# Level 2

MATHEMATICS

Content Descriptors with Learning Goals / Indicators and Proficiencies

Level 2

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 2 Sequence

Number and Algebra – Number and Place value descriptors are in sequence for teaching and learning.

Patterns and Algebra descriptor learning goals and proficiencies need to be integrated and developed consistently within Number and linked to Geometry.

Measurement and Geometry - all sub-strands can become units of work, descriptor goals and proficiencies are sequenced by order of introduction to the concept.

Statistics and Probability – Is not a unit of work, rather the Chance descriptor and Data descriptor should be introduced and revisited frequently throughout each term, related to other curriculum areas and other content within mathematics.

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)

· Rec

ognise 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 2

Measurement and Geometry

AusVELS Content Descriptors

Using units of measurement

· Compare and order several shapes and objects based on length, area, volume and capacity using appropriate uniform informal units (ACMMG037)

· Compare masses of objects using balance scales (ACMMG038)

· Tell time to the quarter hour, using the language of 'past' and 'to' (ACMMG039)

· Name and order months and seasons (ACMMG040)

· Use a calendar to identify the date and determine the number of days in each month (ACMMG041)

Shape

· Describe and draw two dimensional shapes, with and without digital technologies (ACMMG042)

· Describe the features of three dimensional Objects (ACMMG043)

Location and transformation

· Interpret simple maps of familiar locations and identify the relative positions of key features (ACMMG044)

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 shapes and determine attributes of length, area, volume and capacity (R)

· Order shapes by one attribute, discuss using the language of larger, smaller, longer, shorter, least, most, same, equal value (F)

· comparing lengths using finger length, hand span or a piece of string (U)

· estimate and use MAB minis and longs to measure length in centimetres (PS)

· compare areas using the palm of the hand or a stone (R)

· Use uniform grid paper to find the area in squares of given shapes (U)

· Estimate and compare capacities using a range of containers and pouring materials (U)

· Estimate and stack containers with uniform cubes (unifix or MAB minis) to find volume in cubes (U)

· Compare capacity of a range of containers using the same uniform cubes (R & U)

· Show Understanding by using length, area, volume and capacity in problem solving contexts (U)

· Show understanding by making and drawing models and recording measurements of length, area, volume and capacity of objects (U)

· Estimate mass by hefting, using the language of heavier, lighter and similar when comparing objects (R)

· using balance scales to determine whether the mass of different objects is more, less or about the same, or to find out how many single items are need to balance a different item (PS)

· Use a uniform object to compare items to on the balance scale (U)

· Estimate and compare objects using uniform metric weights to compare items on the balance scale (U)

· Show fluency through using units iteratively to compare measurements (F)

· Identify and mark quarters on the analogue clock (U)

· Identifying the quarter after an hour as quarter past an hour and the quarter before an hour as quarter to an hour (U)

· Link the quarter past as 15 minutes and the quarter to as 45 minutes (U)

· describe the characteristics of ‘quarter past’ times on an analogue clock, and identifying that the small hand is pointing just past the number and the big hand is pointing to the three (R)

· describe the characteristics of ‘quarter to’ times on an analogue clock, and identifying that the small hand is pointing almost to the next number and the big hand is pointing to the nine (R)

· Show fluency through describing and comparing time durations (F)

· Problem Solving using time and formulating problems from authentic situations to represent quarter past and quarter to (PS)

· Identify and describe features of a calendar (U)

· Create a calendar to represent a specific time period (PS)

· Understand the current use of Seasons representing a specific time period

· Use a calendar as an ongoing tool to record time and specific events each month (U)

· Use a calendar to predict time frames by month, week and days (PS)

· Use a calendar to identify the date and to determine the number of days in each month (PS & R)

· use calendars to locate specific information, such as finding a given date on a calendar and saying what day it is, and identifying personally or culturally specific days (F)

· investigate the seasons used by Aboriginal people and compare them to those used in Western society, recognising the connection to weather patterns (U)

