Adding fractions inquiry
This 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. I am a recent convert to this prompt. My initial reaction was to have misgivings about the prompt's potential to sow misconceptions in students' minds or to focus their thinking on the operations rather than underlying concepts. However, in a recent lesson study with two departmental colleagues (Helen Hindle and Hugh Salter), I have changed my mind completely. Indeed, the value of the prompt lies precisely in the way it exposes students' misconceptions and procedural thinking that already exist. 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, June 2014
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 (see below).
The students' questions and comments about the prompt (below) 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 to 9/18 or should we use the 'rule' derived from the denominators of the unit fractions (add them for the numerator; multiply for the denominator)?
One class with lower prior attainment that was part of the lesson study posed meaningful questions and made insightful observations (below). The 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.
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 below 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.
The inquiry in a grade 4 classroom
These are the initial thoughts and questions of Amanda Klahn's grade 4 PYP class. Impressively, one student has extended the procedure of adding and multiplying the denominators to a case (one fifth add one seventh) in which they are in the form n and (n + 2). While the fractions remain unit fractions, the procedure continues to give the correct answer.
Another student's presentation (below) shows the use of manipulatives (fraction bricks) to explain the calculations in the prompt. There are more rich mathematical results from the inquiry on the class blog.
At the time of the inquiry, Amanda taught at the Western Academy of Beijing, China.
Creating lines of inquiry
These are the questions and observations of a year 7 mixed attainment class in an inner-city comprehensive school. They show 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 or find more examples, which meant, in this case, to continue to find the sum of unit fractions. Four students preferred to consolidate their knowledge of equivalent fractions by selecting a matching task prepared by the teacher.
In the second lesson, students began to create their own lines of inquiry by changing features of the prompt. For example, they explored the results of changing the numerator (see below).
Others changed the difference between the denominators (from one to two or three) or found the sum of three 'consecutive' unit fractions. The inquiry ended with students giving presentations about the patterns they had noticed. The teacher, using a number line, also contributed an explanation of how three unit fractions were consecutive in a different way to three integers. Indeed, the class resolved not to use the term 'consecutive' in the context of fractions unless the denominators were the same and the numerators had a difference of one. In this definition, a third and a quarter are consecutive, but a third, a quarter and a fifth are not.
The inquiry with a mixed attainment class
After year 7 students asked the questions and comments above, the teacher offered them six regulatory cards . A third of the students were confident enough in adding fractions to Make up more examples or attempt to answer the problem about how the right-hand side continues. A few students requested an explanation; the teacher invited them to the front for a short tutorial. The others selected either Work with another student or Ask the teacher for something to do.
The teacher had three levels of tasks linked to a number line: marking fractions on a number line, finding the fraction of a number to develop an understanding of equivalent fractions, and adding fractions. At each stage, students were required to make up their own examples to assess their level of confidence.
The students who set out to answer the problem concluded, with the teachers help, that the general form of the fraction on the right-hand side (with n being, for them, the denominator of the first fraction) is:
At the start of the second lesson, the same students explained on a number line why the equations in the prompt are correct. They then decided to change the prompt and made the numerators two and then three, proceeding to adapt the generalisation. Other students made up more difficult equations involving the addition and subtraction of fractions with co-prime denominators, which they took great delight in presenting to their peers.
Students' questions to extend the inquiry
These questions come from year 7 students at Haverstock School (Camden, London, UK). They were expressed in the following ways:
(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:
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.