The Awfullest Australian Curriculum Number Lines

As regular readers will know, and have been ignoring, we’ve been steadily working through ACARA’s new mathematics curriculum, compiling the annoying-or-way-worse 1-liner content descriptions and elaborations. No one is reading it of course, because that would be nuts. Or, it would cause the reader to go nuts. But, we’ll continue. In for a penny, in for a pounding. Continue reading “The Awfullest Australian Curriculum Number Lines”

Applied Mathematics is Bad Mathematics

Yes, the the title is clickbait, but it is not our clickbait. It’s the title of an interesting and semi-provocative 1981 article by Paul Halmos. Halmos’s article came to mind after a brief conversation recently, about applied mathematics in Australia. As with pretty much everything Halmos wrote, it seemed worth sharing.* Continue reading “Applied Mathematics is Bad Mathematics”

Funny Numbers

The leadership of my high school in the early 70s was, I presume, pretty typical for a suburban Melbourne school. It was not unkind, but it was authoritative and distant. Except for one guy, a deputy principal or something, who was a complete asshole. This one guy was always angry, always yelling at some kid for something, and forever stomping down the hallways. He also always held his head at a weird and pronounced tilt. And so, the students’ name for this asshole was 43 Degrees. Continue reading “Funny Numbers”

David de Carvalho and the Annoying Taco Kids

Last week we wrote about the mushy Australian puff piece on PISA clown, Andreas Schleicher. Readers may recall that Schleicher was critical of “Australia’s shallow Curriculum”. Schleicher says nothing of substance, simply advocating, ad nauseam,

“teaching fewer things at greater depth”.

The Australian piece also briefly quoted Ben Jensen and Mailie Ross, from some consultancy group called Learning First. In the same issue of The Australian, Jensen and Ross had an opinion piece strongly criticising the Australian Curriculum in the opposite direction:

“The Australian curriculum, however, is not a high-quality, knowledge-rich curriculum. It doesn’t guarantee the knowledge students are supposed to learn … Instead, it is a skills-based curriculum; the standards for students to achieve are skills-based. A skills-based curriculum includes knowledge but isn’t specific about what knowledge should be taught, so there is no guarantee of what will be taught in each year level, let alone across the curriculum.

[The Curriculum should] make it clear what knowledge and skills students have the right to learn in order to participate productively in life. Be honest and acknowledge that the Australian curriculum does not offer this clarity.”

Jensen and Ross’s piece is not great, in particular arguing too loosely on the basis of vague generalities. But, notwithstanding the vagueness of both pieces, there is a clear conflict about what the Australian Curriculum is, and what it should be. Luckily, we have ACARA’s CEO, the all-wise David de Carvalho, to resolve the conflict. Continue reading “David de Carvalho and the Annoying Taco Kids”

The CAS Betrayal

This post will take the form of Betrayal, with a sequence of five stories going backwards in time.

STORY 5

Last year, I was asked by an acquaintance, let’s call him Rob, to take a look at the draft of a mathematics article he was writing. Rob’s article was in rough form but it was interesting, a nice application of trigonometry and calculus, suitable and good reading for a strong senior school student. One line, however, grabbed my attention. Having wound up with a vicious trig integral, Rob confidently proclaimed,

“This is definitely a case for CAS”.

It wasn’t. Continue reading “The CAS Betrayal”

Australian Mathematics Curriculum Awfulnesses

This is our post for noting local nonsense in the new Australian Mathematics Curriculum (downloads here). It is a list of the content descriptions and the elaborations that appear to be deficient/wrong/misplaced/weird/whatever. It includes anything that has come to our attention and has sufficiently annoyed us and/or others. (This post supersedes a previous post, and the suggestions already made have been incorporated below.) There is no intention to be comprehensive, which would probably entail transcribing 80% of the thing. We’ll just keep adding over time, as awfulnesses come to our attention. Of course criticisms and suggestions and discussion are always welcome, and can be made in the comments.

Note that this post is for the identification of local awfulnesses only. Capturing the global, systemic awfulness of the Curriculum, including the many appalling delays and omissions, would require a massive effort, which we have no intention of doing again, and which is also not required: most of the posts listed here still apply in large part, and often in full. The Curriculum excerpts are also not accompanied by any discussion of the awfulness; we’ll update with links to comments and other sources as seems worthwhile.

The (ostensibly mandatory) content descriptions are flush, and the (ostensibly optional) elaborations are indented. The content description linked to a given elaboration is only included if the content is there on its own demerit.

Foundation: Number, Algebra
Year 1: Number, Algebra
Year 2: Number, Algebra
Year 3: Number, Algebra
Year 4: Number, Algebra
Year 5: Number, Algebra
Year 6: Number, Algebra
Year 7: Number, Algebra
Year 8: Number, Algebra
Year 9: Number, Algebra, Space
Year 10: Number, Algebra
Year 10 Optional: Number, Algebra

Foundation Number

name, represent and order numbers including zero to at least 20, using physical and virtual materials and numerals (AC9MFN01) (Link)(Link)

establishing the language and process of counting, understanding that each object must be counted only once, that the arrangement of objects does not affect how many there are, and that the last number counted answers the question of “How many?”; for example, saying numbers in sequence while playing and performing actions (AC9MFN03) (11/09/22)

partitioning collections of up to 10 objects in different ways and saying the part-part-whole relationship; for example, partitioning a collection of 6 counters into 4 counters and 2 counters and saying, “6 is 4 and 2 more, it’s 2 and 4””, then partitioning the same collection into 5 and one or 3 and 3 (AC9MFN04) (11/09/22)

represent practical situations involving addition, subtraction and quantification with physical and virtual materials and use counting or subitising strategies (AC9MFN05)

representing addition and subtraction situations found in leaf games involving sets of objects used to tell stories, such as games from the Warlpiri Peoples of Yuendumu in the Northern Territory (AC9MFN05) (11/09/22)

represent practical situations involving equal sharing and grouping with physical and virtual materials and use counting or subitising strategies (AC9MFN06)

representing situations expressed in First Nations Australians’ stories, such as “Tiddalick, the greedy frog”, that describe additive situations and their connections to Country/Place (AC9MFN05) (11/09/22)

exploring instructive games of First Nations Australians that involve sharing; for example, playing Yangamini of the Tiwi Peoples of Bathurst Island to investigate and discuss equal sharing (AC9MFN06) (11/09/22)

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Foundation Algebra

recognise, copy and continue repeating patterns represented in different ways (AC9MFA01) (22/09/22)

recognising repeating patterns used at home and in daily activities to help make tasks easier or to solve problems; for example, setting the table to eat (AC9MFA01) (22/09/22)

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Year 1 Number

recognise, represent and order numbers to at least 120 using physical and virtual materials, numerals, number lines and charts (AC9M1N01) (14/09/22)

partition one- and two-digit numbers in different ways using physical and virtual materials, including partitioning two-digit numbers into tens and ones (AC9M1N02) (14/09/22)

building knowledge and understanding of the part-part-whole facts to 10, using physical and virtual materials; for example, using virtual ten-frames through a digital app or website to identify pairs of numbers that combine to make 10 (AC9M1N02) (14/09/22)

using physical and virtual materials to partition numbers into counts of tens and ones; for example, recognise 35 as 3 tens and 5 ones or as 2 tens and 15 ones (AC9M1N02) (14/09/22)

using part-part-whole reasoning and physical or virtual materials to represent 24, then partitioning 24 in different ways and recording the partitions using numbers; for example, 10, 10 and 4 combine to make 24 or 10 and 14 combine to make 24 (AC9M1N02) (14/09/22)

add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies (AC9M1N04) (14/09/22)

developing and using strategies for one-digit addition and subtraction based on part-part-whole relationships for each of the numbers to 10 and subitising with physical and virtual materials; for example, 8 and 6 is the same as 8 and 2 and 4 (AC9M1N04) (14/09/22)

creating and performing addition and subtraction stories told through First Nations Australians’ dances (AC9M1N04) (14/09/22)

use mathematical modelling to solve practical problems involving additive situations, including simple money transactions; represent the situations with diagrams, physical and virtual materials, and use calculation strategies to solve the problem (AC9M1N05) (14/09/22)

use mathematical modelling to solve practical problems involving equal sharing and grouping; represent the situations with diagrams, physical and virtual materials, and use calculation strategies to solve the problem (AC9M1N06) (14/09/22)

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Year 1 Algebra

recognise, continue and create repeating patterns with numbers, symbols, shapes and objects, identifying the repeating unit (AC9M1A02) (23/09/22)

considering how the making of shell or seed necklaces by First Nations Australians includes practices such as sorting shells and beads based on colour, size and shape, and creating a repeating pattern sequence (AC9M1A02) (23/09/22)

