Graphs of functions inquiry
Mathematical inquiry processes: Identify and describe properties; extend to other cases. Conceptual field of inquiry: Graphs and equations; gradient; turning point; equations of circles; points of intersection; transformations.
In the form shown in the prompt, the teacher can lead a discussion to draw out the students' existing knowledge and then build upon it.
Jane's original task is more open than the prompt. Her initial starting point in classroom inquiry has been:
Take two points (0,8) and (2,4) and investigate.
Jane explains that she would ask students how they can use the points and what they can find out from them. Depending on the students' responses, Jane might provide relevant mathematical terms and use some or all of the questions that follow:
Can you find a linear function that goes through the points?
Can you find a quadratic (cubic) function that goes through the points? Are there any others? How do you know?
Can you find a circle related to these points?
Where do the curves intersect?
The questions provide a structure if students are struggling. Jane also gives them the set of possible solution curves that are shown in the prompt as further support.
Lines of inquiry
Having used the inquiry many times, Jane has compiled a list of lines of inquiry that regularly arise in classrooms:
1. Find the distance and midpoint of the line between the two coordinates and its perpendicular bisector.
2. Find the equations of the straight line and parabola.
3. Deduce the equation of the curve, which can be worked out using simultaneous equations and differentiation.
4. Find the equation of the circle whose centre is (2,4) and circumference passes through (0,8).
The circle could be included in an alternative prompt (see illustration), although the statement is then only partially true.
Introducing a circle to the inquiry also gives the teacher the opportunity to discuss why a circle is not a function.
5. Find quartic and quintic functions whose graphs pass through the two points.
6. Find a 'family' of cubic equations that pass through the two points and deduce their general form.
7. Work out the points of intersection of the line, curves and circle.
8. Choose two other coordinates and find functions whose graphs pass through the points.
During their inquiries, Jane encourages students to use Desmos to explore equations, find curves, check conjectures and verify findings.