# Number and Algebra Levels F-10A

MATHEMATICS

Scope and Sequence

Number and Algebra Foundation to Level 10

Introduction

What is a Scope and Sequence?

scope

sequence

The breadth and depth of content to be covered in a curriculum at any one time (e.g. week, term, year, over a student’s school life.) All that you do in a given period.

The order in which content is presented to learners over time. The order in which you do it.

Together a scope and sequence of learning bring order to the delivery of content, supporting the maximising of student learning and offering sustained opportunities for learning. Without a considered scope and sequence there is the risk of ad hoc content delivery and the missing of significant learning.

http://activated.act.edu.au/ectl/design/scope_and_sequence.htm

Why does a school need a scope and sequence?

An agreed Scope and Sequence for a Learning Area, provides a sound basis for a school being able to offer a guaranteed and viable curriculum by addressing gaps in students’ leaning and eliminating unnecessary repetition. A shared Scope and Sequence within a school enables teachers to have clarity about the knowledge, skills and dispositions that students will acquire in their learning and what they need to learn next. A Scope and Sequence supports teachers with effective unit and lesson planning and enables teachers to maintain a developmental focus on student learning as students progress through the school.

The Mathematics Scope and Sequence developed by WMR

This document has been developed to support schools with the transition to AusVELS Mathematics for 2013. While it provides examples of yearly overviews and learning sequences based on the content descriptors in the Australian Curriculum, it is not a complete curriculum. Each individual school can use the documents as a basis for developing a guaranteed and viable curriculum that caters for the needs of their school community.

Levels Foundation to 10A each include a set of learning goals/ intentions for each content sub-strand intended to provide a user friendly guide to the essential learnings around which teachers and teams could base their unit and lesson development.

Proficiency strands are listed next to each learning goal / intention as a guide only and teachers / teams are encouraged to consider all proficiencies equally whilst planning units and lessons. Where a particular proficiency is not listed for a content sub-strand teachers and teams should endeavour to contextualise the learning goals to address these proficiencies. Please note the following:

Sequence of teaching

The learning goals/intentions are listed adjacent to the content descriptions to assist teachers when developing a teaching program. They are not necessarily in the order to be taught – teachers /teams will make their own decisions regarding this. The third column has been included to assist teams to develop ideas for unit planning.

A sample Scope and Sequence Overview is also provided for each of the year levels from F to 10A. The number of weeks given to each unit in the overview acts as a guide and the total number of weeks allows for the many interruptions in a typical school year.

Links between the Learning Goals/Intentions and the proficiency strands

(a) The Learning Goals/Intentions have been identified to relate most closely to one of the four proficiency strands (shown in 3 below). This identification is shown in brackets at the end of each Learning Goal/Intention:

· Understanding is identified by (U)

· Fluency is identified by (F)

· Problem Solving is identified by (PS)

· Reasoning is identified by (R)

(b) In this document there are less Problem Solving and Reasoning proficiency strands identified than those for Understanding and Fluency. Should teachers wish to include more of these proficiencies in their curriculum, they are encouraged to emphasise them when teaching, and to develop appropriate learning tasks.

Proficiency strands

The proficiency strands describe the actions in which students can engage when learning and using the content. While not all proficiency strands apply to every content description, they indicate the breadth of mathematical actions that teachers can emphasise. The proficiencies listed next to each learning goal / intention are examples of how students might achieve the goal or what they have demonstrated by achieving the goal but are dependent on the context in which the learning takes place.

Understanding

Students build a robust knowledge of adaptable and transferable mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas. They develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build understanding when they connect related ideas, when they represent concepts in different ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and when they interpret mathematical information.

Fluency

Students develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily. Students are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and regularly use facts, and when they can manipulate expressions and equations to find solutions.

Problem Solving

Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.

Reasoning

Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.

Useful references for teams and teachers to use when planning units of work and lessons include the following:

· Ultranet Design Space – DEECD Big Ideas in Number Maps - 128428217

· Ultranet design Space – Mathematics eBookboxes - 66512121

· Teaching Mathematics Foundations to Middle Years

Dianne Siemon, Kim Beswick, Kathy Brady, Julie Clark, Rhonda Faragher and Elizabeth Warren

· Mathematics Domain Page DEECD

· Building Numeracy – George Booker

· Teaching Primary Mathematics George Booker, Denise Bond, Len Sparrow, Paul Swan

· What We Know About Mathematics Teaching and Learning- MCREL

· WMR Numeracy Design Space 106126201

· Acara Scope and Sequence Documents http://www.australiancurriculum.edu.au/Download

· VCAA – resources http://www.vcaa.vic.edu.au/Pages/foundation10/curriculum/index.aspx

Please note: Teachers will be required to join each Ultranet design space before being able to access the resource. The number associated with each space should be entered into the search box in ‘available design spaces’ in order to find the space.

Foundation Level

Number and Algebra

AusVELS Content Descriptors

Number and place value

· Establish understanding of the language and processes of counting by naming numbers in sequences, initially to and from 20, moving from any starting point(ACMNA001)

· Connect number names, numerals and quantities, including zero, initially up to 10 and then beyond (ACMNA002)

· Subitise small collections of objects (ACMNA003)

· Compare, order and make correspondences between collections, initially to 20, and explain reasoning (ACMNA289)

· Represent practical situations to model addition and sharing (ACMNA004)

Patterns and Algebra

· Sort and classify familiar objects and explain the basis for these classifications. Copy, continue and create patterns with objects and drawings (ACMNA005)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Know the names of the numbers (F)

· Count orally forwards and backwards (initially to and from 20) (F)

· Counting from any starting point to and beyond 20 (F)

· Understand and connect names, numerals and quantities (U)

· Match the names to the numbers and quantities (F)

· Understand and connecting names, numerals and quantities (U)

· Subitise up to 10 (F)

· Understand and connecting names, numerals and quantities (U)

· Compare larger and smaller of two numbers (R)

· Order 3 or more numbers with explanation (R)

· Fluent in counting numbers in sequences readily, continuing patterns, and comparing objects directly

