This article is about some lessons that haven’t happened yet, inspired by a serendipitous realisation during a lesson that has. In a previous post, “No Teachers Are Really Young”, I described some other lessons on graphs of quadratic equations with the same class. That post provides some context for this one and I would recommend you read it if you have the time and haven’t already done so.

Having grown up before the digital revolution took hold and entered teaching in my mid-life years, I find I’m selective about which new technologies and software I use in my classroom, only doing so when there is a clear and significant advantage over traditional media. When I do take the plunge I’m not terribly confident or competent: in particular, my typing is slow, whether on a physical or an on-screen keyboard.

This combination of caution and sluggishness caused both me and my learners to notice some strange goings on while I was entering the equation of a quadratic curve in real time using Desmos software. Take a few moments to imagine what happens when you type

y = x^{2} + 4x + 3

at a snail’s pace.

My long-suffering tech-speedy learners and I were able to witness, discuss and describe – as I painstakingly proceeded, key-by-key – a straight line with gradient 1 metamorphose by bending itself at the origin to become a parabola with a minimum turning point at the origin, then disappear (why?) before returning 4 rungs up the y-axis, darting down to the left at the touch of the x key then, at last, after another disappearing act, assume it’s rightful shape and position. (You might want to open up Desmos and try this now for yourself – if you’re not familiar with it, it’s free to access online or as an app and simple to use.)

We spent a few more minutes predicting what would happen at each keystroke as I entered another quadratic (who knows which terms we used?). Then we moved on with the planned learning for that day and, soon afterwards, to a completely new topic. After all, we have a whole course to get through, don’t we?

My pupils are now on study-leave, less than a week away from their National 5 exam, part of their first taste of the annual external examination diet administered by the Scottish Qualifications Authority after four years of secondary-school mathematics learning, building upon seven years at primary school and one or two at nursery. I finally have some time to reflect on how I can use this brief, chance learning episode to plan for my National 5 learners of the future who, as my current crop did, will have to begin to grapple with different algebraic forms and their associated graphs.

So I sat down this morning and planned a learning sequence around the graph of

y = x^{2} + 4x + 3

based on the idea of slow-typing and predicting the resulting graph at each keystroke. (The required prior learning should, hopefully, be apparent to readers; it is, for the most part, consistent with our course-plan but may need modification in different learning contexts).

Show pupils the set of equations opposite.

Ask them to describe the graph of each equation using appropriate mathematical language. After each equation has been discussed enter it in Desmos and reveal the graph. Here are some questions that can be asked before, during or after displaying each graph.

What features of the graph can be known simply by looking at the equation?

How does each successive graph relate to the one before?

Which algebraic features cause the graph to:

- change shape
- move relative to the axes while maintaining the same shape.

Another approach, once learners have become familiar with this process, would be to have them sketch each graph before it is revealed. By allocating each of the sub-equations appropriately, this might lend itself to paired or group work leading to whole-class collaboration and discussion.

Here is the image that results when all five equations have been entered.

Display this image without the equations.

Provide the four sub-equations and the completed equation separately, in non-colour-coded form. Which equation represents each line or curve shown?

In what order would the individual lines or curves have appeared while entering the desired quadratic equation?

Recall that in the original lesson the full quadratic equation was entered on a single line, resulting in a graph that transformed with almost every keystroke before settling on its final form. Using either the single or multiple line approach to entering, displaying and discussing individual equations, sub-equations, lines and curves, explore what happens when we change the order of terms within the original quadratic expression. For example

y = 4x + x^{2} + 3

or

y = 3 + 4x + x^{2} (think carefully about this one.)

Continuing either in a teacher-lead manner or, if you are so inclined (as I increasingly am), taking a more inquiry-based approach, we might next ask

“In what other algebraic forms could we write the quadratic expression?

How would entering the graph using these forms change how it looks as we enter each character in Desmos?”

Let’s try the factorised form

y = (x + 3)(x + 1)

Then, changing the order in turn,

y = (x + 1)(x + 3);

y = (3 + x)(x + 1);

y = (1 + x)(3 + x)

What about completed-square form?

y = (x + 2)^{2} – 1

Pupils could try experimenting with the order of terms for themselves here, taking care to preserve the essential form. Again imagine, predict and then check how the display will change as the equation is built up keystroke-by-keystroke.

As extension, or maybe at some later stage, classes could begin to investigate what happens when we transform the original quadratic. (In Scotland this is not strictly a National 5 topic as far as polynomial functions are concerned, although it is relevant to trigonometric functions and their graphs). If each of the above graphs and equations represents y = f(x),

What would y = 2f(x) look like, both algebraically (in all its different forms) and graphically?

What about -2f(x)?

What’s the same and what’s different at each sub-step compared to what happened with y = f(x)?

My soon-to-be-senior learners will progress to Higher, some to Advanced Higher and some, I hope, beyond in their personal mathematical journeys. They will learn about new relationships, figures and functions in both their algebraic and graphical representations. And, whether with these or other learners, I will continue to have the privilege of helping them make connections between existing and new concepts and skills. I’m excited about using these activities with my classes and about adapting them for topics involving higher-order polynomials, trigonometric, exponential and rational functions, not to mention the other conic sections and circles.

Thank you for joining me to reflect on learning, teaching and doing mathematics. I hope you’ve enjoyed reading this post as much as I’ve enjoyed writing it.

I think this is a truly inspiring commentary about the importance of having that time to reflect on what was in effect “accidental task design”. Had you subsequently not taken the time to consider the implications of what you inadvertently stumbled upon with your class I and I’m sure others would never have necessarily looked at a task in this way. I really like the idea of building up graph transformations visually in bits and then hopefully guiding students towards a proper understanding of completed square form for example. I think with the excuse of time constraints etc we tend to rush or in some cases even ignore graphical methods completely. This results in linear relationships, simultaneous equations, quadratics, factorising, completing the square becoming abstract and somewhat disjoint. Equally I don’t know if I’d have come up with the method you have described on my own so it is a great catalyst for me on a different way to approach these topics, thanks.

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