Adding fractions inquiry
Mathematical inquiry processes: Identify and create patterns; conjecture and generalise. Conceptual field of inquiry: Addition and subtraction of fractions.
The prompt was devised by Mark Greenaway (an Advanced Skills Teacher in Suffolk, UK) to encourage students to analyse the sum of two unit fractions in which the denominators are in the form n and n + 1.
Initial questions and observations
The teacher can assess students' level of understanding of fractions through their initial questions and observations about the prompt. The board below, which was constructed by a year 7 mixed attainment class, shows that at least some students have sound prior knowledge. The early phases of the inquiry might, therefore, involve spreading that knowledge and ensuring students are secure in finding the sum of two fractions, rather than involve a teacher explanation.
Some students are already beginning to speculate about the next case and the rules for finding the sum of a quarter and a fifth. They suggest two different rules:
Add two to the numerator and six to the denominator, giving 9/18; and
Add two to the numerator and double the denominator, giving 9/24.
That neither are correct intrigues students. In fact, the denominator increases in a quadratic sequence: 6, 12, 20, 30, ...
Teachers might have misgivings about the prompt's potential to sow misconceptions or to focus students' thinking on the operations rather than underlying concepts.
For example, students often notice that the sum and product of the denominators on the left-hand side of the equation give, respectively, the numerator and denominator on the right-hand side. They might try to generalise the two 'rules' to other cases without realising that they only work in the case of unit fractions.
However, the value of the prompt lies precisely in the way it exposes students' misconceptions and procedural thinking that already exist.
Noticeably, the board above does not feature the comment that regularly occurs at the initial stage of the inquiry: "The first answer is wrong because 1 + 1 = 2 and 2 + 3 = 5, so it should be 2/5." In bringing misconceptions to the surface, the inquiry gives teachers the chance to tackle them (see 'Tackling misconceptions' below).
Ultimately, the inquiry could lead into deductive algebraic proof in all years of secondary or high school. For unit fractions, the teacher could introduce:
1/n + 1/(n + 1) = n + 1/n(n +1 ) + n/n(n + 1) = n + (n + 1)/n(n + 1) .
Students might then be expected to construct other proofs for when the numerators are any integer and equal (x/n + x/(n + 1)) or any integer and different (x/n + (x + k)/(n + 1)).
Matthew Bernstein, a teacher of a grade 5/6 class at the Fred Varley Public School (Markham, Ontario), posted these pictures on twitter. He describes how the student-driven inquiry developed:
Students used Google Jamboard (we are still hybrid) to make observations and ask some great questions regarding what they saw.
Within the lots of ideas, there were two that they were interested in exploring: (1) If you add the denominators together it makes the numerator; (2) Can you always multiply the denominators of fractions together in an equation to get a common one?
Students were very curious about this and the class was excited to explore these questions together in small groups to either prove or disprove them. They really enjoyed finding patterns to see if they could be generalized. Students used Mathigon Polypad to help represent ideas visually.
There is no doubt that it helped that most were familiar with the notion of needing to find a common denominator, but I think that's what made the task that much richer for them. I do think this could be done if students were unfamiliar but it would go in a different direction.
The inquiry in a grade 4 classroom
The first picture shows the initial thoughts and questions of Amanda Klahn's grade 4 PYP class at the Western Academy of Beijing, China.
Students are starting to look for patterns linking the two fractions with their sum. They have found the rules for adding and multiplying the denominators to get the numerator and denominator respectively in the sum.
A student has extended the rules to a new case (one fifth and one seventh) in which the denominators are in the form n and (n + 2).
While the fractions remain unit fractions, the rules continue to give the correct answer - in the new case, 1/5 + 1/7 = (5 + 7)/(5 x 7) = 12/35 .
Other students' presentations show the use of manipulatives (fraction bricks) to explain the calculations in the prompt. The final picture shows a student has changed the numerator to two, going on to find the sum of 2/14 and 2/15 .
