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Fraction of a number inquiry

Helen Hindle
(a Lead Practitioner in Brighton, UK) 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 (above) 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, for instance, they had a deeper understanding of fractions and a clearer idea of the aims than in other lessons.
Mathematical note
One pathway of the inquiry leads students into improper fractions. They can find it difficult to believe that, for example, 56/7 of 2 = 2/7 of 56.

Helen Hindle
has an excellent 
website on developing a growth mindset in maths. You can follow her on twitter @HelenHindle1. Helen's account of the lesson study can be found here.

Questions and observations
These are the initial questions and observations of a year 8 mixed attainment class. Two inquiry pathways that emerged from the final observation (bottom right) were, firstly, to create more questions that equal 28; and, secondly, to find more equalities involving different denominators.
Below 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 (on twitter @misscostamaths), reports that the students enjoyed the inquiry that developed over the lesson.