# Level 8

MATHEMATICS

Content Descriptors with Learning Goals / Indicators and Proficiencies

Level 8

All Content Strands

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.

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

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?

· 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.

eBookbox: Working with Numbers

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

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?

· 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.

eBookbox Sets and Real Numbers

Level 8 Number and Algebra

Patterns and Algebra and Linear and Non-Linear Relationships

AusVELS Content Descriptors

Patterns and Algebra

· Extend and apply the distributive law to the expansion of algebraic expressions (ACMNA190)

· Factorise algebraic expressions by identifying numerical factors (ACMNA191)

· Simplify algebraic expressions involving the four operations (ACMNA192)

Linear and Non-Linear Relationships

· Plot linear relationships on the Cartesian plane with and without the use of digital technologies(ACMNA193)

· Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution(ACMNA194)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

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?

· Expand and simplify one bracket expressions eg 2(a +7) = 2a+14 (F)

· Use the Area Model to expand algebraic expressions. (F)

· List all factors of an algebraic term. (F)

· Recognise that factorising is the opposite of expanding. (U)

· Identify the highest common factor of algebraic expressions. (F)

· Gather like terms. (F)

· Factorise an expression by taking out the highest common factor. (F)

· Use the Area Model to factorise algebraic expressions. (F)

· Plot points on the Cartesian plane.(F)

· Complete a table of values, plot the data and discuss the resulting linear relationship. (U)

· Plot points from a linear relationship and describe the shape, steepness and where it cuts the y axis.(F,R)

· Find the rule for a linear relationship. (R)

· Use variables to symbolise simple linear equations and use a variety of strategies to solve them. (F)

· Solve equations using concrete materials, such as the balance model, and explain the need to do the same thing to each side of the equation. (U)

· Use strategies, such as backtracking and guess, check and improve to solve equations. (F)

· Apply solving linear equations to real life problems and discuss the resultant findings. (P)

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.

eBookbox: Linear and Non-Linear Functions

Level 8 Measurement and Geometry

Units of Measurement

AusVELS Content Descriptors

· Choose appropriate units of measurement for area and volume and convert from one unit to another(ACMMG195)

· Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites (ACMMG196)

· Investigate the relationship between features of circles such as circumference, area, radius and diameter. Use formulas to solve problems involving circumference and area (ACMMG197)

· Develop the formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volume(ACMMG198)

· Solve problems involving duration, including using 12- and 24-hour time within a single time zone(ACMMG199)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Students will:

· Distinguish between area and volume and choose the appropriate units of measurement for each. (U)

· Convert between units of area and between units of volume. (F)

· Name and determine the perimeter and area of parallelograms, rhombuses and kites. (F)

· Determine the circumference and area of a circle by direct measurement. (R)

· Demonstrate that by knowing circumference of a circle (its perimeter) we can determine its radius which in turn, can help me find its diameter and area. Or knowing its radius, I can find the area, circumference and diameter. (U, F, R)

· Explain how, what and why Pi is used in equations related to circles. (U)

· Know how the formulae for all 3D shapes are related and variations of Length x Width x Height.(U)

· Solve problems involving duration, including using 12- and 24-hour time within a single time zone. (PS)

· Convert between 12 and 14 hour time and across time zones. (U)

· Determine the arrival time given a flight time and time zones.(R)

Achievement Standard:

By the end of Level 8, students convert between units of measurement for area and volume. They perform calculations to determine perimeter and area of parallelograms, rhombuses and kites. They name the features of circles and calculate the areas and circumferences of circles. Students solve problems relating to the volume of prisms. They make sense of time duration in real applications.

eBookbox: Measurement: Circles and 3D objects

Level 8 Measurement and Geometry

Geometric Reasoning

AusVELS Content Descriptors

· Define congruence of plane shapes using transformations. (ACMMG200)

· Develop the conditions for congruence of triangles. (ACMMG201)

· Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning. (ACMMG202)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Describe transformations including: translations, rotations and reflections. (F)

· Define congruence of plane shapes using transformations. (R)

· Use the conditions for congruence of triangles including, congruence (SSS, SAS, ASA and RHS), and demonstrating which conditions do not prescribe congruence (ASS, AAA). (R)

· Use coordinates to describe the transformation. (F)

· Describe properties of quadrilaterals including squares, rectangles, parallelograms, rhombuses, trapeziums and kites. (U)

· Determine the sum of internal angles of a polygon, using triangles. (F)

· Solve problems using the sum of internal angles for triangles and other polygons. (F)

· Determine lines of symmetry in a given shape. (U)

· Identify properties related to side lengths, parallel sides, angles, diagonals and symmetry. (R)

Unit Development Ideas

Achievement Standard:

By the end of Year 8, students identify conditions for the congruence of triangles and deduce the properties of quadrilaterals.

eBookbox: Angles, Shapes and Transformations

Level 8 Statistics and Probability

Chance and Data Representations

AusVELS Content Descriptors

Chance

· Identify complementary events and use the sum of probabilities to solve problems. (ACMSP204)

· Describe events using language of 'at least', exclusive 'or' (A or B but not both), inclusive 'or' (A or B or both) and 'and'. (ACMSP205)

· Represent events in two-way tables and Venn diagrams and solve related problems. (ACMSP292)

Data Representations

· Investigate techniques for collecting data, including census, sampling and observation(ACMSP284)

· Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes. (ACMSP206)

· Explore the variation of means and proportions of random samples drawn from the same population. (ACMSP293)

· Investigate the effect of individual data values, including outliers, on the mean and median. (ACMSP207)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Demonstrate that probabilities range between 0 to 1 by convention and that calculating the probability of an event allows the probability of its complement to be identified. (R)

· Identify the complement of familiar events (eg the complement of getting a head on a coin is getting a tail, the complement of winning a game is not winning the game). (R)

· Calculate probabilities for sample spaces for single-step experiments (eg drawing a marble from a bag with 2 black and 3 white marbles with replacement. (F)

· Pose ‘and’, ‘or’, ‘not’ probability questions about objects or people. (R)

· Show that representing data in Venn diagrams or two-way tables facilitates the calculation of probabilities. (U)

· Use Venn diagrams and two-way tables to calculate probabilities for events satisfying ‘and’, ‘or’, ‘given’ and ‘not’ conditions. (R)

· Collect data to answer the questions using Venn diagrams or two-way tables. (P)

· Know the difference between a sample and a census and when each might be appropriate. (U)

· Be able to create, implement and interpret survey data through sampling techniques. (P,R)

· Use sample properties (for example mean, median, range) to predict characteristics of the population acknowledging uncertainty. (U)

· Use displays of data to explore and investigate effects. (R)

Achievement Standard:

By the end of Year 8, students choose appropriate language to describe events and experiments. Students model authentic situations with two-way tables and Venn diagrams. They explain issues related to the collection of data and the effect of outliers on means and medians in that data.

eBookbox: Sets and Real Numbers

Probability, inferences and Conjecture