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Year 2 Number

recognise, represent and order numbers to at least 1000 using physical and virtual materials, numerals and number lines (AC9M2N01) (15/09/22)

recognising missing numbers on different number lines; for example, a number line with 1800 on one end and 220 on the other, with every decade numbered (AC9M2N01) (15/09/22)

collecting large quantities of materials for recycling; for example, ring pulls, bottle tops and bread tags, and grouping them into ones, tens and hundreds; using the materials to show different representations of two- and three-digit numbers (AC9M2N01) (15/09/22)

comparing the digits of a number with materials grouped into hundreds, tens and ones, and explaining the meaning of each of the digits in the materials (AC9M2N02) (15/09/22)

add and subtract one- and two-digit numbers, representing problems using number sentences and solve using part-part-whole reasoning and a variety of calculation strategies (AC9M2N04) (15/09/22)

using strategies such as doubles, near doubles, part-part-whole knowledge to 10, bridging tens and partitioning to mentally solve problems involving two-digit numbers; for example, calculating 56 + 37 by thinking 5 tens and 3 tens is 8 tens, 6 + 7 = 6 + 4 + 3 is one 10 and 3, and so the result is 9 tens and 3, 93 (AC9M2N04) (15/09/22)

representing addition and subtraction problems using a bar model and  writing a number sentence, explaining how each number in the sentence is connected to the situation (AC9M2N04) (15/09/22)

using mental strategies and informal written jottings to help keep track of the numbers when solving addition and subtraction problems involving two-digit numbers and recognising that zero added to a number leaves the number unchanged; for example, in calculating 34 + 20 = 54, 3 tens add 2 tens is 5 tens which is 50, and 4 ones add zero ones is 4 ones which is 4, so the result is 50 + 4 = 54 (AC9M2N04) (15/09/22)

using First Nations Australians’ stories and dances to understand the balance and connection between addition and subtraction, representing relationships as number sentences (AC9M2N04) (15/09/22)

multiply and divide by one-digit numbers using repeated addition, equal grouping, arrays, and partitioning to support a variety of calculation strategies (AC9M2N05) (15/09/22)

use mathematical modelling to solve practical problems involving additive and multiplicative situations, including money transactions; represent situations and choose calculation strategies; interpret and communicate solutions in terms of the situation (AC9M2N06) (15/09/22)

modelling practical problems by interpreting an everyday additive or multiplicative situation; for example, making a number of purchases at a store and deciding whether to use addition or subtraction, multiplication or division to solve the problem and justifying the choice of operation; for example, “I used subtraction to solve this problem as I knew the total and one of the parts, so I needed to subtract to find the missing part” (AC9M2N06) (15/09/22)

modelling and solving the problem “How many days are there left in this year?” by using a calendar (AC9M2N06) (15/09/22)

modelling problems involving equal grouping and sharing in First Nations Australian children’s instructive games; for example, Yangamini from the Tiwi Island Peoples, representing relationships  with a number sentence and interpret and communicate solutions in terms of the context (AC9M2N06) (15/09/22)

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Year 2 Algebra

recognising additive patterns in the environment on Country/Place and in First Nations Australians’ material culture; representing them using drawings, coloured counters and numbers (AC9M2A01) (24/09/22)

recall and demonstrate proficiency with multiplication facts for twos; extend and apply facts to develop the related division facts using doubling and halving(AC9M2A03) (24/09/22)

using ten-frames or materials such as connecting cubes to develop and record addition and subtraction strategies including doubles, near doubles, counting on, combinations to 10 and bridging to 10, explaining patterns and connections noticed within the facts(AC9M2A03) (24/09/22) 

recognising and relating terms such as double, twice and multiply by 2, halve and divide by 2 using physical and virtual materials; for example, colouring numbers on a hundreds chart to represent doubles and use to recognise halves; recognising the doubling pattern and applying to find related facts such as for 8 twos think 2 eights(AC9M2A03) (24/09/22)

establishing an understanding of doubles and near doubles using physical or virtual manipulatives; for example, using manipulatives to establish that doubling 5 gives you 10 then extending this doubling fact to respond to the question, “How can you use this fact to double 6 or double 4?” (AC9M2A03) (24/09/22)

develop fluency with doubling and halving numbers within 20 using physical or virtual materials and playing doubling and halving games; for example, using a physical or virtual dice and choosing whether to double or halve to reach a target number (AC9M2A03) (24/09/22)

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Year 3 Number

comparing the Hindu-Arabic numeral system to other numeral systems; for example, investigating the Japanese numeral system,  一、十、百、千、万 (AC9M3N01) (16/09/22)

comparing, reading and writing the numbers involved in the more than 60 000 years of First Peoples of Australia’s presence on the Australian continent through time scales relating to pre-colonisation and post-colonisation (AC9M3N01) (16/09/22)

using partitioning and part-part-whole models and the inverse relationship between addition and subtraction to solve addition or subtraction problems, making informal written “jottings” to keep track of the numbers if necessary (AC9M3N03) (16/09/22)

using physical or virtual grouped materials or diagrams to make proportional models of numbers to assist in calculations; for example, to calculate 214 + 325 representing 214 as 2 groups of 100, one group of 10 and 4 ones and 325 as 3 groups of 100, 2 groups of 10 and 5 ones resulting in 5 groups of 100, 3 groups of 10 and 9 ones which is 539 (AC9M3N03) (16/09/22)

choosing between standard and non-standard place value partitions to assist with calculations; for example, to solve 485 + 365, thinking of 365 as 350 + 15, then adding the parts, 485 + 15 = 500, 500 + 350 = 850 (AC9M3N03) (16/09/22)

solving subtraction problems efficiently by adding or subtracting a constant amount to both numbers to create an easier calculation; for example, 534 – 395 adding 5 to both numbers to make 539 – 400 = 139 (AC9M3N03) (16/09/22)

applying knowledge of place value to assist in calculations when solving problems involving larger numbers; for example, calculating the total crowd numbers for an agricultural show that lasts a week (AC9M3N03) (16/09/22)

multiply and divide one- and two-digit numbers, representing problems using number sentences, diagrams and arrays, and using a variety of calculation strategies (AC9M3N04) (16/09/22)

applying knowledge of numbers and the properties of operations using a variety of ways to represent multiplication or division number sentences; for example, using a Think Board to show different ways of visualising 8 x 4, such as an array, a diagram and as a worded problem (AC9M3N04) (16/09/22)

using part-part-whole and comparative models to visually represent multiplicative relationships and choosing whether to use multiplication or division to solve problems (AC9M3N04) (16/09/22)

formulating connected multiplication and division expressions by representing situations from First Nations Australians’ cultural stories and dances about how they care for Country/Place such as turtle egg gathering using number sentences (AC9M3N04) (16/09/22)

estimate the quantity of objects in collections and make estimates when solving problems to determine the reasonableness of calculations (AC9M3N05) (16/09/22)

estimating how much space a grid paper representation of a large number such as 20 200 will take up on the wall and how much paper will be required (AC9M3N05) (16/09/22)

use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts; formulate problems using number sentences and choose calculation strategies, using digital tools where appropriate; interpret and communicate solutions in terms of the situation (AC9M3N06) (16/09/22)

modelling additive problems using a bar model to represent the problem; for example, “I had 75 tomatoes and then picked some more, now I have 138. How many did I pick?” (AC9M3N06) (16/09/22)

modelling practical multiplicative situations using materials or a diagram to represent the problem; for example, if 4 tomato plants each have 6 tomatoes, deciding whether to use an addition or multiplication number sentence, explaining how each number in their number sentence is connected to the situation (AC9M3N06) (16/09/22)

follow and create algorithms involving a sequence of steps and decisions to investigate numbers; describe any emerging patterns (AC9M3N07) (16/09/22)

following or creating an algorithm to generate number patterns formed by doubling and halving using technology to assist where appropriate; identifying and describing emerging patterns (AC9M3N07) (16/09/22)

following or creating an algorithm that determines whether a given number is a multiple of 2, 5 or 10, identifying and discussing emerging patterns (AC9M3N07) (16/09/22)

creating an algorithm as a set of instructions that a classmate can follow to generate multiples of 3 using the rule “To multiply by 3 you double the number and add on one more of the number”; for example, for 3 threes you double 3 and add on 3 to get 9, for 3 fours you double 4 and add one more 4 to get 12 … (AC9M3N07) (16/09/22)

creating a sorting algorithm that will sort a collection of 5 cent and 10 cent coins and providing the total value of the collection by applying knowledge of multiples of 5 and 10 (AC9M3N07) (16/09/22)

