Describe probabilities using fractions, decimals and percentages; recognise that probabilities lie on numerical scales of 0–1 or 0%–100%; use estimation to assign probabilities that events occur in a given context, using common fractions, percentages and decimals
<ul>
<li>recognising that the probability of an event occurring can be represented numerically as either a number ranging from zero to one or a percentage from 0% to 100% where zero or 0% means it will not happen and one or 100% means it is certain to happen</li>
<li>using a scale of zero to one or 0% to 100% to estimate chances of events</li>
<li>listing the different possible outcomes for rolling a dice and using a scale to locate the relative probability by considering the chance of more than or less than for each possible event, for example, the probability of getting a number greater than 4</li>
<li>recognising the language used to describe situations involving uncertainty, such as what it means to be ‘lucky’, a ‘75% chance’ of rain or a ‘1-in-100 years’ flood</li>
<li>exploring Aboriginal and/or Torres Strait Islander children’s instructive games, such as Weme from the Warlpiri Peoples of Central Australia, to investigate and assign probabilities that events will occur, indicating their estimated likelihood</li>
</ul>
Recognise and use combinations of transformations to create tessellations and other geometric patterns, using dynamic geometry software where appropriate
<ul>
<li>using digital tools to create tessellations of shapes, including paver and tiling patterns, describing the transformations used and discussing why these shapes tessellate; and identifying shapes or combinations of shapes that will or will not tessellate, answering questions such as ‘Do all triangles tessellate?’</li>
<li>designing a school or brand logo using the transformation of one or more shapes and describing the transformations used</li>
<li>using dynamic geometry software and digital tools to experiment with transformations, for example, to demonstrate when the order of transformations produces different results; and experimenting with transformations and their application to fractals</li>
<li>designing an algorithm as a set of instructions to transform a shape, including getting back to where you started from; for example, programming a robot to move around the plane using instructions for movements, such as 2 down, 3 to the right, and combinations of these to transform shapes</li>
<li>investigating symmetry, transformation and tessellation in different shapes on Country/Place, including rock formations, insects, and land and sea animals, discussing the purpose or role symmetry plays in their physical structure</li>
</ul>
Locate points in the 4 quadrants of the Cartesian plane; describe changes to the coordinates when a point is moved to a different position in the plane
<ul>
<li>understanding that the Cartesian plane provides a graphical or visual way of describing location with respect to a fixed origin</li>
<li>understanding that the axes are number lines that can have different scales, including fractions and decimals, depending on purpose</li>
<li>understanding that the horizontal coordinate is written first and is changed if there is a move to the left or right, whereas a move up or down will change the vertical coordinate</li>
<li>using the Cartesian plane to draw lines and polygons, listing coordinates in the correct order to complete a polygon</li>
<li>investigating and connecting land or star maps used by Aboriginal and Torres Strait Islander Peoples with the Cartesian plane through a graphical or visual way of describing location</li>
</ul>
Identify the relationships between angles on a straight line, angles at a point and vertically opposite angles; use these to determine unknown angles, communicating reasoning
<ul>
<li>using protractors or dynamic geometry software to measure and generalise about the size of angles formed when lines are crossed, and combinations of angles that meet at a point, including combinations that form right or straight angles</li>
<li>demonstrating the meaning of language associated with properties of angles, including right, complementary, complement, straight, supplement, vertically opposite, and angles at a point</li>
<li>using the properties of supplementary and complementary angles to represent spatial situations with number sentences and solving to find the size of unknown angles</li>
</ul>
Measure, calculate and compare elapsed time; interpret and use timetables and itineraries to plan activities and determine the duration of events and journeys
<ul>
<li>planning a trip involving one or more modes of public transport</li>
<li>developing a timetable of daily activities for a planned event, for example, a sports carnival</li>
<li>investigating different ways duration is represented in timetables and using different timetables to plan a journey</li>
</ul>
Establish the formula for the area of a rectangle and use it to solve practical problems
<ul>
<li>using the relationship between the length and area of square units and the array structure to derive a formula for calculating the area of a rectangle from the lengths of its sides</li>
<li>using one-centimetre grid paper to construct a variety of rectangles, recording the side lengths and the related areas of the rectangles in a table to establish the formula for the area of a rectangle by recognising the relationship between the length of the sides and its calculated area</li>
<li>solving problems involving the comparison of lengths and areas using appropriate units</li>
<li>investigating the connection between the perimeters of different rectangles with the same area and between the areas of rectangles with the same perimeter</li>
</ul>
Convert between common metric units of length, mass and capacity; choose and use decimal representations of metric measurements relevant to the context of a problem
