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<\/a><\/p>\nIf this looks like pure Greek to you, take heart!\u00a0 Read this article first, then go back and give this question a try!<\/p>\n
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Fractions and mathematical thinking<\/h2>\n
Part of the problem with fractions is that folks learn the basics in grade school, long before the human brain is ready for abstraction.\u00a0\u00a0 Folks learn mechanical procedures, which are the mental equivalent of muscle memory.\u00a0 The trouble is that five, ten, fifteen years after grade school, things get blurry.\u00a0 The problem with a rote mechanical procedure is that, you don\u2019t have it done just right, you often have no way to correct it.\u00a0 Folks get confused with their blurry mechanical procedures, and they naively conclude, \u201cI guess I just can\u2019t do fractions<\/em>!\u201d\u00a0 It is unconscionable that middle school and high school math teachers allow so many intelligent students to draw such conclusions, but there it is.<\/p>\n In fact, as with everything in mathematics, you can\u2019t just learn a procedure.\u00a0 You have to push yourself to understand the why-questions.\u00a0 In the best of all possible worlds, in high school, when you brain can handle abstraction, you should be taught all the \u201cwhy\u2019s\u201d about fractions, but this often doesn\u2019t happen.\u00a0 \u00a0I will cover some of that here.<\/p>\n <\/p>\n First of all, any fraction, in and of itself, is an act of division.\u00a0 In any fraction, the numerator<\/strong> (the top) is divided by the denominator<\/strong> (the bottom).\u00a0 This is particularly easy to see in an improper fraction<\/strong> (a fraction in which the numerator is larger than the denominator); for example, the improper fraction 20\/5 means twenty divide by five, so it equals 4.<\/p>\n This is a little less intuitive for a more typical fraction, such as 3\/5.\u00a0 What exactly does it mean to divide three by five?\u00a0 Well, think of it this way.\u00a0 Suppose you had three pies, and these three pies had to be divided equally among five hungry individuals.\u00a0 The way to do this would be to cut each of the three pies into five equal pieces, and then give each person three of those pieces.\u00a0 Thus, each person would get exactly 3\/5 of a pie.<\/p>\n The pie metaphor is very important. The fraction 3\/5 also means that we slice the pie in five pieces and then select three of these five.<\/p>\n It\u2019s very important to think about this when you are reasoning with fractions, because whenever you can involve the visual portion of your brain, you understand math more deeply.<\/p>\n This also bring up another important thing to understand about any fraction.\u00a0 Mathematicians are, in some sense, a bit casual about writing one number over another.\u00a0 One thing that 3\/5 means is that we have three of these things called a fifth.\u00a0 More properly, it would be written as the number 3 times the fraction 1\/5, but as a kind of abbreviation, mathematicians just write this as 3\/5.\u00a0 This means you always need to recognize, for any fraction, that the numerator is how many of the things you have, and these things are one over the denominator.\u00a0 Thus<\/p>\n If you can have all three of these perspectives at once\u2014the division perspective, the slices of pie visual, and the numerator as a multiplier\u2014then the logic of fractions will really start to make sense.<\/p>\n Notice that, right away, we could compare the size of two different fractions.\u00a0 If two fraction have the same denominator and different numerators, then the one with the larger numerator has the larger value: 5\/7 > 4\/7.\u00a0 We have more of the same things.\u00a0 By contrast, if two fraction have the same numerator and different denominators, the one with the larger denominator has a smaller value, because we are dividing by more.\u00a0 If we divide the same three pies equally among eight people, everyone gets slightly less pie than they would have gotten if the same pies had been divided equal among seven people: 3\/8 < 3\/7.<\/p>\n <\/p>\n It should be clear that if we take 3 of any thing, and add 5 of that same thing, we have 8 of that thing.\u00a0 This basic idea is known in mathematics as the Distributive Law.\u00a0 It doesn\u2019t matter what the \u201cthing\u201d is.\u00a0 We could add 3 eggs to 5 eggs, or $3 to $5, or 3 people to 5 people, or 3 days to 5 days.\u00a0 In each case, the sum would be 8.\u00a0 Even though we could have an infinite number of different individual items, the formal mathematical process is the same.<\/p>\n Well, when we have 3\/11 and 5\/11, then we have three of a thing and five of the same thing, so it must be true that we can add them in exactly the same way we add everything else.<\/p>\n 3\/11 + 5\/11 = 8\/11<\/p>\n Similarly,<\/p>\n 3\/7 + 2\/7 = 5\/7<\/p>\n 1\/9 + 7\/9 = 8\/9<\/p>\n Also,<\/p>\n 9\/13 \u2013 7\/13 = 6\/13<\/p>\n 4\/5 \u2013 3\/5 = 1\/5<\/p>\n We can add or subtract fractions that have the same denominator in exactly the same way we add or subtract eggs or dollar or people or anything else.\u00a0 It\u2019s exactly the same formal process, and once again, this formal process is known as the Distributive Law.<\/p>\n That\u2019s great when the two fractions that we to have to add or subtract happen to have the same denominator.\u00a0 Often, though, we are not quite so lucky.<\/p>\n Here, we need to go back to the idea of fractions as division.\u00a0 Anything (other than zero) divided by itself equals one.\u00a0 This immediately implies that if we make a fraction in which the numerators equals the denominator, that fraction will have to equal 1.\u00a0 We could make any an infinite number of different fractions equal to 1, if we so chose.\u00a0 Of course, one of the many remarkable properties of one is that we can multiply anything by it, and the value of thing doesn\u2019t change.\u00a0 Multiplication by one preserves the identity of any quantity.<\/p>\n Well, suppose we\u00a0 had to perform the following subtraction.<\/p>\n Notice that we know that the first fraction is larger, because it has a smaller denominator.\u00a0 Here, we were not lucky enough to get two fractions with the same denominator.\u00a0 Instead, we need to form common denominators<\/strong>, so we can do ordinary subtraction.\u00a0 Here, we can multiply the first fraction by 4\/4 (a fraction that equals one), and multiply the second by 3\/3.\u00a0 Since each is multiplied by 1, neither changes its value.\u00a0 Then we can subtract.<\/p>\n For simple cases, we can just use the two denominators to make our something-over-itself fractions.\u00a0 With slightly larger numbers, it pays to find the Least Common Denominator (LCD). The Magoosh math lessons discuss the process of finding the LCD for even large numbers.\u00a0 Those are the basics of adding and subtracting fractions.<\/p>\n <\/p>\n Here, we covered some basic ideas about fractions as well as fraction addition and subtraction.\u00a0 If you had any insights while reading, you may want to give the practice question at the top another look before moving on to the explanations below.\u00a0 In the next post<\/a>, we will cover fraction multiplication & division.<\/p>\n <\/p>\n 1) Subtract (1\/6) from both sides.<\/p>\nWays to think about fractions<\/h2>\n
<\/a><\/p>\n<\/a><\/p>\nAdding and subtracting<\/h2>\n
<\/a><\/p>\n<\/a><\/p>\nSummary<\/h2>\n
<\/a><\/p>\nPractice problem explanation<\/h2>\n