Measurement and Geometry F-10A

Mathematics

Scope and Sequence

Measurement and Geometry Foundation to Level 10

Introduction

What is a Scope and Sequence?

scope

The breadth and depth of content to be covered in a curriculum at any one time (e.g. week, term, year, over a student’s school life.) All that you do in a given period.

sequence

The order in which content is presented to learners over time. The order in which you do it.

Together a scope and sequence of learning bring order to the delivery of content, supporting the maximising of student learning and offering sustained opportunities for learning. Without a considered scope and sequence there is the risk of ad hoc content delivery and the missing of significant learning.

http://activated.act.edu.au/ectl/design/scope_and_sequence.htm

Why does a school need a scope and sequence?

An agreed Scope and Sequence for a Learning Area, provides a sound basis for a school being able to offer a guaranteed and viable curriculum by addressing gaps in students’ leaning and eliminating unnecessary repetition. A shared Scope and Sequence within a school enables teachers to have clarity about the knowledge, skills and dispositions that students will acquire in their learning and what they need to learn next. A Scope and Sequence supports teachers with effective unit and lesson planning and enables teachers to maintain a developmental focus on student learning as students progress through the school.

The Mathematics Scope and Sequence developed by WMR

This document has been developed to support schools with the transition to AusVELS Mathematics for 2013. While it provides examples of yearly overviews and learning sequences based on the content descriptors in the Australian Curriculum, it is not a complete curriculum. Each individual school can use the documents as a basis for developing a guaranteed and viable curriculum that caters for the needs of their school community.

Levels Foundation to 10A each include a set of learning goals/ intentions for each content sub-strand intended to provide a user friendly guide to the essential learnings around which teachers and teams could base their unit and lesson development.

Proficiency strands are listed next to each learning goal / intention as a guide only and teachers / teams are encouraged to consider all proficiencies equally whilst planning units and lessons. Where a particular proficiency is not listed for a content sub-strand teachers and teams should endeavour to contextualise the learning goals to address these proficiencies. Please note the following:

Sequence of teaching

The learning goals/intentions are listed adjacent to the content descriptions to assist teachers when developing a teaching program. They are not necessarily in the order to be taught – teachers /teams will make their own decisions regarding this. The third column has been included to assist teams to develop ideas for unit planning.

A sample Scope and Sequence Overview is also provided for each of the year levels from F to 10A. The number of weeks given to each unit in the overview acts as a guide and the total number of weeks allows for the many interruptions in a typical school year.

Links between the Learning Goals/Intentions and the proficiency strands

(a) The Learning Goals/Intentions have been identified to relate most closely to one of the four proficiency strands (shown in 3 below). This identification is shown in brackets at the end of each Learning Goal/Intention:

· Understanding is identified by (U)

· Fluency is identified by (F)

· Problem Solving is identified by (PS)

· Reasoning is identified by (R)

(b) In this document there are less Problem Solving and Reasoning proficiency strands identified than those for Understanding and Fluency. Should teachers wish to include more of these proficiencies in their curriculum, they are encouraged to emphasise them when teaching, and to develop appropriate learning tasks.

Proficiency strands

The proficiency strands describe the actions in which students can engage when learning and using the content. While not all proficiency strands apply to every content description, they indicate the breadth of mathematical actions that teachers can emphasise. The proficiencies listed next to each learning goal / intention are examples of how students might achieve the goal or what they have demonstrated by achieving the goal but are dependent on the context in which the learning takes place.

Understanding

Students build a robust knowledge of adaptable and transferable mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas. They develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build understanding when they connect related ideas, when they represent concepts in different ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and when they interpret mathematical information.

Fluency

Students develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily. Students are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and regularly use facts, and when they can manipulate expressions and equations to find solutions.

Problem Solving

Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.

Reasoning

Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.

