YEAR 6 MATHS FOCUS
NUMBERS AND ALGEBRA
Whole Numbers
Addition & Subtraction
Multiplication & Division
Fractions & Decimals
Patterns & Algebra
Whole Numbers
WHOLE NUMBERS
OUTCOME
A student:
MA3-4NA:
orders, reads and represents integers of any size and describes properties of whole numbers
| TEACHING POINTS |
Students could investigate further the properties of square and triangular numbers, such as all square numbers have an odd number of factors, while all non-square numbers have an even number of factors; when two consecutive triangular numbers are added together, the result is always a square number. |
LANGUAGE | Students should be able to communicate using the following language: number line, whole number, zero, positive number, negative number, integer, prime number, composite number, factor, square number, triangular number. |
| Words such as ‘square’ have more than one grammatical use in mathematics, eg draw a square (noun), square three (verb), square numbers (adjective) and square metres (adjective). |
Expectations of Attainment
Investigate everyday situations that use integers; locate and represent these numbers on a number line (ACMNA124) | recognise the location of negative whole numbers in relation to zero and place them on a number line |
| use the term ‘integers’ to describe positive and negative whole numbers and zero {Literacy} | |
| interpret integers in everyday contexts, e.g. temperature | |
| investigate negative whole numbers and the number patterns created when counting backwards on a calculator | |
| – recognise that negative whole numbers can result from subtraction {Reasoning} | |
| – ask ‘What if’ questions, e.g. ‘What happens if we subtract a larger number from a smaller number on a calculator?’ {Communicating, Literacy, Critical and creative thinking} |
Identify and describe properties of prime, composite, square and triangular numbers (ACMNA122) | determine whether a number is prime, composite or neither |
| – explain whether a whole number is prime, composite or neither by finding the number of factors, eg ’13 has two factors (1 and 13) and therefore is prime’, ’21 has more than two factors (1, 3, 7, 21) and therefore is composite’, ‘1 is neither prime nor composite as it has only one factor, itself’ {Communicating, Reasoning} | |
| – explain why a prime number, when modelled as an array, can have only one row {Communicating, Reasoning} | |
| model square and triangular numbers and record each number group in numerical and diagrammatic form {Literacy} | |
| – explain how square and triangular numbers are created {Communicating, Reasoning, Critical and creative thinking} | |
| – explore square and triangular numbers using arrays, grid paper or digital technologies {Communicating, Problem Solving, Information and communication technology capability} | |
| – recognise and explain the relationship between the way each pattern of numbers is created and the name of the number group {Communicating, Reasoning, Literacy} |
Addition & Subtraction
ADDITION & SUBTRACTION
OUTCOME
A student:
MA3-5NA:
selects and applies appropriate strategies for addition and subtraction with counting numbers of any size
STRATEGIES | Written strategies using informal mental strategies (empty number line):
The difference can be shifted one unit to the left on an empty number line, so that 8000 − 673 becomes 7999 − 672, which is an easier subtraction to calculate.
|
Written strategies using a formal algorithm (decomposition method):
|
TEACHING POINTS | In Stage 3, mental strategies need to be continually reinforced.
|
| Students may find recording (writing out) informal mental strategies to be more efficient than using formal written algorithms, particularly in the case of subtraction. |
| LANGUAGE | Students should be able to communicate using the following language: plus, sum, add, addition, increase, minus, the difference between, subtract, subtraction, decrease, equals, is equal to, operation, digit. When solving word problems, students should be encouraged to write a few key words on the left-hand side of the equals sign to identify what is being found in each step of their working, eg ‘amount to pay = …’, ‘change = …’. Refer also to language in Addition and Subtraction 1. |
Expectations of Attainment
Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving addition and subtraction with whole numbers (ACMNA123) | solve addition and subtraction word problems involving whole numbers of any size, including problems that require more than one operation, eg ‘I have saved $40 000 to buy a new car. The basic model costs $36 118 and I add tinted windows for $860 and Bluetooth connectivity for $1376. How much money will I have left over?’ {Critical and creative thinking} |
| – select and apply appropriate mental and written strategies, with and without the use of digital technologies, to solve unfamiliar problems {Problem Solving, Literacy, Information and communication technology capability, Critical and creative thinking} | |
| – explain how an answer was obtained for an addition or subtraction problem and justify the selected calculation method {Communicating, Problem Solving, Reasoning, Critical and creative thinking} | |
| – reflect on their chosen method of solution for a problem, considering whether it can be improved {Communicating, Reasoning, Critical and creative thinking} | |
| – give reasons why a calculator was useful when solving a problem {Communicating, Reasoning} | |
| record the strategy used to solve addition and subtraction word problems {Literacy} | |
| – use selected words to describe each step of the solution process {Communicating, Problem Solving, Literacy} |
Use estimation and rounding to check the reasonableness of answers to calculations (ACMNA099) | round numbers appropriately when obtaining estimates to numerical calculations |
| use estimation to check the reasonableness of answers to addition and subtraction calculations, e.g. 1438 + 129 is about 1440 + 130 {Critical and creative thinking} |
Create simple financial plans (ACMNA106) | use knowledge of addition and subtraction facts to create a financial plan, such as a budget, eg organise a class celebration on a budget of $60 for all expenses {Personal and social capability, Work and enterprise} |
| – record numerical data in a simple spreadsheet {Communicating, Information and communication technology capability} | |
| – give reasons for selecting, prioritising and deleting items when creating a budget {Communicating, Reasoning, Critical and creative thinking, Personal and social capability, Work and enterprise} |
Multiplication & Division
MULTIPLICATION & DIVISION
OUTCOME
A student:
MA3-6NA:
selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation
TEACHING POINTS | Students could extend their recall of number facts beyond the multiplication facts to 10 × 10 by also memorising multiples of numbers such as 11, 12, 15, 20 and 25, or by utilising mental strategies, eg ’14 × 6 is 10 sixes plus 4 sixes’. |
