# Level 7

MATHEMATICS

Content Descriptors with Learning Goals / Indicators and Proficiencies

Level 7

All Content Strands

Introduction - Mathematics Scope and Sequence Documents

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 7 Number and Algebra

Number and Place Value

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· 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)

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?

· 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)

· 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.

eBookbox: Working with Numbers

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

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?

· 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)

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

· 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)

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

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

· 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)

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

· Recognise and solve problems involving simple ratios (ACMNA173)

· 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)

Money and Financial Mathematics

· 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.

eBookbox: Common fractions decimals and Percentages

Linking fractions decimals and percentages

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

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?

· 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.

eBookboxes: Working with Patterns

Level 7 Measurement and Geometry

Using units of measurement

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving (ACMMG159)

· Explain the difference between perimeter and area, and their respective units.(U)

· Deduce the formula for the area of a rectangle by counting square units and finding a pattern. (R)

· Deduce the formula for the area of triangles and parallelograms using visual constructions. (R)

· Calculate the perimeter and area of rectangles, triangles and parallelograms. (F)

· Solve problems involving the area of rectangles, triangles and parallelograms, and the surface area of related prisms. (PS)

· Calculate volumes of rectangular prisms (ACMMG160)

· Choose the best unit to use when measuring volume. (U)

· Use cubic units when estimating the volume of 3D shapes (F)

· Use concrete materials to deduce the formula for the volume of cubes and other rectangular prisms. (R)

· Calculate the volume and surface area of cubes and other rectangular prisms.(F)

Achievement Standard

By the end of Level 7, students use formulas for the area and perimeter of rectangles and calculate volumes of rectangular prisms.

eBookbox: Measuring the world around us

Level 7 Measurement and Geometry

Shape & Location and Transformation

AusVELS Content Descriptors

Shape

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Draw different views of prisms and solids formed from combinations of prisms (ACMMG161)

· Draw isometric diagrams of prisms and solids formed from combinations of prisms. (F)

· Draw plan and elevation views (front and side) of prisms and solids formed from combinations of prisms. (F)

Location and Transformation

· Perform the following transformations on 2D shapes: translation, reflection in an axis, and rotation in multiples of 900. (F)

· Describe translation, reflection in an axis, and rotation in multiples of 900 using coordinates. (U)

· Create patterns with combinations of translations, reflections and rotations, including using digital technologies. (PS)

· Identify line and rotational symmetries. (U)

· Describe translations, reflections in an axis, and rotations of multiples of 90° on the Cartesian plane using coordinates. Identify line and rotational symmetries (ACMMG181)

Achievement Standard:

By the end of Level 7, students describe different views of three dimensional objects. They represent transformations in the Cartesian plane.

eBookbox: Transforming and Visualising

Level 7 Measurement and Geometry

Geometric Reasoning

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

· Deduce and then describe the conditions for two lines to be parallel. (R)

· Construct a pair of parallel lines with a transversal intersecting with them using concrete materials or geometry software. (F)

· Define and classify pairs of angles as complementary, supplementary, adjacent and vertically opposite. (U)

· Define and classify alternate, corresponding and co-interior angles. (U)

· Deduce the missing angle in a parallel/transversal line situation. (R)

· Deduce and verify the angle sum of a triangle using concrete materials. (R)

· Calculate the missing angle in a triangle. (F)

· Deduce the angle sum of a quadrilateral from knowing the angle sum of a triangle.(R)

· Calculate the missing angle in a quadrilateral.(F)

· Justify and classify triangles as scalene, isosceles or equilateral according to their side properties. (R)

· Justify and classify triangles as right-angled, obtuse-angled and acute angled according to their angle properties. (R)

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

Unit Development Ideas

· Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning (ACMMG164)

· Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal(ACMMG163)

· Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral (ACMMG166)

· Classify triangles according to their side and angle properties and describe quadrilaterals (ACMMG165)

Achievement Standard:

By the end of Level 7, students solve simple numerical problems involving angles formed by a transversal crossing two parallel lines. Students classify triangles and quadrilaterals. They name the types of angles formed by a transversal crossing parallel line.

eBookboxes: Properties of shapes and solids, Transforming and visualising

Level 7 Statistics and Probability

Data Representation and Interpretation and Chance

AusVELS Content Descriptors

Data Representation and Interpretation

· Identify and investigate issues involving numerical data collected from primary and secondary sources (ACMSP169)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· Construct and compare a range of data displays including stem-and-leaf plots and dot plots (ACMSP170)

· Describe the difference between primary and secondary data. (U)

· Collect, organise and describe numerical data collected from primary sources. (PS)

· Analyse numerical data collected from secondary sources. (PS)

· Explain why some data representations are more appropriate than others for particular data sets. (U)

· Construct and compare data displays including ordered stem and leaf plots, and dot plots. (F)

· Calculate mean, median, mode and range for sets of data. Interpret these statistics in the context of data (ACMSP171)

· Describe and interpret data displays using median, mean and range (ACMSP172)

· Calculate the mean, median, mode and range for sets of data. (F)

· Explain and interpret data, including referring to the mean, median, mode and range of the data. (R)

· Compare data sets from real life, including using the location of the mean and median on graphs. (R)

· Describe how outliers may affect the comparison of data sets when the mean, median and range are used.(U)

Chance

· Construct sample spaces for single-step experiments with equally likely outcomes(ACMSP167)

· Assign probabilities to the outcomes of events and determine probabilities for events (ACMSP168)

· Define key terms, including probability, sample space, favourable outcomes, trial, events and experiments. (U)

· Identify equally likely outcomes and outcomes that are not equally likely.(F)

· Construct sample spaces for single-step experiments with equally likely outcomes. (U)

· Calculate the probability of an event as a decimal, fraction or percentage. (F)

Achievement Standard:

By the end of Level 7, students determine the sample space for simple experiments with equally likely outcomes and assign probabilities to those outcomes. They calculate mean, mode, median and range for data sets. They construct stem and leaf plots and dot plots.

eBookbox: Working with Data