Useful references for teams and teachers to use when planning units of work and lessons include the following:

· Ultranet Design Space – DEECD Big Ideas in Number Maps - 128428217

· Ultranet design Space – Mathematics eBookboxes - 66512121

· Teaching Mathematics Foundations to Middle Years

Dianne Siemon, Kim Beswick, Kathy Brady, Julie Clark, Rhonda Faragher and Elizabeth Warren

· Mathematics Domain Page DEECD

· Building Numeracy – George Booker

· Teaching Primary Mathematics George Booker, Denise Bond, Len Sparrow, Paul Swan

· What We Know About Mathematics Teaching and Learning- MCREL

· WMR Numeracy Design Space 106126201

· Acara Scope and Sequence Documents http://www.australiancurriculum.edu.au/Download

· VCAA – resources http://www.vcaa.vic.edu.au/Pages/foundation10/curriculum/index.aspx

Please note: Teachers will be required to join each Ultranet design space before being able to access the resource. The number associated with each space should be entered into the search box in ‘available design spaces’ in order to find the space.

Foundation Level

Measurement and Geometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

How is the essential learning developed into units of work? How do students make connections between the learning goals and those included in other content strands or sub-strands? How do we ensure students become proficient in fluency, understanding, reasoning and problem solving?

Using units of measurement

· Use direct and indirect comparisons to decide which is longer, heavier or holds more, and explain reasoning in everyday language (ACMMG006)

· Compare and order the duration of events using the everyday language of time (ACMMG007

· Connect days of the week to familiar events and actions (ACMMG008)

Shape

· Sort, describe and name familiar two-dimensional shapes and three-dimensional objects in the environment (ACMMG009)

Location and Transformation

· Describe position and movement (ACMMG010)

Students will:

· Compare longer, shorter, same and explain reasoning (R)

· Compare heavier, lighter, same and explain reasoning (R)

· Compare which holds more, less or same and explain reasoning (R)

· Be Fluent in comparing objects directly (F)

· Reason by explaining comparisons of quantities and explaining processes for indirect comparison of measurement (R)

· Know and use the language of time (F)

· Compare and order the duration of events using the language of time (F)

· Be Fluent in sequencing and comparing the duration of events (F)

· Reason by explaining comparisons of time (R)

· Connect days of the week to familiar events and actions (U)

· Understand connections between days of the week and familiar events (U)

Students will:

· Sort and describe squares, circles, triangles, rectangles, spheres and cubes (U)

· Show Understanding by connecting names with objects (U)

· Problem Solve through sorting, describe and classify objects (PS)

· Reason by comparing and naming the shapes and attributes of objects (R)

Students will:

· Know the everyday language of location and direction (F)

· Understand and use the language of location and direction (U)

· Follow and give simple directions using the language of location and direction (PS)

Achievement Standard: They compare objects using mass, length and capacity. Students connect events and the days of the week. They explain the order and duration of events. They use appropriate language to describe location. They group objects based on common characteristics and sort shapes and objects.

Level 1

Measurement and Geometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Using units of measurement

· Measure and compare the lengths and capacities of pairs of objects using uniform informal units (ACMMG019)

· Tell time to the half hour (ACMMG020)

· Describe duration using months, weeks, days and hours (ACMMG021)

Shape

· Recognise and classify familiar two dimensional shapes and three dimensional objects using obvious features (ACMMG022)

Location and transformation

· Give and follow directions to familiar locations (ACMMG023)

Students will:

· Use uniform informal units to compare longer, shorter, same and explain reasoning (R)

· Use uniform informal units to compare heavier, lighter, same and explain reasoning (R)

· Use uniform informal units to compare which holds more, less or same and explain reasoning (R)

· Understand that in order to compare objects, the unit of measurement must be the same size (U)

· Be Fluent in comparing objects directly (F)

· Reason by explaining comparisons of quantities and explaining processes for indirect comparison of measurement (R)

· To know and use the language of time (F)

· Compare and order the duration of events using the language of time (PS)

· read time on analogue and digital clocks and observing the characteristics of half hour times (F)

· Be Fluent in sequencing and comparing the duration of events (F)

· Reason by explaining comparisons of time (R)

· Connect months, weeks and days to familiar events and actions (U)

· Describe durations using months, weeks, days and hours (U)

· Understand connections between days, weeks and months

· Understand and order relative size of months, weeks, days and hours

· Sort and describe squares, circles, triangles, rectangles, spheres and cubes (F)

· Connect shape names to everyday objects (F)

· Explore geometric features and describe shapes and objects using everyday words such as 'corners', 'edges' and 'faces' (R)

· Show Understanding by connecting names with objects

· Problem Solve through sorting, describe and classify objects by corners, edges and faces