| The simplest multiplication word problems relate to rates, eg ‘If four students earn $3 each, how much do they have all together?’ Another type of problem is related to ratio and uses language such as ‘twice as many as’ and ‘six times as many as’. | |
| An ‘operation’ is a mathematical process. The four basic operations are addition, subtraction, multiplication and division. Other operations include raising a number to a power and taking a root of a number. An ‘operator’ is a symbol that indicates the type of operation, eg +, –, × and ÷. | |
| Refer also to background information in Multiplication and Division 1. |
Language | Students should be able to communicate using the following language: multiply, multiplied by, product, multiplication, multiplication facts, area, thousands, hundreds, tens, ones, double, multiple, factor, divide, divided by, quotient, division, halve, remainder, fraction, decimal, equals, strategy, digit, estimate, speed, per, operations, order of operations, grouping symbols, brackets, number sentence, is the same as. |
| When solving word problems, students should be encouraged to write a few key words on the left-hand side of the equals sign to identify what is being found in each step of their working, eg ‘cost of goldfish = …’, ‘cost of plants = …’, ‘total cost = …’. | |
| Grouping symbols’ is a collective term used to describe brackets [ ], parentheses ( ) and braces { }. The term ‘brackets’ is often used in place of ‘parentheses’. | |
| Often in mathematics when grouping symbols have one level of nesting, the inner pair is parentheses ( ) and the outer pair is brackets [ ], eg 360÷[4×(20−11)]. |
Expectations of Attainment
Select and apply efficient mental and written strategies, and appropriate digital technologies, to solve problems involving multiplication and division with whole numbers (ACMNA123) | select and use efficient mental and written strategies, and digital technologies, to multiply whole numbers of up to four digits by one- and two-digit numbers |
| select and use efficient mental and written strategies, and digital technologies, to divide whole numbers of up to four digits by a one-digit divisor, including where there is a remainder | |
| – estimate solutions to problems and check to justify solutions {Problem Solving, Reasoning, Critical and creative thinking} | |
| use mental strategies to multiply and divide numbers by 10, 100, 1000 and their multiples | |
| solve word problems involving multiplication and division, e.g. ‘A recipe requires 3 cups of flour for 10 people. How many cups of flour are required for 40 people?’ {Critical and creative thinking} | |
| – use appropriate language to compare quantities, e.g. ‘twice as much as’, ‘half as much as’ {Communicating, Critical and creative thinking} | |
| – use a table or similar organiser to record methods used to solve problems {Communicating, Problem Solving, Information and communication technology capability} | |
| recognise symbols used to record speed in kilometres per hour, e.g. 80 km/h {Literacy} | |
| solve simple problems involving speed, e.g. ‘How long would it take to travel 600 km if the average speed for the trip is 75 km/h?’ {Critical and creative thinking} |
Explore the use of brackets and the order of operations to write number sentences (ACMNA134) | use the term ‘operations’ to describe collectively the processes of addition, subtraction, multiplication and division |
| investigate and establish the order of operations using real-life contexts, e.g. ‘I buy six goldfish costing $10 each and two water plants costing $4 each. What is the total cost?’; this can be represented by the number sentence 6 × 10 + 2 × 4 but, to obtain the total cost, multiplication must be performed before addition {Literacy, Critical and creative thinking, Work and enterprise} | |
| – write number sentences to represent real-life situations {Communicating, Problem Solving, Literacy} | |
| recognise that the grouping symbols ( ) and [ ] are used in number sentences to indicate operations that must be performed first {Literacy} | |
| recognise that if more than one pair of grouping symbols are used, the operation within the innermost grouping symbols is performed first | |
| perform calculations involving grouping symbols without the use of digital technologies, e.g. 5+(2×3)=5+6 =11 (2+3)×(16−9)=5×7 =35 3+[20÷(9−5)]=3+[20÷4]=3+5 =8 | |
| apply the order of operations to perform calculations involving mixed operations and grouping symbols, without the use of digital technologies, e.g. 32+2−4=34−4 =30 addition and subtraction only, therefore work from left to 32÷2×4=16×4 =64 multiplication and division only, therefore work from left to right 32÷(2×4)=32÷8 =4 perform operation in grouping symbols first (32+2)×4=34×4 =136 perform operation in grouping symbols first 32+2×4=32+8 =40 perform multiplication before addition {Work and enterprise} | |
| – investigate whether different digital technologies apply the order of operations {Reasoning, Information and communication technology capability, Critical and creative thinking} | |
| recognise when grouping symbols are not necessary, eg 32 + (2 × 4) has the same answer as 32 + 2 × 4 |
Fractions & Decimals
FRACTIONS & DECIMALS
OUTCOME
A student:
MA3-7NA:
compares, orders and calculates with fractions, decimals and percentages
TEACHING POINTS | In Stage 3 Fractions and Decimals, students study fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100. A unit fraction is any proper fraction in which the numerator is 1, eg 12, 13, 14, 15, … |
| The process of writing a fraction in its ‘simplest form’ involves reducing the fraction to its lowest equivalent form. In Stage 4, this is referred to as ‘simplifying’ a fraction. | |
| When subtracting mixed numerals, working with the whole-number parts separately from the fractional parts can lead to difficulties, particularly where the subtraction of the fractional parts results in a negative value, e.g. in the calculation of 2 1/3−1 5/6, 1/3−5/6 results in a negative value. |
LANGUAGE | Students should be able to communicate using the following language: whole, equal parts, half, quarter, eighth, third, sixth, twelfth, fifth, tenth, hundredth, thousandth, fraction, numerator, denominator, mixed numeral, whole number, number line, proper fraction, improper fraction, is equal to, equivalent, ascending order, descending order, simplest form, decimal, decimal point, digit, round to, decimal places, dollars, cents, best buy, percent, percentage, discount, sale price. |
| The decimal 1.12 is read as ‘one point one two’ and not ‘one point twelve’. | |
| The word ‘cent’ is derived from the Latin word centum, meaning ‘one hundred’. ‘Percent’ means ‘out of one hundred’ or ‘hundredths’. | |
| A ‘terminating’ decimal has a finite number of decimal places, eg 3.25 (2 decimal places), 18.421 (3 decimal places). |
Expectations of Attainment
Compare fractions with related denominators and locate and represent them on a number line (ACMNA125) | model, compare and represent fractions with denominator of 2, 3, 4, 5, 6, 8, 10, 12 and 100 of a whole object, a whole shape and a collection of objects |
compare the relative size of fractions drawn on the same diagram, e.g.
