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The Improving Mathematics Education in Schools (TIMES) Project SETS AND VENN DIAGRAMS A guide for teachers - Years 7–8 NUMBER AND ALGEBRA Module 1 June 2011 78 YEARS
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The Improving Mathematics Education in Schools (TIMES) Project SETS AND VENN DIAGRAMS A guide for teachers - Years 7–8 NUMBER AND ALGEBRA Module 1 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge 78 YEARS
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{1} A guide for teachers SETS AND VENN DIAGRAMS ASSUMED KNOWLEDGE • Addition and subtraction of whole numbers. • Familiarity with the English words ‘and’, ‘or’, ‘not’, ‘all’, ‘if…then’. MOTIVATION In all sorts of situations we classify objects into sets of similar objects and count them. This procedure is the most basic motivation for learning the whole numbers and learning how to add and subtract them. Such counting quickly throws up situations that may at first seem contradictory. ‘Last June, there were 15 windy days and 20 rainy days, yet 5 days were neither windy nor rainy.’ How can this be, when June only has 30 days? A Venn diagram, and the language of sets, easily sorts this out. Let W be the set of windy days, and R be the set of rainy days. Let E be the set of days in June. Then W and 3; together have size 25, so the overlap between W and R is 10.; The Venn diagram opposite displays the whole situation. E W R 5 10 10 5 The purpose of this module is to introduce language for talking about sets, and some notation for setting out calculations, so that counting problems such as this can be sorted out. The Venn diagram makes the situation easy to visualise.
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The Improving Mathematics Education in Schools (TIMES) Project CONTENT DESCRIBING AND NAMING SETS A set is just a collection of objects, but we need some new words and symbols and diagrams to be able to talk sensibly about sets. In our ordinary language, we try to make sense of the world we live in by classifying collections of things. English has many words for such collections. For example, we speak of ‘a flock of birds’, ‘a herd of cattle’, ‘a swarm of bees’ and ‘a colony of ants’. We do a similar thing in mathematics, and classify numbers, geometrical figures and other things into collections that we call sets. The objects in these sets are called the elements of the set. Describing a set A set can be described by listing all of its elements. For example, S = { 1, 3, 5, 7, 9 }, which we read as ‘S is the set whose elements are 1, 3, 5, 7 and 9’. The five elements of the set are separated by commas, and the list is enclosed between curly brackets. A set can also be described by writing a description of its elements between curly brackets. Thus the set S above can also be written as S = { odd whole numbers less than 10 }, which we read as ‘S is the set of odd whole numbers less than 10’. A set must be well defined. This means that our description of the elements of a set is clear and unambiguous. For example, { tall people } is not a set, because people tend to disagree about what ‘tall’ means. An example of a well-defined set is T = { letters in the English alphabet }. {2}
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