· Reason by comparing and naming the shapes and attributes of objects

· Know the everyday language of location and direction (U)

· follow and give simple directions using the language of location and direction (PS)

· Understand the language of location and direction (U)

· Understand that people need to give and follow directions to and from a place, and that this involves turns, direction and distance (U)

· Understand the meaning and importance of words such as ‘clockwise’, ‘anticlockwise’, ‘forward’ and ‘under’ when giving and following directions interpreting and following directions around familiar locations (U)

Achievement Standard: They recognise Australian coins according to their value. Students explain time durations. They describe two dimensional shapes and three dimensional objects. Students order objects based on lengths and capacities using informal units. They tell time to the half hour. They use the language of direction to move from place to place.

Level 2

Measurement and Geometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Using units of measurement

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

· Compare masses of objects using balance scales (ACMMG038)

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

· Name and order months and seasons (ACMMG040)

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

Shape

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

· Describe the features of three dimensional Objects (ACMMG043)

Location and transformation

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

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

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

Students will:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

· Identify and describe features of a calendar (U)

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

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

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

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

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

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

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

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

· Identify and name 2 dimensional shapes (F)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

· Investigate one step slides using models (PS)

· Investigate one step flips using models (PS)

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

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

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

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

· understand that objects can be moved but changing position does not alter an object’s size or features (U)

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

Level 3

Measurement and Geometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Using units of measurement

· Measure, order and compare objects using familiar metric units of length, mass and capacity(ACMMG061)

· Tell time to the minute and investigate the relationship between units of time (ACMMG062)

Shape

· Make models of three-dimensional objects and describe key features (ACMMG063)

Location and transformation

· Create and interpret simple grid maps to show position and pathways (ACMMG065)

· Identify symmetry in the environment (ACMMG066)

Geometric reasoning

· Identify angles as measures of turn and compare angle sizes in everyday situations (ACMMG064)

Students will:

· Recognise the importance of using common units of measurement. (R)

· Recognise and use centimetres and metres, grams and kilograms, and millilitres and litres. (F)

· Tell and write time to the minute. (F)

· Understand, use and order units of time. (U,F)

· Use nets to make three-dimensional objects and identify faces, edges and vertices. (P)

· Create simple maps. (P)

· Use simple maps such as theme park or zoo maps. (U)

· Identify symmetrical patterns, pictures and shapes. (P)

· Identify comparative sizes of angles in everyday situations including the hands on a clock. (R)

Achievement Standard: By the end of Level 3, students identify symmetry in the environment. They match positions on maps with given information. Students recognise angles in real situations. They interpret and compare data displays. Students use metric units for length, mass and capacity. They tell time to the nearest minute. Students make models of three-dimensional objects.

Level 4

Measurement and Geometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Using units of measurement

Use scaled instruments to measure and compare lengths, masses, capacities and temperatures(ACMMG084)

Compare objects using familiar metric units of area and volume (ACMMG290)

· Convert between units of time (ACMMG085)

· Use am and pm notation and solve simple time problems (ACMMG086)

Shape

· Compare the areas of regular and irregular shapes by informal means (ACMMG087)

· Compare and describe two dimensional shapes that result from combining and splitting common shapes, with and without the use of digital technologies(ACMMG088)

Location and transformation

· Use simple scales, legends and directions to interpret information contained in basic maps(ACMMG090)

· Create symmetrical patterns, pictures and shapes with and without digital technologies (ACMMG091)

Geometric reasoning

· Compare angles and classify them as equal to, greater than or less than a right angle (ACMMG089)

Students will:

· Read and interpreting scales on a range of measuring instruments. (U)

· Comparing areas using centimeter grid paper and volume using centicubes and litres in authentic contexts. (R)

· Convert between units of time including hours to minutes and weeks to days and vise versa. (F)

· Apply am and pm appropriately. (R)

· Calculate elapsed time problems using counting on strategies. (F)

· Comparing areas using informal means such as centimeter grid paper or tiles. (R)

· Identifying common two-dimensional shapes that are part of a composite shape by re-creating it from these shapes. (R)

· Understand and use scales and directions in maps. (U)

· Create symmetrical patterns, pictures and shapes. (P)

· Apply Acute, Right and Obtuse correctly to angles in everyday situations. (F)

Achievement Standard: By the end of Level 4, students compare areas of regular and irregular shapes using informal units. They solve problems involving time duration. They interpret information contained in maps. Students use scaled instruments to measure temperatures, lengths, shapes and objects. They convert between units of time. Students create symmetrical shapes and patterns. They classify angles in relation to a right angle.