{Reasoning, Critical and creative thinking} | |
| compare and order simple fractions with related denominators using strategies such as diagrams, the number line, or equivalent fractions, e.g. write 3/5, 3/10, 1 1/10, 4/5 and 7/10 in ascending order | |
find equivalent fractions by re-dividing the whole, using diagrams and number lines, e.g. | |
| record equivalent fractions using diagrams and numerals | |
| develop mental strategies for generating equivalent fractions, such as multiplying or dividing the numerator and the denominator by the same number, e.g. 1/4=(1×2)/(4×2)=(1×3)/(4×3)=(1×4)/(4×4)= …, i.e. 1/4=2/8=3/12=4/16= … | |
| – explain or demonstrate why two fractions are or are not equivalent {Communicating, Reasoning, Critical and creative thinking} | |
| write fractions in their ‘simplest form’ by dividing the numerator and the denominator by a common factor, e.g. 4/16=(4÷4)/(16÷4)=1/4 | |
| – recognise that a fraction in its simplest form represents the same value as the original fraction {Reasoning} | |
| – apply knowledge of equivalent fractions to convert between units of time, e.g. 15 minutes is the same as 15/60 of an hour, which is the same as 1/4 of an hour (Problem Solving) |
Solve problems involving addition and subtraction of fractions with the same or related denominators (ACMNA126) | add and subtract fractions, including mixed numerals, where one denominator is the same as, or a multiple of, the other, e.g. 2/3+1/6, 2 3/8−1 1/2, 2 3/8−3/4 |
| – convert an answer that is an improper fraction to a mixed numeral {Communicating} | |
| – use knowledge of equivalence to simplify answers when adding and subtracting fractions {Communicating, Reasoning} | |
| – recognise that improper fractions may sometimes make calculations involving mixed numerals easier {Communicating} | |
| solve word problems involving the addition and subtraction of fractions where one denominator is the same as, or a multiple of, the other, eg ‘I ate 1/8 of a cake and my friend ate 1/4 of the cake. What fraction of the cake remains?’ {Literacy Critical and creative thinking} | |
| multiply simple fractions by whole numbers using repeated addition, leading to a rule, e.g. 2/5×3=2/5+2/5+2/5=6/5=1 1/5 leading to 2/5×3=(2×3)/5=6/5=1 1/5 {Critical and creative thinking} |
Find a simple fraction of a quantity where the result is a whole number, with and without the use of digital technologies (ACMNA127) | calculate unit fractions of collections, with and without the use of digital technologies, e.g. calculate 1/5 of 30 {Information and communication technology capability} |
| – describe the connection between finding a unit fraction of a collection and the operation of division {Communicating, Problem Solving, Critical and creative thinking} | |
| calculate a simple fraction of a collection/quantity, with and without the use of digital technologies, e.g. calculate 2/5 of 30 {Information and communication technology capability} | |
| – explain how unit fractions can be used in the calculation of simple fractions of collections/quantities, e.g. ‘To calculate 3/8 of a quantity, I found 1/8 of the collection first and then multiplied by 3’ {Communicating, Reasoning, Critical and creative thinking} | |
| solve word problems involving a fraction of a collection/quantity {Literacy} |
Add and subtract decimals, with and without the use of digital technologies, and use estimation and rounding to check the reasonableness of answers (ACMNA128) | add and subtract decimals with the same number of decimal places, with and without the use of digital technologies {Information and communication technology capability} |
| add and subtract decimals with a different number of decimal places, with and without the use of digital technologies {Information and communication technology capability} | |
| – relate decimals to fractions to aid mental strategies {Communicating} | |
| round a number of up to three decimal places to the nearest whole number | |
| use estimation and rounding to check the reasonableness of answers when adding and subtracting decimals {Critical and creative thinking} | |
| – describe situations where the estimation of calculations with decimals may be useful, eg to check the total cost of multiple items when shopping {Communicating, Problem Solving} | |
| solve word problems involving the addition and subtraction of decimals, with and without the use of digital technologies, including those involving money {Personal and social capability, Work and enterprise} | |
| – use selected words to describe each step of the solution process {Communicating, Problem Solving, Literacy} | |
| – interpret a calculator display in the context of the problem, e.g. 2.6 means $2.60 {Communicating} |
Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without the use of digital technologies (ACMNA129) | use mental strategies to multiply simple decimals by single-digit numbers, e.g. 3.5 × 2 |
| multiply decimals of up to three decimal places by whole numbers of up to two digits, with and without the use of digital technologies, e.g. ‘I measured three desks. Each desk was 1.25 m in length, so the total length is 3 × 1.25 = 3.75 m’ {Information and communication technology capability} | |
| divide decimals by a one-digit whole number where the result is a terminating decimal, e.g. 5.25 ÷ 5 = 1.05 | |
| solve word problems involving the multiplication and division of decimals, including those involving money, eg determine the ‘best buy’ for different-sized cartons of cans of soft drink {Personal and social capability, Work and enterprise, Critical and creative thinking} |
Multiply and divide decimals by powers of 10 (ACMNA130) | recognise the number patterns formed when decimals are multiplied and divided by 10, 100 and 1000 {Critical and creative thinking} |
| multiply and divide decimals by 10, 100 and 1000 | |
| – use a calculator to explore the effect of multiplying and dividing decimals by multiples of 10 {Reasoning} |
Patterns & Algebra
PATTERNS & ALGEBRA
OUTCOME
A student:
MA3-8NA:
analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane
Teaching Points | In Stage 2, students found the value of the next term in a pattern by performing an operation on the previous term. In Stage 3, they need to connect the value of a particular term in the pattern with its position in the pattern. This is best achieved through a table of values. Students need to see a connection between the two numbers in each column and should describe the pattern in terms of the operation that is performed on the position in the pattern to obtain the value of the term. Describing a pattern by the operation(s) performed on the ‘position in the pattern’ is more powerful than describing it as an operation performed on the previous term in the pattern, as it allows any term (eg the 100th term) to be calculated without needing to find the value of the term before it. The concept of relating the number in the top row of a table of values to the number in the bottom row forms the basis for work in Linear and Non-Linear Relationships in Stage 4 and Stage 5. |