· Reason by explaining comparisons of quantities, creating patterns, and explaining processes for indirect comparisons

· Real world (authentic) problems modelling addition and sharing (U)

· Represent the problem using pictures, numbers or words (F)

· Convert between pictures, numbers and words (story problems) (PS)

· Problem Solve using materials to model authentic problems, sort objects, use familiar counting

sequences to solve unfamiliar problems, and discussing the reasonableness of the answer

Students will:

· Sort and classify objects with justification of the classification (U)

· Copy patterns with explanation of the repeating elements (F)

· Continue patterns with explanation of the repeating elements (F)

· Create patterns with explanation of the repeating elements (R)

· Be Fluent when counting numbers in sequences readily, continuing patterns, and comparing objects directly (F)

· Reason when explaining comparisons of quantities, creating patterns, and explain processes for indirect comparisons (R)

Unit Development Ideas

How is the essential learning developed into units of work? How do students make connections between the learning goals and those included in other content strands or sub-strands? How do we ensure students become proficient in fluency, understanding, reasoning and problem solving?

Achievement Standard

Students make connections between number names, numerals and quantities up to 10. Students count to and from 20 and order small collections.

Level 1

Number and Algebra

AusVELS Content Descriptors

Number and place value

· Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count by twos, fives and tens starting from zero (ACMNA012)

· Recognise, model, read, write and order numbers to at least 100. Locate these numbers on a number line (ACMNA013)

· Count collections to 100 by partitioning numbers using place value (ACMNA014)

· Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015)

Fractions and decimals

· Recognise and describe one half as one of two equal parts of a whole. (ACMNA016)

Money and financial mathematics

· Recognise, describe and order Australian coins according to their value (ACMNA017)

Patterns and Algebra

· Investigate and describe number patterns formed by skip counting and patterns with objects (ACMNA018)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Count forwards and backwards to and from 100 from any number (F)

· Count orally by 2’s, 5’s, 10’s to and beyond 100 starting at 0 (F)

· Understand and connect names, numerals and quantities

· Order Number sequences (U)

· Use number lines and number grids to count forwards, backwards and by patterns (F)

· Model counting using number lines (F)

· Fluent in counting numbers in sequences readily, continuing patterns, and comparing objects directly

· Reason by explaining comparisons of quantities, creating patterns, and explaining processes for indirect comparisons

· Bundle ones to create ten and tens to create 100 (F)

· Create, name and order teen numbers as 10 and some more (F)

· understanding partitioning of numbers and the importance of grouping in tens

· understanding two digit numbers as comprised of tens and ones/units

· model with materials, number lines and number grids addition and subtraction within and including 10 (F)

· Model with materials, number lines and number grids addition and subtractions within and including 20 (F)

· Discuss and compare strategies for addition and subtraction (R)

· develop a range of mental strategies for addition and subtraction problems (F)

· Real world (authentic) problems modelling addition and subtraction (U)

· Represent the problem using pictures, numbers or words (F)

· Convert between pictures, numbers and words (story problems) (PS)

· Problem Solve using materials to model authentic problems, sort objects, use familiar counting

sequences to solve unfamiliar problems, and discussing the reasonableness of the answer

· Model the partitioning of one whole into two equal parts (U)

· Model the partitioning of a group into two equal parts (U)

· Name equal parts as halves, one equal part as one half (R)

· sharing a collection of readily available materials into two equal portions (F)

· Show reasoning by splitting an object into two equal pieces and describing how the pieces are equal

· Recognise Australian coins according to their value (F)

· Order Australian coins according to their value (F)

· Describe and order Australian coins according to their value (R)

· showing that coins are different in other countries by comparing Asian coins to Australian coins (U)

· Understanding that the value of Australian coins is not related to size

· Show reasoning by describing the features of coins that make it possible to identify them

· Identify and say patterns 1’s forwards and backwards from any point to 100 (F)

· Identify and say patterns 2’s to 50, 5’s to 100 and 10’s to and beyond 100 (F)

· Identify and say patterns with explanation of the repeating elements (R)

· Investigate and model patterns on a number grid and a number line as ‘skip counting’ (PS)

· Create patterns with explanation of the repeating elements (R)

· Be Fluent when counting numbers in sequences readily, continuing patterns, and comparing objects directly

Reason when explaining comparisons of quantities, creating patterns, and explain processes for indirect comparisons

Unit Development Ideas

How is the essential learning developed into units of work? How do students make connections between the learning goals and those included in other content strands or sub-strands? How do we ensure students become proficient in fluency, understanding, reasoning and problem solving?

Achievement Standard: Students describe number sequences resulting from skip counting by 2s, 5s and 10s. They identify representations of one half. Students count to and from 100 and locate numbers on a number line. They carry out simple additions and subtractions using counting strategies. They partition numbers using place value. They continue simple patterns involving numbers and objects. They recognise Australian coins according to their value.

Level 2

Number and Algebra

AusVELS Content Descriptors

Number and place value

· Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and ten from any starting point, then moving to other sequences. (ACMNA026)

· Recognise, model, represent and order numbers to at least 1000 (ACMNA027)

· Group, partition and rearrange collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting (ACMNA028)

· Explore the connection between addition and subtraction (ACMNA029)

· Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

· Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031)

· Recognise and represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032)

Fractions and decimals

· Recognise and interpret common uses of halves, quarters and eighths of shapes and collections (ACMNA033)

Money and financial mathematics

· Count and order small collections of Australian coins and notes according to their value (ACMNA034)

Patterns and algebra

· Describe patterns with numbers and identify missing elements (ACMNA035)

· Solve problems by using number sentences for addition or subtraction (ACMNA036)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Count orally forwards and backwards by 1’s from any number beyond 100 to 1000 (F)

· Counting orally by 2’s, 5’s and 10’s to 100 and beyond (F)

· Count backwards by 10’s from 3 digit multiple of 10 (F)

· Count by 100’s to 1000 (F)

· Investigate and describe patterns in number sequences, such as adding 10 always results in the same final digit (R)

· Understand and describe final digit patterns of familiar patterns (U)