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Year 3 Algebra

recognise and explain the connection between addition and subtraction as inverse operations, apply to partition numbers and find unknown values in number sentences (AC9M3A01) (25/09/22)

using the inverse relationship between addition and subtraction to find unknown values with a calculator; for example, representing the problem, “Peter had some money and then spent $375, now he has $158 left. How much did Peter have to start with?” as □ – $375 = $158 and solving the problem using $375 + $158 = $533 ; solving 27 + □ = 63 using subtraction, □ = 63 – 27 or by counting on; 27, 37, 47, 57, 60, 63, so add 3 tens and 6 ones, so □ = 36 (AC9M3A01) (25/09/22)

exploring First Nations Australians’ stories and dances that show the connection between addition and subtraction, representing this as a number sentence and discussing how this conveys important information about balance in processes on Country/Place (AC9M3A01) (25/09/22)

using partitioning to develop and record facts systematically; for example, “How many ways can 12 monkeys be spread among 2 trees?”, 12 = 12 + 0, 12 = 11 + 1, 12 = 10 + 2, 12 = 9 + 3, …; explaining how they know they have found all possible partitions (AC9M3A02) (25/09/22)

understanding basic addition and related subtraction facts and using extensions to these facts; for example, 6 + 6 = 12, 16 + 6 = 22, 6 + 7 = 13, 16 + 7 = 23, and 60 + 60 = 120, 600 + 600 = 1200 (AC9M3A02) (25/09/22)

recall and demonstrate proficiency with multiplication facts for 3, 4, 5 and 10; extend and apply facts to develop the related division facts (AC9M3A03) (25/09/22)

using concrete or virtual materials, groups and repeated addition to recognise patterns and establish the  3, 4, 5 and 10 multiplication facts; using the language of “3 groups of 2 equals 6” to develop into “3 twos are 6” and extend to establish the 3 x 10 multiplication facts and related division facts (AC9M3A03) (25/09/22)

recognising that when they multiply a number by 5, the resulting number will either end in a 5 or a zero; using a calculator or spreadsheet to generate a list of the multiples of 5 to develop the multiplication and related division facts for fives (AC9M3A03) (25/09/22)

practising calculating and deriving multiplication facts for 3, 4, 5 and 10, explaining and recalling the patterns in them and using them to derive related division facts (AC9M3A03) (25/09/22)

systematically exploring algorithms used for repeated addition, comparing and describing what is happening, and using them to establish the multiplication facts for 3, 4, 5 and 10; for example, following the sequence of steps, the decisions being made and the resulting solution, recognising and generalising any emerging patterns (AC9M3A03) (25/09/22)

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Year 4 Number

recognising that one is the same as ten-tenths and one-tenth is the same as 10 hundredths and using this relationship to rename decimals; for example, renaming 0.25 as two-tenths and five-hundredths or twenty-five-hundredths (AC9M4N01) (17/09/22)

making models of measurement attributes to show the relationship between the base unit and parts of the unit; for example, 1.5 metres is one metre and five-tenths of the next metre; 1.75 units is one unit and seventy-five hundredths of the next unit (AC9M4N01) (17/09/22)

following an algorithm consisting of a flow chart with a series of instructions and decisions to determine whether a number is even or odd; using the algorithm to identify which elements of a set of numbers are divisible by 2 (AC9M4N02) (17/09/22)

cutting objects such as oranges or sandwiches into quarters and counting by quarters to find the total number, and saying the counting sequence: one-quarter, two-quarters, three-quarters, four-quarters or one-whole, five-quarters or one-and-one-quarter, six-quarters or one-and-two-quarters… eight-quarters or two-wholes… (AC9M4N04) (17/09/22)

using a number line to represent and count in tenths, recognising that 10 tenths is equivalent to one (AC9M4N04) (17/09/22)

using physical or virtual materials to demonstrate the multiplicative relationship between the places (AC9M4N05) (17/09/22)

using a calculator or other digital tools to recognise and develop an understanding of the effect of multiplying or dividing numbers by 10s, 100s and 1,000s, recording sequences in a place value chart, in a table or spreadsheet, generalising the patterns noticed and applying them to solve multiplicative problems without a calculator (AC9M4N05) (17/09/22)

develop efficient strategies and use appropriate digital tools for solving problems involving addition and subtraction, and multiplication and division where there is no remainder (AC9M4N06) (17/09/22)

using and choosing efficient calculation strategies for addition and subtraction problems involving larger numbers; for example, place value partitioning, inverse relationship, compatible numbers, jump strategies, bridging tens, splitting one or more numbers, extensions to basic facts, algorithms and digital tools where appropriate (AC9M4N06) (17/09/22)

using physical or virtual materials to demonstrate doubling and halving strategies for solving multiplication problems; for example, for 5 x 18, using the fact that double 5 is 10 and half of 18 is 9; or using 10 x 18 = 180 and halve 180 is 90; applying the associative property of multiplication, where 5 x 18 becomes 5 x 2 x 9, then 5 x 2 x 9 = 10 x 9 = 90 so that 5 x 18 = 90 (AC9M4N06) (17/09/22)

using an array to represent a multiplication problem, connecting the idea of how many groups and how many in each group with the rows and columns of the array, and writing an associated number sentence (AC9M4N06) (17/09/22)

using materials or a diagram to solve a multiplication or division problem, by writing a number sentence, and explaining what each of the numbers within the number sentence refers to (AC9M4N06) (17/09/22)

representing a multiplicative situation using materials, array diagrams and/or a bar model, and writing multiplication and/or division number sentences, based on whether the number of groups, the number per group or the total is missing, and explaining how each number in their number sentence is connected to the situation (AC9M4N06) (17/09/22)

using place value partitioning, basic facts and an area or region model to represent and solve multiplication problems, such as 16 × 4, thinking 10 × 4 and 6 × 4, 40 + 24 = 64 or a double, double strategy where double 16 is 32, double this is 64, so 16 x 4 is 64 (AC9M4N06) (17/09/22)

using materials or diagrams to develop and explain division strategies; for example, finding thirds, using the inverse relationship to turn division into a multiplication (AC9M4N06) (17/09/22)

using proficiency with basic facts to estimate the result of a calculation and say what amounts the answer will be between; for example, 5 packets of biscuits at $2.60 each will cost between $10 and $15 as 5 x $2 = $10 and 5 x $3 = $15 (AC9M4N07) (17/09/22)

recognising the effect of rounding in addition and multiplication calculations; rounding both numbers up, both numbers down and one number up and one number down, and explaining which is the best approximation and why (AC9M4N07) (17/09/22)

use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts; formulate the problems using number sentences and choose efficient calculation strategies, using digital tools where appropriate; interpret and communicate solutions in terms of the situation (AC9M4N08) (17/09/22)

modelling and solving multiplication problems involving money, such as buying 5 toy scooters for $96 each, using efficient mental strategies and written jottings to keep track if needed; for example, rounding $96 up to $100 and subtracting 5 x $4 = $20, so 5 x $96 is the same as 5 x $100 less $20, giving the answer $500 – $20 = $480 (AC9M4N08) (17/09/22)

modelling situations by formulating comparison problems using number sentences, comparison models and arrays; for example, “Ariana read 16 books for the readathon; Maryam read 4 times as many books. How many books did Maryam read?” using the expression 4 x 16 and using place value partitioning, basic facts and an array, thinking 4 x 10 = 40 and 4 x 6 = 24, so 4 x 16 can be written as 40 + 24 = 64 (AC9M4N08) (17/09/22)

follow and create algorithms involving a sequence of steps and decisions that use addition or multiplication to generate sets of numbers; identify and describe any emerging patterns (AC9M4N09) (17/09/22)

creating an algorithm that will generate number sequences involving multiples of one to 10 using digital tools to assist, identifying and explaining emerging patterns, recognising that number sequences can be extended indefinitely (AC9M4N09) (17/09/22)

creating a basic flow chart that represents an algorithm that will generate a sequence of numbers using multiplication by a constant term; using a calculator to model and follow the algorithm, and record the sequence of numbers generated; checking results and describing any emerging patterns (AC9M4N09) (17/09/22)

using a multiplication formula in a spreadsheet and the “fill down” function to generate a sequence of numbers; for example, entering the number one in the cell A1, using “fill down” to cell A100, entering the formula “ = A1*4 “ in the cell B1 and using the “fill down” function to generate a sequence of 100 numbers; describing emerging patterns (AC9M4N09) (17/09/22)

creating an algorithm that will generate number sequences involving multiples of one to 10 using digital tools to assist, identifying and explaining emerging patterns, recognising that number sequences can be extended indefinitely (AC9M4N09) (17/09/22)

