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LODGE Education Training & Consultancy Services © 2005

Primary School

Mathematics Syllabus Framework2005

Edited by: Phil Lodge

LODGE Education Training & Consultancy Services

This document compilation remains the property of LODGE Education Training & Consultancy Services © 2005,

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IntroductionExplanatory Notes:

This document is the: A Primary School Mathematics Syllabus Framework.

The basis of this syllabus is the Ohio, Academic Content Standards K-7 Mathematics Benchmark and Indicators.

The School has adapted the Ohio program and matched it with the Victorian Essential Learning Standards which are also included for each Level.

This is a prescribed program content syllabus for all year level teachers but is not restrictive. Teachers are encouraged and expected to provide learning experiences in addition to and beyond what has been prescribed after the essential aspects of this document have been introduced, reinforced and assessed.

This framework document is to be used as a resource for planning learning and teaching programs and for assessing student performance against the Standards.

Assessment Protocols are also included at the end of each year level program content section. These are to be used as a reference for determining student performance in Primary School Mathematics to ensure rigour, uniformity and consistency in teacher judgements.

The syllabus is a collection of statements and examples which specify: Skills, knowledge and understandings (SKUs) which should be taught at each year

level and; Behavioural expectations for students – which the teacher has facilitated and

encouraged as a consequence of instruction and learning experiences provided.

Enquiries regarding this document should be directed to:

Phillip Lodge: LODGE Education Training & Consultative ServicesPhone: Australia - 03 9789 4080, Email: [email protected]


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mailto:[email protected]


Maths

LEVEL 1.......................................................................................................................................5Learning focus..................................................................................................................5Standards..........................................................................................................................5

Prep............................................................................................................................ 7Number Dimension....................................................................................................7Measurement, Chance and Data Dimension...........................................................8Space Dimension.......................................................................................................8Working Mathematically Dimension........................................................................9ASSESSMENT PROTOCOLS (Year Prep Level 1).................................................10

LEVEL 2.....................................................................................................................................13Learning focus................................................................................................................13Standards........................................................................................................................14

Year One...................................................................................................................16Number Dimension..................................................................................................16Measurement , Chance and Data Dimension........................................................18Space Dimension.....................................................................................................18Working Mathematically Dimension......................................................................19ASSESSMENT PROTOCOLS (Year 1 Level 2).......................................................20

Number -....................................................................................................................20Year Two.................................................................................................................. 22Number Dimension..................................................................................................22Measurement, Chance and Data Dimension.........................................................23Space Dimension.....................................................................................................24Working Mathematically Dimension......................................................................25ASSESSMENT PROTOCOLS (Year 2 Level 2).......................................................26


Year Three................................................................................................................32Number Dimension..................................................................................................32Measurement, Chance and Data Dimension.........................................................34Space Dimension.....................................................................................................35Working Mathematically Dimension......................................................................35ASSESSMENT PROTOCOLS (Year 3 Level 3).......................................................37Year Four..................................................................................................................40Number Dimension..................................................................................................40Measurement Dimension........................................................................................41Space Dimension.....................................................................................................42Working Mathematically Dimension......................................................................43ASSESSMENT PROTOCOLS (Year 4 Level 3).......................................................43


Year Five.................................................................................................................. 52


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Number.....................................................................................................................52Space........................................................................................................................54Location................................................................................................................... 55Measurement and Data...........................................................................................55ASSESSMENT PROTOCOLS (Year 5 Level 4).......................................................58Year Six....................................................................................................................60Number.....................................................................................................................60Space........................................................................................................................62Location................................................................................................................... 62Measurement and Data...........................................................................................63Assessment Protocols (Year 6 Level 4)................................................................65

LEVEL 5.....................................................................................................................................68Year Seven...............................................................................................................68Number Dimension..................................................................................................68Measurement, Chance and Data Dimension.........................................................68Space Dimension.....................................................................................................70Working Mathematically Dimension..........................................................................71


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LEVEL 1Learning focus

In Number and Space, students manipulate concrete and visual models to develop understanding of the fundamental mathematical concepts and objects of natural number, numeral, shape and location. They relate counting of discrete objects in sets to spatial patterns and arrangements of 1 to 20 objects with physical, visual and written representations including numerals. They apply number to establish sequence and order with respect to the elements of sets and model addition and subtraction by grouping together or by moving apart elements of sets. They manipulate everyday objects to identify and describe the features of common two- and three-dimensional shapes that correspond to the spatial concepts of point, line, boundary, face, interior and exterior. They follow simple instructions for the location of objects and movement from one place to another in familiar situations. Students learn fundamental concepts related to Measurement, chance and data in situations where they need to measure and compare length, capacity, mass, time and temperature using descriptive terms such as hot or fuller than and/or by counting of informal units such as the length of a row of paperclips. They learn to make and check rough estimates of quantitative measurements. Students begin to recognise unpredictability and uncertainty in chance events such as a game of ‘Snakes and Ladders' and identify and gather data required for a birthday party. Students learn about fundamental aspects of Structure and Working mathematically by matching elements of different sets according to given instructions, such as one-to-one correspondence in a simple card game of memory, or a many-to-one correspondence between the students in a class and the first letter of their name. They explore patterns in number and space by manipulating objects according to simple rules and test the truth or otherwise of simple conjectures with respect to number, shape, pattern, measurement and data, simple time structures and the sequence of daily events. Students work with calculators to check the results of simple addition and subtraction and use drawing tools and geometry software to create and colour simple two-dimensional geometric shapes and visual patterns and composite objects based on these shapes. In learning activities at Level 1, Structure provides notions of set, logic, function and algebra fundamental to the development of mathematical concepts, skills and processes in Number, Space, Measurement, chance and data and Working mathematically. The related s for Structure are embedded across these dimensions to underpin an integrated approach to student learning.

Standards

Number

At Level 1 students:

construct small sets of objects and elements according to simple descriptions and form correspondences between these sets based on simple relationships.

use one-to-one correspondence to identify when two sets are equal in size or when one set is larger than another set or smaller than another set.

form collections of sets of equal size. place sets in sequence of increasing size and use the numbers 0 to 20 to count

and to determine the size of a given set, including zero for the empty set. describe the position of an element in an ordered set using ordinal numbers up

to ten. use materials to model addition and subtraction by the aggregation (grouping

together) and dis-aggregation (moving apart) of elements in sets.


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http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#informal

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#order

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#numeral

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add and subtract by counting forward and backward using the natural numbers from 0 to 20.

Space


recognise, copy and draw points, lines and simple free-hand curves and identify interior and exterior, edges; basic two-dimensional shapes such as triangles, circles and squares and basic three-dimensional solids and objects such as boxes and balls.

use attributes of shapes to construct small sets of geometric objects according to simple descriptions and form correspondences between these sets based on simple relationships.

place and orientate shapes according to simple descriptions of relative location such as next to, beside, in front of, behind, over, under, and give and follow simple directions for locating an object and for movement from one place to another over a short distance.

develop and follow simple instructions to move and place shapes and objects in familiar situations in relation to what they can see, and to move themselves from one place to another.

Measurement, Chance and Data


measure and compare length, area, capacity and mass in relation to various familiar objects that are seen and handled using descriptive terms and/or informal units such as the length of a line segment using steps or paces, simple area covered such as a shape by two handprints, the capacity of containers such as half a glass of water, the weight of common objects such as a heavy schoolbag and duration such as the number of days until a birthday.

recognise the flow and continuity of time and the use of natural cycles such as day/night, the seasons, and informal units such as heartbeats and hand claps at regular intervals to segment and describe the passage of time.

recognise and respond to unpredictability and variability in events, such as getting or not getting a certain number on the roll of a die in a game.

identify and describe the outcomes of simple chance events such as the toss of a coin, and collect and display these using simple pictograph data related to their own activities which may include games or events such as a birthday party.

Working Mathematically


make and test simple conjectures such as ‘the larger an object the heavier it is', ‘it is likely to rain after school today' and ‘nine is four more than five'.

make rough estimates and check their work with respect to computations and constructions in Number, Space, Measurement, chance and data and Structure.

devise and follow ways of recording computations involving the use of materials, mental calculations and the digit keys and +, - and = keys on a four function calculator.

use drawing tools such as simple shape templates and geometry software to draw points, lines, shapes and simple patterns and to copy a picture of a simple composite shape such as a child's sketch of a house using these shapes.

In this domain, s for the Structure dimension are introduced at Level 3.


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Prep

Number DimensionNumber and Number Systems

Meaning of Operations

Computation and Estimation

1. Compare and order whole numbers to 20.

2. Explain rules of counting, such as each object should be counted once and that order does not change the number.

3. Count to twenty; e.g., in play situations or while reading number books.

4. Determine “how many” in sets (groups) of 10 or fewer objects.

5. Relate, read and write numerals for numbers (0 to 20).

6. Construct multiple sets of objects each containing the same number of objects.

7. Compare the number of objects in two or more sets when one set has one or two more, or one or two fewer objects.

8. Represent and use whole numbers in flexible ways, including relating, composing and decomposing numbers; e.g., 5 marbles can be 2 red and 3 green or 1 red and 4 green.

9. Identify and state the value of all coins.

10. Model and represent addition as combining sets and counting on, and subtraction as take-away and comparison. For example:

a. Combine and separate small sets of objects in contextual situations; e.g., add or subtract one, two, or another small amount.

b. Count on (forward) and count back (backward) on a number line between 0 and 10.

11. Demonstrate joining multiple groups of objects, each containing the same number of objects; e.g., combining 3 bags of candy, each containing 2 pieces.

12. Partition or share a small set of objects into groups of equal size; e.g., sharing 6 stickers equally among 3 children.

13. Recognize the number or quantity of sets up to 5 without counting; i.e conservation, e.g., recognize without counting the dot arrangement on a domino as 5.


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Measurement, Chance and Data DimensionMeasurement

Units

Use Measurement Techniques and Tools

Data Collection

Statistical Methods

Probability

1.Identify units of time (day, week, month, year) and compare calendar elements; e.g., weeks are longer than days.

2.Compare and order objects of different lengths, areas, weights and capacities; and use relative terms, such as longer, shorter, bigger, smaller, heavier, lighter, more and less.

3. Measure length and volume (capacity) using uniform objects in the environment. For example, find:

a. how many paper clips long is a pencil;b. how many small containers it takes to fill one big container using sand, rice, beans.

4. Order events based on time. For example:

a. activities that take a long or short time;b. review what we do first, next, last;

c. recall what we did or plan to do yesterday, today, tomorrow.

5. Tell the time using analogue and digital clocks – hour (o’clock) and half hour.

1. Gather and sort data in response to questions posed by teacher and students; e.g., how many sisters and brothers, what color shoes.

2. Arrange objects in a floor or table graph according to attributes, such as use, size, color or shape.

3. Select the category or categories that have the most or fewest objects in a floor or table graph.

4. Identify and describe the outcomes of simple chance events such as the toss of a coin.

5. Collect and display data using simple pictograph data related to their own activities which may include games or events such as a birthday party.

Space DimensionCharacteristics and Properties

1.Identify and sort two-dimensional shapes and three-dimensional objects. For example:

a. Identify and describe two-dimensional figures and three-dimensional objects from the


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Spatial Relationships

environment using the child’s own vocabulary.

b. Sort shapes and objects into groups based on student-defined categories.

c. Select all shapes or objects of one type from a group.

d. Build two-dimensional figures using paper shapes or tangrams; build simple three-dimensional objects using blocks.

2. Name and demonstrate the relative position of objects as follows:

a. place objects over, under, inside, outside, on, beside, between, above, below, on top of, upside-down, behind, in back of, in front of;

b. describe placement of objects with terms, such as on, inside, outside, above, below, over, under, beside, between, in front of, behind.

Working Mathematically DimensionUse Patterns, Relations and Functions

Use Algebraic Representations

1. Sort, classify and order objects by size, number and other properties. For example:

a. Identify how objects are alike and different.b. Order three events or objects according to a given

attribute, such as time or size.c. Recognize and explain how objects can be

classified in more than one way.d. Identify what attribute was used to sort groups of

objects that have already been sorted.

2. Identify, create, extend and copy sequences of sounds (such as musical notes), shapes (such as buttons, leaves or blocks), motions (such as hops or skips), and numbers from 1 to 10.