<ul>
<li>recognising the significance of the prefixes in units of measurement</li>
<li>identifying and using the correct operations when converting between units including millimetres, centimetres, metres, kilometres, milligrams, grams, kilograms, tonnes, millilitres, litres, kilolitres and megalitres</li>
<li>recognising the equivalence of measurements, such as 1.25 metres is the same as 125 centimetres</li>
</ul>
Find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations
<ul>
<li>using brackets and the order of operations to write number sentences and appreciating the need for an agreed set of rules to complete multiple operations within the same number sentence; for example, for 40 ÷ 2 × (4 + 6) = □, you solve what is in the brackets first then complete the number sentence from left to right as there is no hierarchy between division and multiplication</li>
<li>constructing equivalent number sentences involving brackets and combinations of the 4 operations, and explaining the need to have shared agreement on the order of operations when solving problems involving more than one operation to have unique solutions</li>
<li>finding pairs of unknown values in numerical equations that make the equation hold true; for example, listing possible combinations of natural numbers that make this statement true: 6 + 4 × 8 = 6 × Δ + □</li>
</ul>
Recognise and use rules that generate visually growing patterns and number patterns involving rational numbers
<ul>
<li>investigating patterns such as the number of tiles in a geometric pattern, or the number of dots or other shapes in successive repeats of a strip or border pattern, looking for patterns in the way the numbers increase or decrease</li>
<li>using a calculator or spreadsheet to experiment with number patterns that result from multiplying or dividing; for example, 1 ÷ 9, 2 ÷ 9, 3 ÷ 9 …, 210 × 11, 211 × 11, 212 × 11 …, 111 × 11, 222 × 11, 333 × 11 …, or 100 ÷ 99, 101 ÷ 99, 102 ÷ 99 …</li>
<li>creating an extended number sequence that represents an additive pattern using decimals; for example, representing the additive pattern formed as students pay their $2.50 for an incursion as 2.50, 5.00, 7.50, 10.00, 12.50, 15.00, 17.50 …</li>
<li>investigating the number of regions created by successive folds of a sheet of paper (one fold, 2 regions; 2 folds, 4 regions; 3 folds, 8 regions) and describing the pattern using everyday language</li>
<li>creating a pattern sequence with materials, writing the associated number sequence and then describing the sequence with a rule so someone else can replicate it with different materials; for example, using matchsticks or toothpicks to create a growing pattern of triangles using 3 for one triangle, 5 for 2 triangles and 7 for 3 triangles and describing the pattern as ‘Multiply the number of triangles by 2 and then add one for the extra toothpick in the first triangle’</li>
</ul>
Use mathematical modelling to solve practical problems involving rational numbers and percentages, including in financial contexts; formulate the problems, choosing operations and using efficient mental and written calculation strategies, and using digital tools where appropriate; interpret and communicate solutions in terms of the situation, justifying the choices made
<ul>
<li>modelling practical situations involving percentages using efficient calculation strategies to find solutions, such as mental calculations, spreadsheets, calculators or a variety of informal jottings, and interpreting the results in terms of the situation, for example, purchasing items during a sale</li>
<li>modelling situations involving earning money and budgeting, asking questions such as ‘Can I afford it?’, ‘Do I need it?’ and ‘How much do I need to save for it?’ and developing a savings plan or budget for an upcoming event or personal purchase</li>
<li>modelling and solving the problem of creating a budget for a class excursion or family holiday, using the internet to research costs and expenses, and representing the budget in a spreadsheet, creating and using formulas to calculate totals</li>
</ul>
Approximate numerical solutions to problems involving rational numbers and percentages, using appropriate estimation strategies
<ul>
<li>using familiar fractions, decimals and percentages to approximate calculations, such as 0.3 of 180 is about 1/3 of 180, or 52% is about 1/2</li>
<li>choosing appropriate estimation strategies including rounding to the nearest whole number, knowledge of multiples of 2, 5 or 10 and partitioning numbers, in contexts such as measuring or cost per unit</li>
<li>recognising the effect of rounding on calculations involving fractions or decimals and saying what numbers the answer will be between</li>
<li>recognising the usefulness of estimation to check calculations for contexts such as dividing wood into a number of lengths, calculating cost per unit, reducing a recipe or dividing the cost of dinner for a group into individual amounts</li>
<li>verifying solutions by estimating percentages in suitable contexts, such as discounts, using common percentages of 10%, 25%, 30%, 50% and 1%</li>
<li>investigating estimation strategies to make decisions about steam cooking in ground ovens by Aboriginal and/or Torres Strait Islander Peoples, including catering for different numbers of people and resources needed for cooking</li>
</ul>
Solve problems that require finding a familiar fraction, decimal or percentage of a quantity, including percentage discounts, choosing efficient calculation strategies with and without digital tools
<ul>
<li>explaining how 1/3 of a quantity can be achieved by dividing by 3, and how knowledge of 1/3 of a quantity can be used to find 2/3 or 4/3 of the same quantity, using situations involving money, length, duration, mass or capacity</li>
