Fractions and sets

This task is about sharing a set and naming fractions of the set.

a)	Draw lines to show how to share 8 cats equally between 2 families. What is \(1 \over 2\) of 8? ______ What is \(1 \over 4\) of 8? ______

b)	Draw lines to show how to share 12 apples equally between 4 friends. What is \(1 \over 4\) of 12? ______ What is \(3 \over 4\) of 12? ______

c)	Draw lines to show how to share 18 eggs equally between 3 children. What is \(1 \over 3\) of 18? ______ What is \(2 \over 3\) of 18? ______

Teaching and learning:

This resource is about partitioning and finding fractions of quantities (and sets).

Students should first have an understanding of fractions as part-whole relationships before beginning to explore how to find the fraction of a quantity (fractions as operators).

Diagnostic and formative information:

Most students who attempted to show how to partition in part i) drew lines to correctly divide the set into the parts required. Across all three questions an average of 24% of students gave the correct solution for the unit fraction, but gave an incorrect solution for the non-unit fraction (11% of students gave the same solution for both fractions).

	Common error	Likely misconception
a) ii) b) ii) c) ii)	2 4 3	Students do not have an understanding of fractions as part-whole relationships, and simply applied a method of using the denominator to name the fractional part, e.g., ¹/₂ of 8 is 2; or ¹/₃ of 18 is 3, irrespective of the quantity the fractional part is the denominator.
a) iii) b) iii) c) iii)	4 [a) ii) 4] 3 [c) ii) 3] 6 [c) ii) 6]	Students have calculated correctly for the unit fraction, but could not solve for the non unit fraction. Some believe that only the denominator is used to find the fraction, e.g., ¹/₄ of 12 is 4 and ³/₄ of 12 is 4 implying that ³/₄ is no different from ¹/₄ .
b) iii) c) iii)	4 or 6 [b) ii) 3] 3 or 9 [c) ii) 6]	As above, but students are aware that finding the non-unit fraction of a quantity yields a different solution.

Based on a representative sample of 183 students

Most students who displayed misunderstandings indicated that they did not see how the top and bottom numbers of a fraction represent a part-whole relationship. They may know some benchmark fractions but did not realise how non-unit fractions are built up from unit fractions. This idea of the part-whole nature of fractions is an important base from which to explore fractions as operators.

Next steps:

Students who could not find the unit fraction of the set (including using the denominator as the answer) need to have more experiences partitioning sets and naming the parts (unit fraction) they have created. Ask them to show how they could share the set amongst x [a number of] people –if required give them materials to model this, but encourage them to work with images and towards visualisation. Students beginning to understand fractions should be encouraged to use words to describe the parts, and delay the fractional notation until they have developed some understanding of what fractions represent.

Students who could find the solution for problem with unit fractions only need to explore fractions as part-whole relationships, looking at the different representations of simple fractions (unit and non-unit fractions), and to see how non-unit fractions are built up.

			Y4 (11/2006)
a)	i) ii) iii)	Some way of partitioning the 8 cats into 2 equal groups 4 2	very easy very easy easy
b)	i) ii) iii)	Some way of partitioning the 12 apples cats into 4 equal groups 3 9	easy moderate difficult
c)	i) ii) iii)	Some way of partitioning the 18 eggs into 3 equal groups 6 12	easy moderate difficult

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Fractions and sets

Fractions and sets