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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. 

Prompt sheet
Promethean flipchart    download
Smartboard notebook   download

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.

Mark Greenaway runs his own highly-recommended website, which contains a huge range of ideas for teaching mathematics. Mark can be followed on twitter @suffolkmaths.
Training teachers of the future through inquiry
Nichola Sowinska
devised the prompt above for a two-day course she was running at the London Metropolitan University (UK). The students were on the Subject K
nowledge Enhancement programme prior to starting a teacher training course. The initial questions they posed about the prompt gave Nichola an insight into their levels of mathematical knowledge and creativity.
Nichola reports that the students assumed the equation was quadratic before realising in discussion that the power of x did not have to be two. The inquiry developed, with the use of @Desmos, into exploring quadratic equations fully and analysing the differences between graphs of equations with even and odd powers.
Jamal King, one of the students, thought the course was very well delivered: "The style of learning, being different to the traditional, really encouraged the class to engage in discussions to find solutions." Nichola adds that the success of the inquiry in the context of teacher training was twofold: "It opened the students' eyes to a different pedagogical approach. It also allowed them to put the formulas they remembered from school into a deeper context and understand links which they had not had the opportunity to make before." Reflecting on the times she has used inquiry, Nichola remarked: "Every time I teach in this way I am always surprised at what my students can do."

Nichola Sowinska
is a maths teacher in Peterborough (UK). She is studying for a Masters degree at the Institute of Education, London. You can follow Nichola on twitter @NixxSunshine.
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.
Extending the prompt to quadratic equations
An extension to the solving equations prompt is provided by the template involving quadratic equations. Questions and comments that arise about this
two-part prompt include:
•  Do numbers go in the boxes?
•  Can all the numbers be the same?
•  Can they be decimals or fractions?
•  What if we use a 'minus'?
•  Are the two sides really equal?
•  Are they equal whatever numbers you use?
Presenting the equation in both forms requires students to consider whether it is possible to "go both ways." Older secondary school students will often have an idea of expanding brackets. If not, students can request teacher instruction. I use the grid method initially so students can clearly see the provenance of the different parts of the solution. The grid method is useful in other ways. Firstly, it helps when discussing the expansion of brackets with younger students because they can appreciate the links between arithmetic and algebraic operations. Secondly, it facilitates an analysis of the lower half of the prompt, which can be more problematic for students than the upper part. Through using the grid "in reverse", students begin to see links between the two parts of the prompt and develop a nascent understanding of factorisation.
These are the initial thoughts of a year 10 high ability class. The teacher reports that instead of the expected questions, the students followed the lead of two groups by putting numbers into the boxes. The class had met the expansion of two linear expressions before and set about reminding themselves how to do that. The inquiry soon developed onto factorisation. Through analysis with the grids, the class began to appreciate the links between the numerical values in the equation.
A further extension might involve working "backwards" from the second part of the equation template (left). Again, an understanding of solving quadratic equations by completing the square can begin in the exploration of this prompt.