· Show understanding by connecting knowledge of time from smallest time frame to largest time frame (U)

· Identify and name 2 dimensional shapes (F)

· Compare and describe 2D shapes to develop the vocabulary of straight and curved lines, edges, points and corners, number of sides (U)

· identifying key features of squares, rectangles, triangles, kites, rhombuses and circles, such as straight lines or curved lines, and counting the edges and corners (R)

· Show reasoning by sorting shapes by key features and describing the categories

· Identify and name known 3 dimensional shapes such as cube, cone, sphere, pyramid (F)

· identifying geometric features of known shapes such as the number of faces, corners or edges (U)

· Investigate and identifying geometric features of prisms and other 3D shapes through the use of models, such as the number of faces, corners or edges (R)

· Use Pictures, Numbers, Words to show what is known about 3D shapes (PS)

· Show reasoning by describing connections between 2D and 3D representations

· Understand and use directional language left and right, up (top) and down (bottom) on a page and transfer this to reading a map

· Use directional language of beside, between, next to, to tell directions on a map (PS)

· construct arrangements of objects from a set of directions (PS)

· Use problem solving strategies to plan problems for others by planning routes on maps and recording the directions (PS)

· Through problem solving match transformations with their original shape (PS)

· Understand that we use representations of objects and their positions, such as on maps, to allow us to receive and give directions and to describe place (U)

· Investigate the effect of onestep slides and flips with and without digital technologies (ACMMG045)

· Identify and describe half and quarter turns (ACMMG046) l

· Investigate one step slides using models (PS)

· Investigate one step flips using models (PS)

· Show understanding by using Pictures, Numbers, Words to show, describe and compare slides and flips (U)

· Students model half and quarter turns related to analogue clock (F)

· predict and reproduce a pattern based around half and quarter turns of a shape and sketching the next element in the pattern (PS)

· Investigate the real world application of flips, slides and turns (PS)

· understand that objects can be moved but changing position does not alter

· an object’s size or features (U)

Achievement Standard: Students recognise the features of three dimensional objects. They interpret simple maps of familiar locations. They explain the effects of one-step transformations. Students make sense of collected information. Students order shapes and objects using informal units. They tell time to the quarter hour and use a calendar to identify the date and the months included in seasons. They draw two dimensional shapes.

.

Level 2

Statistics and Probability

AusVELS Content Descriptors

Chance

· Identify practical activities and everyday events that involve chance. Describe outcomes as ‘likely’ or ‘unlikely’ and identify some events as ‘certain’ or ‘impossible’ (ACMSP047)

Data representation and interpretation

· Identify a question of interest based on one categorical variable. (ACMSP048)

· Collect, check and classify data (ACMSP049)

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?

· Create a probability meter to place everyday events to match the language of chance (R)

· Identify practical activities and everyday events that involve chance. (U)

· Describe outcomes as ‘likely’ or ‘unlikely’ and identify some events as ‘certain’ or ‘impossible’ (U)

· classify a list of everyday events according to how likely they are to happen, using the language of chance, and explaining reasoning

· Make conjectures about chance events and test these (PS)

· Show fluency through listing possible outcomes of chance events (F)

· Formulate problems from authentic situations, (PS)

· determine questions for gathering data (R)

· determine methods for gathering and recording data – including a table (F)

· Gather data relevant to the question (F)

· recognise the usefulness of tally marks (F)

· identifying categories of data and using them to sort data (R)

· check and classify data (F)

· Create displays of data using lists, table and picture graphs and interpret them (ACMSP050)

· create picture graphs to represent data using one to one correspondence (PS)

· create displays of data using lists, table and picture graphs (PS)

· compare the usefulness of different data displays (R)

· show reasoning by creating and interpreting simple representations of data (R)

Achievement Standard: Students make sense of collected information. They describe outcomes for everyday events. Students collect data from relevant questions

to create lists, tables and picture graphs.