# Algebra in Action: Patterns, Generalisations and Relationships

Mathematics

Level 7Level 8Level 9 This learning sequence encourages students to develop their understanding of patterns and algebra. Pattern recognition supports the development of computational thinking. Throughout this learning sequence students are exposed to increasingly more complex patterns and relationships. With each pattern, students identify commonalities and differences and make reasonable predictions. Students describe relationships using variables in spoken, written and symbolic form, and make sense of them in context.

Throughout this learning sequence students develop their algebraic thinking skills by analysing patterns, exploring generalisations and relationships. Students use their knowledge of equivalence to find unknown quantities and identify and describe relationships by building rules.

This learning sequence aims to build students’ capacity 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.

 Big understandings Pattern awareness supports early algebraic thinking. Algebraic thinking is about understanding mathematical structure and exploring generalisations. Algebraic notation is used to express generalisations that allow us to solve problems accurately and efficiently.

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.

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. Rules, Rules, Rules

Suggested Learning Intentions

• To use a pattern to find an unknown
• To determine an unknown using pronumerals
• To explore, construct and explain generalisations between two variables

Explore
• 3. The Snail Problem

Suggested Learning Intentions

• To use a pattern to find an unknown
• To describe an unknown using pronumerals
• To solve linear and non-linear equations using algebraic and graphical techniques
• To compare and contrast the elements of linear and non-linear equations
• To successfully apply a range of problem-solving strategies to a given task

Explore
• 2. Garden Beds

Suggested Learning Intentions

• To use a pattern to find an unknown
• To describe an unknown using pronumerals
• To explore, construct and explain generalisations between two variables
• To investigate, interpret and analyse graphs
• To connect algebraic expressions and their graphical representations
• To understand and justify that there is more than one way to describe a generalisation

Explore
• 4. Chocolate Boxes

Suggested Learning Intentions

• To use a pattern to find an unknown
• To describe an unknown using pronumerals
• To explore, construct and explain generalisations between two variables
• To complete a mathematical investigation involving variables
• To successfully apply a range of problem-solving strategies to a given task

Explore
• Prior knowledge

Throughout this learning sequence students will develop their algebraic thinking skills by analysing patterns and exploring generalisations and relationships. 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 number patterns (patterns of numbers changing, successively, through the repeated application of one or more operations).

Students use their knowledge of equivalent number sentences to find unknown quantities and identify and describe relationships by constructing simple rules. Students also explore creating and solving algebraic equations and further develop their graphing skills. Students who require additional support with understanding the various features of graphs and linear functions may benefit from the following two resources:

• Cartesian plane interactive: This resource supports students to understand that each point on a Cartesian plane is represented by a set of coordinates. This interactive activity allows students to drag points across the Cartesian plane and to build shapes.
• Equation of a line: slope and intercept form (coordinate geometry): This resource supports students to compare and contrast graphs of different functions. Students learn to recognise the important features of graphs, such as positive and negative gradients.

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 employs the following teaching strategies:

• Concrete manipulatives
• Student choice
• Structured lessons
• Collaborative Learning
• Multiple exposures
• Questioning
• Feedback
• Metacognition
• Differentiated teaching

Vocabulary

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

 Table of values Coordinates Expression Graphing Rule Plotting Generalisation Axis Equivalent expressions Intercept Multiple representation Gradient Equation Quadratic equation Linear equation Dilation Non-linear equation Reflection Substitution Domain Variable Range

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 7

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

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

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 8 and addresses the following content descriptions:

 Content description Stage Mathematics Plot linear relationships on the Cartesian plane with and without the use of digital technologies (VCMNA283) Garden Beds The Snail Problem Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution (VCMNA284) Rules, Rules, Rules Garden Beds The Snail Problem Plot graphs of non-linear real life data with and without the use of digital technologies, and interpret and analyse these graphs (VCMNA285) Garden Beds The Snail Problem Chocolate Boxes Simplify algebraic expressions involving the four operations (VCMNA281) Chocolate Boxes Critical and Creative Thinking Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Rules, Rules, Rules Garden Beds The Snail Problem Chocolate Boxes

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 simplify a variety of algebraic expressions and connect expansion and factorisation of linear expressions.
• Students solve linear equations and graph linear relationships on the Cartesian plane.

Level 9

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. It is primarily designed for Level 8, but also addresses the following content descriptions from Level 9:

 Content description Stage Mathematics Sketch linear graphs using the coordinates of two points and solve linear equations (VCMNA310) Garden Beds The Snail Problem Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations (VCMNA311) The Snail Problem

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum in Mathematics Level 9:

• Students develop familiarity with a broader range of non-linear and linear functions and relations, and related algebra and graphs.
• Students graph linear relations and solve linear equations, using tables of values, graphs, and algebra.

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 7 Level 8 Level 9 Number patterns and algebraic thinking Generalising patterns / Number properties Representing unknowns Algebraic expressions

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