WitCH 62: Video Killed the Proficiency Star

It’s not fun, but we gotta do it.

Yesterday, we wrote about Ofsted’s review of mathematics education. We wrote that it is a great review, and it is a great review. It is perfect in its reactionaryism. But, alas, there’s also a video.

Below is Hannah Stoten, one of the review authors and future kidnap victim, launching the review. An excellent review, and a well-spoken author. What could go wrong?

 

UPDATE (01/06/21)

Well, that’s not good:

Nonetheless, we’re willing to assume that Hannah is innocent here, and plans to kidnap her and bring her to Australia are proceeding apace.

RatS 12: Sit Up and Think of England

Good news. We’re giving ACARA, and our readers, the night off. No painful reading tonight; just painful reality.

Ofsted is the UK’s ACARA-ish organisation, although the “ish” hides the fact that Ofsted appears to be competent. Last week, Oftsed published a review into mathematics education. We’re sure we are missing something, because the review appears to be important, clear and correct.

Reportedly the work of Hannah Stoten,* the document lays out in a clear and methodical manner what a mathematics education entails, and thus the nature of a proper mathematics curriculum. Here is how the review gets going:

How the review classifies mathematics curriculum content

For this review, we have classified mathematical curriculum content into declarative, procedural and conditional knowledge.

Declarative knowledge is static in nature and consists of facts, formulae, concepts, principles and rules.

All content in this category can be prefaced with the sentence stem ‘I know that’.

Procedural knowledge is recalled as a sequence of steps. The category includes methods, algorithms and procedures: everything from long division, ways of setting out calculations in workbooks to the familiar step-by-step approaches to solving quadratic equations.

All content in this category can be prefaced by the sentence stem ‘I know how’.

Conditional knowledge gives pupils the ability to reason and solve problems. Useful combinations of declarative and procedural knowledge are transformed into strategies when pupils learn to match the problem types that they can be used for.

All content in this category can be prefaced by the sentence stem ‘I know when’.

When pupils learn and use declarative, procedural and conditional knowledge, their knowledge of relationships between concepts develops over time. This knowledge is classified within the ‘type 2’ sub-category of content (see table below). For example, recognition of the deep mathematical structures of problems and their connection to core strategies is the type 2 form of conditional knowledge.

Summary table of content categories considered in the review:

Category Type 1 Type 2
Declarative "I know that" Facts and formulae Relationship between facts (conceptual understanding)
Procedural "I know how" Methods Relationship between facts, procedures and missing facts (principles/mechanisms)
Conditional "I know when" Strategies Relationship between information, strategies and missing information (reasoning)

Is this perfect? Of course not. It would be easy to nitpick over borderline calls. But as a basic guide to building and analysing a curriculum, it is beautifully simple and clear. As guides should be. And as ACARA’s Wheel of Death most definitely is not.

There’s plenty more we could quote. Like the whole damn thing. But we’ll restrain ourselves, and give just a few more. Here’s a note on “core knowledge”:

Foundational knowledge, particularly proficiency in number, gives pupils the ability to progress through the curriculum at increasing rates later on. The path of learning that begins with a diligent focus on core declarative and procedural knowledge is not a straight line, therefore, but a curve. This is a function of the curriculum’s intelligent design. For example, in countries where pupils do well, pupils are able to attempt more advanced aspects of multiplication and division in Year 4 if they have been given more time on basic arithmetic in Year 1. This may explain why successful curriculum approaches tend to emphasise core knowledge early on.

So, arithmetic skills are kind of important, especially early on. Who would have guessed?

Here’s an early comment on problem-solving:

Problem-solving requires pupils to hold a line of thought. It is not easy to learn, rehearse or experience if the facts and methods that form part of a strategy for solving a problem type are unfamiliar and take up too much working memory. For example, pupils are unlikely to be able to solve an area word problem that requires them to multiply 2 lengths with different units of measurement if they do not realise that the question asks them to use a strategy to find an area. They are also unlikely to be successful if they do not know many number bonds, unit measurement facts, conversion formula or an efficient method of multiplication to automaticity. Therefore, the initial focus of any sequence of learning should be that pupils are familiar with the facts and methods that will form the strategies taught and applied later in the topic sequence.

What’s this? Give the kids the knowledge and skills and techniques before having them embark on problem-solving? Are these people nuts?