· Developing fluency and confidence with numbers and calculations by saying number sequences (F)

· Fluent in counting numbers in sequences readily, continuing patterns, and comparing objects directly (F)

· Reason by explaining comparisons of quantities, creating patterns, and explaining processes for indirect comparisons (R)

· Modelling numbers with materials and diagrams (U)

· recognising there are different ways of representing numbers (U)

· Order numbers in ascending and descending order (F)

· Compare numbers on a number line (U)

· identifying patterns going beyond 100 (R)

· developing fluency with writing numbers in meaningful

contexts (F)

· use materials to model and represent numbers understanding three digit numbers as comprised of hundreds, tens and ones/units (U)

· demonstrate and use models to compare value of numbers in base 10 places (U)

· Use Place Value chart bundling sticks, sticks of Unifix or MAB to model partitioning of 3 digit numbers (F)

· Investigate and discover strategies for efficient counting such as counting up by multiples of 10 then some more on a number line (PS)

· Show Understanding of expanding numbers in 2 and 3 digit numbers through modeling, drawing visuals and describing what is happening with the hundreds, tens and ones (U)

· Real world (authentic) problems modeling addition and sharing with 2 digit numbers (U)

· Represent problems using pictures, numbers and words (F)

· Convert between pictures, numbers and words (story problems) (PS)

· Create addition and subtraction story problems (PS)

· Create, model and record efficient methods for addition and subtraction (R)

· Use number lines and parts and totals models to show the connections between addition and subtraction (U)

· use counting-on to identify the missing element in an

additive problem (F)

· becoming fluent with partitioning numbers to understand the connection between addition and subtraction (F)

· becoming fluent with a range of mental strategies for addition and subtraction problems, such as commutativity for addition, building to 10, doubles, 10 facts and adding 10 (F)

· Problem Solve using materials to model authentic problems, sort objects, use familiar counting

sequences to solve unfamiliar problems, and discussing the reasonableness of the answer (PS)

· Show reasoning by explaining strategies and understanding of connections between addition and subtraction (R)

· Use skip counting on a number line to find ‘four twos’ as an additive concept (F)

· represent array problems with available materials and develop the language of rows and columns (F)

· Use skip counting to count groups of and rows or columns in arrays (F)

· represent array problems with available materials and explain reasoning (R)

· Show understanding by visualising a group of objects as a unit and using this to calculate the number of objects in several identical groups (U)

· dividing the class or a collection of objects into equal sized groups (U)

· identify the difference between dividing a set of objects into three equal groups and dividing the same set of objects into groups of three (R & U)

· develop the language of equal sharing and ‘how many?’ (U)

· Show Understanding by modelling and creating word problems to represent division concepts (U)

· Making models of common fractions such as ½ ¼ 1/3 (F)

· recognise that sets of objects can be partitioned in different ways to demonstrate fractions (U)

· Show understanding by relating the number of parts to the size of a fraction

· Naming the fractions with connection to the sequence words (F)

· identify equivalent values in collections of coins or notes, such as two five cent coins having the same value as one 10cent coin (F)

· counting collections of coins or notes to make up a particular value, such as that shown on a price tag (U)

· Solving real life money problems using 5 cent and 10 cent combination

· Showing Reasoning through justifying solutions and finding alternate representations

· describe a pattern created by skip counting and representing the pattern on a number line (R)

· investigating features of number patterns resulting from adding twos, fives or 10s, 3’s and 4’s (PS)

· Use reasoning to make connections between and describe number patterns

· representing a word problem as a number sentence (PS)

· writing a word problem to represent a number sentence (PS)

· Interpret between Pictures, Numbers and Words to show understanding of addition and subtraction (U)

Solve authentic addition and subtraction problems, explaining strategies and reasoning (R)

Unit Development Ideas

How is the essential learning developed into units of work? How do students make connections between the learning goals and those included in other content strands or sub-strands? How do we ensure students become proficient in fluency, understanding, reasoning and problem solving?

Achievement Standard: Students recognise increasing and decreasing number sequences involving 2s, 3s and 5s. They represent multiplication and division by grouping into sets. Students count to and from 1000. They perform simple addition and subtraction calculations using a range of strategies. They divide collections and shapes into halves, quarters and eighths. They associate collections of Australian coins with their value. Students identify the missing element in a number sequence.

Level 3 Number and Algebra

Achievement Standard: By the end of Level 3, students recognise the connection between addition and subtraction and solve problems using efficient strategies for multiplication. They model and represent unit fractions. They represent money values in various ways. Students count to and from 10 000. They classify numbers as either odd or even. They recall addition and multiplication facts for single digit numbers. Students correctly count out change from financial transactions. They continue number patterns involving addition and subtraction.

Level 4

Number and Algebra

AusVELS Content Descriptors

Number and place value

· Investigate the conditions required for a number to be odd or even and identify odd and even numbers(ACMNA051)

· Recognise, model, represent and order numbers to at least 10 000 (ACMNA052)

· Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053)

· Recognise and explain the connection between addition and subtraction (ACMNA054)

· Recall addition facts for single-digit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation(ACMNA055)

· Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)

· Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies(ACMNA057)

Fractions and decimals

· Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole(ACMNA058)

Money and financial mathematics

· Represent money values in multiple ways and count the change required for simple transactions to the nearest five cents (ACMNA059)

Patterns and algebra

· Describe, continue, and create number patterns resulting from performing addition or subtraction(ACMNA060)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Explain if and why a number is odd or even. (U)

· Read, write and order numbers to 10 000. (F)

· Rename larger numbers using place value and using other non place value partitions. (F)

· Partition larger numbers into place value parts to use to assist calculations (Partial sums, Partial products). (R,F)

· Understand and model unit fractions. Find and relate unit fractions of a group through sharing (such as shared between (÷) 3 is 1/3)

· Count by ½’s, 1/3’s and 1`/4’s to 1. (F)

· Solve addition and subtraction problems using money as a context using strategies such as counting on for addition and counting up for subtraction. (F)

· Understand and apply skip counting number patterns in solving problems involving repeated addition and repeated subtraction (U,F)