There are more rich mathematical results from the inquiry on the class blog.
The inquiry in a mixed attainment class
The picture shows the questions and observations of a year 7 mixed attainment class in an inner-city comprehensive school in the UK. They reveal a wide variety of prior knowledge and approaches to the prompt. At least one student evidently knows how to add fractions, another has a partial recollection that a common denominator is required, and another perpetuates the misconception that you add numerators and denominators separately. Other students prefer to speculate about the sum of a quarter and a fifth by extending the pattern from the two examples in the prompt.
When given the choice of six regulatory cards, the class required an explanation of how to add fractions, which the teacher orchestrated by drawing on the knowledge that already existed in the classroom. The students then opted either to practise a procedure (adding fractions) or find more examples by summing unit fractions.
The first lesson ended with a pair of students presenting the general form of the fraction on the right-hand side of each equation, with n being the denominator of the first fraction: (2n + 1)/n(n + 1).
At the start of the second lesson, other students explained on a number line why the equations in the prompt are correct. The class then created their own lines of inquiry by changing features of the prompt.
Lines of inquiry
(1) Changing the numerator
How do the results change when the numerator is greater than one? For example, 2/3 + 2/4 = 14/12 and 2/4 + 2/5 = 18/20
What if the numerators have a difference of one? For example, 2/3 + 3/4 = 17/12 and 2/4 + 3/5 = 22/20
How does the general form change for the new cases?
(2) Changing the difference between the denominators
How do the results change when the difference between the denominators is greater than one? For example, 1/2 + 1/4 = 6/8 and 1/3 + 1/5 = 8/15 or 1/2 + 1/5 = 7/10 and 1/3 + 1/6 = 9/18
How does the general form change in each case?
(3) Summing three 'consecutive' unit fractions
Can you find the sum of three 'consecutive' unit fractions? What about 1/2 + 1/3 + 1/4? The teacher, using a number line, contributed an explanation of how three unit fractions were not consecutive in the same way as three positive integers. Indeed, the class resolved not to use the term 'consecutive' in the context of fractions unless they (or their equivalents) had the same denominators and numerators with a difference of one. In this definition, a third, a quarter and a fifth are not consecutive.
The inquiry ended with students giving presentations about their findings and the patterns they had noticed.
Engagement and creativity through inquiry
Emmy Bennett, a teacher of mathematics at Priory School, Edgbaston (UK), used the adding fractions prompt to initiate inquiries with her two year 7 classes. The pupils responded in highly creative ways and developed multiple lines of inquiry. Emmy reports:
After the success of my initial inquiry lesson with a year 9 class (see Challenge through inquiry), I decided to try the adding fractions prompt with my two year seven classes. In both classes, the pupils started by discussing the prompt in pairs. Then we shared ideas and decided where to go with the inquiry.
With one class they spent quite a bit of time deciding if the prompt was true and some pupils chose to practice adding fractions after some examples. The picture shows the different strands of the inquiry. The pupils explored some equivalent fractions and were enthusiastic to notice all the properties of the initial prompt as they could.
In the other year seven lesson pupils were interested in finding more examples or changing the prompt to find other patterns.
For this lesson I asked pupils who found more examples to write them on the whiteboard as we went along. (I’m lucky enough to have three whiteboards at the front of my classroom). The pupils loved this and, at one point, there were eight pupils writing on the boards.
One pupil was really interested in looking at examples when the difference and product of two fractions are the same. He called it a 'maths hack' and initially said, 'It doesn’t work when the denominators are two apart.' However, he kept going and noticed that the numerator of the difference became the difference of the initial denominators. The picture shows the record of the pupils’ inquiry.
All the pupils in the two classes were fully engaged throughout the lessons. Unfortunately, I did this inquiry on the last day of term so we couldn't spend more time on it, but some pupils said they were going to explore more at home. It was an absolute joy to teach in this way and I can’t wait to try more inquiries in the future.