| The notion of locating position and plotting coordinates is established in the Position substrand in Stage 2 Measurement and Geometry. It is further developed in this substrand to include negative numbers and the use of the four-quadrant number plane. | |
| The Cartesian plane (commonly referred to as the ‘number plane’) is named after the French philosopher and mathematician René Descartes (1596–1650), who was one of the first to develop analytical geometry on the number plane. On the number plane, the ‘coordinates of a point’ refers to the ordered pair (x,y) describing the horizontal position x first, followed by the vertical position y The Cartesian plane is applied in real-world contexts, eg when determining the incline (slope) of a road between two points. The Cartesian plane is used in algebra in Stages 4 to 6 to describe patterns and relationships between numbers. |
| Language | Students should be able to communicate using the following language: pattern, increase, decrease, term, value, table of values, rule, position in pattern, value of term, number plane (Cartesian plane), horizontal axis (x-axis), vertical axis (y-axis), axes, quadrant, intersect, point of intersection, right angles, origin, coordinates, point, plot. |
Expectations of Attainment
Continue and create sequences involving whole numbers, fractions and decimals; describe the rule used to create the sequence (ACMNA133) | continue and create number patterns, with and without the use of digital technologies, using whole numbers, fractions and decimals, e.g. 14, 18, 116, … or 1.25, 2.5, 5, … {Information and communication technology capability, Critical and creative thinking} |
| – describe how number patterns have been created and how they can be continued {Communicating, Problem Solving, Critical and creative thinking} | |
| create simple geometric patterns using concrete materials, e.g. △,△△,△△△,△△△△, …{Literacy} | |
complete a table of values for a geometric pattern and describe the pattern in words, e.g.
{Critical and creative thinking, Literacy} | |
| – describe the number pattern in a variety of ways and record descriptions using words, e.g. ‘It looks like the multiplication facts for four’ | |
| – determine the rule to describe the pattern by relating the bottom number to the top number in a table, e.g. ‘You multiply the number of squares by four to get the number of matches’ | |
| – use the rule to calculate the corresponding value for a larger number, e.g. ‘How many matches are needed to create 100 squares?’ | |
Complete a table of values for number patterns involving one operation (including patterns that decrease) and describe the pattern in words, e.g. describe the pattern in a variety of ways and record descriptions in words, eg ‘It goes up by ones, starting from four’ determine a rule to describe the pattern from the table, eg ‘To get the value of the term, you add three to the position in the pattern’ use the rule to calculate the value of the term for a large position number, eg ‘What is the 55th term of the pattern?’ {Literacy, Critical and creative thinking} | |
| – explain why it is useful to describe the rule for a pattern by describing the connection between the ‘position in the pattern’ and the ‘value of the term’ {Communicating, Reasoning, Literacy, Critical and creative thinking} | |
| – interpret explanations written by peers and teachers that accurately describe geometric and number patterns {Communicating, Literacy, Critical and creative thinking} | |
| make generalisations about numbers and number relationships, eg ‘If you add a number and then subtract the same number, the result is the number you started with’ {Critical and creative thinking} |
Introduce the cartesian-coordinate-system using all four quadrants (ACMMG143) | recognise that the number plane (Cartesian plane) is a visual way of describing location on a grid |
recognise that the number plane consists of a horizontal axis (x-axis) and a vertical axis (y-axis), creating four quadrants e.g.  | |
| – recognise that the horizontal axis and the vertical axis meet at right angles {Reasoning} | |
| identify the point of intersection of the two axes as the origin, having coordinates (0, 0) {Literacy} | |
| plot and label points, given coordinates, in all four quadrants of the number plane {Literacy} | |
| – plot a sequence of coordinates to create a picture {Communicating, Literacy} | |
| identify and record the coordinates of given points in all four quadrants of the number plane | |
| – recognise that the order of coordinates is important when locating points on the number plane, eg (2, 3) is a location different from (3, 2) {Communicating} |
MEASUREMENT AND GEOMETRY
Length
Area
Volume & Capacity
Mass
Time
3D Space
2D Space
Angles
Position
Length
LENGTH
OUTCOME
A student:
MA3-9MG:
selects and uses the appropriate unit and device to measure lengths and distances, calculates perimeters, and converts between units of length
TEACHING POINTS | When students are able to measure efficiently and effectively using formal units, they should be encouraged to apply their knowledge and skills in a variety of contexts. Following this, they should be encouraged to generalise their method for calculating the perimeters of squares, rectangles and triangles. |
| When recording measurements, a space should be left between the number and the abbreviated unit, e.g. 3 cm, not 3cm. |
| LANGUAGE | Students should be able to communicate using the following language: length, distance, kilometre, metre, centimetre, millimetre, perimeter, dimensions, width. |
Expectations of Attainment
Connect decimal representations to the metric system (ACMMG135) | recognise the equivalence of whole-number and decimal representations of measurements of length, e.g. 165 cm is the same as 1.65 m |
| interpret decimal notation for lengths and distances, e.g. 13.5 cm is 13 centimetres and 5 millimetres | |
| record lengths and distances using decimal notation to three decimal places, e.g. 2.753 km |
Convert between common metric units of length (ACMMG136) | convert between metres and kilometres |
| convert between millimetres, centimetres and metres to compare lengths and distances | |
