Shawki reports on how the fast-moving inquiry developed:

Questioning

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.

Exploration

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.

Conjecture

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".

Reasoning

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.