Sum and product of fractions inquiry
Mathematical inquiry processes: Search for examples that satisfy the condition; conjecture, generalise and prove. Conceptual field of inquiry: Addition and multiplication of fractions (extended to subtraction and division).
This prompt originated in a problem that Andy Strickland (a mathematics 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 diagram.
The diagrams 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:
Questions and observations
These are the responses of a year 10 mixed attainment class at Brittons Academy in Rainham (UK) 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 and Lead Practitioner in the mathematics department, reports that "the students were brilliant."
Year 9 students generated the ideas above in discussion about the prompt. The picture was posted on twitter by their teacher Aine Carroll. The inquiry has advanced a long way with the appearance of improper fractions in the form (n + x)/n and (n + x)/x.
Inquiry for deeper understanding
Amanda James, a teacher of mathematics at Varndean School (Brighton, UK), 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. The first additional prompt stated that, "Finding the fraction of something makes it smaller."
Amanda designed the second additional prompt (below) 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 then tried to express their findings in a general form by using symbols.
Number into algebra through inquiry
The questions about the prompt (above) come from a year 7 extension class at Magdalen College School (Oxford, UK). This record of inquiry covers two 35-minute lessons. Students found a pair of fractions that both simplify to two, whereupon the teacher guided them to search for other fractions that satisfy the condition in the prompt. In the second lesson, students generated different algebraic expressions with the teacher ensuring the class upheld mathematical conventions (see below).
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 because it gives students the experience of being real mathematicians, something which is far too rarely the case in schools today. They loved it."
Rachel Mahoney (a mathematics teacher at Carre's Grammar School in Sleaford (Lincolnshire, UK) blogs about a structured inquiry she ran with her year 8 class. She reports that the students were "engaged and excellent conversations took place."
Connecting concepts through inquiry
Ann Macdonald, a secondary school teacher of mathematics, 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!)
New line of inquiry
The difference of two fractions equals their quotient.
Silena, a year 9 student, developed a new line of inquiry when she set out to find two fractions whose difference and quotient are equal. Adopting the algebraic approach she had used at the end of the main inquiry into the sum and product of two fractions, Silena deduced a formula for the relationship between a, b, c and d when the difference and quotient of a/b and c/d are equal (see below).
Silena used the formula to find values of a, b, c and d that satisfied the prompt. In her first example a = 25, b = 6, c = 5 and d = 3. The difference of the two fractions (25/6 and 5/3) equals their quotient.
A second and third example derived from the formula also satisfy the requirement for the difference and quotient to be equal. However, a fourth example did not work. The difference and quotient of 81/16 and 9/8 (a = 81, b = 16, c = 9 and d = 8) are not equal.
The fourth example led Silena to give more conditions for the relationship between a, b, c and d:
a = c2
b = 2d
c and d are both odd or even, c > d.
It is the last condition that example 4 does not meet because c = 8 and d = 9.
By applying conditions 1 and 2 to the algebraic fractions, we can deduce that c - 2 = d, which explains why c and d have to be both odd or even in condition 3.
Year 8 students' exploration and reasoning
Other lines of inquiry
Mark Richards, a teacher of mathematics at Lancaster Girls' Grammar School (UK), 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."
Extending the inquiry
These supplementary prompts can extend the inquiry:
The difference of two fractions equals their product.
The difference of two fractions equals their quotient.
The general and the particular in mathematical inquiry
Daniel Walker (a teacher in north London, UK) contacted the Inquiry Maths website with this prompt:The prompt gives a specific case of the general statement relating to the difference and product of two fractions (above). This prompt leads to a different type of inquiry that moves from a particular example to a general conjecture. The movement towards generalisation is opposite to the one followed by the main inquiry on this page (at least in its initial stages). A general statement leads to search for a specific example that confirms the contention in the prompt. As Artigue and Baptist say here, mathematical inquiry "can lead to a universal result (showing that all objects belonging to the same category share a given property), but it can also lead to an existential result (showing that there exists at least one object fulfilling a set of given conditions)."
Reflections on first-time inquiry
Alison Browning (who was a secondary school mathematics teacher at Varndean School, Brighton, at the time of the lessons) used the prompt for her first inquiry. She chose her year 8 mixed attainment class and this was also the students' first inquiry. She recorded her reflections on the course of the inquiry here.