| – explain and use the relationship between the size of a unit and the number of units needed to assist in determining whether multiplication or division is required when converting between units, eg ‘More metres than kilometres will be needed to measure the same distance, and so to convert from kilometres to metres, I need to multiply’ {Communicating, Reasoning, Critical and creative thinking} |
Solve problems involving the comparison of lengths using appropriate units (ACMMG137) | investigate and compare perimeters of rectangles with the same area {Critical and creative thinking} |
| – determine the number of different rectangles that can be formed using whole-number dimensions for a given area {Problem Solving, Reasoning, Critical and creative thinking} | |
| solve a variety of problems involving length and perimeter, including problems involving different units of length, e.g. ‘Find the total length of three items measuring 5 mm, 20 cm and 1.2 m’ {Critical and creative thinking} |
Area
AREA
OUTCOME
A student:
MA3-10MG:
selects and uses the appropriate unit to calculate areas, including areas of squares, rectangles and triangles
TEACHING POINTS | Students should have a clear understanding of the distinction between perimeter and area. |
| It is important in Stage 3 that students establish a real reference for the square kilometre and the hectare, eg locating an area of one square kilometre or an area of one hectare on a local map. | |
| When students are able to measure efficiently and effectively using formal units, they should be encouraged to apply their knowledge and skills in a variety of contexts. | |
| Students could be encouraged to find more efficient ways of counting when determining area, such as finding how many squares in one row and multiplying this by the number of rows. They should then begin to generalise their methods to calculate the areas of rectangles (including squares) and triangles. | |
| When generalising their methods to calculate areas, students in Stage 3 should use words. Algebraic formulas for areas are not introduced until Stage 4. |
| LANGUAGE | Students should be able to communicate using the following language: area, square centimetre, square metre, dimensions, length, width, base (of triangle), perpendicular height. |
EXPECTATIONS OF ATTAINMENT
Solve problems involving the comparison of areas using appropriate units (ACMMG137) | investigate the area of a triangle by comparing the area of a given triangle to the area of the rectangle of the same length and perpendicular height, eg use a copy of the given triangle with the given triangle to form a rectangle |
| – explain the relationship between the area of a triangle and the area of the rectangle of the same length and perpendicular height {Communicating, Reasoning, Critical and creative thinking} | |
| establish the relationship between the base length, perpendicular height and area of a triangle {Critical and creative thinking} | |
| record, using words, the method for finding the area of any triangle, e.g. ‘Area of triangle = 1/2 × base × perpendicular height’ {Literacy} | |
| investigate and compare the areas of rectangles that have the same perimeter, e.g. compare the areas of all possible rectangles with whole-number dimensions and a perimeter of 20 centimetres {Critical and creative thinking} | |
| – determine the number of different rectangles that can be formed using whole-number dimensions for a given perimeter {Problem Solving, Reasoning, Critical and creative thinking} | |
| solve a variety of problems involving the areas of rectangles (including squares) and triangles {Critical and creative thinking} |
Volume & Capacity
VOLUME & CAPACITY
OUTCOME
A student:
MA3-11MG:
selects and uses the appropriate unit to estimate, measure and calculate volumes and capacities, and converts between units of capacity
TEACHING POINTS | The attribute of volume is the amount of space occupied by an object or substance and is usually measured in cubic units, eg cubic centimetres (cm^3) and cubic metres (m^3). |
| Capacity refers to the amount a container can hold and is measured in units, such as millilitres (mL), litres (L) and kilolitres (kL). Capacity is only used in relation to containers and generally refers to liquid measurement. The capacity of a closed container will be slightly less than its volume – capacity is based on the inside dimensions, while volume is determined by the outside dimensions of the container. It is not necessary to refer to these definitions with students (capacity is not taught as a concept separate from volume until Stage 4). | |
| Once students are able to measure efficiently and effectively using formal units, they could use centimetre cubes to construct rectangular prisms, counting the number of cubes to determine volume, and then begin to generalise their method for calculating the volume. | |
| The cubic metre can be related to the metre as a unit to measure length and the square metre as a unit to measure area. It is important that students are given opportunities to reflect on their understanding of length and area so that they can use this to calculate volume. |
LANGUAGE | Students should be able to communicate using the following language: capacity, container, litre, millilitre, volume, dimensions, length, width, height, layers, cubic centimetre, cubic metre. |
| The abbreviation m^3 is read as ‘cubic metre(s)’ and not ‘metre(s) cubed’. |
EXPECTATIONS OF ATTAINMENT
Connect volume and capacity and their units of measurement (ACMMG138) | select the appropriate unit to measure volume and capacity |
| demonstrate that a cube of side 10 cm will displace 1 litre of water | |
| demonstrate, by using a medicine cup, that a cube of side 1 cm will displace 1 mL of water | |
| equate 1 cubic centimetre to 1 millilitre and 1000 cubic centimetres to 1 litre | |
| find the volumes of irregular solids in cubic centimetres using a displacement strategy |
Connect decimal representations to the metric system (ACMMG135) | recognise the equivalence of whole-number and decimal representations of measurements of capacities, e.g. 375 mL is the same as 0.375 L |
| interpret decimal notation for volumes and capacities, e.g. 8.7 L is the same as 8 litres and 700 millilitres | |
| record volume and capacity using decimal notation to three decimal places, e.g. 1.275 L |
Convert between common metric units of capacity (ACMMG136) | convert between millilitres and litres |
| – explain and use the relationship between the size of a unit and the number of units needed to assist in determining whether multiplication or division is required when converting between units, e.g. ‘Fewer litres than millilitres will be needed to measure the same capacity, and so to convert from millilitres to litres, I need to divide’ {Communicating, Reasoning, Critical and creative thinking} |
