Algebraic fractions inquiry

The prompt

Mathematical inquiry processes: Identify structure; test cases; generalise and prove. Conceptual field of inquiry: Sum and difference of fractions; difference of two squares; algebraic manipulation.

The prompt was given to a year 12 (grade 11) class at the start of their A-level course to assess and develop their mathematical reasoning and creativity. It shows the difference between the squares of the sum and difference of two fractions. One fraction is the reciprocal of the other. In the prompt, the fractions are 3/2 and 2/3 and the difference of the squares is four. The teacher displayed the equation on the board and asked What do you notice? and What do you wonder?

After five minutes of thinking time, students discussed their ideas in pairs for another five minutes before feeding back to the class. Having listened to the discussions, the teacher attempted to orchestrate the feedback so that contributions developed from the particular to the more general:

  • The equation is true if you work out the sum and difference in the brackets and square them.

  • We notice the two and three are one apart. Would it work with other digits that are one apart, such as three and four or four and five?

  • The digits are one apart. We wondered what would happen if they were two apart. If you use four and two, for example, then the answer is four.

  • We notice the digits are two, three, and four. Would the next one be three, and four for the fractions and five for the answer?

  • Does it have something to do with the difference of two squares?

  • Is the answer always four? Or does it change with the fractions?

  • We wondered if we could use algebra to prove the solution follows a pattern. Maybe it is always four.

The students' questions and observations led into different lines of inquiry. During the course of the inquiry, they realised that any fraction and its reciprocal on the left-hand side of the equation gives four as the solution. Some students moved on from particular cases to attempt a proof of a generalisation (see the example in 'Mathematical notes').

October 2021

Mathematical notes

The illustration below shows the proof that one student constructed for the case when the difference between the numerator and denominator is one. She proved the difference of the two squares is always four.

See the mathematical notes for proofs in other lines of the inquiry, including when the difference between the numerator and denominator is more than one and when using the difference of two squares.