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Year 4 Algebra

find unknown values in numerical equations involving addition and subtraction, using the properties of numbers and operations (AC9M4A01) (26/09/22)

demonstrating the commutative properties of addition using materials, diagrams and number lines; for example, using number lines to demonstrate that 5 + 2 = 2 + 5, demonstrating that 2 + 2 + 3 = 7 and 2 + 3 + 2 = 7 and 3 + 2 + 2 = 7 (AC9M4A01) (26/09/22)

using balance scales and informal uniform units to create addition or subtraction number sentences showing equivalence, such as 7 + 8 = 6 + 9, and to find unknowns in equivalent number sentences, such as 6 + 8 = □ + 10 (AC9M4A01) (26/09/22)

using relational thinking and knowledge of equivalent number sentences to explain whether equations involving addition or subtraction are true; for example, explaining that 27 – 14 = 17 – 4 is true and using a number line to show the common difference is 13 (AC9M4A01) (26/09/22)

using part-part-whole diagrams or bar models to recognise and explain the inverse relationship between addition and subtraction, using this to make calculations easier; for example, solving 27 + □ = 63 using subtraction, □ = 63 – 27 (AC9M4A01) (26/09/22)

using arrays on grid paper or created with blocks or counters to develop, represent and explain patterns in the 10 x 10 multiplication facts; using the arrays to explain the related division facts (AC9M4A02) (26/09/22)

using materials or diagrams to develop and record multiplication strategies such as doubling, halving, commutativity, and adding one more or subtracting from a group to reach a known fact; for example, creating multiples of 3 on grid paper and doubling to find multiples of 6; recording and explaining the connections to the x3 and x6 multiplication facts: 3, 6, 9, … doubled is 6, 12, 18, … (AC9M4A02) (26/09/22)

using known multiplication facts for 2, 3, 5 and 10 to establish multiplication facts for 4, 6, 7, 8 and 9 in different ways; for example, using multiples of 10 to establish the multiples of 9 as “to multiply a number by 9 you multiply by 10 then take the number away”; 9 x 4 = 10 x 4 – 4, so 9 x 4 is 40 – 4 = 36; using multiple of 3 as “to multiply a number by 9 you multiply by 3, and then multiply the result by 3 again” (AC9M4A02) (26/09/22)

using arrays and known multiplication facts for twos and fives to develop the multiplication facts for sevens, applying the distributive property of multiplication; for example, when finding 6 x 7, knowing that 7 is made up of 2 and 5, and using an array to show that 6 x 7 is the same as 6 x 2 + 6 x 5 = 12 + 30 which is 42 (AC9M4A02) (26/09/22)

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Year 5 Number

using a certain number of blocks to form different rectangles and using these to list all possible factors for that number; for example, 12 blocks can form the following rectangles: 1 x 12, 2 x 6, and 3 x 4 (AC9M5N02) (18/02/22)

researching divisibility tests and explaining each rule using materials; for example, using base-10 blocks to test if numbers are divisible by 2, 5 and 10 (AC9M5N02) (18/02/22)

demonstrating and reasoning that all multiples can be formed by combining or regrouping; for example, multiples of 7 can be formed by combining a multiple of 2 with the corresponding multiple of 5; 3 x 7 = 3 x 2 + 3 x 5, and 4 x 7 = 4 x 2 + 4 x 5 (AC9M5N02) (18/02/22)

using pattern blocks to represent equivalent fractions; selecting one block or a combination of blocks to represent one whole, and making a design with shapes; recording the fractions to justify the total (AC9M5N03) (18/02/22)

creating a fraction wall from paper tape to model and compare a range of different fractions with related denominators; using the model to play fraction wall games (AC9M5N03) (18/02/22)

connecting a fraction wall model and a number line model of fractions to say how they are the same and how they are different; for example, explaining \boldsymbol{\color{OliveGreen} \frac14} on a fraction wall represents the area of one-quarter of the whole while on the number line \boldsymbol{\color{OliveGreen} \frac14} is identified as a point that is one-quarter of the distance between zero and one (AC9M5N03) (18/02/22)

using an understanding of factors and multiples as well as equivalence to recognise efficient methods for the location of fractions with related denominators on parallel number lines; for example, explaining on parallel number lines that \boldsymbol{\color{OliveGreen} \frac2{10}} is located at the same position on a parallel number line as \boldsymbol{\color{OliveGreen} \frac15} because \boldsymbol{\color{OliveGreen} \frac15} is equivalent to \boldsymbol{\color{OliveGreen} \frac2{10}} (AC9M5N03) (18/02/22)

creating a model by subdividing a collection of materials, such as blocks or money, to connect decimals and percentage equivalents of tenths and commonly used fractions \boldsymbol{\color{OliveGreen} \frac12}, \boldsymbol{\color{OliveGreen} \frac13} and \boldsymbol{\color{OliveGreen} \frac34}; for example, one-tenth or 0.1 represents 10% and one half or 0.5 represents 50%; recognising that 60% is 10% more than 50% (AC9M5N04) (18/02/22)

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 (AC9M5N04) (18/02/22)

solve problems involving addition and subtraction of fractions with the same or related denominators, using different strategies (AC9M5N05) (18/02/22)

using different ways to add and subtract fractional amounts by subdividing different models of measurement attributes; for example, adding half an hour and three-quarters of an hour using a clock face, adding a \boldsymbol{\color{OliveGreen} \frac34} cup of flour and a \boldsymbol{\color{OliveGreen} \frac14} cup of flour, subtracting \boldsymbol{\color{OliveGreen} \frac34} of a metre from \boldsymbol{\color{OliveGreen} 2\frac14}  metres (AC9M5N05) (18/02/22)

representing and solving addition and subtraction problems involving fractions by using jumps on a number line, bar models or making diagrams of fractions as parts of shapes (AC9M5N05) (18/02/22)

using materials, diagrams, number lines or arrays to show and explain that fraction number sentences can be rewritten in equivalent forms without changing the quantity; for example, \boldsymbol{\color{OliveGreen} \frac12 + \frac14} is the same as \boldsymbol{\color{OliveGreen} \frac24 + \frac14} (AC9M5N05) (18/02/22)

solve problems involving multiplication of larger numbers by one- or two-digit numbers, choosing efficient calculation strategies and using digital tools where appropriate; check the reasonableness of answers (AC9M5N06) (18/02/22)

solving multiplication problems such as 253 x 4 using a doubling strategy; for example, 2 x 253 = 506 and 2 x 506 = 1012 (AC9M5N06) (18/02/22)

solving multiplication problems like 15 x 16 by thinking of factors of both numbers, 5 = 3 x 5, 16 = 2 x 8; rearranging the factors to make the calculation easier, 5 x 2 = 10, 3 x 8 = 24 and 10 x 24 = 240 (AC9M5N06) (18/02/22)

solve problems involving division, choosing efficient strategies and using digital tools where appropriate; interpret any remainder according to the context and express results as a whole number, decimal or fraction (AC9M5N07) (18/02/22)

solving division problems mentally like 72 divided by 9, 72 ÷ 9, by thinking, “how many 9s make 72”, ? x 9 = 72 or “share 72 equally 9 ways” (AC9M5N07) (18/02/22)

using the fact that equivalent division calculations result if both numbers are divided by the same factor (AC9M5N07) (18/02/22)

check and explain the reasonableness of solutions to problems including financial contexts using estimation strategies appropriate to the context (AC9M5N08) (18/02/22)

recognising the effect of rounding addition, subtraction, multiplication and division calculations, rounding both numbers up, both numbers down, and one number up and one number down; explaining which estimation is the best approximation and why (AC9M5N08) (18/02/22)

use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts; formulate the problems, choosing operations and efficient calculation strategies, using digital tools where appropriate; interpret and communicate solutions in terms of the situation (AC9M5N09) (18/02/22)

modelling an everyday situation and determining which operations can be used to solve it using materials, diagrams, arrays and/or bar models to represent the problem ; formulating the situation as a number sentence and justifying their choice of operations in relation to the situation (AC9M5N09) (18/02/22)

modelling a series of contextual problems, deciding whether an exact answer or an approximate calculation is appropriate; explaining their reasoning in relation to the context and the numbers involved (AC9M5N09) (18/02/22)

modelling financial situations such as creating financial plans; for example, creating a budget for a class fundraising event, using a spreadsheet to tabulate data and perform calculations (AC9M5N09) (18/02/22)

investigating mathematical models involving combinations of operations can be used to represent songs, stories and/or dances of First Nations Australians (AC9M5N09) (18/02/22)