Mass
MASS
OUTCOME
A student:
MA3-12MG:
selects and uses the appropriate unit and device to measure the masses of objects, and converts between units of mass
| TEACHING POINTS | One litre of water has a mass of one kilogram and a volume of 1000 cubic centimetres. While the relationship between volume and capacity is constant for all substances, the same volumes of substances other than water may have different masses, e.g. 1 litre of oil is lighter than 1 litre of water, which in turn is lighter than 1 litre of honey. This can be demonstrated using digital scales. See also Year Five |
| LANGUAGE | Students should be able to communicate using the following language: mass, measure, scales, tonne, kilogram, gram. |
| As the terms ‘weigh’ and ‘weight’ are common in everyday usage, they can be accepted in student language should they arise. Weight is a force that changes with gravity, while mass remains constant. |
EXPECTATIONS OF ATTAINMENT
Connect decimal representations to the metric system (ACMMG135) | recognise the equivalence of whole-number and decimal representations of measurements of mass, e.g. 3 kg 250 g is the same as 3.25 kg |
| interpret decimal notation for masses, e.g. 2.08 kg is the same as 2 kilograms and 80 grams | |
| measure mass using scales and record using decimal notation of up to three decimal places, e.g. 0.875 kg |
Convert between common metric units of mass (ACMMG136) | convert between kilograms and grams and between kilograms and tonnes |
| – explain and use the relationship between the size of a unit and the number of units needed to assist in determining whether multiplication or division is required when converting between units, e.g. ‘More grams than kilograms will be needed to measure the same mass, and so to convert from kilograms to grams, I need to multiply’ {Communicating, Reasoning, Critical and creative thinking} | |
| solve problems involving different units of mass, e.g. find the total mass of three items weighing 50 g, 750 g and 2.5 kg {Critical and creative thinking} | |
| relate the mass of one litre of water to one kilogram |
Time
TIME
OUTCOME
A student:
MA3-13MG:
uses 24-hour time and am and pm notation in real-life situations, and constructs timelines
TEACHING POINTS | Australia is divided into three time zones. In non-daylight saving periods, time in Queensland, New South Wales, Victoria and Tasmania is Eastern Standard Time (EST), time in South Australia and the Northern Territory is half an hour behind EST, and time in Western Australia is two hours behind EST. |
| Typically, 24-hour time is recorded without the use of the colon (:), e.g. 3:45 pm is written as 1545 or 1545 h and read as ‘fifteen forty-five hours’. |
| LANGUAGE | Students should be able to communicate using the following language: timetable, timeline, scale, 12-hour time, 24-hour time, hour, minute, second, am (notation), pm (notation). |
EXPECTATIONS OF ATTAINMENT
Interpret and use timetables (ACMMG139) | read, interpret and use timetables from real-life situations, including those involving 24-hour time {Literacy, Personal and social capability} |
| use bus, train, ferry and airline timetables, including those accessed on the internet, to prepare simple travel itineraries {Literacy, Information and communication technology capability, Personal and social capability} | |
| – interpret timetable information to solve unfamiliar problems using a variety of strategies {Problem Solving, Literacy Critical and creative thinking} |
Draw and interpret timelines using a given scale | determine a suitable scale and draw an accurate timeline using the scale, e.g. represent events using a many-to-one scale of 1 cm = 10 years |
| interpret a given timeline using the given scale {Literacy} |
3D Space
THREE-DIMENSIONAL SPACE
OUTCOME
A student:
MA3-14MG:
identifies three-dimensional objects, including prisms and pyramids, on the basis of their properties, and visualises, sketches and constructs them given drawings of different views
| TEACHING POINTS | In Stage 3, students are continuing to develop their skills of visual imagery, including the ability to perceive and hold an appropriate mental image of an object or arrangement, and to predict the orientation or shape of an object that has been moved or altered. Also see Year 5 |
| LANGUAGE | Students should be able to communicate using the following language: object, shape, three-dimensional object (3D object), prism, cube, pyramid, base, uniform cross-section, face, edge, vertex (vertices), top view, front view, side view, net. |
EXPECTATIONS OF ATTAINMENT
Construct simple prisms and pyramids (ACMMG140) | create prisms and pyramids using a variety of materials, e.g. plasticine, paper or cardboard nets, connecting cubes |
| – construct as many rectangular prisms as possible using a given number of connecting cubes {Problem Solving, Critical and creative thinking} | |
| create skeletal models of prisms and pyramids, e.g. using toothpicks and modelling clay or straws and tape {Critical and creative thinking} | |
| – connect the edges of prisms and pyramids with the construction of their skeletal models {Problem Solving} | |
| construct three-dimensional models of prisms and pyramids and sketch the front, side and top views | |
| – describe to another student how to construct or draw a three-dimensional object {Communicating, Literacy} | |
| construct three-dimensional models of prisms and pyramids, given drawings of different views |
2D Space
TWO-DIMENSIONAL SPACE
OUTCOME
A student:
MA3-15MG:
manipulates, classifies and draws two-dimensional shapes, including equilateral, isosceles and scalene triangles, and describes their properties
| TEACHING POINTS | When drawing diagonals, students need to be careful that the endpoints of their diagonals pass through the vertices of the shape. |
LANGUAGE | Students should be able to communicate using the following language: shape, two-dimensional shape (2D shape), circle, centre, radius, diameter, circumference, sector, semicircle, quadrant, triangle, equilateral triangle, isosceles triangle, scalene triangle, right-angled triangle, quadrilateral, parallelogram, rectangle, rhombus, square, trapezium, kite, pentagon, hexagon, octagon, regular shape, irregular shape, diagonal, vertex (vertices), line (axis) of symmetry, translate, reflect, rotate, clockwise, anti-clockwise. |
| A diagonal of a two-dimensional shape is an interval joining two non-adjacent vertices of the shape. The diagonals of a convex two-dimensional shape lie inside the figure. |
EXPECTATIONS OF ATTAINMENT