One last one, on “positive attitude”:

Pupils are more likely to develop a positive attitude towards mathematics if they are successful in it, especially if they are aware of their success. However, teachers should be wary of the temptation to invert this causal pathway by, for example, substituting fun games into lessons as a way of fostering enjoyment and motivation. This is because using games as a learning activity can lead to less learning rather than more.

Some pupils become anxious about mathematics. It is not the nature of the subject but failure to acquire knowledge that is at the root of the anxiety pathway. The origins of this anxiety may have even been present at the start of a pupil’s academic journey. However, if teachers ensure that anxious pupils acquire core mathematical knowledge and start to experience success, those pupils will begin to associate the subject with enjoyment and motivation.

It’s hard to believe, but they seem to be suggesting that to get a kid to like mathematics you should get them good at it, rather than pretending they’re good at it. Crazy, crazy stuff.

Read the whole damn thing. We haven’t read it all yet, and yeah, we’ll probably find something in there that annoys us (because we’re that type). But we haven’t found it yet. It is a great, great document.

UPDATE (31/05/21)

A couple of colleagues, Simon the Likeable and The Hot Dog Man, indicated that they were puzzled by the Review. They both wondered if perhaps what Hannah Stoten is saying isn’t really simple. Indeed, they are correct. In a nutshell, this is Stoten’s message:

The last 50+ years have been a complete screw up. Forget about them, and start again.

UPDATE 26/07/21

It would appear that UK maths ed academics have just about as much sense as their Australian colleagues. The Association of Mathematics Education Teachers is the UK’s professional body for the trainers of mathematics teachers. And, AMET has submitted a formal complaint to Ofsted about Stoten and Wren’s review. Well, they would, wouldn’t they?

It’s a safe bet that any review which pisses off the likes of AMET (or MERGA) is on the right track. AMET’s complaint is laughably thin twaddle, which Greg Ashman has demolished in fine style.

 

*) (Update 31/05/21) And Steve Wren. Efforts are already underway to kidnap them both. No one tell them.

ACARA Crash 10: Dividing is Conquered

This Crash is a companion to, and overlaps with, the previous Crash, on multiplication. It is from Year 5 and Year 6 Number. and is, as near as we can tell, the sum of the instruction on techniques of division for F-6.

ACHIEVEMENT STANDARD (YEAR 5)

They apply knowledge of multiplication facts and efficient strategies to … divide by single-digit numbers, interpreting any remainder in the context of the problem.

CONTENT (YEAR 5)

choose efficient strategies to represent and solve division problems, using basic facts, place value, the inverse relationship between multiplication and division and digital tools where appropriate. Interpret any remainder according to the context and express results as a mixed fraction or decimal

ELABORATIONS

developing and choosing efficient strategies and using appropriate digital technologies to solve multiplicative problems involving multiplication of large numbers by one- and two-digit numbers

solving multiplication problems such as 253 x 4 using a doubling strategy, for example, 253 + 253 = 506, 506 + 506 = 1012

solving multiplication problems like 15 x 16 by thinking of factors of both numbers, 15 = 3 x 5, 16 = 2 x 8; rearranging the factors to make the calculation easier, 5 x 2 = 10, 3 x 8 = 24, 10 x 24 = 240

using an array model to show place value partitioning to solve multiplication, such as 324 x 8, thinking 300 x 8 = 2400, 20 x 8 = 160, 4 x 8 = 32 then adding the parts, 2400 + 160 + 32 = 2592; connecting the parts of the array to a standard written algorithm

investigating the use of digital tools to solve multiplicative situations managed by First Nations Ranger Groups and other groups to care for Country/Place including population growth of native and feral animals such as comparing rabbits or cane toads with platypus or koalas, or the monitoring of water volume usage in communities

LEVEL DESCRIPTION (YEAR 6)

use all four arithmetic operations with natural numbers of any size

ACHIEVEMENT STANDARD (YEAR 6)

Students apply knowledge of place value, multiplication and addition facts to operate with decimals.