· Understand subtraction is the inverse of addition (U)

· Be able to write addition and subtraction fact family number sentences for a set of numbers. (F)

· Apply a range of mental and written strategies to solve the result of addition and subtraction calculations such as partial sums and compensation. (U,F)

· Know multiplication facts for 2’s, 3’s, 5’s and 10’s. (F)

· Relate skip counting to multiplication to division. (U)

· Use a range of strategies for multiplication such as the area model and the partitioning of numbers.(U,F)

· Partition money amounts using place value denominations ($100’s, $10’s, $1’s) and also with other denomination partitions.(F)

· Calculate change using strategies such as counting up. (F)

· Know addition facts and strategies such as compliments to ten, doubles and near doubles and counting by tens and ones forwards and backwards. (F)

Unit Development Ideas

AusVELS Content Descriptors

Number and place value

· Investigate and use the properties of odd and even numbers (ACMNA071)

· Recognise, represent and order numbers to at least tens of thousands (ACMNA072)

· Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems (ACMNA073)

· Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (ACMNA074)

· Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)

· Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder(ACMNA076)

Fractions and decimals

1.

· Investigate equivalent fractions used in contexts(ACMNA077)

· Count by quarters halves and thirds, including with mixed numerals. Locate and represent these fractions on a number line (ACMNA078)

· Recognise that the place value system can be extended to tenths and hundredths. Make connections between fractions and decimal notation(ACMNA079)

·

Money and financial mathematics

· Solve problems involving purchases and the calculation of change to the nearest five cents with and without digital technologies (ACMNA080)

Patterns and algebra

· Explore and describe number patterns resulting from performing multiplication (ACMNA081)

· Solve word problems by using number sentences involving multiplication or division where there is no remainder (ACMNA082)

· Use equivalent number sentences involving addition and subtraction to find unknown quantities(ACMNA083)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Explain why a number is odd or even. (U)

· Relate the properties of odd or even to authentic contexts. (R)

· Read, write and order numbers to 10 000’s. (F)

· Rename larger numbers using place value and using other non place value partitions. (F)

· Partition larger numbers into place value parts to use to assist calculations (Partial sums, Partial products). (R,F)

· Apply a range of mental strategies to solve and/or estimate the result of calculations. (F)

· Understand the number patterns for the multiples of 3 - 9 and use in assisting with determining multiplication facts. (U)

· Know multiplication facts to 10x10. (F)

· Be able to relate all four fact family number facts for multiplication and division to any multiplication or division number sentence. (U)

· Use a range of techniques for multiplication such as the area model and the partitioning of numbers.(F)

· Understand the relationship between ½, 2/4 and 4/8 in authentic contexts and other equivalent fractions. (U)

· Count by ½’s, 1/3’s and 1`/4’s with and without a number line. (F)

· Understand the place value system into the hundredths using real life contexts such as money. (U,R)

· Relate fractions as another way of representing division.

· Relate fractions to decimals through fractions of 100 (1 metre, $1.00, 100 piece block of chocolate). (R)

· Solve addition and subtraction problems using money as a context using strategies such as counting on for addition and counting up for subtraction. (F)

· Understand and apply the multiplication (skip counting) number patterns in solving multiplication and resulting products. (U,F)

· Apply a variety of strategies for multiplication and division including mental strategies for problem solving. (F)

· Rearrange subtraction number sentences into addition and vise versa to solve for unknown quantities. (F)

Achievement Standard: By the end of Level 4, students choose appropriate strategies for calculations involving multiplication and division. They recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places. Students solve simple purchasing problems. They identify unknown quantities in number sentences. They describe number patterns resulting from multiplication. Students use the properties of odd and even numbers. They recall multiplication facts to 10 x 10 and related division facts. Students locate familiar fractions on a number line. They continue number sequences involving multiples of single digit numbers.

Level 5

Number and Algebra

AusVELS Content Descriptors

Number and place value

· Identify and describe factors and multiples of whole numbers and use them to solve problems(ACMNA098)

· Use estimation and rounding to check the reasonableness of answers to calculations(ACMNA099)

· Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100)

· Solve problems involving division by a one digit number, including those that result in a remainder(ACMNA101)

· Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)

Fractions and decimals

· Compare and order common unit fractions and locate and represent them on a number line(ACMNA102)

· Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator (ACMNA103)

· Recognise that the place value system can be extended beyond hundredths (ACMNA104)

· Compare, order and represent decimals(ACMNA105)

Money and financial mathematics

· Create simple financial plans (ACMNA106)

· Patterns and algebra

· Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107)

· Use equivalent number sentences involving multiplication and division to find unknown quantities (ACMNA121)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Identify factors and multiples of whole numbers. (F)

· Know the difference between factors and multiples. (U)

· Identify the patterns involved in identifying factors and multiples. (R)

· Apply a range of mental strategies to estimate the result of calculations and know the usefulness of this. (F)

· Use a range of techniques for multiplication such as the area model and the partitioning of numbers.(F)

· Apply the distributive law and represent arrays to model multiplication. (F,R)

· Solve real life division problems that result in a remainder. (F)

· Use a range of written and mental calculation strategies to solve division problems. (F)

· Interpret the resulting remainder in real contexts. (R)

· Understand, model and order unit fractions. (U)

· Model and therefore add and subtract fractions with similar denominators, such as using the area model for adding and subtracting fractions. (U,R,F)

· Understand the place value system into the thousandths and smaller, including in real life situations such as mass and volume. (R)

· Know how to represent decimals in different ways, such as words, numbers, fractions and models and compare relative sizes of decimals. ((U)

· Develop financial plans for suitable relevant real life situations. (P)

· Find, continue and create number patterns using fractions, decimals and whole numbers. (R)

· Make relevant fact family number sentences for multiplication and division with known and unknown multipliers and quotients. (F)

Achievement Standard: By the end of Level 5, students solve simple problems involving the four operations using a range of strategies. They check the reasonableness of answers using estimation and rounding. Students identify and describe factors and multiples. They explain plans for simple budgets. Students order decimals and unit fractions and locate them on number lines. They add and subtract fractions with the same denominator. Students continue patterns by adding and subtracting fractions and decimals. They find unknown quantities in number sentences.