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Year 5 Algebra

using materials or diagrams to develop and explain division strategies, such as halving, using the inverse relationship to turn division into a multiplication (AC9M5A01) (27/09/22)

using arrays, multiplication tables, and physical and virtual materials to develop families of facts; for example, 3 x 4 = 12, 4 x 3 = 12, 12 ÷ 3 = 4 and 12 ÷ 4 = 3 (AC9M5A01) (27/09/22)

demonstrating multiplicative partitioning using materials, diagrams or arrays and recording 2 multiplication and 2 division facts for each grouping; 4 x 6 = 24, 6 x 4 = 24, 24 ÷ 4 = 6 and 24 ÷ 6 = 4; explaining how each is different from and connected to groups in the materials, diagrams or arrays (AC9M5A01) (27/09/22)

using materials, diagrams or arrays to recognise and explain the inverse relationship between multiplication and division; for example, solving 240 ÷ 20 = □ by thinking 20 x □ = 240; using the inverse to make calculations easier; for example, solving 17 x □ = 221 using division, □ = 221 ÷ 17 (AC9M5A01) (27/09/22)

find unknown values in numerical equations involving multiplication and division using the properties of numbers and operations (AC9M5A02) (27/09/22)

using knowledge of equivalent number sentences to form and find unknown values in numerical equations; for example, given that 3 x 5 = 15 and 30 ÷ 2 = 15 then 3 x 5 = 30 ÷ 2 therefore the solution to 3 x 5 = 30 ÷ □ is 2 (AC9M5A02) (27/09/22)

using relational thinking, an understanding of equivalence and number properties to determine and reason about numerical equations; for example, explaining whether an equation involving equivalent multiplication number sentences is true, such as 15 ÷ 3 = 30 ÷ 6 (AC9M5A02) (27/09/22)

using materials, diagrams and arrays to demonstrate that multiplication is associative and commutative but division is not; for example, using arrays to demonstrate that 2 x 3 = 3 x 2 but 6 ÷ 3 does not equal 3 ÷ 6; demonstrating that 2 x 2 x 3 = 12 and 2 x 3 x 2 = 12 and 3 x 2 x 2 = 12; understanding that 8 ÷ 2 ÷ 2 = (8 ÷ 2) ÷ 2 = 2 but 8 ÷ (2 ÷ 2) = 8 ÷ 1 = 8 (AC9M5A02) (27/09/22)

constructing equivalent number sentences involving multiplication to form a numerical equation, and applying knowledge of factors, multiples and the associative property to find unknown values in numerical equations; for example, considering 3 x 4 = 12 and knowing 2 x 2 = 4 then 3 x 4 can be written as 3 x (2 x 2) and using the associative property (3 x 2) x 2 so 3 x 4 = 6 x 2 and so 6 is the solution to 3 x 4 = □ x 2 (AC9M5A02) (27/09/22)

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Year 6 Number

recognise situations, including financial contexts, that use integers; locate and represent integers on a number line and as coordinates on the Cartesian plane (AC9M6N01) (Link) (09/09/22)

using the definition of a prime number to explain why one is not a prime number (AC9M6N02) (Link)(Link)

representing composite numbers as a product of their factors, including prime factors when necessary and using this form to simplify calculations involving multiplication such as 15 x 16 as 5 x 3 x 4 x 4 which can be rearranged to simplify calculation to 5 x 4 x 3 x 4 = 20 x 12 (AC9M6N02) (Link)

using spreadsheets to list all the numbers that have up to 3 factors, using combinations of only the first 3 prime numbers, recognise any emerging patterns, making conjectures and experimenting with other combinations (AC9M6N02) (Link)

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 (AC9M6N03) (09/09/22)

representing fractions on the same number line, paying attention to relative position, and using this to explain relationships between denominators (AC9M6N03) (09/09/22)

explaining equivalence and order between fractions using number lines, drawings and models (AC9M6N03) (09/09/22)

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, \boldsymbol{\color{OliveGreen} \frac58} is greater than \boldsymbol{\color{OliveGreen} \frac12} can be written as \boldsymbol{\color{OliveGreen} \frac58 > \frac12}, because half of 8 is 4; \boldsymbol{\color{OliveGreen} \frac16} is less than \boldsymbol{\color{OliveGreen} \frac14}, because 6>4 and can be written as \boldsymbol{\color{OliveGreen} \frac16 < \frac14} (AC9M6N03) (09/09/22)

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 (AC9M6N04) (09/09/22)

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 hundredth (AC9M6N04) (09/09/22)

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 (AC9M6N04) (09/09/22)

solve problems involving addition and subtraction of fractions using knowledge of equivalent fractions (AC9M6N05) (09/09/22)

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 (AC9M6N05) (09/09/22)

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 (AC9M6N05) (09/09/22)

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 (AC9M6N06) (09/09/22)

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 (AC9M6N06) (09/09/22)

explaining the effect of multiplying or dividing a decimal by 10, 100, 1000 … in terms of place value and not the decimal point shifting (AC9M6N06) (09/09/22)

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 (AC9M6N07) (09/09/22)

representing a situation with a mathematical expression; for example, numbers and symbols such as \boldsymbol{\color{OliveGreen} \frac14 \times 24}, that involve finding a familiar fraction or percentage of a quantity; using mental strategies or a calculator and explaining the result in terms of the situation in question (AC9M6N07) (09/09/22)

approximate numerical solutions to problems involving rational numbers and percentages, including financial contexts, using appropriate estimation strategies (AC9M6N08) (09/09/22)

using familiar fractions, decimals and percentages to approximate calculations, such as, 0.3 of 180 is about a \boldsymbol{\color{OliveGreen} \frac13} of 180 or 52% is about a \boldsymbol{\color{OliveGreen} \frac12} (AC9M6N08) (09/09/22)

recognising the effect of rounding on calculations involving fractions or decimals and saying what numbers the answer will be between (AC9M6N08) (09/09/22)

investigating estimation strategies to make decisions about steam cooking in ground ovens by First Nations Australians, including catering for different numbers of people and resources needed for cooking (AC9M6N08) (09/09/22)

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 terms of the situation, justifying the choices made (AC9M6N09) (09/09/22)

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 (AC9M6N09) (09/09/22)

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 (AC9M6N09) (09/09/22)

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Year 6 Algebra

recognise and use rules that generate visually growing patterns and number patterns involving rational numbers (AC9M6A01) (28/09/22)

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 x 11, 211 x 11, 212 x 11…, 111 x 11, 222 x 11, 333 x 11…, or 100 ÷ 99, 101 ÷ 99, 102 ÷ 99… (AC9M6A01) (28/09/22)

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 (AC9M6A01) (28/09/22)

find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations (AC9M6A02) (28/09/22)

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 x (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 (AC9M6A02) (28/09/22)

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 x 8 = 6 x Δ + □ (AC9M6A02) (28/09/22)

applying knowledge of inverse operations and number properties to create equivalent number sentences; removing one of the numbers and replacing it with a symbol, then swapping with a classmate to find the unknown values (AC9M6A02) (28/09/22)

create and use algorithms involving a sequence of steps and decisions that use rules to generate sets of numbers; identify, interpret and explain emerging patterns (AC9M6A03) (28/09/22)

using an algorithm to create extended number sequences involving rational numbers, using a rule and digital tools, explaining any emerging patterns (AC9M6A03) (28/09/22)

designing an algorithm to model operations, using the concept of input and output, describing and explaining relationships and any emerging patterns; for example, using function machines to model operations and recognising and comparing additive and multiplicative relationships (AC9M6A03) (28/09/22)

designing an algorithm or writing a simple program to generate a sequence of numbers based on the user’s input and a chosen operation, discussing any emerging patterns; for example, generating a sequence of numbers and comparing how quickly the sequences are growing in comparison to each other using the rule adding 2 to the input number compared to multiplying the input number by 2 (AC9M6A03) (28/09/22)

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Year 7 Number

describe the relationship between perfect square numbers and square roots, and use squares of numbers and square roots of perfect square numbers to solve problems (AC9M7N01) (12/09/22)

using the relationship between perfect square numbers and their square roots to determine the perimeter of a square tiled floor using square tiles; for example, an area of floor with 144 square tiles has a perimeter of 48 tile lengths (AC9M7N01) (12/09/22)

solving problems involving lowest common multiples and greatest common divisors (highest common factors) for pairs of natural numbers by comparing their prime factorisation (AC9M7N02) (12/09/22)

investigating equivalence of fractions using common multiples and a fraction wall, diagrams or a number line to show that a fraction such as \boldsymbol{\color{OliveGreen}\frac23} is equivalent to \boldsymbol{\color{OliveGreen}\frac46} and \boldsymbol{\color{OliveGreen}\frac69} and therefore \boldsymbol{\color{OliveGreen}\frac23< \frac56} (AC9M7N04) (12/09/22)

applying and explaining the equivalence between fraction, decimal and percentage representations of rational numbers; for example, 16%, 0.16, \boldsymbol{\color{OliveGreen}\frac{16}{100}} and \boldsymbol{\color{OliveGreen}\frac{4}{25}}, using manipulatives, number lines or diagrams (AC9M7N04) (12/09/22)

use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation strategies (AC9M7N06) (12/09/22)

solving addition and subtraction problems involving fractions and decimals; for example, using rectangular arrays with dimensions equal to the denominators, algebra tiles, digital tools or informal jottings (AC9M7N06) (12/09/22)

developing efficient strategies with appropriate use of the commutative and associative properties, place value, patterning, and multiplication facts to solve multiplication and division problems involving fractions and decimals; for example, using the commutative property to calculate \boldsymbol{\color{OliveGreen}\frac23} of \boldsymbol{\color{OliveGreen}\frac12} giving \boldsymbol{\color{OliveGreen}\frac12} of \boldsymbol{\color{OliveGreen}\frac23=\frac13} (AC9M7N06) (12/09/22)

solving multiplicative problems involving fractions and decimals using fraction walls, rectangular arrays, algebra tiles, calculators or informal jottings (AC9M7N06) (12/09/22)