Level 5

Measurement and Geometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Using units of measurement

Choose appropriate units of measurement for length, area, volume, capacity and mass(ACMMG108)

Calculate the perimeter and area of rectangles using familiar metric units (ACMMG109)

Compare 12- and 24-hour time systems and convert between them (ACMMG110)

Shape

· Connect three-dimensional objects with their nets and other two-dimensional representations(ACMMG111)

Location and transformation

· Use a grid reference system to describe locations. Describe routes using landmarks and directional language (ACMMG113)

· Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries (ACMMG114)

· Apply the enlargement transformation to familiar two dimensional shapes and explore the properties of the resulting image compared with the original(ACMMG115)

Geometric reasoning

· Estimate, measure and compare angles using degrees. Construct angles using a protractor(ACMMG112)

Students will:

· Choose, use and describe appropriate units of measurement for length, area, volume, capacity and mass. (U,R)

· Use a variety of strategies in calculating the area and perimeter of rectangles. (F,R)

· Compare 12- and 24-hour time systems and convert between them. (F)

· Make and use nets of a variety of 3D shapes. (R)

· Use 2D representations of 3D shapes. ®

· Understand and use common grids in everyday situations such as maps. (U)

· Describe routes using the language of direction. (U)

· identify and describe the lines and rotational symmetries of a range of two-dimensional shapes, by manually cutting, folding and turning shapes and by using digital technologies. (P)

· identify the effects of transformations by manually flipping, sliding and turning two-dimensional shapes and by using digital technologies. (R)

· Enlarge 2D figures using manual and digital technologies. (F)

· Identify and measure the angles in a figure. (F)

· Estimate and compare angles in a figure. (R)

Achievement Standard: By the end of Level 5, students connect three-dimensional objects with their two-dimensional representations. They describe transformations of two-dimensional shapes and identify line and rotational symmetry. They use appropriate units of measurement for length, area, volume, capacity and mass, and calculate perimeter and area of rectangles. They convert between 12 and 24 hour time. Students use a grid reference system to locate landmarks. They measure and construct different angles.

Level 6

Measurement and Geometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Using units of measurement

· Connect decimal representations to the metric system (ACMMG135)

· Convert between common metric units of length, mass and capacity (ACMMG136)

· Solve problems involving the comparison of lengths and areas using appropriate units(ACMMG137)

· Connect volume and capacity and their units of measurement (ACMMG138)

· Interpret and use timetables (ACMMG139)

Shape

· Construct simple prisms and pyramids (ACMMG140)

Location and transformation

· Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies (ACMMG142)

· Introduce the Cartesian coordinate system using all four quadrants (ACMMG143)

Geometric reasoning

Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (ACMMG141)

Students will:

· Understand the connection between the Base 10 System of numbers and the Decimal system of measurement. (U)

· Convert between different decimal units. (F)

· Be able to use different units of measurement in appropriate contexts to solve everyday problems involving length and area. (P)

· Use volume and capacity in everyday situations. (F)

· Use common timetables such as public transport. (F)

· Use a variety of materials to construct and deconstruct prisms and pyramids. (P)

· identify the effects of a combination of transformations by flipping, sliding and turning two-dimensional shapes. (R)

· Use and relate the Cartesian Plane to everyday situations. (U)

· Identify the four quadrants of a Cartesian plane and plot points into all four quadrants. (F)

· Identify, define and measure right, acute, obtuse, straight and reflex angles. (F)

· Identify vertically opposite angles and use to determine unknown angles such as intersecting roads. (U,R)

Achievement Standard: By the end of Level 6, students connect decimal representations to the metric system and choose appropriate units of measurement to perform a calculation. They make connections between capacity and volume. They solve problems involving length and area. They interpret timetables. Students describe combinations of transformations. They solve problems using the properties of angles. Students locate an ordered pair in any one of the four quadrants on the Cartesian plane. They construct simple prisms and pyramids.