3. Describe orally the pattern of a given sequence.

4. Model a problem situation using physical materials.


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ASSESSMENT PROTOCOLS (Year Prep Level 1)Abbreviation and Terminology Key:In Italics: ‘Primary Assessment Measure’ – to be initial and foremost measurement criteria for assessing student performance against Assessment Focus.# SKUs: Skills, Knowledge and Understandings detailed in this Syllabus Framework. To be regarded by teacher when making ‘on-balance’ judgements about student performance.

1. The Assessment Protocols are not a complete assessment tool for classroom teachers.2. The Protocols do not cover all dimensions of the Standards or the school Mathematics Syllabus. 3. Those dimensions that are not included in the Protocols, as well as more detailed aspects of the

Mathematics Syllabus, will be assessed cumulatively by the classroom teacher and form a basis on which the classroom teacher will make final, on-balance judgements about a student’s performance.

4. The Performance Measures in these Protocols are the essential and readily assessable aspects of the Number and Measurement and Chance and Data Dimensions. These are the two dimensions which are required for systemic school data processing and student reporting purposes.

5. The intention of these Protocols is to provide a consistent framework for student assessment in these two dimensions across all year levels of the school.

6. Teachers at each year level will devise and apply Common Assessment Tasks which specifically focus on the aspects of mathematics detailed in these protocols.

MathematicsASSESSMENT FOCUS PERFORMANCE MEASURES Weight

1 Number - Knowledge and understanding of relationships between ordinal and cardinal number, numerals and numeration.

Operations- addition, Subtraction.

Counts, recognises, models and writes numerals to 20, and words to 10.

#SKU Implications Makes, counts, records and estimates size of

small collection of objects. Orders and compares collections of objects. Counts forwards and backwards to 20. Performs and describes simple mental

calculations – “more than, less than, bigger than, smaller than, longer than, shorter than, shortest” etc..

Copies and repeats counting patterns. Models and describes appropriately groupings

of objects – e.g. “three groups of two”. Understands concept of ‘0’..

Solve simple addition and subtraction problems within the counting range of 0 –20.

#SKU Implications Count on to solve addition problems, modelling

with materials. Count back to solve subtraction problems,

modelling with materials. Verbally explains mathematical strategies

involved.

50%


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Operations-multiplication, division

Working Mathematically – Operations and Applications) – apply knowledge of number operations and processes

Spatial Awareness – awareness of relative relationships and position.

Properties of Shapes

Joins multiple groups, shares into equal groups a small number of objects.

#SKU Implications Can join together a small number of equal

groups to find total. Share a small number of objects equally.

Can discuss, describe and model addition and subtraction concepts.

#SKU Implications Verbally explains mathematical strategies

involved. Draws conclusions based on logical reasoning. Uses concrete materials, objects, etc to model

addition and subtraction concept. Describes consequences of adding to or taking

away from as more or less, bigger or smaller etc.

Describes collections of objects as more or less than, and can equalise by adding to or taking away.

Appropriately describes objects relative to position or proximity.

#SKU Implications Conceptualisation of over, under, next to after,

near, far, inside, outside etc.

Demonstrates relationship between characteristics of 2 and 3 D shapes.

#SKU Implications. Draw and name 2D shapes Use 2D paper shapes to represent 3D objects. Cover given area with designated shape.

15%

5%

2 Measurement, Chance and Data – is aware of basic time, linear and area size and relationships in life situations.

Can describe, discuss and apply understanding of time, size and length to every day situations.

#SKU Implications Understands concept of time and relationship to

clocks for daily activities e.g. Play, lunch, home time, dinner, bedtime etc.

Understands relativity of hours, days and weeks.

Recognises analogue time on the hour and half past.

Is aware of months and seasons. Aware of age in terms of years. Can model and describe objects and areas in

terms of long, longer, short, small etc. Understands and can describe area in special

describers – wide, long, little, big, smaller than etc.

20%


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Understands the concept of what happens next, what might or could happen next e.g. toss of coin, repetition as in patterns/sounds/actions etc.

Can interpret pictographs.

#SKU Implications Contribute to simple class data collection and

graphing. Answer questions based on data shown in

graphs.

Recognise chance or variability in events.

#SKU Implications Can use language of chance in everyday

situations.

5%

5%


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LEVEL 2

Learning focus In Number and Space students work with arrays of objects and base-10 models (units, longs, flats and cubes) to identify, order and model the counting numbers up to 1000. By using these materials they develop understanding of patterns in the numbers sequence mentally, by hand and using calculators (constant addition facility) to skip count up to 100 and count on and count back. They solve simple addition and subtraction using natural numbers to 100. They use groups of like materials to develop the notion of multiplication as repeated addition, and division as repeated subtraction of a set into equal-sized groups and represent these as rectangular arrays. They use regular geometric objects divided into equal segments and sets partitioned into arrays of their elements. To develop the concept of simple fraction and common fractions as parts of a whole unit. Students investigate the characteristics of simple shapes and solids with respect to similarity, symmetry and the application of simple transformations. They learn to devise and follow instructions in the forms of oral directions, informal maps and diagrams to locate a range of items or create routes to various places in and around their local environment. As students increase the variety and range of objects and events used in Measurement, chance and data activities, they use informal units to measure length, area and volume, and start to recognise the importance of formal units for consistency in measurement. They recognise time patterns and cycles (second, minute, hour, day, sunrise and sunset, weekend and week) and the features of the calendar such as days of the week, date and month in practical situations related to their everyday family and school life, including telling the time using analogue and digital clocks. They learn to identify and collect data to answer posed questions, and then use simple graphical displays such as bar graphs and pictographs to organise and present the data. Students learn to recognise variability in chance events and describe qualitatively the likelihood and relative likelihood of everyday events using terms such as unlikely and almost certain, and more likely or less likely. Students learn about Structure and Working mathematically by creating and manipulating sets of numbers, shapes, objects and patterns according to given criteria and they use a combination of every-day language and mathematical statements involving numerals, operations, connectives and relations to describe their mathematical working and results. Students test the truth or otherwise of conjectures by attempting to find examples or counter-examples and exploring special cases. They develop and consolidate their understanding of the commutative and associative properties for addition and multiplication. They model number, patterns, motion and data found in stories, familiar settings and daily activities using physical materials, diagrams and maps. During these activities students carry out well defined sequences of steps such as, using blocks, to complete a pattern in a design context, following a recipe, or preparing for school. Students use four-function calculators to check related computations and solutions to simple number sentences and equations and the accuracy of approximations to these computations. Structure provides for the further development of key constructs of set, logic, function and algebra that are fundamental to the development of mathematical concepts, skills and processes in Number, Space, Measurement, chance and data and Working mathematically. The related s for Structure are embedded across these dimensions to underpin an integrated approach to student learning.


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http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#associative

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#commutative

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#counter

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#example

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#relation

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#connective

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#symmetry

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#similarity

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Standards

Number

At Level 2:

students partition a collection (the universal set) into distinct subsets according to simple criteria, and recognise when one set is a subset of another set.

construct and use sets of size 1, 10 and 100 to model place value and order natural numbers from 0 to 1000.

describe simple fractions in terms of equal sized parts of a whole object,

such as of a pizza (part-whole relationship), and collections such as of a set of 20 coloured pencils (subset-set relationship).

use linear and rectangular arrays of like objects, and natural numbers, to skip count by 2s, 4s and 5s from zero to one hundred and to count to 1000 by 1s, 10s and 100s starting from any natural number.

add and subtract one- and two-digit numbers by counting on and counting back. mentally compute simple addition and subtraction calculations involving one- or

two-digit natural numbers using number facts such as complement to 10, doubles and near doubles.

describe and calculate simple multiplication as repeated addition such as 3 x 5 = 5 + 5 + 5; and division as repeated subtraction, such as 8 divided between 4, and as partitioning of a set into equal-sized subsets.

use commutative and associative properties of addition and multiplication in mental computation.

Space


recognise lines, surfaces and planes, corners and boundaries; familiar two-dimensional shapes including rectangles, rhombuses and hexagons, and three-dimensional shapes and objects including pyramids, cones, and cylinders.

partition a collection of geometric shapes, such as a set of attribute blocks, into distinct subsets according to simple criteria, and recognise when one set of shapes is a subset of another set of shapes.

recognise and describe symmetry, asymmetry, and congruence in these shapes and objects.

accurately draw simple two-dimensional shapes by hand and construct, copy and combine these shapes using drawing tools and geometry software.

apply simple transformations to shapes (flips, turns, slides and enlargements) and depict both original and transformed shape together.

specify location as a relative position, including left and right, and interpret simple networks, diagrams and maps involving a small number of points, objects or locations.



use physical models, money and diagrams to illustrate ordered and unordered sets of numbers, shapes, objects and data and carry out related computations and manipulations.


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http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#transformation

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#congruence

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#asymmetry

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#partition


make, describe and compare measurements, characteristics or outcomes such as length, width and height; mass; certain, likely, unlikely and impossible.

develop and apply criteria for the use of informal units, including non-uniform measures, such as hand-span, and uniform measures, such as icy-pole sticks for length; use formal units such as hour for time or litre for capacity; and use the units for length (metre), mass (kilogram) and time (second).

describe temperature qualitatively and informally measure and compare areas as enclosed space or space covered.

judge relative capacity of familiar objects and containers by eye and make informal comparisons of weight by hefting.

recognise the key elements of the calendar and place in sequence days, weeks and months.

describe common and familiar time patterns and such as the time, duration and day of regular sport training and tell the time to hours and half-hours using an analog clock, and to hours and minutes using a digital clock.

make predictions for the likelihood of outcomes of simple random and non-random events, using qualitative descriptors for more likely or less likely, such as such as whether 6's are harder to roll than 2's on a die.

collect simple categorical and numerical data (count of frequency) and present this data using pictographs and simple bar graphs and make predictions about the outcome of chance experiments in response to queries.



test simple conjectures by finding examples, counter-examples and special cases and informally decide whether a conjecture is likely to be true in general.

use place value to enter and read displayed numbers on a calculator. use a four-function calculator, including use of the constant addition function and

x key, to check the accuracy of mental and written estimations and approximations and solutions to simple number sentences and equations.

In this domain, s for the Structure dimension are introduced at Level 3.


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Year One


1. Use ordinal numbers to order objects; e.g., first, second, third.

2. Recognize and generate equivalent forms for the same number using physical models, words and number expressions; e.g., concept of ten is described by “10 blocks,” full tens frame, numeral 10, 5 + 5, 15 - 5, one less than 11, my brother’s age.

3. Read and write the numerals for numbers to 100.

4. Count forward to 100, count backwards from 100, and count forward or backward starting at any number between 1 and 100.

5. Use place value concepts to represent whole numbers using numerals, words, expanded notation and physical models with ones and tens. For example:

a. Develop a system to group and count by twos, fives and tens.

b. Identify patterns and groupings in a 100’s chart and relate to place value concepts.

c. Recognize the first digit of a two-digit number as the most important to indicate size of a number and the nearness to 10 or 100.

6. Identify and state the value of coins in our monetary system.

7. Determine the value of a small collection of coins to a total value up to one dollar.

8. Show different combinations of coins that have the same value.

9. Represent commonly used fractions using words and physical models for halves recognizing fractions are represented by equal size parts of a whole and of a set of objects.

10.Model, represent and explain addition as combining sets (part + part = whole) and counting on. For example:

a. Model and explain addition using physical materials in contextual situations.

b. Draw pictures to model addition.


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c. Write number sentences to represent addition. d. Explain that adding two whole numbers yields a

larger whole number.

11.Model, represent and explain subtraction as take-away and comparison.

For example:a. Model and explain subtraction using physical

materials in contextual situations.b. Draw pictures to model subtraction.c. Write number sentences to represent subtraction.d. Explain that subtraction of whole numbers yields an

answer smaller than the original number.

12. Use conventional symbols to represent the operations of addition and subtraction.

13. Model and represent multiplication as repeated addition and rectangular arrays in contextual situations; e.g., four people will be at my party and if I want to give 3 balloons to each person, how many balloons will I need to buy?

14. Model and represent division as sharing equally in contextual situations; e.g., sharing lollies.

15. Demonstrate that equal means “the same as” using visual representations.

16. Develop strategies for basic addition facts, such as:a. counting all; b. counting on;c. one more, two more;d. doubles;e. doubles plus or minus one;f. make ten;g. using tens frames;h. identity property (adding zero).

17. Develop strategies for basic subtraction facts, such as:a. relating to addition (for example, think of 7 - 3 = ?

as “3 plus ? equals 7”);b. one less, two less;c. all but one (for example, 8 - 7, 5 - 4);d. using tens frames;e. missing addends.


