Solving simultaneous equations inquiry
Mathematical inquiry processes: Identify structure and notice properties; create example sets and generalise. Conceptual field of inquiry: Simultaneous equations; algebraic proof.
Students might require a structured approach to the prompt. They often overlook its key feature - that is, the three numbers in each equation are consecutive terms of a linear sequence. In the case of the prompt, we see 2, 5, 8 and 3, 7, 11. If the class does not notice the sequences in the orientation phase, the teacher will have draw them to the students' attention. The inquiry might continue with students attempting to find solutions for x and y that simultaneously 'work' for both. Whether or not they find x = -1 and y = -2, the teacher should be prepared to lead an explanation of an algebraic method for solving simultaneous equations.
If the class is inexperienced at inquiry or the teacher prefers a more accessible prompt, then the equations below might be better. In the alternative prompt, the coefficients of y are the same. This can facilitate the development of the elimination method as students 'see' that the differences between the left-hand and right-hand sides are 2x and 2 respectively and, thereby, intuitively eliminate the second variable. Some have difficulty appreciating that the differences are in fact 2x and -2 when calculated formally.
When the simultaneous equations are formed of three consecutive terms of linear sequences, the solutions will always be x = -1 and y = 2. (See a proof of this below.) Students can verify this common solution for simultaneous equations of the type in the alternative prompt (in which the coefficients of y are the same) through forming and solving their own examples.
Lines of inquiry
Once classes are comfortable with solving simultaneous equations, the inquiry tends to become fast-moving and multi-faceted as individuals or groups work on their own questions. Changes to the prompt have included:
Descending linear sequences (or one ascending and one descending);
Using sequences of other types, particularly quadratic and Fibonacci. Why are the solutions for the latter always x = 1 and y = 1?;
Changing both signs to subtraction or to addition and subtraction;
Representing the equations on a graph (What is the relevance of the point of intersection?); and
Using x2 or y2 or both.
Craig Barton, TES adviser for secondary maths, tried out the inquiry with his year 10 class. He tweeted about the lesson (below) and posted a picture of the approach taken by one student. She inquired into sequences that use the recurrence relation for the Fibonacci sequence: xn = xn-1 + xn-2. Her theory states that simultaneous equations using any three consecutive numbers in such a sequence give the solution x = 1 and y = 1.
Since using the prompt in 2013 at the time Inquiry Maths was the subject of his podcast, Craig Barton has adopted a different approach to the teaching of mathematics based on the application of cognitive science (see the EEF review of the research). He now takes a critical position towards Inquiry Maths, decrying the time wasted compared to the supposed efficiency of explicit instruction.
Our reply to Craig notes that his new approach impoverishes the mathematics classroom by removing, amongst other things, the excitement evident in his tweet.
This is a proof from a year 12 student studying the A-level course. An alternative proof is given in the mathematical notes.