<li>investigating percentage discounts of 10%, 25% and 50% in an online toy sale, using their equivalent decimal representations of 0.10, 0.25 and 0.50 to calculate the amount of discount on sale items, with and without digital tools</li>
<li>linking percentages to their decimal equivalent of tenths and hundredths and using these to determine percentage discounts; for example, finding a 30% discount by using its equivalence to 0.3, dividing by 10 and multiplying the result by 3 to give 30%</li>
<li>explaining the equivalence between percentages and fractions, for example,33 1/3% and 1/3, keeping to percentages that are equivalent to fractions with small denominators such as 66 2/3% and 12.5%</li>
<li>representing a situation with a mathematical expression, for example, numbers and symbols such as 1/4 × 24, that involves finding a familiar fraction or percentage of a quantity; and using mental strategies or a calculator and explaining the result in terms of the situation in question</li>
</ul>
Multiply and divide decimals by multiples of powers of 10 without a calculator, applying knowledge of place value and proficiency with multiplication facts, using estimation and rounding to check the reasonableness of answers
<ul>
<li>applying place value knowledge, including that the value of the digit is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a decimal by powers of 10</li>
<li>applying and explaining estimation strategies in multiplicative situations involving a decimal greater than one that is multiplied by a two- or three-digit number, using a multiple of 10 or 100 when the situation requires just an estimation</li>
<li>explaining the effect of multiplying or dividing a decimal by 10, 100, 1000 … in terms of place value and not the decimal point shifting</li>
</ul>
Solve problems involving addition and subtraction of fractions using knowledge of equivalent fractions
<ul>
<li>representing addition and subtraction of fractions, using an understanding of equivalent fractions and methods such as jumps on a number line, or diagrams of fractions as parts of shapes</li>
<li>determining the lowest common denominator using an understanding of prime and composite numbers to find equivalent representation of fractions when solving addition and subtraction problems</li>
<li>calculating the addition or subtraction of fractions in the context of real-world problems (for example, using part cups or spoons in a recipe), using the understanding of equivalent fractions</li>
<li>understanding the processes for adding and subtracting fractions with related denominators and fractions as an operator, in preparation for calculating with all fractions; for example, using fraction overlays and number lines to give meaning to adding and subtracting fractions with related and unrelated denominators</li>
</ul>
Apply knowledge of place value to add and subtract decimals, using digital tools where appropriate; use estimation and rounding to check the reasonableness of answers
<ul>
<li>applying estimation strategies to addition and subtraction of decimals to at least thousandths before calculating answers or when a situation requires just an estimation</li>
<li>applying whole-number strategies; for example, using basic facts, place value, partitioning and the inverse relationship between addition and subtraction, and properties of operations to develop meaningful mental strategies for addition and subtraction of decimal numbers to at least hundredths</li>
<li>working additively with linear measurements expressed as decimals up to 2 and 3 decimal places; for example, calculating how far off the world record the athletes were at the last Olympic Games in the women’s long jump or shot-put and comparing school records to the Olympic records</li>
<li>deciding to use a calculator as a calculation strategy for solving additive problems involving decimals that vary in their number of decimal places beyond hundredths; for example, 1.0 − 0.0035 or 2.345 + 1.4999</li>
</ul>
Apply knowledge of equivalence to compare, order and represent common fractions, including halves, thirds and quarters, on the same number line and justify their order
<ul>
<li>applying factors and multiples to fraction denominators (such as halves with quarters, eighths and twelfths, and thirds with sixths, ninths and twelfths) to determine equivalent representations of fractions in order to make comparisons</li>
<li>representing fractions on the same number line, paying attention to relative position, and using this to explain relationships between denominators</li>
<li>explaining equivalence and order between fractions using number lines, drawings and models</li>
<li>comparing and ordering fractions by placing cards on a string line across the room and referring to benchmark fractions to justify their position; for example, 5/8 is greater than 1/2 can be written as 5/8 > 1/2, because half of 8 is 4; 1/6 is less than 1/4, because 6 > 4 and can be written as 1/6 < 1/4</li>
</ul>
Solve problems that require finding a familiar fraction, decimal or percentage of a quantity, including percentage discounts, choosing efficient calculation strategies with and without digital tools
<ul>
<li>explaining how 1/3 of a quantity can be achieved by dividing by 3, and how knowledge of 1/3 of a quantity can be used to find 2/3 or 4/3 of the same quantity, using situations involving money, length, duration, mass or capacity</li>
<li>investigating percentage discounts of 10%, 25% and 50% in an online toy sale, using their equivalent decimal representations of 0.10, 0.25 and 0.50 to calculate the amount of discount on sale items, with and without digital tools</li>
<li>linking percentages to their decimal equivalent of tenths and hundredths and using these to determine percentage discounts; for example, finding a 30% discount by using its equivalence to 0.3, dividing by 10 and multiplying the result by 3 to give 30%</li>