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Measurement , Chance and Data DimensionMeasurement Units


Data Collection

Statistical Methods

Probability

1. Estimate and measure lengths using informal and units.

2.Tell time to the hour on digital and analogue

timepieces.

3.Order a sequence of events with respect to time; e.g., summer, autumn, winter and spring; morning, afternoon and night.

4.Estimate and measure weight using informal - units; e.g., blocks of uniform size.

1. Identify multiple categories for sorting data.

2. Collect and organize data into charts using tally marks.

3. Display data in picture graphs with units of 1.

4. Read and interpret charts, picture graphs as sources of information to identify main ideas, draw conclusions, and make predictions.

5. Construct a question that can be answered by using information from a graph.

6. Arrange three objects by an attribute, such as size or weight, and identify the ordinal position of each object.

7. Answer questions about the number of objects represented in a picture graph; e.g., category with most, how many more in a category, compared to another, how many altogether in two categories.

8. Describe the likelihood of simple events as possible/impossible and more likely/less likely; e.g., when using spinners or number cubes in classroom activities.


1.Identify, compare and sort two-dimensional shapes; i.e., square, circle, oval, triangle, rectangle, and


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hexagon. For example:

a. Recognize and identify triangles independent of position, shape or size;

b. Describe two-dimensional shapes using attributes such as number of sides and number of vertices (corners or angles).

2. Create new shapes by combining or cutting apart existing shapes.

3. Identify the shapes of the faces of three-dimensional objects.

4. Extend the use of location words to include distance (near, far, close to) and directional words (left, right).

5. Copy figures and draw simple two-dimensional shapes from memory.



1. Sort, classify and order objects by two or more attributes, such as color and shape, and explain how objects were sorted.

2. Extend sequences of sounds, shapes or simple number patterns, and create and record similar patterns. For example:a. Analyse and describe patterns with multiple attributes using numbers and shapes; e.g., A, B, a, b, A, B, a, b,…

b. Continue repeating and extending patterns with materials, pictures and geometric items; e.g., XO, XOO, XOOO, XOOOO.

6. Describe orally the basic unit or general plan of a repeating or growing pattern.

7. Solve open sentences by representing an expression in more than one way using the commutative property; e.g., 4 + 5 = 5 + 4 or the number of blue balls plus red balls is the same as the number of red balls plus blue balls (R + B = B + R).

8. Describe orally and model a problem situation using words, objects or number phrase or sentence.


19


ASSESSMENT PROTOCOLS (Year 1 Level 2)Abbreviation and Terminology Key:In Italics: ‘Primary Assessment Measure’ – to be initial and foremost measurement criteria for assessing student performance against Assessment Focus.# SKUs: Skills, Knowledge and Understandings detailed in this Syllabus Framework. To be regarded by teacher when making ‘on-balance’ judgements about student performance.






Assessment Focus Performance Measure Weight1 Number -

Knowledge and understanding of relationships between ordinal and cardinal number, numerals and numeration.



Spatial Awareness – awareness of the properties of shapes and

Understands the composition of and is able to manipulate and apply whole numbers, simple fractions.

#SKU Implications Constructs and uses sets of size 1, 10 and 100 to

model place value and order natural numbers from 0 to 100.

Describes simple fractions ½ in terms of equal sized parts of a whole object, such as ½ of a pizza (part-whole relationship), and collections such as ½ of a set of 10 coloured pencils (subset-set relationship).

Uses linear and rectangular arrays of like objects, and natural numbers, to skip count by 2s and 5s from zero to one hundred

#SKU Implications Adds and subtracts one- and two-digit numbers by

counting on and counting back up to 20. Mentally computes simple addition and subtraction

calculations involving single digit numbers to 10.

#SKU Implications Models multiplication and division concepts with

concrete materials and describes in language as “groups” and “sharing”. (Two groups of three, Share four with two people)

Recognise and use appropriate terminology to describe the attributes of common

50%

20%

20%

10%


20


relative relationships and position of objects.

geometric shapes and solids.

#SKU Implications Recognise lines, corners and boundaries, familiar two-

dimensional shapes including square, triangle, circle, oval, rectangles and hexagons, and three-dimensional shapes and objects including sphere, cone, cube and cylinder.

2 Measurement, Chance and Data -is aware of basic time, linear and area size and relationships in life situations.

Data collection and interpretation

Probability

Extend the range of measurement characteristics and attributes, estimated and measured, using informal and formal units

#SKU Implications Makes, describes and compares measurements,

characteristics or outcomes such as length, width and height; mass and area; certain, likely, unlikely and impossible.

Use of informal measuring units, Uses the standard units for time (hours). Recognises the key elements of the calendar - days,

and months. Tells the time to hours and half-hours using an

analogue clock

Collect data and transpose into and interpret basic graphs

#SKU Implications Collects simple data (count of frequency) and

present this data using pictographs

Devise and conduct simple experiments to test and demonstrate probability (likelihood)

#SKU Implications Makes predictions for the likelihood of outcomes of

simple random and non-random events, using terms such as more likely or less likely.

60%

30%

10%


21


Year Two



1.Use place value concepts to represent, compare and order whole numbers using physical models, numerals and words, with ones, tens and hundreds. For example:

a. Recognize 10 can mean “10 ones” or a single entity (1 ten) through physical models and trading games.

b. Read and write 3-digit numerals (e.g., 243 as two hundred forty three, 24 tens and 3 ones, or 2 hundreds and 43 ones, etc.) and construct models to represent each.

2. Recognize and classify numbers as even or odd.

3. Count money and make change from one dollar using combinations of coins.

4. Represent and write the value of money using the ¢ sign and in decimal form when using the $ sign.

5. Represent fractions (halves, thirds and quarters), using words, numerals and physical models. For example:

a. Recognize that a fractional part can mean different amounts depending on the original quantity.

b. Identify and illustrate parts of a whole and parts of sets of objects.

6. Model, represent and explain subtraction as comparison, take-away and part-to-whole; e.g., solve missing addend problems by counting up or subtracting, such as “I had six footy, my sister gave me more, and I now have ten. How many did she give me?” can be represented as 6 + ? = 10 or 10 - 6 = ?.

7. Model, represent and explain multiplication as repeated addition, rectangular arrays and skip counting.

8. Model, represent and explain division as sharing equally and repeated subtraction.

9. Model and use the commutative property for


22



addition.

10. Demonstrate fluency in addition facts with addends through 9 and corresponding subtractions; e.g., 9 + 9 = 18, 18 – 9 = 9.

11. Add and subtract multiples of 10.

12. Demonstrate multiple strategies for adding and subtracting 2- or 3-digit whole numbers, such as:

i. compatible numbers Eg. 12+20 think 10+2012+20 is about 30

ii. compensatory numbers; Eg. 17-9 is the same as 17-10+1

iii. informal use of commutative and associative properties of addition.

Eg. cummutative- 3+4=74+3=73+4=4+3 associative- (1+2)+3=1+(2+3)

12.Estimate the results of whole number addition and subtraction and judge the reasonableness of the answers.

13.Demonstrate fluency in multiplication tables, 1, 2, 5 and 10 X.

Measurement, Chance and Data DimensionMeasurement

Units


1.Identify and select appropriate units of measure for:

a. length –metres;b. volume (capacity) – litres;c. weight – kilograms;d. time – hours, half-hours (analogue), or minutes

(digital) and time designations, a.m. or p.m.

2. Establish personal or commonly applied informal units of measure to make estimates and comparisons; e.garm span is a metre, a large bottle of lemonade is 2 litres, a litre of water weighs about a kilogram.

3. Tell time to the nearest half hour interval on an analogue clock and by minutes on a digital clock.

4. Estimate and measure the length and weight of common objects.

5. Select and use appropriate formal measurement tools; e.g., a ruler to draw a line, a measuring cup to place 2 cups of rice in a bowl, a scale to weigh a kilogram of sugar.


23


Data Collection

Statistical Methods

Probability

6. Make and test predictions about measurements, using different units to measure the same length or volume.

1. Pose questions, use observations, interviews and surveys to collect data, and organize data in charts, picture graphs and bar graphs.

2. Read, interpret and make comparisons and predictions from data represented in charts, line plots, picture graphs and bar graphs.

3. Read and construct simple timelines to sequence events.

4. Write a few sentences to describe and compare categories of data represented in a chart or graph, and make statements about the data as a whole.

5. Identify untrue or inappropriate statements about a given set of data.

6. Recognize that data may vary from one population to another; e.g., favorite TV shows of students and of parents.

7. List some of the possible outcomes of a simple experiment, and predict whether given outcomes are more, less or equally likely to occur.

8. Use physical models and pictures to represent possible arrangements of 2 or 3 objects.



1. Identify, describe, compare and sort three-dimensional objects (i.e., cubes, spheres, prisms, cones, cylinders and pyramids) according to the shape of the faces or the number of faces, edges or vertices.

2. Predict what new shapes will be formed by combining or cutting apart existing shapes.

3. Recognize two-dimensional shapes and three-dimensional objects from different positions.

4. Identify and determine whether two-dimensional shapes are congruent (same shape and size) or similar (same shape different size) by copying or


24


Transformations and Symmetry

using superposition (lay one thing on top of another).

5. Create and identify two-dimensional figures with line symmetry; e.g., what letter shapes, logos, polygons are symmetrical?



Analyse Change

1. Extend simple number patterns (both repeating and growing patterns), and create similar patterns using different objects, such as using physical materials or shapes to represent numerical patterns.

2. Use patterns to make generalizations and predictions; e.g., determine a missing element in a pattern.

3. Create new patterns with consistent rules or plans, and describe the rule or general plan of existing patterns.

4. Use objects, pictures, numbers and other symbols to represent a problem situation.

5. Understand equivalence and extend the concept to situations involving symbols; e.g., 4 + 5 = 9 and 9 = 4 + 5

6. Use symbols to represent unknown quantities and identify values for symbols in an expression or equation using addition and subtraction; e.g., □ + О = 10, ∆ - 2 = 4.

7. Describe qualitative and quantitative changes. (e.g.Stronger than, faster than, more than less than)


25



1. The Assessment Protocols are not a complete assessment tool for classroom teachers.2. The Protocols do not cover all dimensions of the Standards or the school Mathematics Syllabus. 3. Those dimensions that are not included in the Protocols, as well as more detailed aspects of the Mathematics Syllabus, will be assessed cumulatively by the classroom teacher and form a basis on which the classroom teacher will make final, on-balance judgements about a student’s performance.4. The Performance Measures in these Protocols are the essential and readily assessable aspects of the Number and Measurement and Chance and Data Dimensions. These are the two dimensions which are required for systemic school data processing and student reporting purposes.5. The intention of these Protocols is to provide a consistent framework for student assessment in these two dimensions across all year levels of the school.6. Teachers at each year level will devise and apply Common Assessment Tasks which specifically focus on the aspects of mathematics detailed in these protocols.

ASSESSMENT FOCUS PERFORMANCE MEASURE WEIGHT



Understands the composition of and is able to manipulate and apply whole numbers and vulgar fractions.

#SKU Implications Constructs and uses sets of size 1, 10 and 100 to

model place value and order natural numbers from 0 to 1000.

Describes simple fractions in terms of equal

sized parts of a whole object, such as of a pizza

(part-whole relationship), and collections such as of a set of 20 coloured pencils (subset-set relationship).

Uses linear and rectangular arrays of like objects, and natural numbers, to skip count by 2s, 4s and 5s from zero to one hundred and to count to 1000 by 1s, 10s and 100s starting from any natural number.

#SKU Implications Adds and subtracts one- and two-digit numbers by

counting on and counting back. Mentally computes simple addition and subtraction

calculations involving one- or two-digit numbers using number facts such as complement to 10, doubles and near doubles.

#SKU Implications

50%

20%

20%


26



Spatial Awareness – awareness of the properties of shapes and relative relationships and position of objects.

Describes and calculates simple multiplication as repeated addition such as 3 x 5 = 5 + 5 + 5; and division as repeated subtraction, such as 8 divided between 4.

Recognise and use appropriate terminology to describe the attributes of common geometric shapes and solids.

#SKU Implications Recognise lines, surfaces and planes, corners and

boundaries; familiar two-dimensional shapes including rectangles, rhombuses and hexagons, and three-dimensional shapes and objects including pyramids, cones, and cylinders.

10%



Probability


#SKU Implications Makes describes and compares measurements,

characteristics or outcomes such as length, width and height; mass and area; certain, likely, unlikely and impossible.

