Most of my posts so far have addressed high school level math.\u00a0 This one is for younger students (and their teachers).<\/i><\/p>\n
At another blog that I follow, Kitchen Table Math<\/a>, there has been recent discussion about the \u201cinvert and multiply\u201d rule for dividing fractions and whether it is important and\/or appropriate to teach the reasoning behind this rule.<\/p>\n You know I am going to vote \u201cyes\u201d.\u00a0 But what kind of derivation should we use here?<\/p>\n The rule can be derived algebraically.\u00a0 But that doesn\u2019t really help the grade-school students who are just learning it.\u00a0 It can also be derived in a algebra-ish way.\u00a0 A commenter at Kitchen Table Math pointed to the explanation found here:<\/p>\n http:\/\/www.moveitmaththesource.com\/realfractions\/mrnovaksproof.html<\/a><\/p>\n Mr. Novak\u2019s explanation is elegant and concise. And it certainly answers the question: why do we invert and multiply in order to divide fractions?\u00a0 But it would be nice if students also had an intuitive feel for why this works.<\/p>\n We can build the rule step-by-step in a manner that helps that intuition to develop.<\/p>\n Step 1: Division is for Sharing<\/strong><\/p>\n You have 12 cookies \u2013 each a pretty good size — \u00a0and you need to put them in individual serving bags to hand out at a party.\u00a0 By deciding how many to put in each bag, you also determine how many serving bags you can make.\u00a0 For example, you could put 12 cookies in each bag, but then you would only have enough for 1 bag.\u00a0 On the other hand, it you put 6 cookies in each bag, then you can fill 2 bags because 12\/6 = 2. That seems obvious, but it establishes a method: we are going to divide the total number of cookies by the amount in each bag to find how many bags we can fill.\u00a0 You soon notice that if you put fewer cookies in each bag, you can fill more bags.\u00a0 For example:<\/p>\n So yes, dividing by a smaller number gives a bigger answer.\u00a0 And with 12 cookies, 12 bags is the most you can fill\u2026or is it?\u00a0 I mentioned that these were pretty big cookies.\u00a0 Hmmm\u2026<\/p>\n \u00a0<\/strong>Step 2: Dividing by Fractions (of the \u201cone over\u201d variety)<\/strong><\/p>\n You decide to put make the serving size a \u00bd-cookie.\u00a0 Since that serving size is even smaller, you expect to fill more bags.\u00a0 Though you have not been taught how to divide fractions, you can do this one by thinking:<\/p>\n If I am putting \u00bd in each bag, then each cookie is enough for 2 bags.\u00a0 So 12 cookies fill 24 bags.\u00a0 \u00a0I guess that must mean 12 \u00f7 \u00bd = 24.<\/em><\/p>\n Need to fill more bags?\u00a0 Use smaller portions!\u00a0 Give everyone 1\/3 of a cookie\u2026<\/p>\n You can see where this is going.\u00a0 Let\u2019s continue the chart\u2026<\/p>\n The trend has continued.\u00a0 Dividing by a smaller number gives a bigger answer.\u00a0 And even if you don\u2019t know the rule for dividing by fractions, you can certainly divide by these<\/i> fractions!<\/p>\n To divide by \u201cone over anything\u201d, just multiply by the denominator<\/strong><\/b>.<\/p>\n \u00a0<\/strong>STEP 3: Any fraction at all!<\/strong><\/p>\n Suppose you have those 12 big cookies and you decide to give everyone (2\/3) of a cookie.\u00a0 How many servings can you make?<\/strong><\/i><\/b><\/p>\n We can figure this out using the reasoning we followed above, with one extra twist.<\/p>\n We have already seen that if you want to give everyone just (1\/3) of a cookie, you will have enough to fill 12 \u00d7 3 = 36 bags.\u00a0 But now you decide to give everyone TWO thirds of a cookie.\u00a0 But you have already filled all those bags \u2013 why not just give everyone two of the bags?\u00a0 You will have enough for 36 \u00f7 2 = 18 servings.<\/p>\n Now go back and look at how we got that answer.\u00a0 We don\u2019t have to know an official rule to divide the 12 cookies into servings that each have 2\/3 of a cookie.\u00a0 We can just multiply by 3 and then divide by 2, which is the same as multiplying by 3\/2.<\/p>\n Let\u2019s try another:<\/p>\n Suppose you have 36 pizza pies to sell and you want to give 3\/5 of a pizza to each customer.\u00a0 (No one cuts pizzas into fifths, but don\u2019t worry about that.)\u00a0 How many customers can you serve before running out of pizza? <\/strong><\/i><\/b><\/p>\n Well, if we give everyone ONE fifth of a pie, we can feed 36 \u00d7 5 = 180 customers.\u00a0 Giving everyone THREE fifths means that we can only serve 180 \u00f7 3 = 60 customers.<\/p>\n So to find 36 \u00f7 (3\/5) we multiply by 5 and then divide by 3 which is the same as multiplying by 5\/3.<\/p>\n Which derivation is \u201cbetter\u201d?<\/strong><\/i><\/b><\/p>\n If I could only show students one explanation, I would go with Mr. Novak\u2019s.\u00a0 But if I had time to explore and play, I would start with what you see above.\u00a0 I like that it encourages number sense and intuition.\u00a0 Also, it introduces an idea that students will see again many times: that existing patterns can be extended to cover new situations.<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" Most of my posts so far have addressed high school level math.\u00a0 This one is for younger students (and their teachers). At another blog that I follow, Kitchen Table Math, there has been recent discussion about the \u201cinvert and multiply\u201d … Continue reading \n\n
\n If we take 12 cookies\u2026<\/td>\n And put 12 in each bag\u2026<\/td>\n 12 \u00f7 12 = 1<\/td>\n We can fill\u00a0 1 bag<\/td>\n<\/tr>\n \n If we take 12 cookies\u2026<\/td>\n And put 6 in each bag\u2026<\/td>\n 12 \u00f7 6 = 2<\/td>\n We can fill\u00a0 2 bags<\/td>\n<\/tr>\n \n If we take 12 cookies\u2026<\/td>\n And put 4 in each bag\u2026<\/td>\n 12 \u00f7 4 = 3<\/td>\n We can fill\u00a0 3 bags<\/td>\n<\/tr>\n \n If we take 12 cookies\u2026<\/td>\n And put 2 in each bag\u2026<\/td>\n 12 \u00f7 2 = 6<\/td>\n We can fill\u00a0 6 bags<\/td>\n<\/tr>\n \n If we take 12 cookies\u2026<\/td>\n And put 1 in each bag\u2026<\/td>\n 12 \u00f7 1 = 12<\/td>\n We can fill\u00a0 12 bags<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n
\n If we take 12 cookies\u2026<\/td>\n And put 1\/2 in each bag\u2026<\/td>\n 12 \u00f7 (1\/2) = 24<\/td>\n We can fill\u00a0 24 bags<\/td>\n<\/tr>\n \n If we take 12 cookies\u2026<\/td>\n And put 1\/3 in each bag\u2026<\/td>\n 12 \u00f7 (1\/3) = 36<\/td>\n We can fill\u00a0 36 bags<\/td>\n<\/tr>\n \n If we take 12 cookies\u2026<\/td>\n And put 1\/4 in each bag\u2026<\/td>\n 12 \u00f7 (1\/4) = 48<\/td>\n We can fill\u00a0 48 bags<\/td>\n<\/tr>\n \n If we take 12 cookies\u2026<\/td>\n And put 1\/5 in each bag\u2026<\/td>\n 12 \u00f7 (1\/5) = 60<\/td>\n We can fill\u00a0 60 bags<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n