Level 7 Measurement and Geometry

Using units of measurement

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

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

· Calculate volumes of rectangular prisms (ACMMG160)

Students will:

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

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

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

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

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

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

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

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

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

Achievement Standard

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

Level 7 Measurement and Geometry

Shape & Location and Transformation

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Shape

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

Location and Transformation

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

Students will:

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

Perform the following transformations on 2D shapes: translation, reflection in an axis, and rotation in multiples of 90⁰. (F)
Describe translation, reflection in an axis, and rotation in multiples of 90⁰ using coordinates. (U)
Create patterns with combinations of translations, reflections and rotations, including using digital technologies. (PS)
Identify line and rotational symmetries. (U)

Achievement Standard:

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

Level 7 Measurement and Geometry

Geometric Reasoning

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

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

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

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

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

Students will:

Deduce and then describe the conditions for two lines to be parallel. (R)
Construct a pair of parallel lines with a transversal intersecting with them using concrete materials or geometry software. (F)
Define and classify pairs of angles as complementary, supplementary, adjacent and vertically opposite. (U)
Define and classify alternate, corresponding and co-interior angles. (U)
Deduce the missing angle in a parallel/transversal line situation. (R)

Deduce and verify the angle sum of a triangle using concrete materials. (R)
Calculate the missing angle in a triangle. (F)
Deduce the angle sum of a quadrilateral from knowing the angle sum of a triangle.(R)
Calculate the missing angle in a quadrilateral.(F)

Justify and classify triangles as scalene, isosceles or equilateral according to their side properties. (R)
Justify and classify triangles as right-angled, obtuse-angled and acute angled according to their angle properties. (R)
Describe key features of quadrilaterals including squares rectangles, rhombuses, parallelograms, kites and trapeziums. (U)

Achievement Standard:

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

Level 8 Measurement and Geometry

Units of Measurement

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

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

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

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

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

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

Students will:

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

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

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

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

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

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

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

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

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

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

Achievement Standard:

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

Level 8 Measurement and Geometry

Geometric Reasoning

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

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

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

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

Students will:

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

Define congruence of plane shapes using transformations. (R)

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

Describe properties of quadrilaterals including squares, rectangles, parallelograms, rhombuses, trapeziums and kites. (U)
Determine the sum of internal angles of a polygon, using triangles. (F)
Solve problems using the sum of internal angles for triangles and other polygons. (F)
Determine lines of symmetry in a given shape. (U)
Identify properties related to side lengths, parallel sides, angles, diagonals and symmetry. (R)

Achievement Standard:

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

Level 9 Measurement & Geometry

Units of Measurement

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Calculate the areas of composite shapes (ACMMG216)

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

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

Students will:

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

· estimate the area of a composite shape (R)

· calculate the area of a composite shape (F)

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

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

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

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

Achievement Standard:

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

Level 9 Measurement and Geometry

Geometric Reasoning & Pythagoras and Trigonometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Geometric Reasoning

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

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

Pythagoras and Trigonometry

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

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

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

Students will:

· Explain why two shapes are similar. (R)

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

· Determine ratio and scale factor. (F)

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

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

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

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

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

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

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

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

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

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

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

Achievement Standard:

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

Level 10 Measurement and Geometry

Using units of measurement

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Solve problems involving surface area and volume for a range of prisms, cylinders and composite solids (ACMMG242)

Students will:

l Explain the difference between surface area and volume including the use of square and cubic units(U)

l Use a variety of strategies to calculate surface area and volume of prisms, cylinders, cones, pyramids, spheres and composite solids (R) (F) (U)

l Solve worded problems involving surface area and/or volume calculations and/or comparisons using a variety of strategies (F) (U) (R) (PS)

Achievement Standard

Students solve surface area and volume problems relating to composite solids.