Develops and applies criteria for the use of informal measuring units,

Uses the standard units for length (metre), mass (kilogram) and time (seconds).

Judges relative capacity of familiar objects and containers and make informal comparisons of weight.

Recognises the key elements of the calendar and place in sequence days, weeks and months.

Tells the time to hours and half-hours using an analogue clock, and to hours and minutes using a digital clock.

Collect data and transpose into and interpret basic graphs#SKU Implications

Collects simple data (count of frequency) and present this data using pictographs and simple bar graphs


#SKU Implications Makes predictions for the likelihood of outcomes

of simple random and non-random events, using terms such as more likely or less likely, such as whether 6's are harder to roll than 2's on a die.

50%

30%

20%


27


LEVEL 3

Learning focus In Number students develop their understanding of number value and order from hundredths to tens of thousands. They routinely use multiples to skip count and create number patterns, including using multiples of 10, to explore more fully place value and the operation of multiplication. They work on practical problems in which the complexity of computations extends to include addition and subtraction of three-digit numbers, multiplication by single digits, and division by a single-digit number. Mental computations involve numbers up to 30. They learn to work with equivalent fractions and apply this to the addition and subtraction of simple common fractions. In Space students explore the orientation of lines and classify the key features of two- and three-dimensional shapes and their representations. They develop and follow instructions for the creation of patterns based on simple tessellations and designs, and relate these to work in other domains and to solving practical problems in and around the home. During these activities students become familiar with the concept of angle as a measure of turn and, in conjunction with grid references and the cardinal compass points, use it to specify location with increased precision and work with maps that include a combination of informal and formal directions. In Measurement, chance and data students work in contexts where they use informal (non-uniform and uniform), formal and measurement units to estimate and measure an increasing range of characteristics and attributes of objects and events. They learn to read scales and clocks with accuracy to the major divisions, and extract information from calendars, lists, tables and simple graphical displays when they plan activities and carry out tasks. They observe events in daily life such as recreational games and sports where they recognise natural variability in chance events and order these events from least likely to most likely. Students construct simple frequency graphs from experimental and collected data across the domains and in everyday life, and use simple Karnaugh maps (two-way tables) to display categorical data. In Structure students investigate sets formed by the counting of equal-sized subsets of tens, hundreds and thousands to develop multiplicative thinking, and also partition sets into smaller equal-sized subsets to develop the concepts of division and remainder. Students investigate sets formed by mathematical properties, operations and classifications, for example, equivalent fractions, shape transformations, and outcomes of random events. They recognise and learn to use the commutative and associative properties to support computation, and use simple algorithms for computation. They analyse and classify shapes and solids in terms of their geometric properties. Students develop an appreciation of formal measurement units in length, area and time contexts, and the use of scales and conversions between units. They learn to identify and describe localities and features on local and larger-scale maps and can devise suitable routes between places. Students generate simple number sequences using recursion such as ‘the next term in the sequence is two more than the previous term'. They create and solve simple equations involving integers and simple fractions, and further develop their ability to use mathematical symbols and terminology, for example, decimal form, brackets, the division symbol and inequality symbol, and the logical connectives and, or and not. In Working mathematically students learn to develop conjectures about the concepts of number, space, measurement and chance: the generality of patterns, numbers and shapes; the size and type of numbers resulting from computations; the effects of transformations of shapes; the outcomes of measurements and random experiments; and inferences from collected samples. Students learn to choose suitable representations from a variety of models that represent number, computations, patterns and shapes. Students learn to recognise practical applications of mathematics in daily life, including shopping, travel and time of day, and investigate the historical development of some mathematical concepts and structures. Students use a range of technologies, including


28

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#equation

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#equivalent

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#remainder

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#division

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#angle

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#tessellation

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#fraction


calculators, computer drawing packages and measuring tools, and learn to construct and describe simple algorithms using mathematical conventions.

Standards

Number


use place value to determine the size and order of numbers from hundredths to tens of thousands.

round numbers up and down to the nearest unit, ten, hundred, or thousand. They

compare and order simple common fractions such as . skip count forwards and backwards, from various starting points using multiples of 2,

3, 4, 5, 10 and 100. devise and use algorithms for whole number problems of addition and subtraction

involving three-digit numbers; multiplication by single digits (based on automatic recall of multiplication tables) and multiples and powers of ten; and division by a single-digit divisor (based on inverse relations in multiplication tables).

devise and apply algorithms for the addition and subtraction of numbers to two decimal places.

add and subtract simple common fractions with the assistance of physical models. perform mental computations involving numbers up to 30 accurately and reliably.

Numbers are estimated and ordered to two decimal places. predict the accuracy of estimations for computation and recognise whether these are

likely to be over-estimates or under-estimates.

Space


recognise and describe the orientation of lines as vertical, horizontal and diagonal. describe angle in terms of rotation of line segments which meet at a common end-

point. recognise polygons, prisms and pyramids and their component parts such as edges,

vertices and faces. use and interpret two-dimensional representations of three-dimensional objects or

parts of these objects, for example, nets, cross-sections and simple projections. describe what is seen and not seen of a simple object from different positions. recognise and construct simple tessellations and follow instructions to produce

geometric designs such as tangrams. use and compare ways of locating and identifying places on maps and diagrams. develop and test instructions to specify travel directions and location using compass

directions, N, S, E and W, and grid references such as ‘A5' on a street directory.



extend the range of characteristics and attributes, estimated and measured, using informal and formal units to include angle (simple fractions of a complete turn), temperature and weight.

estimate and measure length, area, volume, mass and time using appropriate instruments.


29

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#tangram

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#shadow

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#crosssection

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#net

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#face

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#vertex

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#edge

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#pyramid

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#prism

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#polygon


recognise and use different units of measurement (informal, formal and metric measures) in appropriate contexts, and interpret linear and circular scales in familiar situations such as measuring weight.

describe and interpret the numbers on analog clocks in relation to the minute and hour hands, and interpret timetables and calendars in relation to familiar events.

rate everyday outcomes in terms of likely occurrence and informally and qualitatively describe the fairness of events.

plan and conduct chance experiments with respect to natural variability and tally results of these experiments.

identify numerical data as discrete or continuous and construct column and bar graphs to display frequency data of ordinal categories.

Structure


construct number collections using counting of composite sets of units such as 2, 3, 4, 5, 10 and 100.

investigate and record sequences of decimal numbers generated using multiplication or division by 10.

partition sets into equal-sized subsets to carry out division and recognise that the sharing of a collection into equal-sized parts frequently leaves a remainder.

identify the set of all possible outcomes of a simple chance event (the event space) and use Karnaugh maps to specify the possible combinations of two attributes.

recognise the importance and meaning of the '=' in mathematical statements and technology displays to indicate the result of a computation and to indicate equivalence.

use the commutative and associative properties in combination to facilitate computations such as 7 + 10 + 13 = 10 + 7 + 13 = 10 + 20. They use the distributive property for multiplication over addition in simple computation.

classify and describe angles, polygons and solids according to their properties. describe and summarise the effects of rotations, reflections, transformations and

shadow projections on shapes with respect to what changes and what does not change (invariance).

identify variables and perform simple operations on variables. construct and solve simple equations involving missing numbers and ‘='. recognise samples as subsets of a set (the population under consideration). organise data into lists and Karnaugh maps.



use brackets to give priority to an operation in a simple sequence of operations. follow and interpret algorithms and methods of approximation used by others. formulate and test conjectures to investigate number (for example, the shapes that

can be used to model common fractions); computations (for example, the nature of the product of even and/or odd numbers); number patterns (for example, the patterns of last digits produced by multiples of a given number); measurement (for example, the relationship between size and capacity of a container); and shapes (for example, the effects of reflections, slides/translations and rotations on the orientation of a shape).

describe and explain why some shapes tessellate, why some shapes have different forms of symmetry, and which solids have nets.

represent depth in drawings and describe ‘what is not seen' in three-dimensional drawings.


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http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#odd

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#even

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#variable

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#shadow

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#reflection

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#rotation

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#decimal

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#continuous

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#discrete


use and interpret physical models, the place-value model, and diagrams to explore the properties of numbers, shapes, and location, and to represent computations and measurements.

develop and apply appropriate methods for collection and presentation of survey and simulation data.

apply number skills to everyday contexts such as shopping, with appropriate rounding to the nearest five cents.

illustrate tiling patterns and stacking of solids. identify familiar places and routes from local and regional maps and diagrams, and

relate daily activities to clock times. describe uses of mathematics in earlier times with respect to different numeration

systems and bases, place value and algorithms for computation. use a calculator to check the accuracy of estimations and computations involving

whole numbers and decimals to two places. use a calculator to identify and classify the form of decimal values that result from

division of natural numbers. use computer software to create shapes, tessellations, maps and diagrams, and to

organise and present data. use a range of mechanical and electronic measuring instruments to support

mathematical development at this level.


31


Year Three



1. Use place value concepts to represent whole numbers and decimals using numerals, words, expanded notation and physical models. For example:

a. Recognize 100 means “10 tens” as well as a single entity (1 hundred) through physical models and trading games.

b. Describe the multiplicative nature of the number system; e.g., the structure of 3205 as 3 x 1000 plus 2 x 100 plus 5 x 1.

c. Model the size of 1000 in multiple ways; e.g., packaging 1000 objects into 10 boxes of 100, modeling a metre with centimetre/millimetre strips or other concrete items.

d. Explain the concept of tenths and hundredths using physical models, such as metric pieces, base ten blocks, decimal squares or money.

2. Skip count forwards and backwards by multiples of 2, 3, 4, 5 and 10 starting at any number.

3. Use mathematical language and symbols to compare and order; e.g., less than, greater than, equal to, not equal to, <, >, =,

4. Count money and make change using coins and notes (round off to nearest 5c).

5. Represent fractions and mixed numbers using words, numerals and physical models.

6. Compare and order commonly used fractions and mixed numbers using number lines, models (such as fraction circles or bars), points of reference (such as more or less than ½ ), and equivalent forms using physical or visual models.

7. Recognize and use decimal and fraction concepts and notations as related ways of representing parts of a whole or a set; e.g., 3 of 10

marbles are red can also be described as and 3

tenths are red.

8. Model, represent and explain multiplication; e.g., repeated addition, skip counting, rectangular arrays


32



and area model. For example:

a. Use conventional mathematical symbols to write equations for word problems involving multiplication.

b. Understand that, unlike addition and subtraction, the factors in multiplication and division may have different units; e.g., 3 boxes of 5 bicuits each.

9. Model, represent and explain division; e.g., sharing equally, repeated subtraction, rectangular arrays and area model. For example:

a. Translate contextual situations involving division into conventional mathematical symbols.

b. Explain how a remainder may impact an answer in a real-world situation; e.g., 14 biscuits being shared by 4 children.

10. Explain and use relationships between operations, such as:

a. relate addition and subtraction as inverse operations;

b. relate multiplication and division as inverse operations;

c. relate addition to multiplication (repeated addition);

d. relate subtraction to division (repeated subtraction).

11. Model and use the commutative and associative properties for addition and multiplication.

12. Add and subtract whole numbers with and without regrouping (up to 3 digits).

13. Demonstrate fluency in multiplication facts up to 10 and corresponding division facts by 2, 3, 4, 5 and 10.

14. Multiply and divide 2- and 3-digit numbers by a single-digit number, without remainders for division.

15. Evaluate the reasonableness of computations based upon operations and the numbers involved; e.g., considering relative size, place value and estimates.


33


Measurement, Chance and Data DimensionMeasurement Units


Data Collection

1. Identify and select appropriate formal and informal units for measuring:

a. length – metres, kilometres and other units of measure as appropriate;

b. volume (capacity) – litres and millilitres;c. weight – grams, or kilograms;d. temperature – degrees (Celsius).

2. Tell time to the nearest 5 minutes and find elapsed time using a calendar or a clock.

3. Read thermometres in Celsius scale.

4. Estimate and measure length, mass and volume (capacity), using metric units, accurate to the nearest or unit as appropriate.

5. Use appropriate measurement tools and techniques to construct a figure or approximate an amount of specified length, mass or volume (capacity); e.g., construct a rectangle with length 2centimetres and width 3 centimetres, fill a measuring cup to the 250 ml. mark.

6. Make estimates for perimetre, area and volume using informal units.

1. Collect and organize data from an experiment, such as recording and classifying observations or measurements, in response to a question posed.