developing efficient strategies with appropriate use of the commutative and associative properties, regrouping or partitioning to solve additive problems involving fractions and decimals (AC9M7N06) (12/09/22)

carry out calculations to solve problems using the representation that makes computations efficient such as 12.5% of 96 is more efficiently calculated as \boldsymbol{\color{OliveGreen}\frac18} of 96, including contexts such as comparing land-use by calculating the total local municipal area set aside for parkland or manufacturing and retail, the amount of protein in daily food intake across several days, or increases/decreases in energy accounts each account cycle (AC9M7N06) (12/09/22)

compare, order and solve problems involving addition and subtraction of integers (AC9M7N07) (Link)

ordering, adding and subtracting integers using a number line (AC9M7N07) (12/09/22)

using diagrams, physical or virtual materials to represent ratios, recognising that ratios express the quantitative relationship between 2 or more groups; for example, using counters or coloured beads to show the ratios 1:4 and 1:1:2 (AC9M7N08) (12/09/22)

use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts; formulate problems, choosing representations and efficient calculation strategies, using digital tools as appropriate; interpret and communicate solutions in terms of the situation, justifying choices made about the representation (AC9M7N09) (Link)

modelling additive situations involving positive and negative quantities; for example, a lift travelling up and down floors in a high-rise apartment where the ground floor is interpreted as zero; in geography when determining altitude above and below sea level (AC9M7N09) (12/09/22)

modelling contexts involving proportion, such as the proportion of students attending the school disco, proportion of bottle cost to recycling refund, proportion of school site that is green space, 55% of Year 7 students attended the end of term function or 23% of the school population voted yes to a change of school uniform; interpreting and communicating answers in terms of the context of the situation (AC9M7N09) (12/09/22)

using mathematical modelling to investigate the proportion of land mass/area of Australian First Nations Peoples’ traditional grain belt compared with Australia’s current grain belt (AC9M7N09) (12/09/22)

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Year 7 Algebra

recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown (AC9M7A01) (28/09/22)

linking variables to attributes and measures being modelled when using formulas, such as the area of a rectangle is equal to the length x width as A = l x w; using p = 6g + b to describe a total of points expressed as goals (worth 6 points) and behinds (worth one point) (AC9M7A01) (28/09/22)

using everyday formulas and their application to contexts on Country/Place, investigating the relationships between variables (AC9M7A01) (28/09/22)

formulate algebraic expressions using constants, variables, operations and brackets (AC9M7A02) (28/09/22)

recognising and applying the concept of variable as something that can change in value, investigating the relationships between variables, and the application to processes on Country/Place, including how cultural expressions of First Nations Australians, such as storytelling, communicate mathematical relationships that can be represented as mathematical expressions (AC9M7A02) (28/09/22)

solve one-variable linear equations with natural number solutions; verify the solution by substitution (AC9M7A03) (29/09/22)

solving equations using concrete materials, the balance model, and backtracking, explaining the process (AC9M7A03) (29/09/22)

describe relationships between variables represented in graphs of functions from authentic data (AC9M7A04) (29/09/22)

using graphs to analyse a building’s electricity or gas usage over a period of time, the value of shares on a stock market, or the temperature during a day, interpreting and discussing the relationships they represent (AC9M7A04) (29/09/22)

using travel graphs to compare the distance travelled to and from school, interpreting and discussing features of travel graphs such as the slope of lines and the meaning of horizontal line segments (AC9M7A04) (29/09/22)

using graphs of evaporation rates to explore and discuss First Nations Australians’ methods of water resource management (AC9M7A04) (29/09/22)

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Year 8 Number

recognise irrational numbers in applied contexts, including square roots and π (AC9M8N01) (Link)

using digital tools to systematically explore contexts or situations that use irrational numbers, such as finding the length of the hypotenuse in a right-angled triangle with the other 2 sides having lengths of one metre or 2 metres and one metre; or given the area of a square, finding the length of the side where the result is irrational; or finding ratios involved with the side lengths of paper sizes A0, A1, A2, A3 and A4 (AC9M8N01) (Link)(Link)

investigating the golden ratio in art and design, and historical approximations to π in different societies (AC9M8N01) (Link)

connecting the ratio between the circumference and diameter of any circle to the irrational value of π using circular objects and string or dynamic drawing software (AC9M8N01) (Link)

establish and apply the exponent laws with positive integer exponents and the zero-exponent, using exponent notation with numbers (AC9M8N02) (Link)

using digital tools to systematically explore the application of the exponent laws; observing that the bases need to be the same (AC9M8N02)

using examples such as \boldsymbol{\color{OliveGreen}\frac{3^{4}}{3^4}=1} and \boldsymbol{\color{OliveGreen} 3^{4-4} = 3^0} to illustrate the necessity that for any non-zero natural number \boldsymbol{\color{OliveGreen} n}, \boldsymbol{\color{OliveGreen} n^0 = 1} (AC9M8N02) (Link)

recognise terminating and recurring decimals, using digital tools as appropriate (AC9M8N03) (Link)

identifying terminating, recurring and non-terminating decimals and choosing their appropriate representations such as \boldsymbol{\color{OliveGreen} \frac13} is represented as \boldsymbol{\color{OliveGreen} 0.\overline{3}} (AC9M8N03) (13/09/22)

use the 4 operations with integers and with rational numbers, choosing and using efficient strategies and digital tools where appropriate (AC9M8N04) (13/09/22)

using patterns to assist in establishing the rules for the multiplication and division of integers (AC9M8N04) (13/09/22)

applying and explaining efficient strategies such as using the commutative or associative property for regrouping, partitioning, place value, patterning, multiplication or division facts to solve problems involving positive and negative integers, fractions and decimals (AC9M8N04) (13/09/22)

use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts; formulate problems, choosing efficient calculation strategies and using digital tools where appropriate; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model (AC9M8N05) (13/09/22)

modelling situations involving weather and environmental contexts including temperature or sea depths by applying operations to positive and negative rational numbers; for example, involving average temperature increases and decreases (AC9M8N05) (13/09/22)

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Year 8 Algebra

create, expand, factorise, rearrange and simplify linear expressions, applying the associative, commutative, identity, distributive and inverse properties (AC9M8A01) (30/09/22)

rearranging and simplifying linear expressions involving variables with integer coefficients and constants; using manipulatives such as algebra tiles to support calculations; for example, using manipulatives to demonstrate that 2x + 4 = 2(x + 2), 3(a – b) = 3a – 3b, or 5(m + 2n) + 3m – 4n = 5m + 10n + 3m – 4n = 8m + 6n (AC9M8A01) (30/09/22)

demonstrating the relationship between factorising and expanding linear expressions using manipulatives, such as algebra tiles or area models, and describing with mathematical language (AC9M8A01) (30/09/22)

using the distributive, associative, commutative, identity and inverse properties to expand and factorise algebraic expressions using strategies such as the area model (AC9M8A01) (30/09/22)

graph linear relations on the Cartesian plane using digital tools where appropriate; solve linear equations and one-variable inequalities using graphical and algebraic techniques; verify solutions by substitution (AC9M8A02) (30/09/22)

completing a table of values, plotting the resulting points on the Cartesian plane and determining whether the relationship is linear (AC9M8A02) (30/09/22)

graphing the linear relationship ax + b = c for given values of a, b and c and identifying from the graph where ax + b < c or where ax + b > c (AC9M8A02) (30/09/22)

solving linear equations of the form ax + b = c and one-variable inequalities of the form ax + b < c or ax + b > c  where a > 0 using inverse operations and digital tools, and checking solutions by substitution (AC9M8A02) (30/09/22)

solving linear equations such as 3x + 7 = 6x – 9, representing these graphically, and verifying solutions by substitution (AC9M8A02) (30/09/22)

use mathematical modelling to solve applied problems involving linear relations , including financial contexts; formulate problems with linear functions, choosing a representation; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model (AC9M8A03) (30/09/22)

modelling patterns on Country/Place and exploring their connections and meaning to linear equations, using the model as a predictive tool and critiquing results by connecting back to Country/Place (AC9M8A03) (30/09/22)

experiment with linear functions and relations using digital tools, making and testing conjectures and generalising emerging patterns (AC9M8A04) (30/09/22)

using graphing software to investigate the effect of systematically varying parameters of linear functions on the corresponding graphs, making and testing conjectures; for example, making a conjecture that if the co-efficient of x is negative, then the line will slope down from left to right (AC9M8A04) (30/09/22)

using graphing software to systematically contrast the graphs of y = 2x, -y = 2x, y = -2x and -y = -2x with those of y < 2x, -y < 2x, y < -2x and -y < -2x and those of  y > 2x, -y > 2x, y > -2x and -y > -2x, making and testing conjectures about sign and direction of the inequality (AC9M8A04) (30/09/22)

using digital tools to investigate integer solutions to equations such as 2x + 3y = 48 (AC9M8A04) (30/09/22)