<li>explaining the equivalence between percentages and fractions, for example,33 1/3% and 1/3, keeping to percentages that are equivalent to fractions with small denominators such as 66 2/3% and 12.5%</li>
<li>representing a situation with a mathematical expression, for example, numbers and symbols such as 1/4 × 24, that involves finding a familiar fraction or percentage of a quantity; and using mental strategies or a calculator and explaining the result in terms of the situation in question</li>
</ul>
Recognise situations, including financial contexts, that use integers; locate and represent integers on a number line and as coordinates on the Cartesian plane
<ul>
<li>extending the number line in the negative direction to locate and represent integers, recognising the difference in location between (−2) and (+2) and their relationship to zero as −2 < 0 < 2</li>
<li>using integers to represent quantities in financial contexts, including the concept of profit and loss for a planned event</li>
<li>using horizontal and vertical number lines to represent and find solutions to everyday problems involving locating and ordering integers around zero (for example, elevators, above and below sea level) and distinguishing a location by referencing the 4 quadrants of the Cartesian plane</li>
<li>recognising that the sign (positive or negative) indicates a direction in relation to zero – for example, 30 metres left of the admin block is (−30) and 20 metres right of the admin block is (+20) – and programming robots to move along a number line that is either horizontal or vertical but not both at the same time</li>
<li>representing the temperatures of the different planets in the solar system, using a diagram of a thermometer that models a vertical number line</li>
</ul>
Recognise that 100% represents the complete whole and use percentages to describe, represent and compare relative size; connect familiar percentages to their decimal and fraction equivalents
<ul>
<li>recognising applications of percentages used in everyday contexts, for example, the bar model used for charging devices indicating the percentage of power remaining, and advertising in retail contexts relating to discounts or sales</li>
<li>creating a model by subdividing a whole (for example, using 10 × 10 grids to represent various percentage amounts) and recognising complementary percentages (such as 30% and 70%) combine to make 100%</li>
<li>creating a model by subdividing a collection of materials, such as blocks or money, to connect decimals and percentage equivalents of tenths and the commonly used fractions 1/2, 1/4, and 3/4; for example, connecting that one-tenth or 0.1 represents 10% and one-half or 0.5 represents 50%, and recognising that 60% of a whole is 10% more of the whole than 50%</li>
<li>using physical and virtual materials to represent the relationship between decimal notation and percentages, for example, 0.3 is 3 out of every 10, which is 30 out of every 100, which is 30%</li>
</ul>
Recognise that probabilities lie on numerical scales of 0 – 1 or 0% – 100% and use estimation to assign probabilities that events occur in a given context, using common fractions, percentages and decimals
Identify the relationships between angles on a straight line, angles at a point and vertically opposite angles; use these to determine unknown angles, communicating reasoning
Convert between common metric units of length, mass and capacity; choose and use decimal representations of metric measurements relevant to the context of a problem
Find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations
Use mathematical modelling to solve practical problems, involving rational numbers and percentages, including in financial contexts; formulate the problems, choosing operations and efficient calculation strategies, and using digital tools where appropriate; interpret and communicate solutions in te
Approximate numerical solutions to problems involving rational numbers and percentages, including financial contexts, using appropriate estimation strategies
Solve problems that require finding a familiar fraction, decimal or percentage of a quantity, including percentage discounts, choosing efficient calculation strategies and using digital tools where appropriate
Multiply and divide decimals by multiples of powers of 10 without a calculator, applying knowledge of place value and proficiency with multiplication facts; using estimation and rounding to check the reasonableness of answers
Apply knowledge of place value to add and subtract decimals, using digital tools where appropriate; use estimation and rounding to check the reasonableness of answers
Apply knowledge of equivalence to compare, order and represent common fractions including halves, thirds and quarters on the same number line and justify their order
Recognise situations, including financial contexts, that use integers; locate and represent integers on a number line and as coordinates on the Cartesian plane
Recognise that 100% represents the complete whole and use percentages to describe, represent and compare relative size; connect familiar percentages to their decimal and fraction equivalents
Recognise and use combinations of transformations to create tessellations and other geometric patterns, using dynamic geometric software where appropriate
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
Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies
Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers
teaching resource
Year 6 Numeracy Exit Tickets – Worksheets
Updated: 03 Oct 2023
23 Numeracy Exit Ticket activities for students to provide evidence of their learning progress.
23 Numeracy Exit Ticket activities for students to provide evidence of their learning progress.
This teaching resource is an assessment tool for students to show evidence of their learning. Use Exit Tickets after a numeracy skill has been taught to show evidence that they have acquired the skill and are ready to move to the next learning goal.
