ACARA Crash 11: Pulped Fractions

We’re still crazy-nuts with work, so, for today, it’s just another fish. This one is from Year 7 Number. and appears to be the sum of fraction arithmetic in Year 7.

LEVEL DESCRIPTION 

As students engage in learning mathematics in Year 7 they … develop their understanding of integer and rational number systems and their fluency with mental calculation, written algorithms, and digital tools and routinely consider the reasonableness of results in context

ACHIEVEMENT STANDARD 

By the end of Year 7, students use all four operations in calculations involving positive fractions and decimals, using the properties of number systems and choosing the computational approach. … They determine equivalent representations of rational numbers and choose from fraction, decimal and percentage forms to assist in computations. They solve problems involving rational numbers, percentages and ratios and explain their choice of representation of rational numbers and results when they model situations, including those in financial contexts.

CONTENT 

determine equivalent fraction, decimal and percentage representations of rational numbers. Locate and represent positive and negative fractions, decimals and mixed numbers on a number line

ELABORATIONS

investigating equivalence of fractions using common multiples and a fraction wall, diagrams or a number line to show that a fraction such as \color{blue}\boldsymbol{\frac23} is equivalent to \color{blue}\boldsymbol{\frac46} and \color{blue}\boldsymbol{\frac69} and therefore \color{blue}\boldsymbol{\frac23 < \frac56}

expressing a fraction in simplest form using common divisors

applying and explaining the equivalence between fraction, decimal and percentage representations of rational numbers, for example, \color{blue}\boldsymbol{16\%, 0.16, \frac{16}{100}} and \color{blue}\boldsymbol{\frac4{25}}, using manipulatives, number lines or diagrams

representing positive and negative fractions and mixed numbers on various intervals of the real number line, for example, from -1 to 1, -10 to 10 and number lines that are not symmetrical about zero or without graduations marked

investigating equivalence in fractions, decimals and percentage forms in the patterns used in the weaving designs of Aboriginal and Torres Strait Islander Peoples

CONTENT

carry out the four operations with fractions and decimals and solve problems involving rational numbers and percentages, choosing representations that are suited to the context and enable efficient computational strategies

ELABORATIONS 

exploring 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

choosing an appropriate numerical representation for a problem so that efficient computations can be made, such as \color{blue}\boldsymbol{12.5\%, \frac{1}{8}, 0.125} or \color{blue}\boldsymbol{\frac{25}{1000}}

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

exploring multiplicative (multiplication and division) problems involving fractions and decimals such as fraction walls, rectangular arrays, algebra tiles, calculators or informal jottings

developing efficient strategies with appropriate use of the commutative and associative properties, regrouping or partitioning to solve additive (addition and subtraction) problems involving fractions and decimals

calculating solutions to problems using the representation that makes computations efficient such as 12.5% of 96  is more efficiently calculated as \color{blue}\boldsymbol{\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

using the digits 0 to 9 as many times as you want to find a value that is 50% of one number and 75% of another using two-digit numbers

CONTENT

model situations (including financial contexts) and solve problems using rational numbers and percentages and digital tools as appropriate. Interpret results in terms of the situation

ELABORATIONS

calculating mentally or with calculator using rational numbers and percentages to find a proportion of a given quantity, for example, 0.2 of total pocket money is spent on bus fares, 55% of Year 7 students attended the end of term function, 23% of the school population voted yes to a change of school uniform

calculating mentally or with calculator using rational numbers and percentages to find a proportion of a given quantity, for example, 0.2 of total pocket money is spent on bus fares,  of Year 7 students attended the end of term function,  of the school population voted yes to a change of school uniform

interpreting tax tables to determine income tax at various levels of income, including overall percentage of income allocated to tax

using modelling contexts to investigate proportion such as proportion of canteen total sales happening on Monday and Friday, proportion of bottle cost to recycling refund, proportion of school site that is green space; interpreting and communicating answers in terms of the context of the situation

expressing profit and loss as a percentage of cost or selling price, comparing the difference

investigating the methods used in retail stores to express discounts, for example, investigating advertising brochures to explore the ways discounts are expressed

investigating the proportion of land mass/area of Aboriginal Peoples’ traditional grain belt compared with Australia’s current grain belt

investigating the nutritional value of grains traditionally cultivated by Aboriginal Peoples in proportion to the grains currently cultivated by Australia’s farmers

11 Replies to “ACARA Crash 11: Pulped Fractions”

  1. “Equivalent” fractions…

    What is so wrong with the word “equal”?

    Actually… the whole document seems to be fixated with taking a simple idea and then rendering it unreadable, un-understandable and overall so misleading that you can’t complain because you don’t know what you are complaining about!

    Except the digital tools of course. They work for ACARA.

    1. RF, I think you have the perfect summary:

      so misleading that you can’t complain because you don’t know what you are complaining about.

      It is so difficult to grasp what the draft curriculum is telling teachers to teach, it is difficult to say exactly how it is wrong. It is definitely wrong, if only because human beings don’t communicate with such language, but to pinpoint the mathematical complaint each time is very difficult.

      For this Crash, for example, I had to work really hard, trying to figure out what constituted “fraction arithmetic” in Year 7 (and Years 6 and 8, which I might have included). In the end I decided to include the first and third content-elaboration combos, not because they seemed to indicate anything beyond trivial fraction arithmetic, but because maybe someone would want to argue that they do.

      1. God help any new teachers trying to understand this. I struggle and have some experience in teaching. In theme with Terry’s Egyptian, comment we need the Rosetta stone.

  2. You probably all know this…but in ancient Egypt, arithmetic with fractions was based on fractions of the form 1/n (2/3 was an exception). I don’t know why they did this, but it is interesting. So, they created tables of 2/n (n being odd) expressed as sums of distinct fractions of the form 1/m.

    Exercise: Express 2/73 as a sum of distinct fractions of the form 1/m. Answer attached.

    Even giving year 7 students the answer and ask them to check it (without a calculator) would challenge many students. But they would learn that there is more to ancient Egypt than pyramids and pharaohs.

  3. “investigating the nutritional value of grains traditionally cultivated by Aboriginal Peoples in proportion to the grains currently cultivated by Australia’s farmers”

    WTF!?

    1. Hope the records were accurate and that the same units of measurement were used…

      Oh, wait…

      There goes that idea.

  4. This obtuse document illustrates the thinking which confuses the teaching of school mathematics, confuses students, and leads to a generation of adults who even after 13 years of school – 11 of them with mandatory mathematics – still believe that 0.15 is bigger than 0.3 because “fifteen is bigger than three”. See for example the preliminary research findings at https://www2.merga.net.au/documents/RP_Steinle_Stacey_1998.pdf

    This is an old document – nearly 25 years – but I would assume that matters have only got worse since. And if the above confusing word-soup from ACARA is any evidence, they’re going to get worse still.

  5. Please tell me the error in dealing with 12.5% is your typo and not published in an official document as equal to 25/1000

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