Solving advanced equations inquiry
Mathematical inquiry processes: Identify structure; create example sets and generalise. Conceptual field of inquiry: Fractional equations with the unknown on both sides; algebraic generalisation.
This prompt is an example of a template that could develop out of the solving equations inquiry. Alternatively, the teacher could give the prompt to a class that already has experience of solving equations. Other templates for an inquiry could include:
If you use the solving equations prompt first, you might introduce the advanced templates by linking them to a student's question or observation from the initial prompt (see below). In this way, students retain ownership of the inquiry process and, in consequence, remain more motivated to develop their mathematical understanding.
Mark Greenaway designed this collection of seven prompts. The first two prompts are more general and could be used to initiate a open inquiry. He used ideas 3 to 7 in the same inquiry, giving out the prompts to students for verification, generalisation, and proof.
Using students' questions to guide the inquiry
Above are the questions and observations of a year 10 class that had experience of carrying out mathematical inquiries. The teacher used a number of the questions to guide the inquiry into the advanced templates. Linking a new template to a specific question is important because students feel they are retaining ownership of the inquiry even when they may not have constructed the template themselves. While the templates emerge in a process of co-construction between teacher and students, they are firmly rooted in students' agency and creativity.
The comment about using variables a, b, c and d leads into finding a general solution for each template. Examples of students' attempts to find general solutions are shown below.