior knowledge

Before you commence this sequence, it is suggested that you ensure your students are familiar with general number operations (addition, subtraction, multiplication, and division) and, in particular, number patterns (numbers changing, successively, through the repeated application of one or more operations).

You can find support for building students’ understanding of these concepts in the Mathematics Curriculum Companion. The Teaching Context and Teaching Ideas related to content descriptions VCMNA138, VCMNA161 and VCMNA219 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

This sequence makes reference to multiple teaching strategies, including modelling 'think-alouds' and scaffolding practices.

More information about modelling 'think-alouds' can be found in the Literacy Teaching Toolkit and from nzmaths. This strategy is used in Paths in the Park and Garden Beds.

The Victorian Department of Education and Training provides information on scaffolding practices, as part of numeracy and maths supports for teachers.

Vocabulary

Depending on the progress made, students should be able to understand and use the following concepts and terms by the end of the learning sequence:

 Table of values Multiple representation Expression Substitution Rule Variable Equation Coordinates Generalisation Graphing Equivalent expressions

You can find definitions of some of these terms in the F-10 Victorian Curriculum Mathematics Glossary.

It is recommended that the explicit teaching of vocabulary occur throughout the 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 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 also published annotated work samples that provide teachers with examples of student learning achievements for each level and 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 Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence (VCMNA219) Paths in the Park Garden Beds Snail Trail

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 specify rules used to generate sequences involving whole numbers, fractions and decimals.

Level 7

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

 Content description Stage Mathematics Introduce the concept of variables as a way of representing numbers as letters (VCMNA251) Paths in the Park Garden Beds Chocolate Boxes Create algebraic expressions and evaluate them by substituting a given value for each variable (VCMNA252) Paths in the Park Garden Beds Chocolate Boxes Extend and apply the laws and properties of arithmetic to algebraic terms and expressions (VCMNA253) Paths in the Park Garden Beds Chocolate Boxes Given coordinates, plot points on the Cartesian Plane, and find coordinates for a given point (VCMNA255) Garden Beds Solve simple linear equations (VCMNA256) Paths in the Park Garden Beds Investigate, interpret and analyse graphs from real life data, including consideration of domain and range (VCMNA257) Garden Beds Critical and Creative Thinking (Level 7 - 8) Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Paths in the Park Garden Beds Snail Trail

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 use variables to represent arbitrary numbers and connect the laws and properties of number to algebra and substitute numbers into algebraic expressions.
• Students develop simple linear models for situations, make predictions based on these models, solve related equations and check their solutions.

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 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) Snail Trail Critical and Creative Thinking Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Paths in the Park Garden Beds Snail Trail

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 estimate answers and solve everyday problems involving rates, ratios and percentages.

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 5 Level 6 to 7 Level 8 Number Patterns and Algebraic Thinking Generalising patterns / Number sentences Representing unknowns Algebraic expression

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