Investigate the diagonals of two-dimensional shapes | identify and name ‘diagonals’ of convex two-dimensional shapes {Literacy} |
| – recognise the endpoints of the diagonals of a shape as the vertices of the shape {Communicating, Literacy} | |
| determine and draw all the diagonals of convex two-dimensional shapes | |
| compare and describe diagonals of different convex two-dimensional shapes | |
| – use measurement to determine which of the special quadrilaterals have diagonals that are equal in length {Problem Solving} | |
| – determine whether any of the diagonals of a particular shape are also lines (axes) of symmetry of the shape {Problem Solving} |
Identify and name parts of circles | create a circle by finding points that are all the same distance from a fixed point (the centre) |
| identify and name parts of a circle, including the centre, radius, diameter, circumference, sector, semicircle and quadrant {Literacy} |
Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies (ACMMG142) | identify whether a two-dimensional shape has been translated, reflected or rotated, or has undergone a number of transformations, e.g. ‘The parallelogram has been rotated clockwise through 90° once and then reflected once’ |
| construct patterns of two-dimensional shapes that involve translations, reflections and rotations using computer software {Information and communication technology capability} | |
| predict the next translation, reflection or rotation in a pattern, eg ‘The arrow is being rotated 90° anti-clockwise each time’ | |
| – choose the correct pattern from a number of options when given information about a combination of transformations {Reasoning, Critical and creative thinking} |
Angles
ANGLES
OUTCOME
A student:
MA3-16MG:
measures and constructs angles, and applies angle relationships to find unknown angles
| TEACHING POINTS | Students should be encouraged to give reasons when finding unknown angles. |
LANGUAGE | Students should be able to communicate using the following language: angle, right angle, straight angle, angles on a straight line, angle of revolution, angles at a point, vertically opposite angles. |
| Two angles at a point are called adjacent if they share a common arm and a common vertex, and lie on opposite sides of the common arm. |
EXPECTATIONS OF ATTAINMENT
Investigate, with and without the use of digital technologies, angles on a straight line, angles at a point, and vertically opposite angles; use the results to find unknown angles(ACMMG141) | identify and name angle types formed by the intersection of straight lines, including right angles, ‘angles on a straight line’, ‘angles at a point’ that form an angle of revolution, and ‘vertically opposite angles’ {Literacy} |
| – recognise right angles, angles on a straight line, and angles of revolution embedded in diagrams {Reasoning} | |
| – identify the vertex and arms of angles formed by intersecting lines {Communicating} | |
| – recognise vertically opposite angles in different orientations and embedded in diagrams {Reasoning} | |
| investigate, with and without the use of digital technologies, adjacent angles that form a right angle and establish that they add to 90° | |
| investigate, with and without the use of digital technologies, adjacent angles on a straight line and establish that they form a straight angle and add to 180° | |
| investigate, with and without the use of digital technologies, angles at a point and establish that they form an angle of revolution and add to 360° | |
| use the results established for adjacent angles that form right angles, straight angles and angles of revolution to find the size of unknown angles in diagrams {Critical and creative thinking} | |
| – explain how the size of an unknown angle in a diagram was calculated {Communicating, Reasoning, Critical and creative thinking} | |
| investigate, with and without the use of digital technologies, vertically opposite angles and establish that they are equal in size | |
| use the equality of vertically opposite angles to find the size of unknown angles in diagrams |
Position
POSITION
OUTCOME
A student:
MA3-17MG:
locates and describes position on maps using a grid-reference system
TEACHING POINTS | In Stage 2, students were introduced to the compass directions north, east, south and west, and north-east, south-east, south-west and north-west. In Stage 3, students are expected to use these compass directions when describing routes between locations on maps. |
| By convention when using grid-reference systems, the horizontal component of direction is named first, followed by the vertical component. This connects with plotting points on the Cartesian plane in Stage 3 Patterns and Algebra, where the horizontal coordinate is recorded first, followed by the vertical coordinate. |
| LANGUAGE | Students should be able to communicate using the following language: position, location, map, plan, street directory, route, grid, grid reference, legend, key, scale, directions, compass, north, east, south, west, north-east, south-east, south-west, north-west. |
EXPECTATIONS OF ATTAINMENT
Use a grid-reference system to describe locations (ACMMG113) | find locations on maps, including maps with legends, given their grid references {Literacy} |
| describe particular locations on grid-referenced maps, including maps with a legend, eg ‘The post office is at E4’ |
Describe routes using landmarks and directional language (ACMMG113) | find a location on a map that is in a given direction from a town or landmark, eg locate a town that is north-east of Broken Hill {Literacy} |
| describe the direction of one location relative to another, eg ‘Darwin is north-west of Sydney’ {Literacy} | |
| follow a sequence of two or more directions, including compass directions, to find and identify a particular location on a map {Literacy} | |
| use a given map to plan and show a route from one location to another, eg draw a possible route to the local park or use an Aboriginal land map to plan a route(Literacy, Aboriginal and Torres Strait Islander histories and cultures} | |
| – use a street directory or online map to find the route to a given location {Problem Solving, Literacy, Information and communication technology capability} | |
| describe a route taken on a map using landmarks and directional language, including compass directions, eg ‘Start at the post office, go west to the supermarket and then go south-west to the park’ {Literacy} |
STATISTICS AND PROBABILITY
Data
Chance
Data
DATA
OUTCOME
A student:
MA3-18SP:
uses appropriate methods to collect data and constructs, interprets and evaluates data displays, including dot plots, line graphs and two-way tables
