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Solving equations inquiry

The prompt acts as a template for students to explore linear equations. It has often led to an inquiry that combines inductive and deductive reasoning in a mutually supportive process.

The inductive side can be developed from a consideration of whether equations of this form always have a solution. As students solve particular cases, the conjecture soon arises that they do (unless the coefficients of x are the same). Deductive reasoning is a feature of the solving process. It can also be evident in students' attempts to solve the general form ax + b = cx + d, giving x in terms of a, b, c and d.

In a structured inquiry, the teacher could start the inquiry by requesting integers to place in the boxes. Even with this closed start, the prompt has the potential to generate various suggestions for further inquiry. Some examples from classes in lower secondary school are:
• Change the operation(s) to subtraction.
• Use fractions instead of integers.
• Re-design the equation with the unknown on one side only.
• Include a second variable on both sides of the equation.
• Re-design the equation to include an algebraic fraction.
• Add a third expression equal to the other two.
2x + 4 = 6x + 8
2x + 4 = 6x - 8
2x - 4 = 6x + 8
2x - 4 = 6x - 8
As students change the operation, they have created sets of equations. The set above gives the solutions, respectively, x = -1, 3, -3 and 1. Students have then gone on to explore solutions for other sets and to explain their solutions.

Richard Goodman (Principal Lecturer at the University of Brighton, UK) contacted inquiry maths to make the following comment about the prompt: "I like the solving equations inquiry, but I am surprised you describe it as closed at the start. My reaction was to close it down a bit further - asking what happens if we know the solution? Or what about using the same two numbers in different order on either side (2x + 5 = 5x + 2)?"

Notes
The inquiry can develop into solving advanced equations (including simultaneous and quadratic equations, and algebraic fractions). See reports on classroom inquiries using advanced templates here.
Lesson notes
Resources

Source of the prompt
Since posting this inquiry, I have come across an excellent paper by Colin Foster on embedding opportunities for students to develop procedural fluency within rich contexts. In the paper, Colin discusses a 'mathematical etude', which he defines as a framework or problem to achieve the twin aims of fluency and exploratory investigation. One 'etude' (right) is almost identical to the inquiry prompt on this page. It seems improbable that I developed the prompt independently of Colin's work. Therefore, it is with great pleasure that I acknowledge the source of the inquiry prompt. Colin also has a website of pictorial prompts that can be used to stimulate inquiry in secondary school classrooms.

Colin Foster is Assistant Professor of Mathematics Education at the University of Nottingham, UK. You can follow him on twitter @colinfoster77.
Multiple inquiry pathways
The picture shows the questions and observations of students at The Bishop Wand Church of England Secondary School and Sixth Form (Sunbury, UK). There are a number of interesting lines of inquiry that could develop:
• Using sequences of numbers (2 4 6 8);
• Using fractions or decimals;
• Finding numbers that give different types of answers, such as negative numbers;
• Deciding if there is always a solution; and
• Establishing a rule for each side of the equation, such as using different numbers on either side (4 4 5 5), reversing the same numbers (6 2 2 6) or using numbers with the same relationship (3 3 2 4).
The final idea could lead into a proof that for equations of the type nx + n = (n - 1)x + (n + 1), the solution is always x = 1. The teacher reports that the lesson involved "impressive and thoughtful inquiries," which included lots more pathways than the few shown in the picture.

You can follow the Maths Department of The Bishop Wand School on twitter @BishopWandMaths.

Classroom inquiry in year 7
These are the questions and comments of a year 7 (grade 6) mixed attainment class in a UK secondary school. As the students were relatively experienced in mathematical inquiry, the teacher decided to run an open inquiry. The questions and comments led to three hours of classroom inquiry.

Finding a general solution
Caitriona Martin's year 8 high attaining class asked the following questions about the prompt:
 Content of the boxes - Could the same number go in the boxes?- Can negative numbers go in the boxes?- Can we use fractions or decimals?- ax + b = bx + a or ax + b = cx + d?- Could the boxes be algebraic terms? Solving the equation - Can you solve the equation by putting numbers in the boxes?- If you put any number in the boxes, can you solve it? Value of x - Can the x's be numbers? Changing the prompt - Could you put in brackets?- Would it work with subtracting, dividing or multiplying?
The inquiry ended with the demonstration below of a general solution for solving an equation in the form ax + b = cx + d . As Caitriona comments, it is a "great example of pupils stretching just beyond their current knowledge", which is precisely the aim of the inquiry prompts.

Questioning and noticing
Amy Flood recorded these questions and observations from her year 8 class at Haverstock School (Camden, UK). The students are attempting to understand the prompt by applying their existing knowledge about equations. Amy was able to assess the students' existing level of knowledge in order to construct a guided inquiry.

Amy is second-in-charge in the mathematics department.

Developing resilience
Alison Browning used the solving equations prompt with her year 7 class. She was particularly interested in developing resilience and asked the students to reflect on how they felt during the different phases of the inquiry. The students drew diagrams to show how they had moved from confusion to understanding and solving their own equations. Alison sent inquiry maths a sample of the diagrams.

Alison Browning is a secondary school maths teacher in the UK and can be followed on twitter @browning_alison.