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