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Year 9 Number

recognise that the real number system includes the rational numbers and the irrational numbers, and solve problems involving real numbers using digital tools (AC9M9N01) (19/09/22)

investigating the real number system by representing the relationships between irrationals, rationals, integers and natural numbers and discussing the difference between exact representations and approximate decimal representations of irrational numbers (AC9M9N01) (19/09/22)

using a real number line to indicate the solution interval for inequalities of the form ax + b < 7 ;
for example, 2x + 7 < 0, or of the form ax = b > c ; for example, 1.2x – 5.4 > 10.8 (AC9M9N01) (19/09/22)

using positive and negative rational numbers to solve problems; for example, for financial planning such as budgeting (AC9M9N01) (19/09/22)

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Year 9 Algebra

apply the exponent laws to numerical expressions with integer exponents and extend to variables (AC9M9A01) (01/10/22)

simplifying and evaluating numerical expressions, involving both positive and negative integer exponents, explaining why; for example, \boldsymbol{\color{OliveGreen}5^{-3}=\frac1{5^3}=\left(\frac15\right)^3=\frac1{125}} and connecting terms of the sequence \boldsymbol{\color{OliveGreen}125, 25, 5, 1, \frac15, \frac1{25}, \frac1{125}\dots} to terms of the sequence  \boldsymbol{\color{OliveGreen}5^3, 5^2, 5^1, 5^0, 5^{-1}, 5^{-2}, 5^{-3} \dots} (AC9M9A01) (01/10/22)

recognising exponents in algebraic expressions and applying the relevant exponent laws and conventions; for example, for any non-zero natural number a, \boldsymbol{\color{OliveGreen}a^0 = 1}, \boldsymbol{\color{OliveGreen}x^1 = x}, \boldsymbol{\color{OliveGreen}r^2 = r \times r}, \boldsymbol{\color{OliveGreen}h^3 = h \times h \times h}, \boldsymbol{\color{OliveGreen}y^4 = y \times y \times y \times y}, and \boldsymbol{\color{OliveGreen}\frac1{w} \times \frac1{w}= \frac1{w^2} = w^{-2}} (AC9M9A01) (01/10/22)

simplify algebraic expressions, expand binomial products and factorise monic quadratic expressions (AC9M9A02) (01/10/22)

expanding combinations of binomials such as \boldsymbol{\color{OliveGreen}(x+7)(x+8)}, \boldsymbol{\color{OliveGreen}(x+7)(x-8)}, \boldsymbol{\color{OliveGreen}(x-7)(x+8)}, \boldsymbol{\color{OliveGreen}(x-7)(x-8)}, to identify expansion and factorisation patterns related to \boldsymbol{\color{OliveGreen}(x+a)(x+b)=x^2 + (a+b)x+ab}, where a and b are integers (AC9M9A02) (01/10/22)

using manipulatives such as algebra tiles or area models to expand or factorise algebraic expressions with readily identifiable binomial factors; for example, \boldsymbol{\color{OliveGreen}(x+1)(x+3)=x^2 +4x+3}, \boldsymbol{\color{OliveGreen}(x-5)^2=x^2 -10x+25} or \boldsymbol{\color{OliveGreen}(x-3)^2+4=x^2 -6x+9 + 4=x^2 -6x+13} (AC9M9A02) (01/10/22)

recognising the relationship between expansion and factorisation, and using digital tools to systematically explore the factorisation of \boldsymbol{\color{OliveGreen}x^2 +mx+n} where m and n are integers (AC9M9A02) (01/10/22)

find the gradient of a line segment, the midpoint of the line interval and the distance between 2 distinct points on the Cartesian plane (AC9M9A03) (01/10/22)

using digital tools and transformations to illustrate that parallel lines in the Cartesian plane have the same gradient and that the relationship between the gradients of pairs of perpendicular lines is that their product is (-1) (AC9M9A03) (01/10/22)

investigating graphical and algebraic techniques for finding the midpoint and gradient of the line segment between 2 points (AC9M9A03) (01/10/22)

using dynamic graphing software and superimposed images; for example, playground equipment, ramps and escalators, to investigate gradients in context and their relationship to rule of a linear function, and interpret gradient as a constant rate of change in linear modelling contexts (AC9M9A03) (01/10/22)

identify and graph quadratic functions, solve quadratic equations graphically and numerically, and solve monic quadratic equations with integer roots algebraically, using graphing software and digital tools as appropriate (AC9M9A04) (01/10/22)

recognising that in a table of values, if the second difference between consecutive values of the dependent variable is constant, then it is a quadratic (AC9M9A04) (01/10/22)

graphing quadratic functions using digital tools and comparing what is the same and what is different between these different functions and their respective graphs; interpreting features of the graphs such as symmetry, turning point, maximum and minimum values, and determining when values of the quadratic function lie within a given range (AC9M9A04) (01/10/22)

solving quadratic equations algebraically and comparing these to graphical solutions (AC9M9A04) (01/10/22)

recognising that the equation \boldsymbol{\color{OliveGreen}x^2 = a}, where \boldsymbol{\color{OliveGreen}a>0}, has 2 solutions, \boldsymbol{\color{OliveGreen}x=\sqrt{a}} and \boldsymbol{\color{OliveGreen}x=-\sqrt{a}}; for example, if  \boldsymbol{\color{OliveGreen}x^2 = 39} then  \boldsymbol{\color{OliveGreen}x = \sqrt{39} = 6.245} correct to 3 decimal places, or  \boldsymbol{\color{OliveGreen}x = -\sqrt{39} = -6.245} correct to 3 decimal places, and representing these graphically (AC9M9A04) (01/10/22)

graphing percentages of illumination of moon phases in relation to First Nations Australians’ understandings that describe the different phases of the moon (AC9M9A04) (01/10/22)

use mathematical modelling to solve applied problems involving change including financial contexts; formulate problems, choosing to use either linear or quadratic functions; interpret solutions in terms of the situation; evaluate the model and report methods and findings (AC9M9A05) (01/10/22)

modelling measurement situations and determining the perimeter and areas of rectangles where the length, l, of the rectangle is a linear function of its width, w, for example, \boldsymbol{\color{OliveGreen}l=w, l=w+5, l=3w,l=2w+7} (AC9M9A05) (01/10/22)

modelling practical contexts using simple quadratic functions, tables and graphs (hand drawn or using digital tools) and algebraically, interpreting features of the graphs such as the turning point and intercepts in context; for example, area, paths of projectiles, parabolic mirrors, satellite dishes (AC9M9A05) (01/10/22)

modelling the hunting techniques of First Nations Australians using quadratic functions and exploring the effect of increasing the number of hunters to catch more prey (AC9M9A05) (01/10/22)

experiment with the effects of the variation of parameters on graphs of related functions, using digital tools, making connections between graphical and algebraic representations, and generalising emerging patterns (AC9M9A06) (01/10/22)

investigating transformations of the graph of \boldsymbol{\color{OliveGreen}y=x} to the graph of  \boldsymbol{\color{OliveGreen}y=ax+b} by systematic variation of a and b and interpreting the effects of these transformations using digital tools; for example, \boldsymbol{\color{OliveGreen}y=x \to y=2x} (vertical enlargement as a>1) \boldsymbol{\color{OliveGreen} \to y=2x-1} (vertical translation) and \boldsymbol{\color{OliveGreen}y=x \to y=\frac12 x} (vertical compression as a<1) \boldsymbol{\color{OliveGreen} \to y= -\frac12 x} (reflection in the horizontal axis) \boldsymbol{\color{OliveGreen} \to y= -\frac12 x + 3} (vertical translation) (AC9M9A06) (01/10/22)

investigating transformations of the parabola \boldsymbol{\color{OliveGreen}y=x^2} in the Cartesian plane using digital tools to determine the relationship between graphical and algebraic representations of quadratic functions, including the completed square form; for example, \boldsymbol{\color{OliveGreen}y=x^2 \to y=\frac13 x^2} (vertical compression as a<1) \boldsymbol{\color{OliveGreen}\to y=\frac13 (x-5)^2}  (horizontal translation)   \boldsymbol{\color{OliveGreen}\to y=\frac13 (x-5)^2 + 7} (vertical translation) or \boldsymbol{\color{OliveGreen}y=x^2 \to y=2 x^2} (vertical enlargement as a>1) \boldsymbol{\color{OliveGreen} \to y=-2 x^2} (reflection in the horizontal axis) \boldsymbol{\color{OliveGreen}\to y=-2 (x+6)^2} (horizontal translation) \boldsymbol{\color{OliveGreen}\to y=-2 (x+6)^2+10} (vertical translation) (AC9M9A06) (01/10/22)

experimenting with digital tools by applying transformations to the graphs of functions, such as reciprocal \boldsymbol{\color{OliveGreen} y=\frac1{x}}, square root \boldsymbol{\color{OliveGreen} y=\sqrt{x}}, cube  \boldsymbol{\color{OliveGreen} y=x^3} and exponential functions, \boldsymbol{\color{OliveGreen} y=2^x, y=\left(\frac12\right)^x}, identifying patterns (AC9M9A06) (01/10/22)

