Completing the square inquiry
Mathematical inquiry processes: Explore; generate examples; conjecture; reason. Conceptual field of inquiry: Completing the square; graphs of quadratic functions; turning point; algebraic manipulation .
Shawki Dayekh, a teacher of mathematics responsible for A-level teaching in his school, devised the prompt for his year 12 (grade 11) class. He wanted the students to explore minimum and maximum turning points by graphing quadratic equations. Why is there a minimum point when a > 1 and a maximum point when a < 1? What is the relationship between the values of a, b and c and the y-coordinate at the turning point? Why are only a and b involved in finding the x-coordinate?
The design of the prompt uses y(max), which, perhaps, is not typically the first case students would meet. Shawki explained that he was hoping the students themselves would initiate a line of inquiry related to y(min). He also saw the potential for students to move from right to left as well as left to right. If y(max) is known, what does that imply about f(x)? Another line of inquiry involved starting with higher order functions, such as ax4 + bx2 + c.
Shawki reports on how the fast-moving inquiry developed:
The students' questions about the prompt (right) showed their existing level of knowledge. We had just covered completing the square and you can see a couple of questions about how the topic fitted in with the prompt. They were trying to apply their knowledge to the new context.
The class was new to inquiry so I directed them a lot at first. We started with students exploring the turning points. They made up different values for a, b and c. I encouraged them to vary the values systematically to see how the turning point changed. One pair, for example, used (1, 2, 3) for (a, b, c) and then (2, 4, 6), (3, 6, 9) and so on (see illustration below). They were getting immediate results on the Desmos graphing calculator and then checking the y-value using the algebraic expression in the prompt.
Reporting back and discussion
After about 20 minutes, one pair reported on using negative values for a, b and c. This led on to a discussion about y(min) and y(max) and their relationship to the variables.
I'm not sure how, but we started to concentrate on the y-intercept. One student suggested that each y(max) or y(min) had to have a corresponding unique value of the y-intercept. This was really exciting. I wrote the conjecture in formal language: "If y(max) or y(min) is given, then c is unique".
Students started to disagree, but based more on instincts and a partial visualisation than reasoning. We stopped the discussion to consider and come up with convincing reasons. After a couple of minutes, one student used the algebraic expression to show us two examples:
a = 0.5, b = 2, c = 4 and y(min) = 2 and
a = 0.25, b= 2, c = 6 and y(min) = 2
These two cases with the same y(min) and different y-intercepts is a counter-example that shows the conjecture is false (see illustration).
I reminded the class that we were assuming that the prompt was correct. It was the perfect time to link the two sides of the prompt and we worked through the algebra by completing the square to finish the lesson.
The next lesson students were still talking about the inquiry. One tried to remember the conjecture, but reversed it: "If c is given, then y(max) or y(min) is unique". This is also false and students were quickly able to suggest equations for a demonstration on Desmos.
A further line of inquiry
Since the inquiry, I have been thinking about the link between the y-intercept and the turning point, rather than just the y-coordinate. If we are given one, does it imply the uniqueness of the other? What if we use the condition that a,b ≠ 0? That is the beauty of inquiry. It opens the door to the depth of mathematics.