One of the biggest ‘why oh why’ questions in secondary school mathematics is the purpose of studying quadratic equations because ‘I am never going to need them in everyday life’. This irritating question is based on a sound premise. The applications generally offered to students to persuade them of the usefulness of quadratics tend to be dropping things from tall buildings, kicking/throwing/shooting, accelerating/braking, or calculating areas. All of these actions in ‘everyday life’ are generally calculated by estimation, rules of thumb, or in the case of areas, some kind of non-standard measuring unit such as the number of tiles or length of carpet.
Plan A for teachers answering this question is to continue the myth that adults go around doing school mathematics on bits of paper in all kinds of day-to-day contexts.
Plan B could be something like ‘no but you do need them to get your exam grades’ – an answer that continues the myth of irrelevance but is at least true.
Plan C: could be something like ‘no but you do need them to help you understand some of the maths you’re going to use in other subjects and employment’.
Plan C was the focus of a parliamentary debate at 3.43pm on 26th June 2003 about quadratic equations. I can imagine an enterprising group of students turning this debate into a humorous sketch, but the messages it contains are valuable. Tony McWalter (MP) starts with:
‘I put this matter on the agenda today because I have been troubled since the president of a teachers’ union suggested a couple of months ago that mathematics might be dropped as a compulsory subject by pupils the age of 14… He cited the quadratic equation as an example of the sort of irrelevant topic that pupils study. I had hoped that the Government would make a robust rebuttal, but there was no defence either of mathematics in general or the quadratic equation in particular.’
In his speech, Tony McWalter gives several reasons why quadratic equations are relevant that seem to me to be important when planning the place and role of quadratics in students’ school experience.
- access to understanding calculus, such as rates of change and marginal effect
- access to understanding science, such as gravity
- understanding that effort and difficulty present rewards beyond what can be achieved by guessing or trial and error
- access to the authentic historical culture of mathematics
Alan Johnson replies by pointing out some further reasons: ‘quadratic equations allow us to analyse the relationships between variable quantities, and they are the tool for understanding variable rates of change.’
Note that I have not said ‘in the National Curriculum’, because it is sadly limited to testable statements:
- develop algebraic and graphical fluency, including understanding linear and simple quadratic functions
- recognise, sketch and produce graphs of … quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane
- use…quadratic graphs to estimate values of y for given values of x and vice versa…
Instead, when we think about the curriculum from the students’ point of view, it is best to embed answers to the ‘why oh why?’ question all the way through and give everyone, including you and your colleagues, a strong sense of shared purpose.
So why are quadratic functions important?
Quadratic functions hold a unique position in the school curriculum. They are functions whose values can be easily calculated from input values, so they are a slight advance on linear functions and provide a significant move away from attachment to straight lines.
They are functions which have variable rates of change, that can be described qualitatively. It is easy to provide students with quadratic functions whose output values can be represented on equal aspect x and y axes, so that understanding rate of change as the gradient of tangent at point is relatively straightforward without worrying about scaling. They therefore provide opportunities to learn about the importance of paying attention to scaling of the y-axis when reasoning graphically about rates of change/gradients, by comparing the different visual appearance of graphs when the scaling changes.
When we translate linear functions it is visually unclear whether the translation was horizontal or vertical. It can also be a mystery to students why we focus on the y intercept in the y = mx + c representation, particularly as the ax + by = c representation gives both intercepts equal importance. The quadratic function clarifies these issues by making it necessary to know the direction of translation, and by presenting a new – hugely important – meaning of intercepts on the x-axis. For all future work with functions and graphs, whether in pure maths or in some application of maths, students will need to focus on these qualitative characteristics, and also rates of change, constants represented by y intercepts, and zeros.
Quadratics are the only functions where students can use fairly accessible algebraic and arithmetical manipulation to show the relationships between input/output values, different algebraic representations, and graphical representations. Other elementary functions are accessible for some, but not all, of these connections.
The study of quadratics, therefore, provides the opportunity to ask a variety of questions about quantitative characteristics of phenomena. Such as questions about increase and decrease, rates of change, upturns and downturns; or questions about achieving certain values, such as zero, and the location of maxima and minima. Within quadratic functions, these questions can be answered at KS3/4 level.
By questioning quadratics, students of higher mathematics also have access to the use of imaginary numbers to solve problems through the necessary invention of i – the square root of -1.
How do we show their purpose in the classroom?
What would your school scheme of work look like throughout KS3 if it had these questions arranged in a coherent fashion so that students’ experience had some meaning, both for those who will go on to study higher mathematics and for those who will use graphs in a wide range of subjects and purposes?
It’s pretty obvious to me that the factorised form of the quadratic (y = k(x-a)(x-b)) and interpretation and sketching of graphs should go hand-in-hand. Then vertical translations and scaling could be used to distinguish between quadratics that had the same roots. Using curiosity, exploration and conjecture with graphical software would be one way to hook students into this, by giving them the intellectual power to create and identify quadratics that are visually very different.
The factorised form also gives access to the area model of the quadratic expression that is historically important and also widely used in textbooks. This approach, focusing on overall characteristics rather than individual points (apart from zeros) also embeds an understanding of functions as objects in their own right, rather than as a way of ‘joining the dots’. Manipulating expressions, by expanding brackets or factorising also has the purpose of moving between different ways to express quadratics, and hence focusing on different characteristics of the function.
This was the kind of development we had in mind when we began to write the curriculum, but much gets lost in the process of reducing conceptual development to brief statements. We imagined that this kind of work could go on through constant use of graphical software and alongside experience with other functions. Functions that can be understood qualitatively, but maybe not algebraically, such as exponentiation, inverse proportion, sinusoidal and so on, asking questions that, for the quadratic, can often be answered exactly using KS3 methods.
So, if I were planning how to tackle quadratics throughout KS3 I would vote for Plan C, and provide a meaningful, coherent and purposeful experience of quadratics (and their associated manipulations and transformations), as a bridge from linear functions and graphs to a world of functions that express all kinds of relations in science and mathematics, and elsewhere.
All the best,
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.