Level 6

Number and Algebra

AusVELS Content Descriptors

Number and place value

1.

· Apply efficient Identify and describe properties of prime, composite, square and triangular numbers(ACMNA122)

· Select and mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)

· Investigate everyday situations that use integers. Locate and represent these numbers on a number line (ACMNA124)

Fractions and decimals

1.

Compare fractions with related denominators and locate and represent them on a number line(ACMNA125)

Solve problems involving addition and subtraction of fractions with the same or related denominators(ACMNA126)

Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies (ACMNA127)

Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers(ACMNA128)

Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies (ACMNA129)

Multiply and divide decimals by powers of 10(ACMNA130)

Make connections between equivalent fractions, decimals and percentages (ACMNA131)

Money and financial mathematics

1.

Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies (ACMNA132)

2.

Patterns and algebra

Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence (ACMNA133)

Explore the use of brackets and order of operations to write number sentences (ACMNA134)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Identify and describe properties of prime, composite, square and triangular numbers. (F)

· Use these properties to create patterns and solve problems. (R)

· Select between and use a variety of written, mental and digital calculation strategies involving the four operations to solve a variety of everyday problems. (F)

· Become familiar with and use the range of integers. (F)

· Use a number line to solve addition and subtraction problems using positive and negative integers. (F)

· Apply integers in everyday situations. (F)

· Understand, model and order fractions with related denominators. (U)

· Model and therefore add and subtract fractions with related denominators, such as using the area model for adding and subtracting fractions. (U,F)

· Find fractional quantities where a group is the whole. (F)

· Apply a variety of strategies to add and subtract decimals. (F)

· Estimate the solution to addition and subtraction problems involving decimals. (R)

· Understand and use as variety of strategies to multiply and divide problems involving decimals in everyday situations (U,F)

· Understand how multiplying and dividing decimals by powers of ten affects the initial decimal. (U)

· Know how to represent decimals in different ways, such as words, numbers, fractions and models and compare relative sizes of decimals. (R)

· Use the understanding of percentages to calculate the sale price of items in everyday situations. (U,F)

· Find, continue and create number patterns using fractions, decimals and whole numbers. (R)

· Find and describe the rule used to create a pattern. (R)

· Understand and apply the rules for completing multiple operations within the same number sentence including brackets (BODMAS,BOMDAS,PEMDAS,etc.) (F)

Unit Development Ideas

Achievement Standard: By the end of Level 6, students recognise the properties of prime, composite, square and triangular numbers. They describe the use of integers in everyday contexts. They solve problems involving all four operations with whole numbers. Students connect fractions, decimals and percentages as different representations of the same number. They solve problems involving the addition and subtraction of related fractions. Students make connections between the powers of 10 and the multiplication and division of decimals. They describe rules used in sequences involving whole numbers, fractions and decimals. Students locate fractions and integers on a number line. They calculate a simple fraction of a quantity. They add, subtract and multiply decimals and divide decimals where the result is rational. Students calculate common percentage discounts on sale items. They write correct number sentences using brackets and order of operations.

Level 7 Number and Algebra

Number and Place Value

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Investigate index notation and represent whole numbers as products of powers of prime numbers (ACMNA149)

· Investigate and use square roots of perfect square numbers (ACMNA150)

· Define and compare prime and composite numbers and explain the difference between them. (U)

· Apply knowledge of factors to strategies for expressing whole numbers as products of powers of prime factors, such as repeated division by prime factors or creating factor trees. (R)

· Solve problems involving lowest common multiples and greatest common divisors (highest common factors) for pairs of whole numbers by comparing their prime factorization (F)

· Use diagrams to investigate perfect square numbers such as 25 and 36 and their square roots. (R)

· Estimate between which two whole numbers a square root lies. (R)

· Use the associative, commutative and distributive laws to show the equivalence of different number sentences, and then apply them when calculating mentally or in writing.(F)

· Apply the associative, commutative and distributive laws when calculating mentally or on paper. (PS)

· Compare and order integers. (U)

· Add and subtract integers. (F)

· Apply the associative, commutative and distributive laws to aid mental and written computation (ACMNA151)

· Compare, order, add and subtract integers (ACMNA280)

Achievement Standard

By the end of Level 7, students solve problems involving the comparison, addition and subtraction of integers. They make the connections between whole numbers and index notation and the relationship between perfect squares and square roots.

Level 7 Number and Algebra

Real Numbers and Money and Financial Mathematics

AusVELS Content Descriptors

Real Numbers

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Compare fractions using equivalence. Locate and represent positive and negative fractions and mixed numbers on a number line (ACMNA152)

· Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153)

· Locate positive and negative fractions and mixed numbers on a number line. (U)

· Compare fractions using equivalence, including using visual representations or concrete materials such as fraction walls or number lines (R)

· Add and subtract fractions, including using visual representations such as fraction walls or rectangular arrays. (F)

· Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)

· Multiply fractions using strategies including visual representations, repeated addition and written. (F)

· Deduce the process for division of fractions as the inverse of multiplication (R)

· Divide fractions using visual representations and written strategies. (F)

· Multiply decimals, using strategies including patterning and repeated addition. (F)

· Divide decimals, using strategies including patterning. (F)

· Calculate one quantity as a fraction of another, using real life examples. (F)

· Express one quantity as a fraction of another, with and without the use of digital technologies (ACMNA155)

· Round decimals to a specified number of decimal places (ACMNA156)

· Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157)

· Find percentages of quantities and express one quantity as a percentage of another, with and without digital technologies. (ACMNA158)

· Know that a fraction can be expressed as a decimal and a percentage, and vice versa. (U)

· Carry out simple conversions between fractions, decimals and percentages, including showing visual representations. (F)

· Calculate percentages of quantities in real life situations, including using multiples of 10% and 25%. (F)

· Calculate one quantity as a percentage of another. (F)

· Calculate a part to part relationship as a ratio. (F)

· Calculate at part to whole relationship as a ratio. (F)

· Calculate proportions of a given ratio such as converting quantities of a recipe for 4 people to one for 6 people. (PS)

· Calculate, estimate and make judgements when shopping. (PS)

· Apply the unitary method to identify the cheapest of several like products. (F)

· Investigate and calculate 'best buys', with and without digital technologies (ACMNA174)

Achievement Standard:

By the end of Level 7, students solve problems involving percentages and all four operations with fractions and decimals. They compare the cost of items to make financial decisions. Students use fractions, decimals and percentages, and their equivalences. They express one quantity as a fraction or percentage of another.