CONTENT (YEAR 6)

apply knowledge of place value and multiplication facts to multiply and divide decimals by natural numbers using efficient strategies and appropriate digital tools. Use estimation and rounding to check the reasonableness of answers

ELABORATIONS

applying place value knowledge such as the value of numbers is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a natural number by a decimal of at least tenths

applying and explaining estimation strategies to multiplicative (multiplication and division) situations involving a natural number that is multiplied or divided by a decimal to at least tenths before calculating answers or when the situation requires just an estimation

deciding to use a calculator in situations that explore multiplication and division of natural numbers being multiplied or divided by a decimal including beyond hundredths

explaining the effect of multiplying or dividing a decimal by 10, 100, 1000… in terms of place value and not the decimal point shifting

ACARA Crash 9: Their Sorrows Shall Be Multiplied

We still have no time for the deep analysis of this shallow nonsense. So, we’ll just continue with the fish.

Below are two content-elaborations combos, from Year 5 and Year 6 Number. As near as we can tell, that’s about the sum of the instruction on techniques of multiplication for F-6.

ACHIEVEMENT STANDARD (YEAR 5)

They apply knowledge of multiplication facts and efficient strategies to multiply large numbers by one-digit and two-digit numbers

CONTENT (YEAR 5)

choose efficient strategies to represent and solve problems involving multiplication of large numbers by one-digit or two-digit numbers using basic facts, place value, properties of operations and digital tools where appropriate, explaining the reasonableness of the answer

ELABORATIONS

interpreting and solving everyday division problems such as, ‘How many buses are needed if there are 436 passengers, and each bus carries 50 people?’, deciding whether to round up or down in order to accommodate the remainder

solving division problems mentally like 72 divided by 9, 72 ÷ 9, by thinking, ‘how many 9 makes 72’, ? x 9 = 72 or ‘share 72 equally 9 ways’

investigating the use of digital technologies to solve multiplicative situations managed by First Nations Ranger Groups and other groups to care for Country/Place including population growth of native and feral animals such as comparing rabbits or cane toads with platypus or koalas, or the monitoring of water volume usage in communities

LEVEL DESCRIPTION (YEAR 6)

use all four arithmetic operations with natural numbers of any size

ACHIEVEMENT STANDARD (YEAR 6)

Students apply knowledge of place value, multiplication and addition facts to operate with decimals.

CONTENT (YEAR 6)

apply knowledge of place value and multiplication facts to multiply and divide decimals by natural numbers using efficient strategies and appropriate digital tools. Use estimation and rounding to check the reasonableness of answers

ELABORATIONS

applying place value knowledge such as the value of numbers is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a natural number by a decimal of at least tenths

applying and explaining estimation strategies to multiplicative (multiplication and division) situations involving a natural number that is multiplied or divided by a decimal to at least tenths before calculating answers or when the situation requires just an estimation

deciding to use a calculator in situations that explore multiplication and division of natural numbers being multiplied or divided by a decimal including beyond hundredths

explaining the effect of multiplying or dividing a decimal by 10, 100, 1000… in terms of place value and not the decimal point shifting

UPDATE (29/50/21)

we’ve just discovered some multiplication techniques tucked inside some division elaborations, as indicated in this companion Crash. The two Crashes should be considered together (and should have been just one Crash, dammit.)

You Got a Problem With That?

We’ll take a day off from bashing the draft curriculum, in order to bash the draft curriculum. This one’s not a Crash post, but it gets to the disfigured heart of the draft.

Yesterday, a good friend and colleague, let’s call him Mr. Big, threw a book at us. By Alexandre Borovik and Tony Gardiner, the book is called The Essence of Mathematics Through Elementary Problems. The book is free to download, and it is beautiful.

There is much to say about this book. It is, unsurprisingly, a collection of problems and solutions. By “elementary”, the authors mean, in the main, in the domain of secondary school mathematics. Note that “elementary” does not equate to “easy”, although there are easy problems as well.

The problems have been chosen with great care. As the authors write, the problems are included for two reasons:

    • they constitute good mathematics
    • they embody in a distilled form the quintessential spirit of elementary mathematics

As indicated by the the Table of Contents, the problems in The Essence of Mathematics are also arranged very carefully, by topic and in a roughly increasing level of conceptual depth, and the book includes interesting and insightful commentary. Their twenty problems and solutions embodying 3 – 1 = 2 is a beautiful illustration.

