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Simultaneous equations inquiry

This prompt can require considerable guidance at first. Students in years 10 and 11 quickly work out that x = -1, but they often overlook the key feature of the prompt - that is, the coefficients of x and y and the sum of the two amounts form a linear sequence. In the case of the prompt, we see 5, 7 and 9 and 3, 7 and 11. Obviously, the coefficient 
of y is the same in both equations to facilitate the use of the elimination method. If the teacher wishes to make the prompt more challenging, then alternatives include those on the right.
      
When the simultaneous equations are formed of linear sequences, then the solutions will always be x = -1 and y = 2. (A proof of this is given in the mathematical notes below.) Students can verify this common solution for simultaneous equations of the type in the prompt through inspection of a number of examples. They are also 
ready to acknowledge that solutions are easier to find if the coefficients of are the same and can explain why it is permissible to multiply the terms in one equation by an integer. Many classes have requested a teacher's explanation for this procedure, but often it has turned out that questioning can elicit a correct understanding from the students themselves.
    
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 = 1 and = 1?);
  • Changing the signs, initially to subtraction (or 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 (perhaps suggested by the teacher).

Notes 
Resources
Promethean flipchart    download

Classroom inquiry 
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.

You can follow Craig on twitter at @tesMaths. He also runs his own website.