Year 7 Number and Algebra

Patterns and Algebra & Linear and Non-Linear Relationships

AusVELS Content Descriptors

Patterns and Algebra

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Identify patterns in the way that numbers increase or decrease. (F)

· Describe mathematical relationships in patterns. (U)

· Define variables and constants. (U)

· Describe a pattern using pronumerals (as variables). (U)

· Know that number patterns can be described using algebra. (U)

Unit Development Ideas

· Introduce the concept of variables as a way of representing numbers using letters (ACMNA175)

· Create algebraic expressions and evaluate them by substituting a given value for each variable (ACMNA176)

· Extend and apply the laws and properties of arithmetic to algebraic terms and expressions (ACMNA177)

· Know that expressions on either side of an equals sign have the same value. (U)

· Create algebraic expressions (using variables and constants) from authentic situations. (PS)

· Substitute numbers into algebraic expressions and authentic formulas to evaluate them. (F)

· Use brackets and the order of operations to write number sentences, and then extend their use to algebraic terms and expressions. (R)

· Use commutative, associative and distributive properties to write number sentences, and then extend their use to algebraic terms and expressions. (R)

· Use algebra to describe a situation described in words, and vice versa. (F)

Linear and Non-Linear Relationships

· Given coordinates, plot points on the Cartesian plane, and find coordinates for a given point (ACMNA178)

· Solve simple linear equations (ACMNA179)

· Specify the location of a point on the Cartesian plane using coordinates. (F)

· Plot points on the Cartesian plane when given coordinates. (U)

· Describe simple patterns (such as linear) from points plotted from a table of integer values.(F)

· Solve linear equations using concrete materials, including using the balance model. (F)

· Describe the need to do the same thing to each side of an equation. (U)

· Check the solution to an equation by substitution. (F)

· Describe situations depicted by graphs of everyday events, including travel graphs. (R)

· Describe the shape and features of a graph. (U)

· Make predictions from graphs of authentic data. (R)

· Investigate, interpret and analyse graphs from authentic data (ACMNA180)

Achievement Standard:

By the end of Level 7, students represent numbers using variables. They connect the laws and properties for numbers to algebra. They interpret simple linear representations and model authentic information. Students solve simple linear equations and evaluate algebraic expressions after numerical substitution. They assign ordered pairs to given points on the Cartesian plane.

Level 8 Number and Algebra

Number and Place Value

AusVELS Content Descriptors

· Use index notation with numbers to establish the index laws with positive integral indices and the zero index (ACMNA182)

· Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies (ACMNA183)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Evaluate numbers expressed as powers of positive integers. (F)

· Know that any number expressed to the power of zero is 1 and why. (R)

· Understand how and why we use index notation. (U)

· Add, subtract, multiply and divide positive and negative numbers using written and digital technologies. (F)

· Develop a range of mental strategies for calculating involving the four operations. (F)

Achievement Standard:

By the end of Year 8, students recognise index laws and apply them to whole numbers. Students use efficient mental and written strategies to carry out the four operations with integers.

Level 8 Number and Algebra

Real Numbers and Financial Maths

AusVELS Content Descriptors

Real Numbers

· Investigate terminating and recurring decimals(ACMNA184)

· Investigate the concept of irrational numbers, including π (ACMNA186)

· Solve problems involving the use of percentages, including percentage increases and decreases, with and without digital technologies (ACMNA187)

· Solve a range of problems involving rates and ratios, with and without digital technologies (ACMNA188)

Financial Maths

· Solve problems involving profit and loss, with and without digital technologies (ACMNA189)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Recognise terminating, recurring and non-terminating decimals and choose their appropriate representations. (F)

· Give examples of terminating, recurring and non-terminating decimals. (F)

· Define and identify rational and irrational numbers and give examples of each. (U)

· Explain that the real number system includes Irrational numbers. (U)

· Explain why the Real Number system includes Irrational numbers.(R)

· Locate the approximate position of an irrational number on a number line. (R)

· Describe certain subsets of the real number and explain their particular properties. Eg. Square numbers, primes, etc (U)

· Use percentages to solve problems, including those involving mark-ups, discounts, profit and loss and GST. (F)

· Develop mental strategies for calculating percentage discounts using 10% as a reference. (F)

· Solve rate and ratio problems using fractions or percentages and chooses the most efficient form to solve a particular problem. (F)

· Express profit and loss as a percentage of cost or selling price, comparing the difference eg. Investigate the methods used in retail stores to express discounts. (F)

Achievement Standard:

By the end of Level 8, students solve everyday problems involving rates, ratios and percentages. They describe rational and irrational numbers. Students solve problems involving profit and loss.

Level 8 Number and Algebra

Patterns and Algebra and Linear and Non-Linear Relationships

Achievement Standard:

By the end of Year 8, students make connections between expanding and factorising algebraic expressions. They simplify a variety of algebraic expressions. They solve linear equations and graph linear relationships on the Cartesian plane.

