Transforming functions and the value of visualisation
Combined transformations of functions can be a challenging topic for students to get to grips with. For many, making the best possible use of the tools available to them is crucial to understanding the intricacies of this subject and not feeling daunted when they finally encounter it in exams.
During our conversations with teachers and mathematicians about these sorts of issues, some common themes have emerged, including how useful it is for students to see various visual representations of a problem.
With this in mind, we asked our maths expert, Simon May, to share his views on function transformations, particularly around the importance of graphing and visualisation to support student learning and understanding.
Grasping the finer details
Simon noted that handheld graphing technology really comes into its own when students are exploring topics such as function transformations, owing to the importance of being able to see what’s happening with any given problem or question.
“If the students have their own graphing handset, then they can do investigations themselves, and the understanding light bulb is more likely to go off. That’s much more effective from a teaching point of view, because students are more likely to retain that new information,” he said.
Having the option to visualise functions can also make a big difference when it comes to students prioritising accuracy and taking steps in the right order to get the correct outcome.
Simon offered the example of a question involving a sine curve being translated vertically by two and stretched horizontally by a factor of a third. Plenty of students will instantly think this should be represented as the function y=sin 3x+2, and that is the correct graph, but Simon’s advice was to modify the format of the function slightly to y=2+sin 3x. This creates the same graph, but gives a clearer picture of what’s happening from a transformation point of view.
He underlined this point by incorporating parentheses to create the function y=sin (3x+2), which produces a new graph and prompts students to focus on what processes are happening inside the parentheses to give the different result.
By examining the outcomes visually, learners can see the importance of understanding the various steps involved in a required transformation. For example, the inclusion of parentheses has created the function, which has only translated the original function horizontally. It can also easily be seen that the horizontal stretch was performed before the translation, thus highlighting the importance of the correct order of transformation.
“It’s a tricky bit of maths that students do struggle with, because unless they can see what’s happening, just approaching it algebraically can be quite daunting,” Simon said.
Making students’ lives easier
Simon also spoke about how, in an exam scenario, students might come across function transformation questions that don’t necessarily require a calculator to answer. When they have graphing technology at their fingertips, however, they can use it to add a visual dimension to the problem, check their answers and gain peace of mind that they’re on the right track.
For example, when modelling a real-life temperature measurement using students don’t necessarily need a graphing calculator to work out the value of A and B. With the initial information they’re given – and pre-existing knowledge about the exponential function, they can ascertain that their calculated answers are correct by plotting the curve and tracing a point.
For students that are uncertain, there’s great value in being able to plot the function as a graph and get visual confirmation from their calculator that everything they have done so far is correct.
Simon pointed out that taking a purely algebraic approach, with no visual representation of the different stages of transforming the function, can lead to serious problems for students’ understanding. The smallest error at any point in the process could have knock-on effects on the later values they find and the final answers they come up with.
“The very fact that they can draw this on the graphic calculator makes their understanding so much stronger, and they’re less likely to make a mistake,” he said.
As teachers already know, many maths exam questions today are designed to test understanding and expect candidates to show their reasoning, not just give the right answer.
In the case of text-heavy questions, students need to decipher the information provided and show that they grasp key concepts, such as how the y-intercept relates to the real-life interpretation of the model.
“Students need to make those connections, and being able to visualise it and draw the graph just makes life easier,” Simon said. “And that’s the feedback you get from students as well: ‘This is so much easier, because I can see what’s going on now.'”
We’ve learned many things from talking to teachers, including that, when students feel positive about the work they’re doing, they learn better and the whole process just runs smoother for everyone.
It all starts with the teacher, of course. If you’re keen to improve your own graphic calculator skills, why not take advantage of a free Casio fx-CG50 training session?