# Level 9

MATHEMATICS

Content Descriptors with Learning Goals / Indicators and Proficiencies

Level 9

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 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

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. (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.

eBookbox: Computation and rational numbers ebookbox

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

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?

· 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.

eBookbox: Linear Equations ebookbox

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)

· 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)

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

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

eBookbox: Introducing Quadratic Equations

Level 9 Measurement & Geometry

Units of Measurement

AusVELS Content Descriptors

· Calculate the areas of composite shapes (ACMMG216)

· Calculate the surface area and volume of cylinders and solve related problems (ACMMG217)

· Solve problems involving the surface area and volume of right prisms (ACMMG218)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· deconstruct a composite shape into simple shapes with the appropriate dimensions (U)

· estimate the area of a composite shape (R)

· calculate the area of a composite shape (F)

· sketch and recognise the net that applies to prisms and cylinders. (R)

· estimate the surface area of a right prism (R)

· calculate the surface area of a cylinders and right prisms (F)

· calculate the volume of a cylinder and right prisms (F)

Achievement Standard:

By the end of Level 9, students calculate areas of shapes and the volume and surface area of right prisms and cylinders.

eBookbox: Measurement: Circles and 3D

Level 9 Measurement and Geometry

Geometric Reasoning & Pythagoras and Trigonometry

AusVELS Content Descriptors

Geometric Reasoning

· Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar (ACMMG220)

· Solve problems using ratio and scale factors in similar figures (ACMMG221)

Pythagoras and Trigonometry

· Investigate Pythagoras’ Theorem and its application to solving simple problems involving right angled triangles (ACMMG222)

· Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles (ACMMG223)

· Apply trigonometry to solve right-angled triangle problems (AMMG224)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· explain why two shapes are similar. (R)

· explain the conditions for similarity of triangles, (ASS, AAA). (R)

· determine ratio and scale factor. (F)

· use similar triangles to solve geometric problems. (U)

· identify the parts of a right angle triangle, Opposite, Adjacent and Hypotenuse. (F)

· explain the relationship between the sides of the right angle triangle. (F)

· calculate the length of the hypotenuse of a right angle triangle. (F)

· calculate the length of a short side of a right angle triangle. (F)

· apply Pythagoras Theorem to real life problems. (PS)

· explain the constancy of the trigonometric ratios for right-angle triangles.

· identify the adjacent, opposite and hypotenuse sides of a right angle triangle. (U)

· identify Sine, Cosine and Tangent Ratios of a triangle. (U)

· determine missing side lengths of the triangle, using the ratios. (F)

· find missing angles in a right angle triangle, using the ratios. (F)

Achievement Standard:

By the end of Level 9, students interpret ratio and scale factors in similar figures. They explain similarity of triangles. Students recognise the connections between similarity and the trigonometric ratios. They use Pythagoras’ Theorem and trigonometry to find unknown sides of right-angled triangles.

eBookbox: Trig Ratios and Pythagoras Theorem ebookbox

Level 9 Statistics and Probability

Chance and Data Representations

AusVELS Content Descriptors

Data Representations

· Investigate reports of surveys in digital media and elsewhere for information on how data were obtained to estimate population means and medians (ACMSP227)

· Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly from secondary sources (ACMSP228)

· Construct back-to-back stem-and-leaf plots and histograms and describe data, using terms including ‘skewed’, ‘symmetric’ and ‘bi modal’ (ACMSP282)

· Compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread (ACMSP283)

Chance

· List all outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays. Assign probabilities to outcomes and determine probabilities for events (ACMSP225)

· Calculate relative frequencies from given or collected data to estimate probabilities of events involving 'and' or 'or' (ACMSP226)

Learning Goals/ Intentions and Proficiencies

Essential Learning

Students will:

Unit Development Ideas

· develop a question that will aid the comparison of two or more sets of data. (U)

· use appropriate investigative techniques to collect data. (PS)

· choose appropriate secondary data (F)

· identify everyday questions and issues involving at least one numerical and at least one categorical variable (R)

· display comparative data, such as back-to-back stem-and-leaf plots and histograms. (F)

· explain comparative data, such as back-to-back stem-and-leaf plots and histograms. (R)

· describe data , using spread, mean, medium, outliers, skewed, symmetric . (U)

· explain the data and draw conclusions. (R)

· evaluate media reports and use statistical knowledge to draw conclusions (U)

· list all the outcomes for a two step chance experiment using a tree diagram. (F)

· assign probabilities, using the tree diagram. (F)

· use the tree diagram to solve problems, including and, or and not. (F)

· solve probability questions, using a tree diagram for events ( without replacement). (F)

· use Venn Diagrams to solve problems with ‘and’, ‘or’, and ‘not’. (PS)

Achievement Standard:

By the end of Level 9, students compare techniques for collecting data in primary and secondary sources. They make sense of the position of the mean and median in skewed, symmetric and bi-modal displays to describe and interpret data. Students calculate relative frequencies to estimate probabilities, list outcomes for two-step experiments and assign probabilities for those outcomes. They construct histograms and back-to-back stem-and-leaf plots.

eBookboxes yet to be developed