The Essence of Mathematics also contains an incredibly important message. Here is the very first problem in the book:

1(a)   Compute for yourself, and learn by heart, the times tables up to 9 × 9.

Regular readers will know exactly where we’re going with this. Chapter 1 of Essential Mathematics is titled Mental Skills, it includes simple written skills as well, and the message is obvious. As the authors write,

The chapter is largely devoted to underlining the need for mastery of a repertoire of instantly available techniques, that can be used mentally, quickly, and flexibly to analyse less familiar problems at sight.

In particular, on their first problem,

Multiplication tables are important for many reasons. They allow us to appreciate directly, at first hand, the efficiency of our miraculous place value system – in which representing any number, and implementing any operation, are reduced to a combined mastery of

(i) the arithmetical behaviour of the ten digits 0–9, and

(ii) the index laws for powers of 10.

Fluency in mental and written arithmetic then leaves the mind free to notice, and to appreciate, the deeper patterns and structures which may be lurking just beneath the surface.

What does all this have to do with ACARA’s draft curriculum? Alas, nothing whatsoever.

The draft curriculum is the antithesis of Essence. The “problems” and “investigations” and “models” in the draft curriculum are anything but well-chosen, being typically sloppy and ill-defined, with no clear direction or purpose. The draft curriculum also displays nothing but contempt for the prior mastery of basic facts and skills required for problem-solving, or anything.

Essence is not a textbook, but the authors clearly see a large role for problem-solving in mathematics education, and, with genuine modesty, they can imagine their book as a natural supplement to a good curriculum. Such a role can mean slow and open-ended learning, or at least open-ended teaching:

Learning mathematics is a long game; and teachers and students need the freedom to digress, to look ahead, and to build slowly over time. 

The value of such digressions and explorations, however, does not negate the primary goal of mathematics education:

Teachers at each stage must be free to recognise that their primary responsibility is not just to improve their students’ performance on the next test, but to establish a firm platform on which subsequent stages can build.

The effect [of political pressures] has been to downgrade the more important challenges which every student should face: namely

    • of developing a robust mastery of new, forward-looking techniques (such as fractions, proportion, and algebra), and
    • of integrating the single steps students have at their disposal into larger, systematic schemes, so that they can begin to tackle and solve simple multi-step problems.

Building systematic schemes out of the mastery of techniques. Or, there’s the alternative:

A didactical and pedagogical framework that is consistent with the essence, and the educational value of elementary mathematics cannot be rooted in false alternatives to mathematics (such as numeracy, or mathematical literacy).

There is problem-solving, and there is “problem-solving”. ACARA is shovelling the latter.

 

UPDATE (28/05/21)

Mrs. Big, AKA Mrs. Uncle Jezza, has given the draft curriculum a very good whack in the comments, below. As part of that, she has noted an excellent quotation that begins the Preface of Essential. The quotation is by Richard Courant and Herbert Robbins, and is from the Preface of their classic What is Mathematics?

“Understanding mathematics cannot be transmitted by painless entertainment … actual contact with the content of living mathematics is necessary. The present book … is not a concession to the dangerous tendency toward dodging all exertion.”

While we’re here, we’ll include another great quote, from the About section of Essential, by John von Neumann:

“Young man, in mathematics you don’t understand things. You just get used to them.”

Understanding is a fine goal, but it can also be a dangerously distracting goal. ACARA’s “deep understanding” is an absurdity.

The ACARA Page

Honestly, it wasn’t our intention to write three hundred posts on ACARA and their appalling draft mathematics curriculum. But, we did. Given that we did, it seems worthwhile having a pinned metapost, so that anybody who wants to can find their way through the jungle. (There’s probably a better way to do this, with a separate blog page or whatever, but we can’t be bothered figuring that out right now.)

So, here we are: the complete works, roughly in reverse chronological order, and laid out as clearly as we can think to do it. It includes older posts and articles, on the current mathematics curriculum (which also sucks) and NAPLAN (which also also sucks).

Continue reading “The ACARA Page”

ACARA Crash 8: Multiple Contusions

OK, roll out the barrel, grab the gun: it’s time for the fish. Somehow we thought this one would take work but, really, there’s nothing to say.

