# Hunting for Patterns and Rules

Mathematics

Level 4Level 5Level 6

This learning sequence encourages students to develop their understanding of patterns and algebra. Pattern recognition supports the development of computational thinking. This sequence provides opportunities for students to build their algebraic thinking by analysing patterns and exploring generalisations and relationships.

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.

 Big understandings Patterns are a predictable arrangement of elements (generalisations) which can be either repeating or growing in nature. The equals sign in mathematics represents a balance between the values, equations or expressions written on either side of it. Symbols (pronumerals) can be used to represent unknowns in algebraic equations. These equations can be manipulated to find the unknown.

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 five flexible stages, including suggested learning intentions.

Overview of stages

• 1. Functions and Relations

Suggested Learning Intentions

• To understand that functions describe the relationships between numbers
• To understand that a relation is a set of inputs and outputs
• To understand that a function is a relation with one output for each input

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• 3. Equivalence

Suggested Learning Intentions

• To understand that expressions on either side of an equal sign must be balanced
• To explain that when a function on one side of an equation is performed an equal function must be performed on the other side to maintain balance

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• 2. Exploring Patterns

Suggested Learning Intentions

• To understand that a pattern is a predictable arrangement of elements
• To understand that patterns are determined by the application of different functions
• To understand the difference between repeated and growing patterns

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• 4. Using Symbols to Recognise Unknowns

Suggested Learning Intentions

• To understand that symbols (pronumerals) can be used to represent a variable or an unknown quantity in an equation
• To understand that symbols can be used to represent an unknown in a pattern or a generalisation
• To understand that these equations can be manipulated to determine the unknown quantity

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• Prior knowledge

Before you commence this sequence, it is recommended that students are familiar with creating simple repeating patterns and that they can describe the difference between two numbers.

The following activity supports students to develop their ability to sequence patterns. This activity is interactive.

You can find support for building students' understanding of pattern-related concepts in the Mathematics Curriculum Companion. The Teaching Context and Teaching Ideas related to content descriptions VCMNA112, VCMNA138, VCMNA139, and VCMNA163 may be particularly useful.

Students who require support with skip counting may benefit from additional practice. This Number Patterns Game provides students with the opportunity for multiple exposures in a fun environment.

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:

 Pattern Expression Repeating Pronumeral Growing Factors Unknown Multiples Table of data Rule Inverse operations Symbol Equivalence Algorithm Generalise Function Equation

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

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 5.

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 4

This sequence addresses content from the Victorian Curriculum in Mathematics, Number and Algebra, Patterns and Algebra. It is primarily designed for Level 5, but also addresses the following content descriptions from Level 4:

 Content description Stage Mathematics Define a simple class of problems and solve them using an effective algorithm that involves a short sequence of steps and decisions (VCMNA164) Functions and Relations Exploring Patterns Equivalence Use equivalent number sentences involving addition and subtraction to find unknown quantities (VCMNA163) Functions and Relations Equivalence Solve word problems by using number sentences involving multiplication or division where there is no remainder (VCMNA162) Functions and Relations Using Symbols to Recognise Unknowns Explore and describe number patterns resulting from performing multiplication (VCMNA161) Functions and Relations Exploring Patterns Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (VCMNA154) Functions and Relations Exploring Patterns Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (VCMNA156) Functions and Relations Exploring Patterns Using Symbols to Recognise Unknowns Critical & Creative Thinking Consider concrete and pictorial models to facilitate thinking, including a range of visualisation strategies (VCCCTM018) Exploring Patterns Equivalence Using Symbols to Recognise Unknowns

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum in Mathematics Level 4:

• Students identify unknown quantities in number sentences
• Students use the properties of odd and even numbers and describe number patterns resulting from multiplication
• Students continue number sequences involving multiples of single-digit numbers and unit fractions and locate them on a number line
• They choose appropriate strategies for calculations involving multiplication and division, with and without the use of digital technology.

Level 5

This sequence addresses content from the Victorian Curriculum in Mathematics, Number and Algebra, Patterns and Algebra. It is primarily designed for Level 5 and addresses the following content descriptions:

 Content description Stage Mathematics Use equivalent number sentences involving multiplication and division to find unknown quantities. (VCMNA193) Functions and Relations Equivalence Using Symbols to Recognise Unknowns Describe, continue, and create pattern with fractions, decimals and whole numbers resulting from addition and subtraction. (VCMNA192) Functions and Relations Exploring Patterns Follow a mathematical algorithm involving branching and repetition (iteration) (VCMNA194) Functions and Relations Exploring Patterns Identify and describe factors and multiples of whole numbers and use them to solve problems (VCMNA181) Functions and Relations Exploring Patterns Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies (VCMNA183) Functions and Relations Exploring Patterns Critical & Creative Thinking Identify and form links and patterns from multiple information sources to generate non-routine ideas and possibilities (VCCCTQ023) Exploring Patterns Investigate thinking processes using visual models and language strategies (VCCCTM029) Exploring Patterns Equivalence Using Symbols to Recognise Unknowns

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

• Students find unknown quantities in number sentences and continue patterns by adding or subtracting fractions and decimals
• Students identify and describe factors and multiples.

Level 6

This sequence addresses content from the Victorian Curriculum in Mathematics, Number and Algebra, Patterns and Algebra. It is primarily designed for Level 5, but also addresses the following content descriptions from Level 6:

 Content description Stage Mathematics Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers and make estimates for these computations (VCMNA209) Functions and Relations Equivalence Using Symbols to Recognise Unknowns Continue and create sequences involving whole numbers, fractions, and decimals. Describe the rule used to create the sequence (VCMNA219) Functions and Relations Exploring Patterns Using Symbols to Recognise Unknowns Explore the use of brackets and order of operations to write number sentences (VCMNA220) Functions and Relations Equivalence Design algorithms involving branching and iteration to solve specific classes of mathematical problems (VCMNA221) Functions and Relations Exploring Patterns Using Symbols to Recognise Unknowns Critical & Creative Thinking Identify and form links and patterns from multiple information sources to generate non-routine ideas and possibilities (VCCCTQ023) Exploring Patterns Investigate thinking processes using visual models and language strategies (VCCCTM029) Exploring Patterns Equivalence Using Symbols to Recognise Unknowns

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 solve problems that involve all four operations with whole numbers and describe the use of integers in everyday contexts
• Students specify rules used to generate sequences involving whole numbers, fractions, and decimals
• Students write number sentences using brackets and order of operations.

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 4 Level 5 Level 6 Number patterns and algebraic thinking Generalising Patterns Generalising patterns Representing Unknowns

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