These Numeracy Exit tickets collect evidence for basic Year 6 curriculum requirements. For the prepopulated exit ticket, use the dropdown arrow on the Download button and select the PDF version of the resource. Choose the editable Word version to create your own class-specific exit ticket.
NOTE: Please note that the Word version of this exit ticket will be blank so that you can create your own class-specific exit ticket.
Describe probabilities using fractions, decimals and percentages; recognise that probabilities lie on numerical scales of 0–1 or 0%–100%; use estimation to assign probabilities that events occur in a given context, using common fractions, percentages and decimals
<ul>
<li>recognising that the probability of an event occurring can be represented numerically as either a number ranging from zero to one or a percentage from 0% to 100% where zero or 0% means it will not happen and one or 100% means it is certain to happen</li>
<li>using a scale of zero to one or 0% to 100% to estimate chances of events</li>
<li>listing the different possible outcomes for rolling a dice and using a scale to locate the relative probability by considering the chance of more than or less than for each possible event, for example, the probability of getting a number greater than 4</li>
<li>recognising the language used to describe situations involving uncertainty, such as what it means to be ‘lucky’, a ‘75% chance’ of rain or a ‘1-in-100 years’ flood</li>
<li>exploring Aboriginal and/or Torres Strait Islander children’s instructive games, such as Weme from the Warlpiri Peoples of Central Australia, to investigate and assign probabilities that events will occur, indicating their estimated likelihood</li>
</ul>
Recognise and use combinations of transformations to create tessellations and other geometric patterns, using dynamic geometry software where appropriate
<ul>
<li>using digital tools to create tessellations of shapes, including paver and tiling patterns, describing the transformations used and discussing why these shapes tessellate; and identifying shapes or combinations of shapes that will or will not tessellate, answering questions such as ‘Do all triangles tessellate?’</li>
<li>designing a school or brand logo using the transformation of one or more shapes and describing the transformations used</li>
<li>using dynamic geometry software and digital tools to experiment with transformations, for example, to demonstrate when the order of transformations produces different results; and experimenting with transformations and their application to fractals</li>
<li>designing an algorithm as a set of instructions to transform a shape, including getting back to where you started from; for example, programming a robot to move around the plane using instructions for movements, such as 2 down, 3 to the right, and combinations of these to transform shapes</li>
<li>investigating symmetry, transformation and tessellation in different shapes on Country/Place, including rock formations, insects, and land and sea animals, discussing the purpose or role symmetry plays in their physical structure</li>
</ul>
Locate points in the 4 quadrants of the Cartesian plane; describe changes to the coordinates when a point is moved to a different position in the plane
<ul>
<li>understanding that the Cartesian plane provides a graphical or visual way of describing location with respect to a fixed origin</li>
<li>understanding that the axes are number lines that can have different scales, including fractions and decimals, depending on purpose</li>
<li>understanding that the horizontal coordinate is written first and is changed if there is a move to the left or right, whereas a move up or down will change the vertical coordinate</li>
<li>using the Cartesian plane to draw lines and polygons, listing coordinates in the correct order to complete a polygon</li>
<li>investigating and connecting land or star maps used by Aboriginal and Torres Strait Islander Peoples with the Cartesian plane through a graphical or visual way of describing location</li>
</ul>
Identify the relationships between angles on a straight line, angles at a point and vertically opposite angles; use these to determine unknown angles, communicating reasoning
<ul>
<li>using protractors or dynamic geometry software to measure and generalise about the size of angles formed when lines are crossed, and combinations of angles that meet at a point, including combinations that form right or straight angles</li>
<li>demonstrating the meaning of language associated with properties of angles, including right, complementary, complement, straight, supplement, vertically opposite, and angles at a point</li>
<li>using the properties of supplementary and complementary angles to represent spatial situations with number sentences and solving to find the size of unknown angles</li>
</ul>
Measure, calculate and compare elapsed time; interpret and use timetables and itineraries to plan activities and determine the duration of events and journeys
<ul>
<li>planning a trip involving one or more modes of public transport</li>
<li>developing a timetable of daily activities for a planned event, for example, a sports carnival</li>
<li>investigating different ways duration is represented in timetables and using different timetables to plan a journey</li>
</ul>
Establish the formula for the area of a rectangle and use it to solve practical problems
<ul>
<li>using the relationship between the length and area of square units and the array structure to derive a formula for calculating the area of a rectangle from the lengths of its sides</li>
<li>using one-centimetre grid paper to construct a variety of rectangles, recording the side lengths and the related areas of the rectangles in a table to establish the formula for the area of a rectangle by recognising the relationship between the length of the sides and its calculated area</li>
