{"id":1426,"date":"2013-08-30T09:02:15","date_gmt":"2013-08-30T09:02:15","guid":{"rendered":"http:\/\/mrreddy.com\/blog\/?p=1426"},"modified":"2024-10-12T14:24:26","modified_gmt":"2024-10-12T14:24:26","slug":"the-multiple-personalities-of-fractions","status":"publish","type":"post","link":"http:\/\/mrreddy.com\/blog\/2013\/08\/the-multiple-personalities-of-fractions\/","title":{"rendered":"The multiple personalities of fractions"},"content":{"rendered":"

The following is an extract of the text I wrote for an ebook<\/a> aimed as a guide for first year TeachFirst maths teachers. The excerpt covers what I see as the major difficulty of grasping fractions – their multiple personalities.<\/p>\n

\n

You know where you are with integers. The number one means one and nothing else. It behaves like a one in every context. Two is the same \u2013 it was born to be a two. Same with three and four and five and any whole number.
\nThat\u2019s not the case with fractions. Fractions come in different forms, they mean different things in different contexts, and they obey different rules most of which aren\u2019t intuitively obvious. The multiple personalities create a minefield for most adults, let alone pupils, and to make it worse there are multiple \u2018multiple personalities\u2019. Let\u2019s take a look at the four sets of multiple personalities.<\/p>\n

    \n
  1. What is the fraction trying to tell you? What does it stand for?<\/strong>
    \n\tFractions have three meanings.<\/p>\n
      \n
    1. Fractions can represent a position on a number line<\/strong>
      \nThis is the most overlooked personality in the maths classroom but it\u2019s absolutely essential to understand that fractions hold a value in the way that whole numbers do. Fractions aren\u2019t simply parts of a whole, they lie on a number line between all the integers. We can order fractions and we can compare their size because of this personality.<\/p>\n

      Three quarters sits on this line between 0 and 1. We can see it\u2019s closer to 1 and that it\u2019s more than \u00bd.<\/p>\n

      \"numberline\"<\/a>\n\t\t<\/li>\n

    2. Fractions are operations<\/strong>
      \nLet\u2019s take \u00be again. At first glance, you might read that as \u201c3 parts out of 4\u201d but it can also be thought of as \u201cthree divided by four\u201d. Fractions can be shorthand for division.<\/p>\n

      \"3_Wholes_Cut_Into_4\"<\/a>\n\t\t<\/li>\n

    3. Fractions are parts of a whole<\/strong>
      \nThis is the obvious one, the one we start teaching, so we\u2019ve left it until last.
      \nHere\u2019s the archetypal image to represent three quarters.
      \n      \"threequarters\"<\/a>\n\t\t<\/li>\n<\/ol>\n
    4. Every fraction has an infinite number of equivalent fractions but only one in simplest form<\/strong>
      \nThe idea that two fractions can be worth the same but involve different numerals is utterly perplexing. What happened to one being one and two being two? To find out that three quarters is equivalent to fifteen twentieths or 51\/68 shakes the foundation of a pupil\u2019s theory of number and can be quite unsettling, not to say confusing.<\/p>\n
      <\/div>\n<\/li>\n
    5. Improper fractions can be written as mixed numbers and mixed numbers can be written as improper fractions.<\/strong>
      \nFractions with a value greater than one have two separate forms \u2013 mixed number and improper fractions \u2013 before we even mention equivalence. Allowing for the possibility of a fraction where the numerator is bigger than the denominator, which is a mental leap in itself for those trained to see fractions as parts of a single whole (\u201cWhat does 4 out of 3 mean?\u201d), when you start dropping in mixed numbers, the perplexity needle enters the \u2018dangerously high levels\u2019 zone.<\/p>\n

      Here are some questions and representations to use for helping pupils understand what improper fractions and mixed numbers are and how to convert between them.<\/p>\n

      \"activity1\"<\/a><\/p>\n

      \"activity2\"<\/a><\/p>\n

      \"activity3\"<\/a><\/p>\n

      \"activity4\"<\/a><\/p>\n<\/li>\n

    6. Fractions, decimals, percentages and ratio<\/strong>
      \nYou can tell by now where this is heading. That fractions can be written as decimals and percentages adds yet another layer to the cake of confusion. Take, for example, one tenth. We know that it can be written as 1\/10, 0.1 or 10% and pupils can recognise that quickly too. They\u2019re likely to draw false conclusions as to how convert between fractions and decimals from this, however, so when you tell them that 1\/5 is not 5% and 20% is not 1\/20 you\u2019re really going to have some explaining to do.<\/p>\n

      And then there\u2019s the cousin of fractions \u2013 ratio \u2013 to contend with. How is that a ratio of 1:5 leads to a fraction of 1\/6 or 5\/6? When is a fraction a fraction and when is it a proportion?\n\t<\/li>\n<\/ol>\n

      Fractions are treacherous territory for students but that\u2019s not to say you can\u2019t teach it well. As with all of the tricky concepts in maths, you need to believe you can deliver. Enter the classroom with extra relish on the days you\u2019re teaching fractions.<\/p>\n

      \n

      The rest of the chapter goes on to talk about specific teaching strategies for fractions. The full ebook, which includes the writers’ collective experience of teaching the 30 must-know maths topics, is available to download to your Kindle (or anything with the Kindle app<\/a> installed). Simply search for ‘Teach First maths’ or visit this Amazon page<\/a>.<\/p>\n

      Important note:<\/strong> While I contributed to the editing of How to Start on Teach First: Maths<\/em> and writing some of the chapters, such as the one above, money from the sales doesn’t come to me. Money from sales goes to the educational charity Teach First, and its reinvestment in this collection of books for new teachers is at the discretion of the charity.<\/p>\n","protected":false},"excerpt":{"rendered":"

      The following is an extract of the text I wrote for an ebook aimed as a guide for first year TeachFirst maths teachers. The excerpt covers what I see as the major difficulty of grasping fractions – their multiple personalities.…<\/p>\n","protected":false},"author":1,"featured_media":1429,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,4,8],"tags":[],"class_list":["post-1426","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-opinion","category-pedagogy","category-teachinglearning-ideas"],"_links":{"self":[{"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/posts\/1426","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/comments?post=1426"}],"version-history":[{"count":18,"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/posts\/1426\/revisions"}],"predecessor-version":[{"id":2324,"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/posts\/1426\/revisions\/2324"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/media\/1429"}],"wp:attachment":[{"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/media?parent=1426"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/categories?post=1426"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/mrreddy.com\/blog\/wp-json\/wp\/v2\/tags?post=1426"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}