In both cases, the points have been plotted correctly, but joined up incorrectly. In the first, the points have been joined together with straight lines. In the second, the graph does curve on both sides, but has a flattened bottom (no sniggering!) where there should be a dip.
When viewed through the lens of arbitrary and necessary, which I wrote about in this previous post, I see some arbitrary things: the ordering of the coordinate pairs (x, then y), the particular quadratic function being plotted and (perhaps arguably) the order of operations. However, there is one particular necessary thing that has not been fully grasped in the above examples - the curvature of the quadratic function. This is a 'Must-Be' Maths fact, true not by convention, but because it is an intrinsic property of quadratic functions.
At the time of first encountering quadratic graphs (as my Year 9 class did today), students are typically early in the process of developing an important conceptual understanding of the distinction between linear and non-linear relationships. It is important, therefore, that students engage with and think deeply about the differences between these.
That a quadratic graph should be drawn as a smooth curve can easily be presented to students as though were an arbitrary, conventional fact. However, I believe doing so will lead to the sorts of mistakes illustrated above. Instead, I offer a suggestion of how a student can come to know and understand this fact more deeply and with certainty.
Let's say that students have correctly plotted the integer points of a quadratic function, such as:
The question now arises 'how do we join these points up?' It is tempting for the teacher to simply instruct pupils to draw a smooth curve that passes through all the points. However, this is presenting a 'Must-Be' maths fact as though it is an arbitrary convention. To students, rather than the curve representing the very nature of a quadratic relationship, this may seem a random choice - something else just to be memorised and easily confused with other types of graphs (eg. scatter or frequency polygons.)
And so, instead I offer students the following questions to help to reach a decision as to how the points should be joined:
This reinforces the fact that the function doesn't only apply to integers, which students may also have assumed from what we have done so far.
Through plotting these as additional points...
...it becomes clear that drawing straight lines between the blue dots will not fit how the function 'behaves' between the integers...
...and the true shape of the graph starts to emerge.
Through trying out some of the values between the integers, students have gained a clearer sense of what the shape of the graph is, and equally what it isn't. Students have truly engaged with the real mathematics of the situation - how a quadratic relationship 'behaves' over a continuous scale.
The arbitrary facts in this example included: the particular quadratic relationship chosen (y = x²), the use of the (x, then y) coordinate plotting system and the use of particular integer values to start with. Putting all these conventions together, it is then a necessary and intrinsic property of the quadratic relationship that the result will be a curved, parabolic graph.
Working this through with students served as a means to model mathematical behaviour. Once the integer points were plotted, we could conjecture as to what the shape should be: perhaps join the points with a straight line, perhaps a curve... but it is through further mathematical investigation, inquiring into these conjectures that a conclusion is reached. Students are shown that, in certain situations, they have the power to 'test' if mathematical facts are true - it is not simply a case of randomly guessing or just accepting what you are told.
Later on in the lesson today, the second misconception in the picture at the top emerged. After just a little prompting from me, students were testing points halfway along their 'flattened bottom' and seeing for themselves how their graph should look.
Perhaps other (all?) misconceptions should be treated as conjectures in the classroom?