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Solving advanced equations inquiry

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:
The inquiry has also been developed into solving quadratic equations by factorisation and completing the square (see box below). If a teacher uses the solving equations prompt first, it is advisable to introduce an advanced template by linking it to a student's question or observation from the initial prompt. In this way, students retain ownership of the inquiry process and, in consequence, remain more motivated to develop their mathematical understanding. 
  
Algebraic fractions
This collection of seven prompts was designed by Mark Greenaway (an advanced skills teacher in Suffolk, UK). 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. 
Looking at idea 7, for example, students can attempt to prove the general equation below is true.
You can follow Mark Greenaway on twitter @suffolkmaths.
Using students' questions to guide the inquiry
Above are the questions and observations of a year 10 (grade 9) 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.
Question Advanced template (general form)
Can you change the order? 
Can you use other operations?
ax - b = cx - d
ax - b = c - dx
a - bx = c - dx
Can you add on boxes?
ax + b = c(dx = e)
a(bx + c) = d(ex + f)
(ax + b)/c = dx + e
(ax + b)/c = (dx + e)/f
Can you use other letters?
 Simultaneous equations
ax + by = c
dx + ey = f
Can you use indices? Quadratic equations
ax2 + bx + c = d
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.