2. Draw and interpret picture graphs in which a symbol or picture represents more than one object.

3. Read, interpret and construct bar graphs with intervals greater than one.

4. Support a conclusion or prediction orally and in writing, using information in a table or graph.

5. Match a set of data with a graphical representation of the data.

6. Translate information from charts, tables, picture graphs, line graphs and bar graphs; e.g., create a bar graph from the information in a chart.


34


Statistical Methods

Probability

7. Analyse and interpret information represented on a timeline.

8. Conduct a simple experiment or simulation of a simple event, record the results in a chart, table or graph, and use the results to draw conclusions about the likelihood of possible outcomes.

9. Use physical models, pictures, diagrams and lists to solve problems involving possible arrangements or combinations of two to four objects.




Visualization and Geometric Models

1. Analyse and describe properties of two-dimensional shapes and three-dimensional objects using terms such as vertex, edge, angle, side and face.

2. Identify and describe the relative size of angles with respect to right angles as follows:

a. Use physical models, like straws, to make different sized angles by opening and closing the sides, not by changing the side lengths.

b. Identify, classify and draw right and straight angles.

3. Find and name locations on a labeled grid or coordinate system; e.g., a map or graph.

4. Draw lines of symmetry to verify symmetrical two-dimensional shapes.

5. Build a three-dimensional model of an object composed of cubes; e.g., construct a model based on an illustration or actual object.


Use Algebraic

1. Identify next number or symbol in sequence or pattern.

2. Analyse and replicate arithmetic sequences with and without a calculator.

3. Use patterns to make predictions, identify relationships, and solve problems.

4. Model problem situations using objects, pictures,


35


Representations

Analyse Change

tables, numbers, letters and other symbols.

5. Write, solve and explain simple mathematical statements, such as 7 + □ > 8 or ∆ + 8 = 10.

6. Express mathematical relationships as equations and inequalities.

7. Create tables to record, organize and Analyse data to discover patterns and rules.

8. Identify and describe quantitative changes, especially those involving addition and subtraction; e.g., the height of water in a glass becoming 1 centimetre lower each week due to evaporation.


36








Year 3ASSESSMENT FOCUS PERFORMANCE MEASURE WEIGHT1 Number -

Knowledge and understanding of relationships between ordinal and cardinal number, numerals and numeration.



Understands the composition of and is able to manipulate whole numbers to 1000, vulgar fractions greater than 1/10 and decimal fractions to two decimal places.

#SKU Implications Use place value to determine the size and order of

numbers from hundredths to thousands. Round numbers up and down to the nearest unit,

ten, hundred, or thousand. They compare and order simple common fractions

such as . Skip count forwards and backwards, from various

starting points using multiples of 2, 3, 4, 5, 10 and 100.

#SKU Implications Devise and solve algorithms for whole number

problems of addition and subtraction involving three-digit numbers; (Requiring ‘trading’ technique 2 digits e.g 263 – 179 =)

Devise and solve algorithms for the addition and subtraction of numbers to two decimal places e.g. Money problems.

Add and subtract simple common fractions with the assistance of physical models.

#SKU Implications Devise and solve algorithms for multiplication by

single digits (based on automatic recall of multiplication tables) and multiples and powers of ten; and division by a single-digit divisor (based on

50%


37


Structure



inverse relations in multiplication tables). Perform mental multiplication and division

computations using multiples of 2, 3, 4, 5 and 10 involving numbers up to 30 accurately and reliably.

Construct number collections using counting of composite sets of units such as 2, 3, 4, 5, 10 and 100.

#SKU Implications Use partition groups into equal-sized subgroups to

carry out division and recognise that the sharing of a collection into equal-sized parts frequently leaves a remainder.

Recognise the importance and meaning of the '=' in mathematical statements to indicate the result of a computation and to indicate equivalence.

Use the distributive property for multiplication over addition in simple computation.

Construct and solve simple equations involving missing numbers and ‘='.

Apply knowledge and understanding of number and mathematical processes and values to every day situations and circumstances.

#SKU Implications Be aware of and use different numeration systems

and bases, place value and algorithms for computation such as the Roman system, binary system.

Use a calculator to check the accuracy of estimations and computations involving whole numbers and decimals to two places.

Use a range of mechanical and electronic measuring instruments to support mathematical development at this level.

Apply number skills to everyday contexts such as shopping.


#SKU Implications Construct physical models, to explore the

properties of numbers, shapes, and location, and to represent computations and measurements.

Identify familiar places and routes from local and regional maps and diagrams.

Recognise right angle – less than, greater than. Describe what is seen and not seen geometric

components of a simple object from different positions. .

Locate and identifying places on maps and diagrams.

Develop instructions to specify travel directions and location using compass directions, N, S, E and W, and grid references such as ‘A5' on a street directory.

10%

5%

5%


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Probability

Extend the range of measurement characteristics and attributes, estimated and measured, using informal and formal units to include angle (simple fractions of a complete turn), temperature and weight.

#SKU Implications Estimate and measure length, area, volume, mass

and time using appropriate instruments. Recognise and use different units of measurement

(informal, formal and standard metric measures) in appropriate contexts.

Describe and interpret the numbers on analogue clocks in relation to the minute and hour hands, and interpret timetables and calendars in relation to familiar events.

Relate daily activities to clock times.

Collect data and transpose into and interpret basic graphs#SKU Implications

Develop and apply appropriate methods for collection and presentation of survey data such as frequency of occurrence.

Construct column and bar graphs to display data.


#SKU Implications Conduct chance experiments with respect to

variability and tally results of these experiments. Makes predictions for the likelihood of outcomes

of simple random and non-random events, using qualitative descriptors such as more likely or less likely, such as whether 6's are harder to roll than 2's on a die.

20%

5%

5%


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Year Four




1.Identify and generate equivalent forms of fractions and decimals. For example:

a. Connect physical, verbal and symbolic representations of fractions, decimals and whole numbers; e.g., ,

, “five tenths,” 0.5, shaded rectangles with half, and five tenths.

b. Understand and explain that ten tenths is the same as one whole in both fraction and decimal form.

2. Use place value structure of the base-ten number system to read, write, represent and compare whole numbers through tens of thoudsands and decimals through hundredths.

3. Round whole numbers to a given place value.

4. Identify and represent factors and multiples of whole numbers through 100, and classify numbers as prime or composite.

5. Use models and points of reference to compare commonly used fractions.

6. Use associative and distributive properties to simplify and perform computations; e.g., use left to right multiplication and the distributive property to find an exact answer without paper and pencil, such as 5 x 47 = 5 x 40 + 5 x 7 = 200 + 35 = 235.

7. Recognize that division may be used to solve different types of problem situations and interpret the meaning of remainders; e.g., situations involving measurement, money.

8. Solve problems involving counting money and making change, using both coins and notes.

9. Estimate the results of computations involving whole numbers, fractions and decimals, using a variety of strategies.

10. Use physical models, visual representations, and


40


paper and pencil to add and subtract decimals and commonly used fractions with like denominators.

11. Develop and explain strategies for performing computations mentally.

12. Analyse and solve multi-step problems involving addition, subtraction, multiplication and division using an organized approach, and verify and interpret results with respect to the original problem.

13. Use a variety of methods and appropriate tools for computing with whole numbers; e.g., mental math, paper and pencil, and calculator.

14.Demonstrate fluency in adding and subtracting whole numbers and in multiplying and dividing whole numbers by 1- and 2-digit numbers and multiples of ten.

Measurement DimensionMeasurement Units


1. Relate the number of units to the size of the units used to measure an object; e.g., compare the relationship between a number of different capacity containers to fill a litre bottle with water.

2. Demonstrate and describe perimetre as surrounding and area as covering a two-dimensional shape, and volume as filling a three-dimensional object.

3.Develop and use strategies to find perimetre using informal units, area using tiles or a grid, and volume using blocks; e.g., count squares to find area of regular or irregular shapes on a grid, layer cubes in a box to find its volume.

4. Identify and select appropriate units and instrument to measure:

perimetre – (metres or centimetres). area – (square centimetres or square metres). volume – (litre, millilitres). Mass – (kilograms, grams)

5. Make simple unit conversions within a measurement system; e.g., millimetres to centimetres, centimetres to metres, kilograms to grams, millilitres to litres.

6. Solve and verify solutions to multi-step problems involving measurement.


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Data Collection

Statistical Methods

Probability

1. Create a plan for collecting data for a specific purpose.

2. Represent and interpret data using tables, bar graphs, line plots and line graphs.

3. Interpret and construct Venn diagrams to sort and describe data.

4. Compare different representations of the same data to evaluate how well each representation shows important aspects of the data, and identify appropriate ways to display the data.

5. Propose and explain interpretations and predictions based on data displayed in tables, charts and graphs.

6. Describe the characteristics of a set of data based on a graphical representation, such as range of the data, clumps of data, and holes in the data.

7. Conduct simple probability experiments and draw conclusions from the results; e.g., rolling number cubes or drawing marbles from a bag.

8. Represent the likelihood of possible outcomes for chance situations; e.g., probability of selecting a red marble from a bag containing 3 red and 5 white marbles.

9. Place events in order of likelihood and use a diagram or appropriate language to compare the chance of each event occurring; e.g., impossible, unlikely, equal, likely, certain.


1. Identify, describe and model intersecting, parallel and perpendicular lines and line segments; e.g., use straws or other material to model lines.

2. Describe, classify, compare and draw model two- and three-dimensional objects using their attributes.

3. Identify similarities and differences of quadrilaterals; e.g., squares, rectangles, parallelograms and trapezoids.

4. Identify and classify triangles based on angle


42




measures (e.g. greater than, less than a right angle), length of sides and orientation.

5. Describe points, lines and planes, and identify models in the environment.

7. Identify and describe congruent shapes use reflections (flips), rotations (turns)



Analyse Change

1. Use models and words to describe, extend and make generalizations of patterns and relationships occurring in computation, numerical patterns, geometry, graphs and other applications.

2. Represent and analyse patterns and functions using words, tables and graphs.

3. Construct a table of values to solve problems associated with a mathematical relationship.

4. Use rules and variables to describe patterns and other relationships.

5. Represent mathematical relationships with equations or inequalities.

6. Describe how a change in one variable affects the value of a related variable; e.g., as one increases the other increases or as one increases the other decreases


The Assessment Protocols are not a complete assessment tool for classroom teachers. The Protocols do not cover all dimensions of the Standards or the school Mathematics Syllabus. Those dimensions that are not included in the Protocols, as well as more detailed aspects of the


The Performance Measures in these Protocols are the essential and readily assessable aspects of the Number and Measurement and Chance and Data Dimensions. These are the two dimensions which are required for systemic school data processing and student reporting purposes.


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The intention of these Protocols is to provide a consistent framework for student assessment in these two dimensions across all year levels of the school.

Teachers at each year level will devise and apply Common Assessment Tasks which specifically focus on the aspects of mathematics detailed in these protocols.



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Structure

Understands the composition of and is able to manipulate whole numbers to 1000, vulgar fractions greater than 1/10 and decimal fractions to two decimal places.

#SKU Implications Use place value to determine the size and order

of numbers from hundredths to tens of thousands

Round numbers up and down to the nearest unit, ten, hundred, or thousand.

Compare and order simple common fractions

such as . Skip count forwards and backwards, from

various starting points using multiples of 2, 3, 4, 5, 10 and 100.

#SKU Implications Devise and solve algorithms for whole number

problems of addition and subtraction involving three-digit numbers (Trading technique to 3 digits);

Devise and solve algorithms for the addition and subtraction of numbers to two decimal places.

Add and subtract simple common fractions with the assistance of physical models.

#SKU Implications Devise and solve algorithms for multiplication by

single digits (based on automatic recall of multiplication tables) and multiples and powers of ten; and division by a single-digit divisor (based on inverse relations in multiplication tables).

Perform mental multiplication and division computations using multiples of 2, 3, 4, 5 and 10 involving numbers up to 30 accurately and reliably.

Construct number collections using counting of composite sets of units such as 2, 3, 4, 5, 10 and 100.

#SKU Implications Use partition groups into equal-sized subgroups to

carry out division and recognise that the sharing of a collection into equal-sized parts frequently leaves a remainder.

Recognise the importance and meaning of the '=' in mathematical statements to indicate the result of a computation and to indicate equivalence.

Use the distributive property for multiplication over addition in simple computation. (6+4+6+2+6+4 = 3X6+2X4+2=)

Classify and describe angles, polygons and solids according to their properties.

Construct and solve simple equations involving missing numbers and ‘='.