<li>solving problems involving the comparison of lengths and areas using appropriate units</li>
<li>investigating the connection between the perimeters of different rectangles with the same area and between the areas of rectangles with the same perimeter</li>
</ul>
Convert between common metric units of length, mass and capacity; choose and use decimal representations of metric measurements relevant to the context of a problem
<ul>
<li>recognising the significance of the prefixes in units of measurement</li>
<li>identifying and using the correct operations when converting between units including millimetres, centimetres, metres, kilometres, milligrams, grams, kilograms, tonnes, millilitres, litres, kilolitres and megalitres</li>
<li>recognising the equivalence of measurements, such as 1.25 metres is the same as 125 centimetres</li>
</ul>
Find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations
<ul>
<li>using brackets and the order of operations to write number sentences and appreciating the need for an agreed set of rules to complete multiple operations within the same number sentence; for example, for 40 ÷ 2 × (4 + 6) = □, you solve what is in the brackets first then complete the number sentence from left to right as there is no hierarchy between division and multiplication</li>
<li>constructing equivalent number sentences involving brackets and combinations of the 4 operations, and explaining the need to have shared agreement on the order of operations when solving problems involving more than one operation to have unique solutions</li>
<li>finding pairs of unknown values in numerical equations that make the equation hold true; for example, listing possible combinations of natural numbers that make this statement true: 6 + 4 × 8 = 6 × Δ + □</li>
</ul>
Recognise and use rules that generate visually growing patterns and number patterns involving rational numbers
<ul>
<li>investigating patterns such as the number of tiles in a geometric pattern, or the number of dots or other shapes in successive repeats of a strip or border pattern, looking for patterns in the way the numbers increase or decrease</li>
<li>using a calculator or spreadsheet to experiment with number patterns that result from multiplying or dividing; for example, 1 ÷ 9, 2 ÷ 9, 3 ÷ 9 …, 210 × 11, 211 × 11, 212 × 11 …, 111 × 11, 222 × 11, 333 × 11 …, or 100 ÷ 99, 101 ÷ 99, 102 ÷ 99 …</li>
<li>creating an extended number sequence that represents an additive pattern using decimals; for example, representing the additive pattern formed as students pay their $2.50 for an incursion as 2.50, 5.00, 7.50, 10.00, 12.50, 15.00, 17.50 …</li>
<li>investigating the number of regions created by successive folds of a sheet of paper (one fold, 2 regions; 2 folds, 4 regions; 3 folds, 8 regions) and describing the pattern using everyday language</li>
<li>creating a pattern sequence with materials, writing the associated number sequence and then describing the sequence with a rule so someone else can replicate it with different materials; for example, using matchsticks or toothpicks to create a growing pattern of triangles using 3 for one triangle, 5 for 2 triangles and 7 for 3 triangles and describing the pattern as ‘Multiply the number of triangles by 2 and then add one for the extra toothpick in the first triangle’</li>
</ul>
Use mathematical modelling to solve practical problems involving rational numbers and percentages, including in financial contexts; formulate the problems, choosing operations and using efficient mental and written calculation strategies, and using digital tools where appropriate; interpret and communicate solutions in terms of the situation, justifying the choices made
<ul>
<li>modelling practical situations involving percentages using efficient calculation strategies to find solutions, such as mental calculations, spreadsheets, calculators or a variety of informal jottings, and interpreting the results in terms of the situation, for example, purchasing items during a sale</li>
<li>modelling situations involving earning money and budgeting, asking questions such as ‘Can I afford it?’, ‘Do I need it?’ and ‘How much do I need to save for it?’ and developing a savings plan or budget for an upcoming event or personal purchase</li>
<li>modelling and solving the problem of creating a budget for a class excursion or family holiday, using the internet to research costs and expenses, and representing the budget in a spreadsheet, creating and using formulas to calculate totals</li>
</ul>
Approximate numerical solutions to problems involving rational numbers and percentages, using appropriate estimation strategies
<ul>
<li>using familiar fractions, decimals and percentages to approximate calculations, such as 0.3 of 180 is about 1/3 of 180, or 52% is about 1/2</li>
<li>choosing appropriate estimation strategies including rounding to the nearest whole number, knowledge of multiples of 2, 5 or 10 and partitioning numbers, in contexts such as measuring or cost per unit</li>
<li>recognising the effect of rounding on calculations involving fractions or decimals and saying what numbers the answer will be between</li>
<li>recognising the usefulness of estimation to check calculations for contexts such as dividing wood into a number of lengths, calculating cost per unit, reducing a recipe or dividing the cost of dinner for a group into individual amounts</li>
<li>verifying solutions by estimating percentages in suitable contexts, such as discounts, using common percentages of 10%, 25%, 30%, 50% and 1%</li>
<li>investigating estimation strategies to make decisions about steam cooking in ground ovens by Aboriginal and/or Torres Strait Islander Peoples, including catering for different numbers of people and resources needed for cooking</li>
</ul>
Solve problems that require finding a familiar fraction, decimal or percentage of a quantity, including percentage discounts, choosing efficient calculation strategies with and without digital tools