Level 9 Number and Algebra

Real Numbers

AusVELS Content Descriptors

· Apply index laws to numerical expressions with integer indices (ACMNA209)

· Extend and apply the index laws to variables, using positive integer indices and the zero index (ACMNA212)

· Express numbers in scientific notation (ACMNA210)

· Investigate very small and very large time scales and intervals (ACMMG219)

NB: Mention only of positive indices but this makes it impossible to refer to scientific notation of very small numbers so Law 5 needs to be included

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Evaluate numbers expressed as powers of positive integers. (U)

· Express an algebraic term in expanded form. (F)

· Express an expanded term in index form. (F)

· Apply the First Index Law. (F)

· Deduce the laws for division and expanding (laws 2 & 3) (R)

· Apply the Second Index Law. (F)

· Apply the Third Index Law. (F)

· Explain the effect of the zero power. (U)

· Apply Index Law 5. (F)

· Combine multiple laws to simplify an expression. (U)

· Recognise that an expression is in its simplest form. (U)

· Express large and small numbers in scientific notation. (F)

· Add and subtract numbers that are in scientific notation (F)

Achievement Standard:

By the end of Level 9, students apply the index laws to numbers and express numbers in scientific notation.

Level 9 Number and Algebra

Linear relationships and Money and financial mathematics

AusVELS Content Descriptors

Linear relationships

Sketch linear graphs using the coordinates of two points and solve linear equations (ACMNA215)

· Find the distance between two points located on a Cartesian plane using a range of strategies, including graphing software (ACMNA214)

· Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (ACMNA294)

· Solve problems involving direct proportion. Explore the relationship between graphs and equations corresponding to simple rate problems (ACMNA208)

Money and financial mathematics

Solve problems involving simple interest (ACMNA211)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Sketch a linear graph given two points (F)

· Sketch a linear graph given the gradient and one point (F)

· Solve linear equations algebraically (F)

· Make predictions based on a linear relationship (R)

· Calculate the distance between to points on a Cartesian plane using a formula (F)

· Calculate the gradient of a line from a graph (F)

· Determine the gradient of a line from an equation (F)

· Calculate the midpoint of a line segment using the formula (F)

· Use graphing software to determine the gradient, midpoint and line length of a line (F)

· Identify variable and constant in a worded linear relationship problem (PS, U)

· Sketch a graph to show the relationship of real world variables (PS)

· Make decisions based on information from a linear graph (R)

· Calculate simple interest (F)

· Graph Total repayments against principal (F)

· Explain the financial impact when factors vary when borrowing or investing (R)

Achievement Standard:

By the end of Level 9, students find the distance between two points on the Cartesian plane and the gradient and midpoint of a line segment. They sketch linear relations. Students solve problems involving simple interest.

Level 9 Number and Algebra

Patterns and Algebra & Non-Linear Relationships

AusVELS Content Descriptors

Non-Linear Relationships

· Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations (ACMNA296)

Patterns and Algebra

· Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213)

· NB: The following are Year 10 Content descriptors but tend to be covered in Year 9 in Victoria

· Factorise algebraic expressions by taking out a common algebraic factor (ACMNA230)

· Expand binomial products and factorise monic quadratic expressions using a variety of strategies (ACMNA233)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Recognise a quadratic pattern by determining second difference (U)

Plot a parabola from an equation. (F)

describe the graphs shape and key features (R)

Identify and sketch a y translation . (F)

Identify and sketch an x translation. (F)

Identify and sketch a reflection. (F)

Identify and sketch a dilation. (F)

Describe the transformation shown on a graph (U)

Connect a graph to it’s equation (U)

Unit Development Ideas

Expand one bracket (F)

Expand two binomial factors(F)

Expand a perfect square (F)

Use the distributive law and the index laws to factorise algebraic expressions (F)

Factorise a quadratic trinomial using sum and product technique (F)

Factorise a quadratic trinomial using by identifying a perfect square (F)

Factorise a quadratic expression using the difference of squares (F)

Choose the appropriate technique to factorise a quadratic (U)

Achievement Standard

By the end of Level 9, students expand binomial expressions and sketch non-linear relations.

Level 10 Number and Algebra

Money and Financial Mathematics

AusVELS Content Descriptors

Money and Financial Maths

· Connect the compound interest formula to repeated applications of simple interest using appropriate digital technologies (ACMNA229)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Solve equations using substitution (F)

· Define compound interest using examples (U)

· Understand the difference between compound interest and simple interest and the context in which each may be used (U)

· Understand the connection between compound interest and simple interest (U)

· Calculate compound interest using a formula (F)

· Transpose equations as required to perform calculations (U)

· Use digital technologies to calculate compound interest (F)

· Decide whether compound interest or simple interest applies to a situation (R)

· Solve authentic problems that involve calculations of compound interest (F, U, R, PS)

Achievement Standard: By the end of Level 10, students recognise the connection between simple and compound interest.

Level 10 Number and Algebra

Patterns and Algebra

AusVELS Content Descriptors

· Factorise algebraic expressions by taking out a common algebraic factor (ACMNA230)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Determine factors of numbers and algebraic terms (U)

· Determine common factors in a group of numbers or algebraic terms (F, U)

· Recognise the highest common factor in a group of numbers or algebraic terms (F)

· Recognise the highest common factor in algebraic expressions (U)

· Factorise an algebraic expression by recognising the highest common factor (number or algebraic term or expression) and dividing each term by this factor (R)

· Simplify number sentences and algebraic expressions using a range of index laws (F)

· Represent large numbers and small numbers using scientific notation (F)

· Explain why index notation is used (U)

· Explain, using indices, the meaning of a negative index (U)

· Simplify algebraic expressions involving positive and negative indices and applying a range of index laws (U) (R)

· Simplify fractions using highest common factors (U)

· Add fractions using common denominators (F)

· Solve a range of linear equations (not fractions) using the four operations (U) (R)

· Solve linear equations, including those with numerical denominators (U) (R) (PS)

· Check solutions to linear equations using substitution (R)

Unit Development Ideas

· Simplify algebraic products and quotients using index laws (ACMNA231)

· Apply the four operations to simple algebraic fractions with numerical denominators (ACMNA232)

· Multiply algebraic terms (U)

· Expand binomial products (U)

· Simplify expressions resulting from expansion of binomial products (U)

· Factorise monic quadratic equations e.g. x2 + 7x + 12 using a variety of strategies (R) (PS)

· Identify common factors including binomial terms in algebraic expressions (U)

· Factorise algebraic expressions with four terms by using grouping in pairs (U) (R)

· Recognise patterns for special binomial products e.g. (a+b)(a-b) and (a+b)2 to expand the products (F) (U)

· Recognise patterns to factorise special cases of quadratic equations e.g. a2 – b2 (F) (U)

· Use the area model to factorise quadratic expressions such as ax2 + bx + c where a = + 1 (U)

· Factorise quadratic expressions using the method of completing the square (U) (R)

· Write linear equations to represent word problems (U) (R) (PS)

· Solve word problems using linear equations (R) (PS) (U)

· Expand binomial products and factorise monic quadratic expressions using a variety of strategies (ACMNA233)

· Substitute values into formulas to determine an unknown (ACMNA234)

Achievement Standard: Students expand binomial expressions and factorise monic quadratic expressions. They find unknown values after substitution into formulas. They perform the four operations with simple algebraic fractions.