50%

10%


45




Apply knowledge and understanding of number and mathematical processes and values to every day situations and circumstances.

#SKU Implications Use brackets to give priority to an operation in a

simple sequence of operations. (1+2)X3..1+(2X3)Be aware of and use different numeration systems

and bases, place value and algorithms for computation such as the Roman system, binary system.

Use a calculator to check the accuracy of estimations and computations involving whole numbers and decimals to two places.

Use a range of mechanical and electronic measuring instruments to support mathematical development at this level.

Apply number skills to everyday contexts such as shopping.


#SKU Implications Construct physical models, to explore the

properties of numbers, shapes, and location, and to represent computations and measurements.

Identify familiar places and routes from local and regional maps and diagrams.

Describe angles in terms of rotation of line segments which meet at a common end-point (acute, obtuse, right angle).

Recognise polygons, prisms and pyramids and their component parts such as edges, vertices and faces.

Describe what are seen and not seen geometric components of a simple object from different positions. .

Locate and identifying places on maps and diagrams.

Develop instructions to specify travel directions and location using compass directions, N, S, E and W, and grid references such as ‘A5' on a street directory.

5%

5%


Extend the range of measurement characteristics and attributes, estimated and measured, using informal and formal units to include angle (simple fractions of a complete turn), temperature and weight.

#SKU Implications Estimate and measure length, area, volume, mass

and time using appropriate instruments. Recognise and use different units of measurement

(informal, formal and standard metric measures) in appropriate contexts.

Tell the time using analogue clocks in increments

20%


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Probability

of 5 minutes and digital to the minute. Translate digital to analogue and analogue to

digital time. Interpret timetables and calendars in relation to

familiar events. Relate daily activities to clock times.

Collect data and transpose into and interpret basic graphs

#SKU Implications Develop and apply appropriate methods for

collection and presentation of survey data such as frequency of occurrence.

Construct column and bar graphs to display data and label components appropriately (title, axis, scale increments).


#SKU Implications Conduct chance experiments with respect to

variability and tally results of these experiments. Makes predictions for the likelihood of outcomes of

simple random and non-random events, using qualitative descriptors such as more likely or less likely, such as whether 6's are harder to roll than 2's on a die.

5%

5%


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LEVEL 4

Learning focusIn Number students work with the size and order of large and small numbers including negative numbers, and rational numbers in fraction and decimal form. They learn to identify natural numbers and their factors as prime, even or odd, and to use decimals, ratios and percentages to represent equivalent forms of common fractions. Students develop and apply mental and written algorithms for the addition, subtraction, multiplication and division of natural numbers; the addition, subtraction, multiplication of decimals (to two decimal places); and the addition, multiplication and subtraction of common fractions. In division of natural numbers, they identify and interpret remainders as fractions and recognise the role that remainders play in developing algorithms for finding factors of numbers and decomposing numbers into their product of powers of prime numbers. They routinely make estimations and approximations in numerical computation and learn to make judgments about the appropriateness, effectiveness and reasonableness of these estimates and approximations. In Space students develop their understanding of mathematical properties of shapes and solids and incorporate concepts of size and scale into their descriptions of these shapes and solids. They work with the concepts of boundary, finite and infinite, in relation to lines, surfaces and shapes, and use these ideas to investigate self-similarity in lines and shapes. Students learn to describe the features of shapes and solids that remain the same (are invariant) or change when the shape or solid is enlarged or reduced. They use the ideas of size, scale and direction to specify the relative positions of places and objects in maps and, in the process, apply concepts of shape and connectedness to represent and interpret simple networks. In Measurement, chance and data students learn to use suitable instruments to measure accurately the characteristics of length, area, volume, capacity, angle, time and temperature in formal and units. They describe time elapsed in hours, minutes and seconds, and in simple decimal subdivisions of a second (tenths and hundredths). They measure the perimetre and surface area of a range of shapes and describe the accuracy and adequacy of their measurements in context. Students learn to distinguish between discrete and continuous measurement data and to apply measures of centre and simple measures of spread to informally describe simple characteristics of the distribution of data in a set. They refine their descriptions of chance (random) events from impossible to certain using mathematical words and fractions between 0 and 1, and follow simple simulations of chance events. They use mechanical and electronic technology to simulate outcomes in a random experiment, and a scientific calculator to calculate the mean of a discrete or continuous data set. They also use technology to create graphs of data sets and find and extract relevant information from a database available on the Internet. In Structure students learn to form and specify sets based on given properties or criteria (for example, sequences generated from small composite units for counting forwards and backwards); representation of natural numbers in different bases; shapes that enclose space in two dimensions and solids in three dimensions; common objects that may be used to make measurements in practical situations; and outcomes in chance experiments. They learn to classify patterns of remainders formed when larger natural numbers are divided by small natural numbers such as 3 or 5; and classify shapes and solids according to their geometric properties. Students sort and classify lines in shapes and solids found in the environment according to their relative orientations, and identify shapes that have reflection symmetry and/or rotation symmetry. They learn to distinguish the amount of turn in angles relative to straight, right and zero angles, and find the shortest paths or routes between places on a map. They use the concept of one-to-one correspondence to construct number lines involving whole numbers, decimals to one decimal place and simple common fractions.


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http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#measureofcentre

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#perimeter

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#connected

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#scale

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#infinite

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#finite

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#approximate

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#power

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#algorithm

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#percentage

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#ratio

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#odd

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#even

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#prime

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#rational

http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#negative


Students draw Venn diagrams and Karnaugh maps to help them test the validity of simple deductive arguments involving the quantifiers none, some and all. They make judgments about the appropriateness, scope and reasonableness of procedures for tasks, and develop simple algorithms involving words, diagrams and mathematical symbols. They generate sequences using recursion (computing the next term from the previous term or terms), and develop function rules for computing terms in sequences depending on their position in the sequence. Students learn to use the identity elements for arithmetic operations on integers and rational numbers, and apply the relationship between identity and inverse to the addition, subtraction and multiplication of integers and fractions. They play and interpret games in which the rules are based on transformations of shapes, or combinations of transformations of shapes, and form a mathematical structure. In Working mathematically using the processes of specialising, exemplifying, justifying and refuting, students learn to make judgments about the truth of conjectures regarding even and prime numbers, remainders, shapes and their properties, and relationships in measurement. They make and test conjectures and generalisations about sets of numbers and shapes and their properties, and develop convincing, principled arguments for propositions. They use concrete materials, diagrams and functions as models to test conjectures relating to the Number, Space and Measurement chance and data dimensions.Students develop, use and refine criteria for collecting, organising and presenting data. They identify situations in everyday life where estimates of numbers and computations are considered appropriate, and investigate the methods used to make these estimates. They apply concepts from Number and Space to develop strategies for games, and identify myths and misconceptions about chance and fairness in everyday situations. They identify key concepts and developments from contexts in the history of mathematics, for example, triangular, square, pentagonal, hexagonal and perfect numbers in classical Greek mathematics. Students develop and use estimation procedures to predict and check the results of computations carried out with technology. They use technology for more complex and extended computations involving a series of operations with small and/or large numbers. They use application programs in a graphics calculator to explore and analyse games and puzzles involving numbers. They carry out a sequence of instructions to draw a shape, solid or net of a solid, and translate these instructions into a language or form that can be used in a computer software package or graphics calculator to represent shapes and solids under transformations.

Standards

Number


comprehend the size and order of small and large numbers (from thousandths to millions), including negative numbers, common fractions and decimals.

accurately estimate the size of fractions and decimals in the vicinity of 0 and 1 relative to 0, ½ and 1.

identify numbers and their factors as square, prime or composite, and interpret these numbers and their factors in terms of the area and the dimensions of their corresponding rectangular geometric arrays.

recognise and evaluate simple powers of natural numbers such as 24 = 16. explain and use mental and written algorithms for the addition, subtraction,

multiplication and division of natural numbers; the addition, subtraction, multiplication of decimals (to two decimal places); and the addition, multiplication and subtraction of common fractions.

represent natural numbers in other bases.


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construct and recognise multiples of integers (including lowest common multiple) and common fractions, and interpret constant multiples of a number as scale factors of the number.

use decimals, ratios and percentages to find equivalent representations of common fractions.

identify and interpret remainders as fractions and recognise the role that remainders play in algorithms for finding the factors of natural numbers.

use repeated division by increasing primes to express numbers as a product of powers of prime numbers, for example, 360 = 23 x 32 x 51.

When using estimates of numbers in computation, students apply strategies appropriate for the situation, in particular, mental computations.

develop and use criteria for deciding if an estimate of a computation is reasonable or not.

Space


identify the mathematical properties of horizontal, vertical, parallel and perpendicular lines in relation to each other; shapes and solids, including prisms, pyramids, cylinders and cones; and incorporate the ideas of angle, size and scale into descriptions of the features of these shapes and solids.

explain the idea of finiteness and non-finiteness in relation to lines and surfaces, and use recursion to investigate the idea of self-similarity of shapes.

make two-dimensional representations of three-dimensional objects. use sketches of shapes and solids to represent the surrounding environment, and

describe in mathematical language their relative sizes in that environment. also use the ideas of size, scale and direction when referring to the relative positions

of places and objects in maps, and demonstrate understanding of shape and connectedness in diagrams of networks.

develop a sequence of instructions for drawing a shape, solid or net of a solid, and adjust these instructions to take account of scale.

formulate and test procedures, expressed in terms of compass points and simple coordinate systems, that describe how to get from one place to other places.

use and interpret conventional symbols and language in activities relating to place, direction, paths and scale in maps.



accurately measure the characteristics of length (including perimetre), area (including surface area), volume, capacity, angle, time (including duration of time) and temperature in formal and units using appropriate instruments and scales.

choose accuracy of measurement relevant to the situation at hand and sufficient to distinguish between the sizes of things with the same characteristic, and find the corresponding difference between these sizes to that accuracy.

refine their descriptions of chance (random) events in the range from impossible to certain using words and fractions or decimals between 0 and 1.

explain the role of symmetry in chance situations and experiments involving equally likely events (for example, that the symmetry inherent in a device used to generate random events may be used to calculate the probability of the outcomes in each event).

comprehend that experimental estimates of probabilities (relative frequencies) converge to the theoretical probability values in the long run.


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comprehend how chance events may be simulated (for example, randomly choosing a birth month by selecting from a shuffled pack of cards without kings), and that simulations provide models (estimates) of situations that are impractical to deal with without using an empirical approach.

classify numerical data as discrete or continuous, and collect, organise, analyse, interpret and represent categorical, ordinal and numerical data in response to planned questions.

attend to the clarity of the questions, sampling techniques, and methods used to present data.

recognise and describe the relationship between measures of centrality and simple measures of spread used to describe and order data in a set.

follow a plan and sequence of instructions involving shapes and measurements to construct an object from prefabricated parts, for example, a piece of furniture or a model car.

calculate and compare the times taken by various people to complete activities. calculate probabilities associated with experiments involving equally likely events

(for example, the probability of each outcome when two die are rolled), and the probabilities of symmetric events in the event space, and the mean, median and range for grouped and ungrouped data.

construct graphs to represent data sets and devise suitable scales (nominal, ordinal or interval) for the reference lines (axes) for the categories and frequencies of the data.

Structure


form and specify sets of numbers, shapes, transformations and data according to given criteria and/or conditions such as equivalence, congruence identify the nature of the set (the population) and data from which samples are drawn as finite or infinite and discrete or continuous.

use Venn diagrams and Karnaugh maps to test the validity of simple deductive arguments involving simple applications of the quantifiers none, some or all to sets.

identify variables and related variables in everyday situations, and explain the ideas of change, dependency and allowable values in relationships between pairs of variables.

interpret sketch graphs involving functions of a single variable. construct rules for sequences using recursion relations and relations that depend on

the position of the term in the sequence. describe the features of shapes or solids that remain the same or change when the

shape or solid is enlarged or reduced. describe general patterns using words, numbers, diagrams and symbols. establish equivalence of simple mathematical expressions involving properties, such

as the distributive property for multiplication over addition, a(b + c ) = ab + ac. identify and apply the identity and inverse elements for the arithmetic operation on

integers and rational numbers, and for simple transformations in space.



explain why a few successful examples are not sufficient to form a generalisation and how a single counter-example suffices to invalidate a generalisation.

make and test conjectures about the generalised forms of numbers in terms of divisors, factors and remainders; shapes and their properties and related measurements.