<ul>
<li>explaining how 1/3 of a quantity can be achieved by dividing by 3, and how knowledge of 1/3 of a quantity can be used to find 2/3 or 4/3 of the same quantity, using situations involving money, length, duration, mass or capacity</li>
<li>investigating percentage discounts of 10%, 25% and 50% in an online toy sale, using their equivalent decimal representations of 0.10, 0.25 and 0.50 to calculate the amount of discount on sale items, with and without digital tools</li>
<li>linking percentages to their decimal equivalent of tenths and hundredths and using these to determine percentage discounts; for example, finding a 30% discount by using its equivalence to 0.3, dividing by 10 and multiplying the result by 3 to give 30%</li>
<li>explaining the equivalence between percentages and fractions, for example,33 1/3% and 1/3, keeping to percentages that are equivalent to fractions with small denominators such as 66 2/3% and 12.5%</li>
<li>representing a situation with a mathematical expression, for example, numbers and symbols such as 1/4 × 24, that involves finding a familiar fraction or percentage of a quantity; and using mental strategies or a calculator and explaining the result in terms of the situation in question</li>
</ul>
Multiply and divide decimals by multiples of powers of 10 without a calculator, applying knowledge of place value and proficiency with multiplication facts, using estimation and rounding to check the reasonableness of answers
<ul>
<li>applying place value knowledge, including that the value of the digit is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a decimal by powers of 10</li>
<li>applying and explaining estimation strategies in multiplicative situations involving a decimal greater than one that is multiplied by a two- or three-digit number, using a multiple of 10 or 100 when the situation requires just an estimation</li>
<li>explaining the effect of multiplying or dividing a decimal by 10, 100, 1000 … in terms of place value and not the decimal point shifting</li>
</ul>
Solve problems involving addition and subtraction of fractions using knowledge of equivalent fractions
<ul>
<li>representing addition and subtraction of fractions, using an understanding of equivalent fractions and methods such as jumps on a number line, or diagrams of fractions as parts of shapes</li>
<li>determining the lowest common denominator using an understanding of prime and composite numbers to find equivalent representation of fractions when solving addition and subtraction problems</li>
<li>calculating the addition or subtraction of fractions in the context of real-world problems (for example, using part cups or spoons in a recipe), using the understanding of equivalent fractions</li>
<li>understanding the processes for adding and subtracting fractions with related denominators and fractions as an operator, in preparation for calculating with all fractions; for example, using fraction overlays and number lines to give meaning to adding and subtracting fractions with related and unrelated denominators</li>
</ul>
Apply knowledge of place value to add and subtract decimals, using digital tools where appropriate; use estimation and rounding to check the reasonableness of answers
<ul>
<li>applying estimation strategies to addition and subtraction of decimals to at least thousandths before calculating answers or when a situation requires just an estimation</li>
<li>applying whole-number strategies; for example, using basic facts, place value, partitioning and the inverse relationship between addition and subtraction, and properties of operations to develop meaningful mental strategies for addition and subtraction of decimal numbers to at least hundredths</li>
<li>working additively with linear measurements expressed as decimals up to 2 and 3 decimal places; for example, calculating how far off the world record the athletes were at the last Olympic Games in the women’s long jump or shot-put and comparing school records to the Olympic records</li>
<li>deciding to use a calculator as a calculation strategy for solving additive problems involving decimals that vary in their number of decimal places beyond hundredths; for example, 1.0 − 0.0035 or 2.345 + 1.4999</li>
</ul>
Apply knowledge of equivalence to compare, order and represent common fractions, including halves, thirds and quarters, on the same number line and justify their order
<ul>
<li>applying factors and multiples to fraction denominators (such as halves with quarters, eighths and twelfths, and thirds with sixths, ninths and twelfths) to determine equivalent representations of fractions in order to make comparisons</li>
<li>representing fractions on the same number line, paying attention to relative position, and using this to explain relationships between denominators</li>
<li>explaining equivalence and order between fractions using number lines, drawings and models</li>
<li>comparing and ordering fractions by placing cards on a string line across the room and referring to benchmark fractions to justify their position; for example, 5/8 is greater than 1/2 can be written as 5/8 > 1/2, because half of 8 is 4; 1/6 is less than 1/4, because 6 > 4 and can be written as 1/6 < 1/4</li>
</ul>
Solve problems that require finding a familiar fraction, decimal or percentage of a quantity, including percentage discounts, choosing efficient calculation strategies with and without digital tools
<ul>
<li>explaining how 1/3 of a quantity can be achieved by dividing by 3, and how knowledge of 1/3 of a quantity can be used to find 2/3 or 4/3 of the same quantity, using situations involving money, length, duration, mass or capacity</li>
<li>investigating percentage discounts of 10%, 25% and 50% in an online toy sale, using their equivalent decimal representations of 0.10, 0.25 and 0.50 to calculate the amount of discount on sale items, with and without digital tools</li>