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Year 9 Space

investigating patterns to reason about nested similar triangles that are aligned on a coordinate plane, connecting ideas of parallel sides and identifying the constancy of ratios of corresponding sides for a given angle (AC9M9SP01) (10/09/22)

apply the enlargement transformation to shapes and objects using dynamic geometry software as appropriate; identify and explain aspects that remain the same and those that change (AC9M9SP02)

using the properties of similarity to solve problems involving enlargement (AC9M9SP02) (10/09/22)

investigating and generalising patterns in length, angle, area and volume when side lengths of shapes and objects are enlarged or dilated by whole and rational numbers; for example, comparing an enlargement of a square and a cube of side length 2 units by a factor of 3 increases the area of the square, \boldsymbol{\color{OliveGreen}2^2} to \boldsymbol{\color{OliveGreen}(3\times 2)^2=9\times 2^2=9}
times the original area and the volume of the cube, \boldsymbol{\color{OliveGreen}2^3} to \boldsymbol{\color{OliveGreen}(3\times 2)^3=27\times 2^3=27}
times the volume 
(AC9M9SP02) (10/09/22)

design, test and refine algorithms involving a sequence of steps and decisions based on geometric constructions and theorems; discuss and evaluate refinements (AC9M9SP03) (Link)

creating an algorithm using pseudocode or flow charts to apply the triangle inequality, or an algorithm to generate Pythagorean triples (AC9M9SP03) (10/09/22)

creating and testing algorithms designed to construct or bisect angles, using pseudocode or flow charts (AC9M9SP03) (10/09/22)

developing an algorithm for an animation of a geometric construction, or a visual proof, evaluating the algorithm using test cases (AC9M9SP03) (10/09/22)

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Year 10 Number

recognise the effect of using approximations of real numbers in repeated calculations and compare the results when using exact representations (AC9M10N01) (20/09/22)

comparing and contrasting the effect of truncation or rounding on the final result of calculations when using approximations of real numbers rather than exact representations (AC9M10N01) (20/09/22)

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Year 10 Algebra

expand, factorise and simplify expressions and solve equations algebraically, applying exponent laws involving products, quotients and powers of variables, and the distributive property (AC9M10A01) (02/10/22)

applying knowledge of exponent laws to algebraic terms and using both positive and negative integral exponents to simplifying algebraic expressions and solve equations algebraically (AC9M10A01) (02/10/22)

solve linear inequalities and simultaneous linear equations in 2 variables; interpret solutions graphically and communicate solutions in terms of the situation (AC9M10A02) (02/10/22)

describing the solution of simultaneous equations within the context of the situation (AC9M10A02) (02/10/22)

testing when a circle of a specified radius has a corresponding area greater than a given value, or whether a point satisfies an inequality; for example, whether the point (3, 5) satisfies \boldsymbol{\color{OliveGreen}2y<x^2} (AC9M10A02) (02/10/22)

investigating the strategies inherent in First Nations Australian children’s instructive games; for example, Weme from the Warlpiri Peoples of central Australia, and their connection to strategies to solve simultaneous linear equations in 2 variables (AC9M10A02) (02/10/22)

recognise the connection between algebraic and graphical representations of exponential relations and solve related exponential equations, using digital tools where appropriate (AC9M10A03) (02/10/22)

recognising that in a table of values, if the ratio between consecutive values of the dependent variable is constant, then it is an exponential relation (AC9M10A03) (02/10/22)

investigating the links between algebraic and graphical representations of exponential functions using graphing software (AC9M10A03) (02/10/22)

using digital tools with symbolic manipulation functionality to systematically explore exponential relations (AC9M10A03) (02/10/22)

investigating First Nations Australian Ranger groups’ and other groups’ programs that attempt to eradicate feral animals for survival of native animals on Country/Place, exploring the competition between feral and native animals and their impact on natural resources by formulating exponential equations for population growth for each animal species (AC9M10A03) (02/10/22)

use mathematical modelling to solve applied problems involving growth and decay, including financial contexts; formulate problems, choosing to apply linear, quadratic or exponential models; interpret solutions in terms of the situation; evaluate and modify models as necessary and report assumptions, methods and findings (AC9M10A04) (02/10/22)

modelling situations that involve working with authentic information, data and interest rates to calculate compound interest and solve related problems (AC9M10A04) (02/10/22)

modelling and formulating situations involving population growths of native animals on Country/Place with varying reproductive behaviour, using exponential equations and critiquing their applicability to real-world situations (AC9M10A04) (02/10/22)

experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns (AC9M10A05) (02/10/22)

applying the graphing zoom functionality of digital tools and systematically refining intervals to identify approximate location of points of intersection of the graphs of 2 functions, such as \boldsymbol{\color{OliveGreen}x^2 = 2^x} (AC9M10A05) (02/10/22)

applying transformations to the graph of \boldsymbol{\color{OliveGreen}x^2 + y^2 =1} (AC9M10A05) (02/10/22)

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Year 10 Optional Number

operations on numbers involving fractional exponents and surds (06/10/22)

explaining that \boldsymbol{\color{OliveGreen}\sqrt{a}= a^{\frac12}=a^{0.5}} for \boldsymbol{\color{OliveGreen}a\geq 0}, generalizing to \boldsymbol{\color{OliveGreen}\sqrt[n]{a}= a^{\frac1{n}}}, and evaluating corresponding expressions; for example, \boldsymbol{\color{OliveGreen}\sqrt{10}= 10^{0.5}\approx 3.162} , \boldsymbol{\color{OliveGreen}2^5=32} so \boldsymbol{\color{OliveGreen}32^{\frac15}=2} (Link)

explaining that \boldsymbol{ \color{OliveGreen} a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^m  =\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}  } and evaluating corresponding expressions; for example, \boldsymbol{ \color{OliveGreen} 8^{\frac{2}{3}}=\left(\sqrt[3]8\right)^2  = 2^2 = 4} and \boldsymbol{ \color{OliveGreen} 8^{\frac{2}{3}}=\sqrt[3]{8^2}  = \sqrt[3]{64} = 4}

showing that \boldsymbol{\color{OliveGreen}\sqrt{a+b}\neq\sqrt{a} + \sqrt{b}} and \boldsymbol{\color{OliveGreen}\sqrt{a-b}\neq\sqrt{a} - \sqrt{b}} for \boldsymbol{\color{OliveGreen}a,b > 0}, for example, \boldsymbol{\color{OliveGreen}\sqrt{16 + 9}=5} but \boldsymbol{\color{OliveGreen}\sqrt{16}+\sqrt{9}=4+3=7}, and \boldsymbol{\color{OliveGreen}\sqrt{16 - 9}=7\approx 2.646}, but \boldsymbol{\color{OliveGreen}\sqrt{16}-\sqrt{9}=4-3=1} (Link)

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Year 10 Optional Algebra

simplification of combinations of linear expressions with rational coefficients and the solution of related equations (03/10/22)

algebraic representations of quadratic functions of the form \boldsymbol{\color{blue} f(x) =ax^2 + bx +c} (03/10/22)

wherea, b, and c are non-zero integers, and their transformation to the form \boldsymbol{\color{blue} f(x) =a(x+h)^2 + k}, where and are non-zero rational numbers, and the solution of related equations (03/10/22)

exploring the use of the unit circle and animations to show the periodic, symmetric, and complementary nature of the sine and cosine functions (03/10/22)

establishing relationships between Pythagoras’ theorem, the unit circle, trigonometric ratios, and angles in half-square triangles and equilateral triangles (03/10/22)

approximating values of the sine and cosine functions from a suitably scaled diagram of the unit circle, and solving equations of the form \boldsymbol{\color{OliveGreen}sin(x)=\frac1{\sqrt2} } and \boldsymbol{\color{OliveGreen}cos(x)=-0.73} over a specified interval graphically (03/10/22)

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Original Digital Sin

About a decade ago, the New York Times ran an opinion piece in which the authors argued for a renewed emphasis on the traditional algorithms for arithmetic. In particular, the authors claimed and lamented an increasing use of calculators as a supposed alternative to proper instruction in the algorithms:

The idea is that competence with algorithms can be substituted for by the use of calculators, and reformists often call for training students in the use of calculators as early as first or second grade.

Keith Devlin wrote a snarky response, including an accusation of straw manning: Continue reading “Original Digital Sin”

We Are the Robots

First, enjoy some great Kraftwerk, because, and just because:

Regular readers may recall Australian reporter Natasha Bita. Natasha did some really excellent stenographic work for ACARA. Natasha also played right along with AMSI’s most recent Chicken Little crusade. It turns out that Natasha is an excellent stenographer even when there’s nothing to stenograph. Continue reading “We Are the Robots”