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use appropriate physical models and graphs when testing the truth of conjectures. design algorithms as models of mathematical processes such as the construction of

an equilateral triangle. engage in a planned investigation involving mathematical modelling and refine a

model in terms of its formulation and interpretation. identify the historical evolution of key mathematical ideas such as the emergence of

negative numbers. identify situations in everyday life where estimates of numbers and computations are

considered appropriate, and investigate the methods used to make these estimates and estimate likelihood from simulations.

collect and analyse data about people's beliefs about fairness in games of chance. use the memory function on a scientific or graphics calculator to do computations

with a series of operations involving small and large numbers, and, with the aid of a scientific or graphics calculator, use estimation procedures to predict and check the results of computations.

use a scientific or graphics calculator to implement algorithms to find factors and prime factors of numbers and to explore facts and puzzles involving numbers.

use graphics calculators or computer-drawing packages and application programs to represent shapes and solids under a range of transformations, and use technology to generate simple simulations of events such as gender and order of children born in a family.

Year Five


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NumberPlace-value and number properties

Read, say, write (figures and words), compare and order whole numbers to 6 digits and numbers with decimal fractions to two decimal places.

Compare and order decimal fractions with equal numbers of places

(e.g. 3.05, 3.01, 4.43, 3.12)

o Represent the structure of whole numbers to 6 digits and decimal numbers to 2 decimal places (e.g. 215634.65 = (2x100,000) +(1x10,000) + (5 × 1000) + (6 × 100) + (3 × 10) + (4 × 1) + (6x1/10)+(5x1/100)

Interpret negative whole numbers and locate them on a scale (e.g. – 5oC on a temperature scale).

Add mentally by rounding to the nearest 10 (e.g. 53 cm + 44 cm; think 50 and 40, plus 7: $75 + $18; think 75 and 20, subtract 2)

Use place-value to extend multiplication and division facts (e.g. 70 m x 50 m; think 7 tens by 5 tens is 35 hundreds: how many 50 g in 450 g? Think 45 divided by 5)

Use number properties to generate multiples quickly and calculate mentally with fractions (e.g. 12 by 5 equals 6 by 10, 7 twelves equals 7 tens and 7 twos).


53


Common Fractions Compare and order common fractions with like denominators e.g. 2/8 and 3/8

Locate fractions on a number line and use it to count in fractional steps

Convert a simple common fraction to a decimal and vice versa

Automatic recall multiplication and division facts

Automatically recall and use multiplication and division facts up to 10 × 10

Estimation Strategies

Use front-end estimation to check computations e.g. 323 km+189 km is approximately 300km+200km. Is approximately 500. 4509 divided by 3 must be greater than 1000.

Round decimals to the nearest whole number to check a computation (e.g. 8.6 × 3.1 is around 9 × 3).

Computation and application of number

Add and subtract decimal numbers with equal numbers of decimal places.

Use written algorithms for the addition, subtraction, multiplication (2 by 2 digits) and division (3 by 1 digit) of natural numbers;

Add, subtract, multiply decimals (to one decimal place.

Divide whole numbers by 1-digit whole numbers without using a calculator and check the answer using multiplication 7)35624 5089 r. 1

Compare various paper-and-pencil methods for ease and efficiency (e.g. Lattice multiplication and the long multiplication algorithm).

Model operations with common fractions

Add and subtract fractions with small like denominators and record these using understood written methods


54


Find fractional parts of discrete collections and quantities

Problem solving Select the relevant information to solve a one process problem or carry out a practical task.

Use a calculator and its memory facility to solve problems involving up to 2 sequential operations.

Generating and investigating number sequences

Use rules which involve a combination of operations to generate a sequence (e.g. Start with 4, multiply the previous term by 2 and subtract 4)

Multiples and factors

Generate and recognise multiples of whole numbers

Find and use factors of whole numbers (e.g. To multiply by 15, multiply by 3 then by 5)

Define and identify prime numbers.

Number sentences involving the four operations, brackets, decimal numbers and fractions.

Complete number sentences (e.g. Complete 16 = . . × (. . + . .) ) and solve number puzzles expressed in words or with missing numbers (e.g. I’m thinking of a number, if I multiply it by 6 and add 3, I get 33).

SpaceShape and space Classify angles as right angles, those greater than

and those less than a right angle

Identify lines in the environment (e.g. horizontal, vertical, perpendicular, parallel)

Use a ruler and a protractor to draw accurate angles (multiples of 45 degrees).

Spatial properties and spatial terms

Analyse, explain and compare the spatial properties of lines, angles, polygons, polyhedra and cross-sections (e.g. This triangular prism has 6 vertices and 5 faces, and 2 of the faces are parallel)

Use standard language to describe and label line, segment, ray, angle, skew, parallel and perpendicular.


55


Label vertex, rays, interior and exterior for an angle.

Use conventions for drawing three-dimensional objects to show the depth dimension (e.g. Draw a cube with some rectangular faces drawn as parallelograms, and dotted lines for hidden edges)

Identify faces of an object with parts of its net and use various nets to construct 3D objects.

Describe how many small cubes were used to make a larger cube (e.g. From a drawing of a cube 4 long × 4 wide × 4 high, consider the ‘hidden’ cubes).

Predict and describe the shapes and movements required to make or continue a spatial pattern

LocationConventional

location language

Give a clear sequence of directions (e.g. Travel north to the fork at G9 on this map, turn right, continue north-east for about 5 kilometres)

Describe turn using fractions of a full turn (quarter, half, three-quarter turn)

Coordinates and

directions

Locate coordinate points on graph paper (e.g. A rectangle has corners at (3, 2), (3, 5), (2, 5), (2, 2))

Describe movements on isometric dot paper or grid paper using NE, NW, SE, SW.

Find given features on an unfamiliar map (e.g. Using the appropriate Melbourne street directory page, find Degraves Street via the index, find some hospitals using the legend)

Apply simple

scale

Interpret a simple scale bar on a map or plan and uses this to calculate distances or lengths.

Measurement and Data

Appropriate attributes and standard units

Choose the appropriate formal units to measure different attributes of the same object (e.g. To compare two people both height and mass could be specified)


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Select the appropriate unit to specify a quantity (e.g. Height of a building, area of a football ground, capacity of a storage tank)

Use known sizes of familiar things and standard measures to help make and improve estimates.

Estimate by comparison with standard units.

Measuring and Drafting

Accurately measure, using appropriate instruments and scales ,the characteristics of:

Length (including perimeter) in metres, centimetres, millimetres,

Area (including surface area), square metres, square centimetres etc.

Volume, (cubic units)Capacity, (Litres, millilitres) Angle, (degrees)Time (including duration of time) analogue,

digital, 24 hour, minutes, seconds etc.Temperature in formal and standard units

(Celsius)

Identify paths between points on a grid or coordinate plane and compare the lengths of the paths; e.g., shortest path, paths of equal length.

Demonstrate understanding of the differences among linear units, square units and cubic units.


Chance

Read, construct and interpret frequency tables, pie graphs and line graphs, bar graphs, Venn Diagrams, Arrow and Tree Diagram.

Select and use a graph that is appropriate for the type of data to be displayed;

Read and interpret increasingly complex displays of data,

Determine appropriate data to be collected to answer questions posed by students or teacher, collect and display data, and clearly communicate findings.

Determine and use the range, mean, median and mode, and explain what each does and does not indicate about the set of data.

List and explain all possible outcomes in a given situation.


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Identify the probability of events within a simple experiment, such as three chances out of eight.

Use 0, 1 and ratios between 0 and 1 to represent the probability of outcomes for an event, and associate the ratio with the likelihood of the outcome.


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Number - In Number students work with the size and order of large and small numbers including negative numbers, and rational numbers in fraction and decimal form.



Understands the composition of and is able to manipulate and apply whole numbers, vulgar fractions and decimal fractions.

#SKU Implications Read write and order numbers from 0.01 to 99 999. Order, compare and rename common fractions and

decimals. Use estimation and rounding off as an aid to verify

answers to algorithms and problem solving situations Investigate and generate number sequences with whole

numbers and decimals. Use a calculator to check answers and to perform multiple

function algorithms.

#SKU Implications Add and subtract whole numbers to 99 999; decimal

fractions with an equal number of places, to hundredths: vulgar fractions with common denominators.

Solve problems incorporating units of measure (linear, volume, mass), time (minutes, hours, days, weeks) involving one process addition or subtraction.

#SKU Implications Multiply whole numbers up to three places by two digit

numbers. Divide three digit numbers by single digit numbers and

record remainders as fraction. Accurately recall all multiplication tables from 2 to 10 time

and +n and –n facts to 20. Solve problems incorporating units of measure (linear,

volume, mass), time (minutes, hours, days, weeks)

50%

20%

20%


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Spatial Awareness – students develop their understanding of mathematical properties of shapes and solids and incorporate concepts of size and scale into their descriptions of these shapes and solids.

involving one process multiplication or division by single digits.


#SKU Implications Describe, name and draw 2D and 3D objects using faces,

edges and vertices Display an awareness of the concepts of transformation

and tessellation Begin to recognise, describe and represent parallel,

perpendicular, horizontal and vertical lines, right angles and angles greater than or less than 90o.

Use scale, direction and coordinates on making and reading informal maps.

Identify lines of symmetry in 2D shapes.

10%


Students learn to use suitable instruments to measure accurately the characteristics of length, area, volume, capacity, angle, time and temperature in formal and standard units..


Probability


#SKU Implications Read and measure time accurately using digital and

analogue clocks Accurately measure area and perimeter of regular shapes,

using CM. and metres Select and use appropriate conventional units of measure

from cm., m., g., kg., ml., and litres. Make judgements about the relative size of objects and

compare them to known standard measurements. Competently use a protractor to measure angles. Compare and measure the volume and mass of objects. Follow a plan and sequence of instructions involving

shapes and measurements to construct an object from prefabricated parts, for example, a model house, a piece of furniture or a model car.

Collect data and transpose into and interpret basic graphs#SKU Implications Collect, organise, and represent numerical data for

analysis and interpretation in response to planned questions.

Construct bar and column graphs to represent data sets and devise suitable scales for the reference lines (axes) for the categories and frequencies of the data.


#SKU Implications Comprehend the concept of probability and conduct

simple experiments (e.g. the toss of a coin) to tally and to validate predictions.

60%

30%

10%


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Year Six

NumberPlace-Value Read, say, write (figures and words), compare and order

whole numbers to 7 digits and numbers with decimal fractions to three decimal places.

Compare and order decimal fractions with unequal numbers of places (e.g. 3.05, 3.001, 4.4, 3.12)

Represent the structure of whole numbers to 7 digits and decimal numbers to 3 decimal places (e.g. 2,175,634.657 = (2x1,000,000) +(1x100,000) + (7x10,000) + (5 × 1000) + (6 × 100) + (3 × 10) + (4 × 1) + (6x1/10) + (5x1/100) + (7X1/1000)

Count in two digit decimal fractional amounts (e.g. 0.25, 0.50, 0.75, 1.00, and so on)

Common Fractions

Compare and order common fractions with related denominators e.g. ¼ and 3/8

Convert a simple common fraction to a percentage and vice versa

Automatic recall Automatically recall and use multiplication and division facts up to 10 × 10

Automatically recall and use simple common fraction addition and subtraction facts

Automatically recall frequently used common fraction, decimal and percentage equivalences

Estimation Strategies

Use front-end estimation to check computations


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Computation and application of number

Use written algorithms for the addition, subtraction, multiplication (4 by2 digits) and division (4 by 1 digit) of natural numbers;

Add, subtract, multiply decimals (to two decimal places

Add, subtract and multiply common fractions.Model operations with common fractions, and develop written methods for carrying out these operations.

Add and subtract fractions with small related denominators (e.g. Fifths and tenths) and record these using understood written methods

Recognise the relationship between division and fractions and interpret a remainder when dividing by a whole number

Problem solving Select the relevant information to solve a problem (up to

3 processes) or carry out a practical task and determine whether additional information is required

Use a calculator and its memory facility to solve problems involving a sequence of operations on decimal numbers, interpreting remainders, negative numbers and overflow displays

Operate with negative whole numbers in everyday situations (e.g. The temperature at Mount Hotham dropped 6oc from 3oc).

Generating and investigating number sequences

Use rules which involve a combination of operations including decimal numbers, to generate a sequence (e.g. Start with 4, multiply the previous term by 2.5 and subtract 4)

Describe and verify rules for number sequences which relate each element to the previous element or elements (e.g. Next term in a Fibonacci sequence) and each element to its position in the sequence (e.g. Tenth term of 1, 4, 9, 16, 25, …).

