Adding fractions inquiry

The prompt

Mathematical inquiry processes: Identify and create patterns; conjecture and generalise. Conceptual field of inquiry: Equivalent fractions; addition and subtraction of fractions.

Mark Greenaway (an Advanced Skills Teacher in Suffolk, UK) devised the prompt for students in year 7 (grade 6). It leads to speculation about the sum of two unit fractions in which the denominators are in the form n and n + 1.

Question, notice, and wonder

The teacher can assess students' level of understanding of fractions from their questions and observations in the orientation phase of the inquiry.

The board below, which shows the responses from a year 7 mixed attainment class, suggests that at least some students have sound prior knowledge. The early phases of the inquiry might, therefore, involve spreading that knowledge through students' explanations, rather than involve teacher instruction.

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, ...

Tackling misconceptions

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.

Proof

Ultimately, the inquiry could lead into algebraic proof in all years of secondary or high school. For unit fractions, the teacher could introduce:

Students might then be expected to construct other proofs when the numerators are any integer (a) or integers with a difference of k (a and a + k):

June 2014

Structured inquiry

A teacher new to inquiry learning might prefer to run a structured inquiry, particularly if the class is not used to exploring a prompt, making conjectures, and explaining ideas.

A structured inquiry starts in the same way as other types of inquiry with students responding to the prompt in a question, notice, and wonder phase. The teacher might consider dealing with one or two responses immediately but, in general, the inquiry will follow the planned lines of inquiry in the Adding fractions structured inquiry (PowerPoint).

The planned steps include:

Explain how to add fractions using a number line;
Extend the prompt and identify patterns and rules;
Explore other pairs of fractions that share a property (see the picture for an example);
Generalise using algebraic notation if appropriate; and
Follow other lines of inquiry.

To ensure students feel their contributions from the start of the inquiry are valued, the teacher should return to them before the end. This will involve deciding whether the questions have been answered and the comments addressed. If any remain open, the class might decide to pursue another line of inquiry.

January 2023

Student-driven inquiry

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 generalised. 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.

June 2022

The inquiry in a grade 4 classroom

The 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.

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). As the fractions are still unit fractions, the rules continue to work - that is, 1/5 + 1/7 = (5 + 7)/(5 x 7) = 12/35 .

Students' presentations (below) show the use of manipulatives (fraction bricks) to explain the calculations in the prompt. In the final picture a student has changed the numerator to two, going on to find the sum of 2/14 and 2/15 .

The inquiry in a mixed attainment class

The picture shows the questions and observations of a year 7 mixed attainment class at Haverstock School (Camden. UK). They reveal a wide variety of prior knowledge and approaches to the prompt. At least one pair of students know how to add fractions, another has a partial recollection that a common denominator is required, and a third 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.

Lines of inquiry

The first lesson ended with a pair of students presenting the general form of the sum of the two fractions, with n being the denominator of the first fraction: (2n + 1)/n(n + 1).

At the start of the second lesson, other students used a number line to explain why the equations in the prompt are correct. The class then created their own lines of inquiry by changing features of the prompt:

(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) Finding the sum of 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, explained that the three unit fractions are not consecutive in the same way as three positive integers. A half, a third, and a quarter are six, four, and three twelfths respectively. As five-twelfths is missing, the fractions are not consecutive. However, the class decided that the digits in the denominators could be described as 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.

Students' questions

(Top) The picture shows the questions and observations from one of Mark Greenaway's classes. (Below) Year 7 students wrote their questions on whiteboards mounted on the walls around the classroom.

Questions to extend the inquiry

Shawki Dayekh, a teacher of mathematics, reports that the following questions came from year 7 students at Haverstock School (Camden, UK) midway through the inquiry:

(1) Would two fraction equations ever be correct if you switched the numerators and denominators?

(2) If you switched the numerators and denominators in a pair of fractions, could the sum of the new pair of fractions equal the sum of the original pair?

These are the extension questions expressed in formal mathematical language:

Alternative prompts

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.

The prompt was devised by Janice Novakowski to challenge pupils' misconceptions about adding fractions. Students justified why the prompt was wrong using diagrams and equivalent fractions.