Level 10 Number and Algebra

Linear and Non-Linear Relationships

AusVELS Content Descriptors

Linear and Non-Linear Relationships

· Solve problems involving linear equations, including those derived from formulas (ACMNA235)

· Solve linear inequalities and graph their solutions on a number line (ACMNA236)

· Solve linear simultaneous equations, using algebraic and graphical techniques including using digital technology (ACMNA237)

· Solve problems involving parallel and perpendicular lines (ACMNA238)

· Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technology as appropriate (ACMNA239)

· Solve linear equations involving simple algebraic fractions (ACMNA240)

· Solve simple quadratic equations using a range of strategies (ACMNA241)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Transpose equations (mathematical and other) in order to solve for a particular unknown

· use substitution as a checking strategy (R) (U)

· solve linear inequalities (F)

· graph linear inequalities and their solutions (U)

· identify word problems that can be represented with simple linear inequalities (U)

· represent word problems using simple linear inequalities (R)

· solve word problems through the use of linear inequalities (PS)

· solve linear equations (F)

· solve pairs of simultaneous equations using a variety of techniques e.g. elimination, graphing, substitution (R) (F)

· identify the pairs of equations in worded problems (U)

· solve worded problems involving simultaneous equations (R) (PS)

· recognise parallel and perpendicular lines from their graphical representation (F)

· identify parallel lines and perpendicular lines using their algebraic representations (F) (U)

· use geometric software to investigate parallel and perpendicular lines (F)

· Identify graphical representations of parabolas, exponential functions and circles (F)

· Match algebraic representations of parabolas, exponential functions and circles to their graphs (U)

· Describe the effect of changing an algebraic expression on its corresponding graph (U)

· Identify intercepts, turning points and transformations from an algebraic expression and a graph (F)

· Sketch graphs of parabolas, exponential functions and circles from their algebraic representation (U)

· Solve a wide range of linear equations including those with simple algebraic fractions (F) (U)

· Check solutions to equations using substitution (F)

· Represent word problems using linear equations (R)

· Solve word problems using linear equations (PS) (R)

· Identify non-linear relationships from their algebraic or graphical representations (F)

· Connect real-life situations to linear and non–linear relationships (U)

· Solve quadratic equations using a variety of strategies (U)

· Factorise quadratic expressions using a variety of strategies including completing the square (U)

· Represent quadratic equations graphically by first solving and/ or factorising (U) (R)

Unit Development Ideas

Achievement Standard: Students make the connections between algebraic and graphical representations of relations. They solve simple quadratic equations and pairs of simultaneous equations. They recognise the relationships between parallel and perpendicular lines.

Level 10A

Year 10A content descriptors indicate optional additional content suitable for development of student mathematical background in preparation for further study of functions, algebra, and calculus; as well as other additional content related to statistics and trigonometry. Teachers can incorporate a selection of this and other additional content in Year 10 mathematics courses, as applicable for extension and enrichment purposes, and to prepare students for subsequent study.

Level 10A Number and Algebra

Real Numbers

AusVELS Content Descriptors

Define rational and irrational numbers and perform operations with surds and fractional indices (ACMNA264)

Use the definition of a logarithm to establish and apply the laws of logarithms (ACMNA265)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Students will:

define a rational number (U)

define an irrational number (U)

simplify expressions involving surds (U)

perform operations (addition, subtraction and multiplication) with surds (F)

represent surds with fractional indices (U)

perform operations with fractional indices (F)

evaluate numeric expressions using index laws (U)

simplify algebraic expressions using index laws (U)

define a logarithm

understand the application of logarithms in real-life situations (U)

understand the relationship between exponential and logarithmic expressions (U)

understand the logarithmic scale and its use (U)

use the laws of logarithms to simplify expressions (R)

Level 10A Number and Algebra

Patterns and Algebra

AusVELS Content Descriptors

· Investigate the concept of a polynomial and apply the factor and remainder theorems to solve problems (ACMNA266)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Students will:

Identify a polynomial expression (F)

Perform long division using numerals (F) (U)

Perform divisions of polynomials using factors and remainders (F) (U)

Level 10A Number and Algebra

Linear and Non-Linear Relationships

AusVELS Content Descriptors

· Solve simple exponential equations (ACMNA270)

· Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (ACMNA267)

· Apply understanding of polynomials to sketch a range of curves and describe the features of these curves from their equation (ACMNA268)

· Factorise monic and nonmonic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts (ACMNA269)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Students will:

· Understand that exponential equations can describe real life data such as population growth (U)

· Solve exponential equations (R) (U)

· Solve problems involving multiplying by a constant term (including negative terms) using a range of strategies (PS) (R)

· Represent parabolas graphically given their algebraic representation (F)

· Represent hyperbolas graphically given their algebraic representation (F)

· Represent circles graphically given their algebraic representation (F)

· Represent exponential functions graphically given their algebraic representation (F)

· Transform graphs as a result of changes to their algebraic representations (U)

· Sketch polynomials efficiently given their algebraic representation (U)

· Describe the features of a polynomial given its algebraic representation (U)

· investigate the features of graphs of polynomials using digital technology (F)

· apply factorisation of a range of quadratic expressions to solve word problems (PS) (U) (R)

· apply the solving of quadratic equations using a variety of strategies to solve word problems (PS) (U) (R)