<li>linking percentages to their decimal equivalent of tenths and hundredths and using these to determine percentage discounts; for example, finding a 30% discount by using its equivalence to 0.3, dividing by 10 and multiplying the result by 3 to give 30%</li>
<li>explaining the equivalence between percentages and fractions, for example,33 1/3% and 1/3, keeping to percentages that are equivalent to fractions with small denominators such as 66 2/3% and 12.5%</li>
<li>representing a situation with a mathematical expression, for example, numbers and symbols such as 1/4 × 24, that involves finding a familiar fraction or percentage of a quantity; and using mental strategies or a calculator and explaining the result in terms of the situation in question</li>
</ul>
Recognise situations, including financial contexts, that use integers; locate and represent integers on a number line and as coordinates on the Cartesian plane
<ul>
<li>extending the number line in the negative direction to locate and represent integers, recognising the difference in location between (−2) and (+2) and their relationship to zero as −2 < 0 < 2</li>
<li>using integers to represent quantities in financial contexts, including the concept of profit and loss for a planned event</li>
<li>using horizontal and vertical number lines to represent and find solutions to everyday problems involving locating and ordering integers around zero (for example, elevators, above and below sea level) and distinguishing a location by referencing the 4 quadrants of the Cartesian plane</li>
<li>recognising that the sign (positive or negative) indicates a direction in relation to zero – for example, 30 metres left of the admin block is (−30) and 20 metres right of the admin block is (+20) – and programming robots to move along a number line that is either horizontal or vertical but not both at the same time</li>
<li>representing the temperatures of the different planets in the solar system, using a diagram of a thermometer that models a vertical number line</li>
</ul>
Recognise that 100% represents the complete whole and use percentages to describe, represent and compare relative size; connect familiar percentages to their decimal and fraction equivalents
<ul>
<li>recognising applications of percentages used in everyday contexts, for example, the bar model used for charging devices indicating the percentage of power remaining, and advertising in retail contexts relating to discounts or sales</li>
<li>creating a model by subdividing a whole (for example, using 10 × 10 grids to represent various percentage amounts) and recognising complementary percentages (such as 30% and 70%) combine to make 100%</li>
<li>creating a model by subdividing a collection of materials, such as blocks or money, to connect decimals and percentage equivalents of tenths and the commonly used fractions 1/2, 1/4, and 3/4; for example, connecting that one-tenth or 0.1 represents 10% and one-half or 0.5 represents 50%, and recognising that 60% of a whole is 10% more of the whole than 50%</li>
<li>using physical and virtual materials to represent the relationship between decimal notation and percentages, for example, 0.3 is 3 out of every 10, which is 30 out of every 100, which is 30%</li>
</ul>
Recognise that probabilities lie on numerical scales of 0 – 1 or 0% – 100% and use estimation to assign probabilities that events occur in a given context, using common fractions, percentages and decimals
Identify the relationships between angles on a straight line, angles at a point and vertically opposite angles; use these to determine unknown angles, communicating reasoning
Convert between common metric units of length, mass and capacity; choose and use decimal representations of metric measurements relevant to the context of a problem
Find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations
Use mathematical modelling to solve practical problems, involving rational numbers and percentages, including in financial contexts; formulate the problems, choosing operations and efficient calculation strategies, and using digital tools where appropriate; interpret and communicate solutions in te
Approximate numerical solutions to problems involving rational numbers and percentages, including financial contexts, using appropriate estimation strategies
Solve problems that require finding a familiar fraction, decimal or percentage of a quantity, including percentage discounts, choosing efficient calculation strategies and using digital tools where appropriate
Multiply and divide decimals by multiples of powers of 10 without a calculator, applying knowledge of place value and proficiency with multiplication facts; using estimation and rounding to check the reasonableness of answers
Apply knowledge of place value to add and subtract decimals, using digital tools where appropriate; use estimation and rounding to check the reasonableness of answers
Apply knowledge of equivalence to compare, order and represent common fractions including halves, thirds and quarters on the same number line and justify their order
Recognise situations, including financial contexts, that use integers; locate and represent integers on a number line and as coordinates on the Cartesian plane
Recognise that 100% represents the complete whole and use percentages to describe, represent and compare relative size; connect familiar percentages to their decimal and fraction equivalents
Recognise and use combinations of transformations to create tessellations and other geometric patterns, using dynamic geometric software where appropriate
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
Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies
Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers
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