| TEACHING POINTS | Data selected for interpretation can include census data, environmental audits of resources such as water and energy, and sports statistics. Also see Year 5 |
| LANGUAGE | Students should be able to communicate using the following language: data, collect, category, display, table, column graph, scale, axes, two-way table, side-by-side column graph, misleading, bias. |
Expectations of Attainment
Interpret and compare a range of data displays, including side-by-side column graphs for two categorical variables (ACMSP147) | interpret data presented in two-way tables {Literacy, Civics and citizenship} |
| create a two-way table to organise data involving two categorical variables | |
| interpret side-by-side column graphs for two categorical variables, eg favourite television show of students in Year 1 compared to that of students in Year 6 {Literacy} | |
| interpret and compare different displays of the same data set to determine the most appropriate display for the data set | |
| – compare the effectiveness of different student-created data displays {Communicating} | |
| – discuss the advantages and disadvantages of different representations of the same data {Communicating, Critical and creative thinking, Ethical understanding} | |
| – explain which display is the most appropriate for interpretation of a particular data set {Communicating, Reasoning, Literacy, Critical and creative thinking} | |
| – compare representations of the same data set in a side-by-side column graph and in a two-way table {Reasoning, Critical and creative thinking} |
Interpret secondary data presented in digital media and elsewhere (ACMSP148) | interpret data representations found in digital media and in factual texts {Literacy, Information and communication technology capability} |
| – interpret tables and graphs from the media and online sources, eg data about different sports teams {Reasoning, Information and communication technology capability, Critical and creative thinking} | |
| – identify and describe conclusions that can be drawn from a particular representation of data {Communicating, Reasoning, Literacy} | |
| critically evaluate data representations found in digital media and related claims {Literacy, Information and communication technology capability. Critical and creative thinking, Personal and social capability, Ethical understanding} | |
| – discuss the messages that those who created a particular data representation might have wanted to convey {Communicating, Literacy, Critical and creative thinking, Personal and social capability, Ethical understanding, Civics and citizenship} | |
| – identify sources of possible bias in representations of data in the media by discussing various influences on data collection and representation, eg who created or paid for the data collection, whether the representation is part of an advertisement {Communicating, Reasoning, Literacy, Critical and creative thinking, Personal and social capability, Ethical understanding} | |
| – identify misleading representations of data in the media, eg broken axes, graphics that are not drawn to scale {Reasoning, Literacy, Information and communication technology capability, Critical and creative thinking, Personal and social capability, Ethical understanding} |
Describe and interpret different data sets in context (ACMSP120) | interpret line graphs using the scales on the axes {Literacy} |
| describe and interpret data presented in tables, dot plots, column graphs and line graphs, eg ‘The graph shows that the heights of all children in the class are between 125 cm and 154 cm’ {Literacy} | |
| – determine the total number of data values represented in dot plots and column graphs, eg find the number of students in the class from a display representing the heights of all children in the class {Problem Solving, Reasoning} | |
| – identify and describe relationships that can be observed in data displays, eg ‘There are four times as many children in Year 5 whose favourite food is noodles compared to children whose favourite food is chicken’ {Communicating, Reasoning, Literacy} | |
| – use information presented in data displays to aid decision making, eg decide how many of each soft drink to buy for a school fundraising activity by collecting and graphing data about favourite soft drinks for the year group or school {Reasoning, Critical and creative thinking} |
Chance
CHANCE
OUTCOME
A student:
MA3-19SP:
conducts chance experiments and assigns probabilities as values between 0 and 1 to describe their outcomes
TEACHING POINTS | Random generators include coins, dice, spinners and digital simulators. |
| As the number of trials in a chance experiment increases, the observed probabilities should become closer in value to the expected probabilities. | |
| Refer also to background information in Chance 1. |
LANGUAGE | Students should be able to communicate using the following language: chance, event, likelihood, equally likely, experiment, outcome, expected outcomes, random, fair, trials, probability, expected probability, observed probability, frequency, expected frequency, observed frequency. |
| The term ‘frequency’ is used in this substrand to describe the number of times a particular outcome occurs in a chance experiment. In Stage 4, students will also use ‘frequency’ to describe the number of times a particular data value occurs in a data set. |
EXPECTATIONS OF ATTAINMENT
Conduct chance experiments with both small and large numbers of trials using appropriate digital technologies (ACMSP145) | assign expected probabilities to outcomes in chance experiments with random generators, including digital simulators, and compare the expected probabilities with the observed probabilities after both small and large numbers of trials {Information and communication technology capability} |
| – determine and discuss the differences between the expected probabilities and the observed probabilities after both small and large numbers of trials {Communicating, Reasoning, Literacy} | |
| – explain what happens to the observed probabilities as the number of trials increases {Communicating, Reasoning, Literacy} | |
| use samples to make predictions about a larger ‘population’ from which the sample comes, eg take a random sample of coloured lollies from a bag, calculate the probability of obtaining each colour of lolly when drawing a lolly from the bag, and use these probabilities and the total number of lollies in the bag to predict the number of each colour of lolly in the bag {Critical and creative thinking} | |
| – discuss whether a prediction about a larger population, from which a sample comes, would be the same if a different sample were used {Communicating, Reasoning, Literacy, Critical and creative thinking} |