It has obviously occurred to ACARA that the benefits of their Glorious Revolution may not be readily apparent to us mathematical peasants. And, one of the things we peasants tend to worry about are the multiplication tables. It is therefore no great surprise that ACARA has addressed this issue in their FAQ:

When and where are the single-digit multiplication facts (timetables) covered in the proposed F–10 Australian Curriculum: Mathematics?

These are explicitly covered at Year 4 in both the achievement standard and content descriptions for the number strand. Work on developing knowledge of addition and multiplication facts and related subtraction and division facts, and fluency with these, takes place throughout the primary years through explicit reference to using number facts when operating, modelling and solving related problems.

Nothing spells sincerity like getting the name wrong.* It’s also very reassuring to hear the kids will be “developing knowledge of … multiplication facts”. It’d of course be plain foolish to grab something huge like 6 x 3 all at once. In Year 4. And, how again will the kids “develop” this knowledge? Oh yeah, “when operating, modelling and solving related problems”. It should work a treat.

That’s the sales pitch. That’s ACARA’s conscious attempt to reassure us peasants that everything’s fine with the “timetables”. How’s it working? Feeling good? Wanna feel worse?

What follows is the relevant part of the Year Achievement Standards, and the Content-Elaboration for “multiplication facts” in Year 4 Algebra.

ACHIEVEMENT STANDARD

By the end of Year 4, students … model situations, including financial contexts, and use … multiplication facts to … multiply and divide numbers efficiently. … They identify patterns in the multiplication facts and use their knowledge of these patterns in efficient strategies for mental calculations. 

CONTENT

recognise, recall and explain patterns in basic multiplication facts up to 10 x 10 and related division facts. Extend and apply these patterns to develop increasingly efficient mental strategies for computation with larger numbers

ELABORATIONS

using arrays on grid paper or created with blocks/counters to develop and explain patterns in the basic multiplication facts; using the arrays to explain the related division facts

using materials or diagrams to develop and record multiplication strategies such as skip counting, doubling, commutativity, and adding one more group to a known fact

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 ten to establish the multiples 9 as ‘to multiply a number by 9 you multiply by 10 then take the number away’; 9 x 4 = 10 x 4 – 4 , 40 – 4 = 36 or using multiple of three as ‘to multiply a number by 9 you multiply by 3, and then multiply the result by 3 again’

using the materials or diagrams to develop and explain division strategies, such as halving, using the inverse relationship to turn division into a multiplication

using known multiplication facts up to 10 x 10 to establish related division facts

 

Alternatively, the kids could just learn the damn things. Starting in, oh, maybe Year 1? But what would we peasants know.

 

*) It has since been semi-corrected to “times-tables”.

ACARA Crash 7: Spread Sheeet

(In keeping with our culturally sensitive ways, the title should be read with a thick Mexican accent.)

We’re working on a non-WitCHlike Crash post, but no way will that be done tonight. Luckily, frequent commenter Glen has flagged some easily postable nonsense, and we can keep the Crash ball rolling.

This A-Crash consists of a Content-Elaboration combo for Year 6 Number:

CONTENT

identify and describe the properties of prime and composite numbers and use to solve problems and simplify calculations

ELABORATIONS

understanding that a prime number has two unique factors of one and itself and hence 1 is not a prime number

testing numbers by using division to distinguish between prime and composite numbers, recording the results on a number chart to identify any patterns

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 \color{blue}\boldsymbol{15 \times 16} as \color{blue}\boldsymbol{5 \times 3 \times 4 \times 4} which can be rearranged to simplify calculation to \color{blue}\boldsymbol{5 \times 4 \times 3 \times 4 =20 \times 12}

using spread sheets to list all of the numbers that have up to three factors using combinations of only the first three prime numbers, recognise any emerging patterns, making conjectures and experimenting with other combinations

understanding that if a number is divisible by a composite number then it is also divisible by the prime factors of that number, for example, 216 is divisible by 8 because the number represented by the last three digits is divisible by 8, and hence 216 is also divisible by 2 and 4, using this to generate algorithms to explore

 

UPDATE (25/05/21)

Thanks, everyone, so far. We’re going nuts with work, so a quick WitCHlike update while the window is open.

0) How can ACARA be so, so, so appallingly bad with their grammar and punctuation? We honestly don’t get it. Is the content descriptor accidentally missing a pronoun, and a comma, and a preposition, or do they genuinely like how it reads?