Helen Hindle, Hugh Salter and Andrew Blair, three teachers at Longhill High School (Brighton, UK), used the prompt to challenge students' misconceptions about adding fractions. The inquiries that developed from the prompt featured hugely valuable discussions in which entrenched notions were challenged and an understanding of the concept of a fraction was reconstructed by students and the teacher.
Andrew Blair reports on the lesson study:
In the lesson study cycle, which involved year 7 classes, I went first. I decided to use a number line as a tool with which to approach the concepts of a fraction and then of adding fractions. Before showing the class the prompt, we started by locating fractions on a number line.
This led immediately to our first misconception about representing 1/6, which one student argued should be placed half way along the number line as six is half of twelve.
Speculation about patterns
The students' questions and comments about the prompt provided a strong foundation for inquiry. In particular, the speculation around the solution to 1/4 + 1/5 motivated the students to request instruction in how to add fractions. Should we continue the sequence 5/6, 7/12 by adding two to the numerator and six to the denominator, giving 9/18? Or should we apply the 'rule' derived from the denominators of the unit fractions? In the latter case, we would find their sum for the numerator and their product for the denominator, giving 9/20 .
Inquiry for all attainment levels
One class with lower prior attainment that was part of the lesson study posed meaningful questions and made insightful observations (see picture). The students' responses show the potential of the prompt to promote questioning and noticing in all classes.
As the inquiries developed, students were taught to link the number of intervals on the number line with the product of the denominators. The students then showed the sum of any two fractions on a number line by using equivalent fractions. So, typically, a student went on to show 1/4 + 1/5 on a number line of length 20, explain why it is equivalent to 5/20 + 4/20, and give the solution 9/20.
Misconceptions identified during the lesson study
1/6 should be placed half way along a line of 12 units because the 'number' six stands for the length along the line. The student who said this had no problem marking a quarter. Thus, while students might have a sense of 1/4, they might not have developed a conceptual understanding.
Having started with number lines of length 12 units, students refused to use a line of six units to show the first calculation in the prompt. This revealed an inability to conceive of a fraction as part of any whole. Once a third was represented by an arrow four units along a line of length 12, students would not accept it could also be shown as two units along a shorter line of length six.
To add two fractions, students claimed, you add the numerators and then the denominators separately. Thus, 1/2 + 1/3 = 2/5. This shows a misconception of fractions as two unrelated 'numbers'.
To add two fractions, you add the denominators to get the numerator in the answer and multiply them to get the denominator. (As students realise during their inquiry, this works for unit fractions, but not when the numerator is greater than one.)
When showing the solution to 1/2 + 1/3 on a number line, students start both fractions at zero, rather than place one fraction after the other (see illustration). This idea that the fraction of a line can only be shown from zero was surprisingly common.
Questions to extend the inquiry
The following questions come from year 7 students at Haverstock School (Camden, London, UK) midway through the inquiry:
(1) "Would it ever be true if you switched the numerators and denominators?" The students could not find any values to make this true.
(2) "If you switched the numerators and denominators in the question, could they be equal?" The students found values for a, b, c and d that satisfy the equation. They realised that ac = bd.
The questions expressed formally would be:
Terry Patterson, a maths teacher in London, contacted Inquiry Maths about a prompt she had devised. Terry's first experience of an inquiry lesson came when she used the prompt with her year 8 class. She commented on the emotional impact an inquiry can have: "The students' questions are moving and revealing. They loved running the lesson. I was quite choked up after my first lesson yesterday - an eye-opener."
The class had low prior attainment in maths and the prompt gave Terry an insight into the students' level of understanding: "Every question they posed revealed the group's bafflement." The questions included:
Why does 1 - 1 = 1?
Why does 2 - 3 = 6?
Is it to do with times tables?
The last question could follow from identifying supposed links between the numerators and denominators - that is, 1 x 1 = 1 and 2 x 3 = 6 respectively. The questions reveal the kinds of misconceptions that are common when students are faced with fractions prompts.