Number sentences involving the four operations, brackets, decimal numbers and fractions.

Construct and verify number sentences (e.g. Write a number sentence with two operations and the number 3/4 e.g. 2 + ¾ of 4 =


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Space

Shape and space

Use a ruler and a protractor to draw accurate angles (multiples of 10 degrees).

Spatial properties and spatial terms.

Use standard language to define geometric vocabulary: vertex, face, altitude, diagonal, isosceles, equilateral, acute, obtuse and other vocabulary as appropriate.

Classify and describe two-dimensional and three-dimensional geometric figures and objects by using their properties; e.g., interior angle measures, perpendicular/parallel sides, congruent angles/sides.

Predict and describe sizes, positions and orientations of two-dimensional shapes after transformations such as reflections, rotations, translations and dilations.

Build three-dimensional objects with cubes, and sketch the two-dimensional representations of each side; i.e., projection sets.

Predict and sketch various nets for 3D objects such as pyramids and prisms

Design and use a template of a figure to make a tessellating pattern and explain why certain types of shapes will or will not tessellate

Use a square grid to reduce or enlarge a two-dimensional shape

Where technology is available, use a computer drawing package to enhance Spacial activities.

LocationConventional location language

Describe turn using fractions of a full turn (quarter, half, three-quarter turn) and degrees (90, 180, 270, 360)

Coordinates and directions

Use a map to plan a journey that visits given points (e.g. A paper round route).

Sketch detailed maps oriented to north with attention to distance and direction or showing accurate relative proportion

Apply simple scale

Make a simple scale drawing of a piece of furniture and explains the scale used


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Measurement and DataAppropriate attributes and standard units

Estimate perimeter or circumference and area for circles, triangles and quadrilaterals, and surface area and volume for prisms and cylinders by:

a) estimating lengths using string or links, areas using tiles or grid, and volumes using cubes;

b) measuring attributes (diameter, side lengths, or heights) and using established formulas for circles, triangles, rectangles, parallelograms and rectangular prisms.

Measuring and Drafting

Understand and describe the difference between surface area and volume.

Use strategies to develop formulas for finding circumference and area of circles.

Determine which measure (perimeter, area, surface area, volume) matches the context for a problem situation; e.g., perimeter is the context for fencing a garden, surface area is the context for painting a room.

Understand the difference between perimeter and area, and demonstrate that two shapes may have the same perimeter, but different areas or may have the same area, but different perimeters.

Use benchmark angles (e.g.; 45º, 90º, 120º) to estimate the measure of angles, and use a tool to measure and draw angles.


Chance

Read, construct and interpret frequency tables, pie graphs and line graphs, bar graphs, Venn Diagrams, Arrow and Tree Diagram.

Select, create and use graphical representations that are appropriate for the type of data collected.

Compare representations of the same data in different types of graphs, such as a bar graph and circle graph.

Understand the different information provided by measures of centre (mean, mode and median) and measures of spread (range).

Make logical inferences from statistical data.

Design an experiment to test a theoretical probability and explain how the results may vary.


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Assessment Protocols (Year 6 Level 4)Abbreviation and Terminology Key:In Italics: ‘Primary Assessment Measure’ – to be initial and foremost measurement criteria for assessing student performance against Assessment Focus.# SKUs: Skills, Knowledge and Understandings detailed in this Syllabus Framework. To be regarded by teacher when making ‘on-balance’ judgements about student performance.







Number - In Number students work with the size and order of large and small numbers including negative numbers, and rational numbers in fraction and decimal form.


Understands the composition of and is able to manipulate and apply whole numbers, vulgar fractions and decimal fractions.

#SKU Implications Read, write and order numbers from three decimal places to

millions. (0.001 – 9 999 999) Order, compare and rename common fractions, decimal

numbers and percentages. Choose and apply mental strategies, including estimation, to

solving problems involving four processes, money and measurement.

Investigate and generate number sequences with whole numbers, fractions and decimals.

Select and use relevant information and appropriate operations to problem solving situations.

Use a calculator to perform a series of operations and to check answers.

identify numbers and their factors as square, prime or composite.

recognise and evaluate simple powers of natural numbers such as 24 = 16.

#SKU Implications Add and subtract whole numbers to 999 999; decimal fractions

to three places, and fractions with common denominators, as well as those requiring simple conversions e.g. ½ + ¼ =.

Use mental and written algorithms for the addition, subtraction of natural numbers; the addition, subtraction, of decimals (to two decimal places); and the addition and subtraction of common fractions.

50%

20%


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http://vels.vcaa.vic.edu.au/essential/discipline/mathematics/glossary.html#composite






Spatial Awareness – students develop their understanding of mathematical properties of shapes and solids and incorporate concepts of size and scale into their descriptions of these shapes and solids.

Solve problems incorporating units of measure (linear, volume, mass), time (minutes, hours, days, weeks) involving multiple processes of addition and subtraction.

#SKU Implications Multiply whole numbers by 2 digit numbers and decimals by

single digits. Recall automatically and accurately all Xn tables from 2X to

12X and number facts involving +n, -n, and division. Divide by single digit numbers and record answers as vulgar

and decimal fractions. Use mental and written algorithms for the multiplication and

division of natural numbers; the multiplication of decimals (to two decimal places); and multiplication of common fractions.

Solve problems incorporating units of measure (linear, volume, mass), time (minutes, hours, days, weeks) involving multiple processes of multiplication division.


#SKU Implications Describe, name and draw 2D and 3D objects using faces,

edges and vertices Display an awareness of the concepts of transformation and

tessellation Begin to recognise, describe and represent parallel,

perpendicular, horizontal and vertical lines, right angles and angles greater than or less than 90o.

Use scale, direction and coordinates on making and reading informal maps.

Identify lines of symmetry in 2D shapes.

20%

10%

Measurement, Chance and Data -students learn to use suitable instruments to measure accurately the characteristics of length, area, volume, capacity, angle, time and temperature in formal and standard units.


#SKU Implications Estimate, read and measure time accurately using digital and

analogue clocks and 24 hour time notation. Accurately measure area and perimeter of regular shapes,

using cm. and metres and hectares in problem solving situations.

Select and use appropriate conventional units of measure from cm., m., g., kg., ml., and litres as well as accurately read simple scales such as protractors and thermometers to the nearest gradation.

Make judgements about the relative size of objects and compare them to known standard measurements.

Competently use measurement instruments such as compass and protractor to measure angles.

Compare and measure the volume and mass of objects and the relationships between them.

Follow a plan and sequence of instructions involving shapes and measurements to construct an object from prefabricated parts, for example, a model house, a piece of furniture or a model car.

60%


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Probability

Collect data and transpose into and interpret basic graphs#SKU Implications Collect, organise, and represent numerical data for analysis

and interpretation in response to planned questions. Construct bar, column, line or pie graphs to represent data sets

and devise suitable scales for the reference lines (axes) for the categories and frequencies of the data. (Apply computer aided drawing programs where appropriate)


#SKU Implications Comprehend the concept of probability and conduct simple

experiments (e.g. the toss of a coin) to tally and to validate predictions.

Apply appropriate language to logically describe probability in social and natural environment e.g. males/females in population sample, eye colour, height etc., weather patters, popular colours/TV programs/Football teams etc.

30%

10%


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LEVEL 5

Year Seven




1. Demonstrate an understanding of place value using powers of 10 and write large numbers in scientific notation.

2. Explain the meaning of exponents that are negative or 0.

3. Describe differences between rational and irrational numbers; e.g., use technology to show that some numbers (rational) can be expressed as terminating or repeating decimals and others (irrational) as non-terminating and non-repeating decimals.

4. Use order of operations and properties to simplify numerical expressions involving integers, fractions and decimals.

5. Explain the meaning and effect of adding, subtracting, multiplying and dividing integers; e.g., how adding two integers can result in a lesser value.

6. Simplify numerical expressions involving integers and use integers to solve real-life problems.

5. Solve problems using the appropriate form of a rational number (fraction, decimal or percent).

6. Develop and Analyse algorithms for computing

with percents and integers, and demonstrate fluency in their use.

9.Represent and solve problem situations that can be modeled by and solved using concepts of absolute value, exponents and square roots (for perfect squares).

Measurement, Chance and Data DimensionMeasurement Units

1.Select appropriate units for measuring derived measurements; e.g., miles per hour, revolutions per minute.

2. Convert units of area and volume within the same measurement system using proportional reasoning


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Data Collection

Statistical Methods

and a reference table when appropriate; e.g., square feet to square yards, cubic metres to cubic centimetres.

3. Estimate a measurement to a greater degree of precision than the tool provides.

4. Solve problems involving proportional relationships and scale factors; e.g., scale models that require unit conversions within the same measurement system.

5. Analyse problem situations involving measurement concepts, select appropriate strategies, and use an organized approach to solve narrative and increasingly complex problems.

6. Use strategies to develop formulas for finding area of trapezoids and volume of cylinders and prisms.

7. Develop strategies to find the area of composite shapes using the areas of triangles, parallelograms, circles and sectors.

8. Understand the difference between surface area and volume and demonstrate that two objects may have the same surface area, but different volumes or may have the same volume, but different surface areas.

10. Describe what happens to the surface area and volume of a three- dimensional object when the measurements of the object are changed; e.g., length of sides are doubled.

1. Read, create and interpret box-and-whisker plots, stem-and-leaf plots, and other types of graphs, when appropriate.

2. Analyze how decisions about graphing affect the graphical representation; e.g., scale, size of classes in a histogram, number of categories in a circle graph.

3. Analyze a set of data by using and comparing combinations of measures of center (mean, mode, median) and measures of spread (range, quartile, interquartile range), and describe how the inclusion or exclusion of outliers affects those measures.

4. Construct opposing arguments based on analysis of the same data, using different graphical representations.


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Probability

5. Compare data from two or more samples to determine how sample selection can influence results.

6. Identify misuses of statistical data in articles, advertisements, and other media.

8. Compute probabilities of compound events; e.g., multiple coin tosses or multiple rolls of number cubes, using such methods as organized lists, tree diagrams and area models.

8. Make predictions based on theoretical probabilities, design and conduct an experiment to test the predictions, compare actual results to predicted results, and explain differences.

Space DimensionCharacteristics and

Properties1. Use proportional reasoning to describe and

express relationships between parts and attributes of similar and congruent figures.

2. Determine sufficient (not necessarily minimal) properties that define a specific two-dimensional figure or three-dimensional object. For example:

a. Determine when one set of figures is a subset of another; e.g., all squares are rectangles.

b. Develop a set of properties that eliminates all but the desired figure; e.g., only squares are quadrilaterals with all sides congruent and all angles congruent.

3. Use and demonstrate understanding of the properties of triangles. For example:

a. Use Pythagorean Theorem to solve problems involving right triangles.

b. Use triangle angle sum relationships to solve problems.

4. Determine necessary conditions for congruence of triangles.

5. Apply properties of congruent or similar triangles to solve problems involving missing lengths and angle measures.


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Visualization and Geometric Models

6. Determine and use scale factors for similar figures to solve problems using proportional reasoning.

7. Identify the line and rotation symmetries of two-dimensional figures to solve problems.

8. Perform translations, reflections, rotations and dilations of two-dimensional figures using a variety of methods (paper folding, tracing, graph paper).

9. Draw representations of three-dimensional geometric objects from different views.



Analyse Change

1. Represent and Analyse patterns, rules and functions with words, tables, graphs and simple variable expressions.

2. Generalize patterns by describing in words how to find the next term.

3. Recognize and explain when numerical patterns are linear or nonlinear progressions; e.g., 1, 3, 5, 7... is linear and 1, 3, 4, 8, 16... is nonlinear.

4. Create visual representations of equation-solving processes that model the use of inverse operations.

5. Represent linear equations by plotting points in the coordinate plane.

6. Represent inequalities on a number line or a coordinate plane.

7. Justify that two forms of an algebraic expression are equivalent, and recognize when an expression is simplified; e.g., 4m = m + m + m + m or a · 5 + 4 = 5a + 4.

8. Use formulas in problem-solving situations.

9. Recognize a variety of uses for variables; e.g., placeholder for an unknown quantity in an equation, generalization for a pattern, formula.

10. Analyse linear and simple nonlinear relationships to explain how a change in one variable results in the change of another.

11. Use graphing calculators or computers to Analyse change; e.g., distance-time relationships.


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