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The sum of two fractions equals their product.

This prompt originated in a problem that Andy Strickland (a maths teacher in Worthing, UK) posed himself: Is it possible to find a pair of fractions to satisfy the statement? Since being turned into a prompt, the statement has provided the starting point for inquiry with all types of secondary school classes. Initial comments and questions include:
  • What do 'sum' and 'product' mean?
  • How do you add and multiply fractions?
  • Does it always work?
  • It will never work (sometimes accompanied by an example).
  • Can you show or prove it is true?
Students are intrigued by this statement because, at first sight, it is not possible. However, that is only because students tend to define fractions in a limited way, concentrating on proper fractions in their exploration. Indeed, proper fractions cannot satisfy the condition in the prompt. Teachers can illustrate an explanation of why not by using a number line or diagram (see box).
  
   
Diagrams to show that the sum and product of two proper fractions are not equal. The diagrams can be used to reason that the sum and product of any two proper fractions will never be equal.
  
With a move onto improper fractions (possibly under the teacher's guidance), the inquiry can develop along different pathways. (At this point, the teacher might choose to rule out the trivial solution of two fractions that both simplify to two.) In classrooms, the inquiry has passed through different phases, such as planning, exploration, generalising and evaluating (see phases of inquiry). Alternatively, it has zig-zagged between inductive and deductive reasoning (see forms of reasoning). Students as young as those in year 8 have been involved in adding and multiplying algebraic fractions. Older students have shown that fractions in the following form satisfy the condition in the prompt:
  
Another inquiry pathway develops by representing the solution set graphically (see the box).
  
Graphing as an inquiry pathway  
Mark Richards wrote to Inquiry Maths about a pathway that could develop from the prompt: "I have been looking at the 'the sum of two fractions is equal to their product' inquiry and am intending to try it in a lesson soon. I was thinking that since the solutions all satisfy x + y = xy which rearranges to y = x/(x-1) it's helpful to draw a graph of possible (x,y) solutions. It can be seen that if negative fractions are not allowed then only improper fractions will have a solution. Also, substituting in x = a/b into the rearranged equation gives y = a/(a-b). It can be proved that these two satisfy the condition and they also give an infinite number of solutions."

Mark Richards is a maths teacher at Lancaster Girls' Grammar School, UK.
  
The inquiry has been extended in classrooms by changing the prompt in the following ways:
   
The difference of two fractions equals their product.
The difference of two fractions equals their quotient.
  

Notes 
Reflections on using the inquiry for the first time by Alison Browning (a secondary school maths teacher who is on twitter @browning_alison)
Resources
Promethean flipchart    download
Smartboard notebook   download

Inquiry discussion
Year 9 students generated the ideas above in discussion about the prompt. The picture was posted on twitter by their teacher Aine Carroll‏ (@MissCarrollMath). The inquiry has advanced a long way with the appearance of improper fractions in the form (n + x)/n and (n + x)/x.
  
Questions and observations
These are the responses of a year 10 mixed attainment class in their first inquiry. Students ask about the fractions that satisfy the statement, speculate about the denominators of those fractions and even, at this early stage, suggest a change to the prompt. Emma Rouse, the class teacher, reports that "the students were brilliant."

Emma is a Lead Practitioner in the maths department at Brittons Academy in Rainham (east London, UK). You can follow her on twitter @Emmaths1618.
  
Number into algebra through inquiry
The questions about the prompt (above) come from a year 7 extension class
at Magdalen College School (Oxford, UK). The
record of inquiry covers two 35-minute lessons. Students find a pair of fractions that both simplify to two, whereupon the teacher guides students to search for other fractions that satisfy the condition in the prompt. In the second lesson, students generate different algebraic expressions (see right for an example) with the teacher ensuring the class learns and upholds mathematical conventions. Over the two lessons, students moved from particular numerical examples to general algebraic expressions. One student summarised the inquiry in the following way: "It's really funny that we started with something that no-one thought worked, but it turned out that it was interesting."

Luke Pearce, the class teacher, commented on running the inquiry: "I was quite excited to manage my first inquiry. It was pretty chaotic at the start, but there is so much scope for real mathematics. Deciding when to end an inquiry is difficult and it could easily have run into more lessons. I love the inquiry process." Luke has blogged about the inquiry here. You can follow him on twitter @discoverymaths.
   
Inquiry for deeper understanding
Amanda James used the prompt with her year 8 class (set 4 out of 5). The questions and comments above show the students' initial ideas. Even though the class seems confident working with fractions, Amanda reports that it became clear the students' lacked a deep understanding of the concept of a fraction. As the inquiry went on, Amanda set more prompts for the students to think about: 
  
Additional prompt 1 to promote reasoning
Finding the fraction of something makes it smaller.
   
Additional prompt 2 to promote fluency and generalisation
  
Amanda reported that the inquiry process led to a deeper understanding of fractions: "The class commented on how much they actually understand what a fraction is now and this gave them the confidence to perform calculations with fractions. We spent more time discussing the meaning of a fraction than
they did actually calculating with them, but their recent assessment indicates that they understood in a deeper way rather than the surface learning of 'adding fractions' or 'finding fractions of an amount'." Towards the end of the inquiry, students started to use improper fractions. They tried to express their findings in a general form by using symbols (see illustration).

Amanda James teaches mathematics at Varndean School, Brighton (UK). You can follow Amanda on twitter at @AJ0573.
   
Connecting concepts through inquiry
Ann Macdonald tried out the prompt with a year 8 class for her first experience of inquiry. She recorded the students' rich questions and comments (above) and sent Inquiry Maths this description of how their mathematical thinking connected to factorisation: 
   
"Normally when I introduce factorising for the first time I give the students a few expanding brackets questions and then ask them to 'write the question' when given an answer. This normally leads to them asking why we sometimes want our expressions with the brackets and sometimes without. My answer never sounds convincing (or convinces!). At this point I usually start thinking that teaching re-arranging formulae before introducing factorising would be a good idea so I can illustrate how factorising helps us change the subject. Yesterday during the inquiry the year 8 class managed this:
Two fractions (a/b and a/c) have the same product and sum when a = b + c ... and then this: a/b x a/c = a2/bc
and, finally, this: a/b + a/c = (ac + ab)/bc.
Two students tried to explain why a2 = ac + ab and we talked about this being difficult to explain. So then I asked them if they knew how to factorise and it seemed they had heard the word but didn't know how. I explained how to factorise the above to give 
a2 = a(c + b) and they could see the point of factorising!
  
The Deputy Headteacher looked in at this point and the students were listening to me in silence. He probably thought I was lecturing them to death. I wish he had been there for the whole lesson so he could see how all the algebra on the board had come from them! I told the kids they had created a board that looked like Sheldon Cooper's and they liked that (well the geeky ones did!)"

Ann Macdonald is a secondary school teacher and second in charge of a maths department in the UK.
   
Year 8 students' exploration and reasoning