1) Yes, the free-floating and otherwise irritating “hence”, the fact that “prime” is undefined is appalling. So is using “1” and “one” in the same sentence to refer to the same thing. So is “two unique factors of one and itself and …”.

2) Possibly John’s guess on the second elaboration is correct. What would be focussed and useful is to take a 12 x 12 table of numbers and cross off the multiples (and circle 1). So, you get the kids to do the sieve of Eratosthenes thing, and emphasise the multiples as composites. You know, a clearly expressed investigation, with clear purposes.

3) This is Year 6, and so we’re not so concerned about “Fundamental theorem of arithmetic” not being mentioned here, although of course both existence and uniqueness of the prime factorisation should have been spelled out, even if only as something to “explore”. It’s way too important to be included as just a “by the way” part of a multiplication trick. As a side point, in regard to our previous Crash post, it is notable when and how “Fundamental theorem” first appears.

4) 15 x 16? Really?

5) We’re guessing the spread sheeet activity was intended to mean using each prime at most once. Given these people can’t write, however, it’s only a guess. But if so, that would be a reasonable exercise, IF you ditched the spread sheeet, and IF you repeated the exercise a few times with varying selections of primes. None of which will happen.

6) It is unbelievably stupid to introduce prime stuff in combination with divisibility tricks. The former is, well, fundamental, and the latter is a base ten game.

7) “The number represented by the last three digits”. Of what? Who talks this way? Who talks this way and expects to be understood?

8) What are the other digits of 216?

9) Even if there were other digits, a number ending in 216 is a really stupid choice to demonstrate divisibility by 8. These things matter.

ACARA Crash 6: Crossed Words

Lots of 'em

Word/Phrase Number of Occurrences Clarification
Investigat* 298
Model* 224
Explor* 261
Solve ... Problem(s) 154

Not lots of 'em

Word/Phrase Number of Occurrences Clarification
Difference of two squares 1 The only occurrence is in Year 10 Optional Number, in the context of surds.
Perfect square(s) 1 The only occurrence is in Year 7 Number, in the context of squares of integers.
Completing the Square 0
Null factor 0

 

Alright, kiddies, this is one that you can play at home. Just grab your handy copy of the Daft Australian Curriculum, and go word searching. For example, you might look up the word “aimless” and, strangely, nothing will occur. On the other hand, look up “effective”/”effectively” and, just as strangely, you will get plenty of hits.

So, go to it. Look up your favourite/anti-favourite mathematical words and phrases, and let us know the number of hits in the comments. We’ll keep track of the results in our handy dandy Lots and Not-Lots tables, above.

Just a few quick notes:

*) Different derivatives of the same root word or phrase should be grouped together.

*) We’ll add clarifying notes on usage of the word/phrase when it seems appropriate.

*) We won’t be checking your puzzling skills very carefully. We’ll simply put up the numbers, and it’s up to others to do the checking. Then, we’ll correct the totals when need be.

Happy hunting.

ACARA Crash 5: Completing the Squander

The previous A-Crash consisted of everything we could find in the Daft Curriculum on the algebraic treatment of polynomials and polynomial equations. This companion A-Crash consists of everything we could find in the Year 10 draft on the same material. 

CONTENT (Year 10)

expand and factorise expressions and apply exponent laws involving products, quotients and powers of variables. Apply to solve equations algebraically

ELABORATIONS

reviewing and connecting exponent laws of numerical expressions with positive and negative integer exponents to exponent laws involving variables

using the distributive law and the exponent laws to expand and factorise algebraic expressions

explaining the relationship between factorisation and expansion

applying knowledge of exponent laws to algebraic terms, and simplifying algebraic expressions using both positive and negative integral exponents to solve equations algebraically

CONTENT (Year 10 Optional Content)

numerical/tabular, graphical and algebraic representations of quadratic functions and their transformations in order to reason about the solutions of \color{blue}\boldsymbol{f(x) = k}

 ELABORATIONS

connecting the expanded and transformed representations

deriving and using the quadratic formula and discriminant to identify the roots of a quadratic function

identifying what can be known about the graph of a quadratic function by considering the coefficients and the discriminant to assist sketching by hand

solving equations and interpreting solutions graphically

recognising that irrational roots of quadratic equations of a single real variable occur in conjugate pairs