Level 10 Measurement and Geometry

Geometric Reasoning

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Formulate proofs involving congruent triangles and angle properties (ACMMG243)

· Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes (ACMMG244)

Students will:

· Identify a shape by its properties (U)

· Identify congruent triangles (U)

· Use congruent triangles and angle properties to prove that a quadrilateral with equal length diagonals bisecting at right angles is a square (R)

· present formal geometric arguments to develop skills in mathematical reasoning and present reasoned arguments (proofs) (R) (U)

· use mathematical language and notation, based on congruence and similarity (U)

· apply an understanding of relationships to deduce properties of geometric figures (for example the base angles of an isosceles triangle are equal) (R) (PS) (U)

· distinguish between a practical demonstration and a proof (for example demonstrating triangles are congruent by placing them on top of each other, as compared to using congruence tests to establish that triangles are congruent) (U) (R)

Achievement Standard: Students apply deductive reasoning to proofs and numerical exercises involving plane shapes. They use triangle and angle properties to prove congruence and similarity.

Level 10 Measurement and Geometry

Pythagoras and Trigonometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Solve rightangled triangle problems including those involving direction and angles of elevation and depression (ACMMG245)

Students will:

· Identify a right angled triangle (F)

· Recall Pythagoras’s Theorem (F)

· Calculate the lengths of sides of triangles using Pythagoras’s Theorem (F) (U)

· Apply Pythagoras’s Theorem to worded problems and real life situations to solve problems (PS) (R) (U)

· Label sides of a triangle according to their location in relation to an angle (e.g. opposite, adjacent ) (F)

· Recall trigonometric relationships (F)

· Use trigonometric relationships to calculate lengths of sides and sizes of angles (F)

· Represent real life situations diagrammatically in order to apply trigonometry to solve a problem (R)

· Apply trigonometric relationships to real life situations to solve problems (PS) (R)

· Solve problems involving angles of elevation and depression using Pythagoras’s Theorem and Trigonometry (R) (PS)

Achievement Standard: Students use trigonometry to calculate unknown angles in rightangled triangles.

Level 10A

Year 10A content descriptors indicate optional additional content suitable for development of student mathematical background in preparation for further study of functions, algebra, and calculus; as well as other additional content related to statistics and trigonometry. Teachers can incorporate a selection of this and other additional content in Year 10 mathematics courses, as applicable for extension and enrichment purposes, and to prepare students for subsequent study.

Level 10A Measurement and Geometry

Using Units of Measurement

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Solve problems involving surface area and volume of right pyramids, right cones, spheres and related composite solids (ACMMG271)

Students will:

Use length, area and volume relationships to solve problems (F) (R)
Calculate simple surface areas and volumes using a variety of strategies (F)
Solve problems involving surface area of right pyramids (R) (PS) (U) (F)
Solve problems involving surface area of right cones (R) (PS) (U) (F)
Solve problems involving surface area of spheres (R) (PS) (U) (F)
Solve problems involving surface area of composite solids (R) (PS) (U) (F)
Solve problems involving volume of right pyramids (R) (PS) (U) (F)
Solve problems involving volume of right cones (R) (PS) (U) (F)
Solve problems involving volume of spheres (R) (PS) (U) (F)
Solve problems involving volume of composite solids (R) (PS) (U) (F)

Level 10A Measurement and Geometry

Geometric Reasoning

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Prove and apply angle and chord properties of circles (ACMMG272)

Students will:

· Describe a variety of parts of circles (U)

· Calculate arc lengths (F)

· Calculate angles and chord lengths using circle theorems (R)

· Prove angle and chord properties of circles (R)

Level 10A Measurement and Geometry

Pythagoras and Trigonometry

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Establish the sine, cosine and area rules for any triangle and solve related problems (ACMMG273)

· Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies (ACMMG274)

· Solve simple trigonometric equations (ACMMG275)

· Apply Pythagoras’ theorem and trigonometry to solving three dimensional problems in rightangled triangles (ACMMG276)

Students will:

apply knowledge of sine, cosine and area rules to authentic problems such as those involving surveying and design (PS) (R) (U) (F)

understand the relationship of the unit circle to trigonometric functions for angles of any magnitude (U)
graph trigonometric functions with and without digital technologies (U)
understand the graphs of trigonometric functions (U)

solve simple trigonometric equations (F)
solve problems related to trigonometric functions as periodic e.g. those describing motion (PS) (R)
represent real-life problems using right angled triangles where appropriate in three dimensions (R) (U)
Solve authentic problems using Pythagoras’s Theorem and Trigonometry (PS) (R) (U) (F)

Comments