Fraction of a number inquiry
Mathematical inquiry processes: Verify, generate more examples; generalise and prove. Conceptual field of inquiry: Fraction of a number; equivalence.
Helen Hindle (Head of Department at Park View School, Haringey, London) adapted the percentages prompt for a lesson study she was leading with two colleagues from local schools. The teachers designed the prompt about finding the fraction of a number to initiate inquiry with their lower attaining classes. Before starting the lesson study, each teacher chose three of their own students on whom to focus observations. Each teacher hypothesised about the students’ reactions to the prompt, their prior and developing conceptual knowledge, and their ability to regulate activity. The teachers ran the inquiry in turn while the two colleagues observed the focus students. In Helen’s class, the students gave the following responses to the prompt:
The numbers are switched around.
It has the same numbers in a different order.
Both denominators are 10.
4 ÷ 10 = 0.4 and 7 ÷ 10 = 0.7.
4/10 of 70 = 28 and 7/10 of 40 = 28 – both are equal and the statement is correct.
In order to support the inquiry process, the teachers prepared four resources offering increasing levels of independence. Afterwards they compared how they had used the resources. In one classroom, the teacher directed each student towards a particular resource; in another, the teacher allowed students to choose one of the four; and in the third, the teacher only introduced the resources after the students had chosen the regulatory cards Ask the teacher for something to do or Use a worksheet or textbook. After Helen’s first lesson, the students filled out a questionnaire, reporting that they had a deeper understanding of fractions and a clearer idea of the aims than in other lessons.
One pathway that the inquiry can take leads students into improper fractions. They can find it difficult to believe that, for example, 56/7 of 2 = 2/7 of 56.
Developing algebraic reasoning
Jordan Wilson (a maths teacher at Longhill High School in Brighton, UK) sent pictures of an inquiry he conducted with his high attaining year 8 class. He explains that, "I used the prompt as a starting point and the class really did engage with it. The majority wanted to prove it and a few of them started using algebraic notation to show how it works. I also got a few of them to use two fractions before the number to see if that worked as an idea. What I will add is how how good inquiries can be for the understanding of a class if delivered well."
Questions and observations
These are the questions and observations of a year 8 mixed attainment class. Two inquiry pathways emerged from the final observation (bottom right): firstly, to create more questions that equal 28; and, secondly, to find more examples involving different denominators.
This is a picture of the initial questions and observations of a year 8 class with high prior attainment at Coleshill School (Warwickshire, UK). One notable idea involves inverting the fractions. The class teacher, Miss L Costa, reports that the students enjoyed the inquiry that developed over the lesson.