eternalsummer Posted October 24, 2016 Share Posted October 24, 2016 So I have one of those "Everything your 3rd Grader Needs to Know" books and I just randomly open it every few days and make sure DS knows or is learning whatever it says. Â It's reassuring. Â Usually. Â Yesterday I opened to division. Â Great! Â He understands division and is making progress on the fact families. Â All is good. Â But I read the definition and now I am questioning everything. Â Â It gives examples like this: 18 divided by 3 means that you've got 18 hats and you are dividing them into sets of 3, so 18 divided by 3 equals 6. Â Then there are pictures with 6 sets of 3 hats each. Â !!!!!!!!!!! Â I always thought (and of course taught, ack) that 18 divided by 3 meant you were dividing 18 things into 3 *sets* - so each set has 6 things. Â Am I wrong? Â does anyone else think like this? Â How did I get through math understanding division, of all things, completely backwards? Â Does it even matter? Quote Link to comment Share on other sites More sharing options...

creekland Posted October 24, 2016 Share Posted October 24, 2016 Division is repeated subtraction. Â If you have 18 / 3 that means you would be subtracting 3 six times, with 6 being your answer. Â It's a shortcut. Â We often think of them interchangeably simply because it is, but at it's basic meaning... that's why the book shows what it does. Â Multiplication is repetitive adding. 2 Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 (edited) You're not wrong, there is more than one way to conceptualize division. Both your way and the book's way are correct and useful conceptualizations. Â Liping Ma's book comparing math education in the US and China brings up the value of understanding and being able to use different conceptualizations of division. I seem to recall there was even a third way of understanding division mentioned in that book. Â Teach your kids both your way and the book's way, it will help them be more flexible problem solvers. Edited October 24, 2016 by maize 3 Quote Link to comment Share on other sites More sharing options...

creekland Posted October 24, 2016 Share Posted October 24, 2016 You're not wrong, there is more than one way to conceptualize division. Both your way and the book's way are correct and useful conceptualizations. Â Liping Ma's book comparing math education in the US and China brings up the value of understanding and being able to use different conceptualizations of division. I seem to recall there was even a third way of understanding division mentioned in that book. Â Teach your kids both your way and the book's way, it will help them be more flexible problem solvers. Â We show both ways at school, and how it all fits with multiplication/adding. Â IRL it rarely matters as long as one understands what is going on overall. Â In my post above, I was explaining why the book the OP was reading explained it as it did. Â Division is a short cut for subtraction just as multiplication is a short cut for adding. Â In the US the OP's original way of looking at it is used far more often IRL. Â "Let's see, 8 slices of pizza divided among 4 people means 2 slices per person." Â That's four groups of two, not two groups of four. Â However, 8/4 means 4 gets subtracted and that can only happen twice - two groups of four. Â That whole deal is really getting into the more petty basics of math and while a neat novelty of understanding, it's not really worth that much IMO. Â I tend to be happy if a student can do 8/4 without picking up their calculator. Â That ability is getting more rare as time moves on. Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 (edited) This talks about both models: http://langfordmath.com/ECEMath/Multiplication/DivModelsText.html Edited October 24, 2016 by maize Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 We show both ways at school, and how it all fits with multiplication/adding. Â IRL it rarely matters as long as one understands what is going on overall. In my post above, I was explaining why the book the OP was reading explained it as it did. Division is a short cut for subtraction just as multiplication is a short cut for adding. Â In the US the OP's original way of looking at it is used far more often IRL. "Let's see, 8 slices of pizza divided among 4 people means 2 slices per person." That's four groups of two, not two groups of four. However, 8/4 means 4 gets subtracted and that can only happen twice - two groups of four. That whole deal is really getting into the more petty basics of math and while a neat novelty of understanding, it's not really worth that much IMO. Â I tend to be happy if a student can do 8/4 without picking up their calculator. That ability is getting more rare as time moves on. I disagree with you on the usefulness of understanding different models. Â Being able to consider a problem from a different but equally valid perspective can be extremely useful. Quote Link to comment Share on other sites More sharing options...

kiana Posted October 24, 2016 Share Posted October 24, 2016 Both ways are correct imo, and as long as you know that if you divide eighteen items into three equal sets they each have six items and if you divide eighteen items into six equal sets they each have three items, you're in fine shape. They're the solution to different word problems. Â You have eighteen cookies and six kids. If you split the cookies equally, how many cookies will each kid have? Three. You have eighteen cookies and you give each kid three. How many kids did you give them to? Six.Â Â What worries me is when a kid can't figure out that they need to use division to answer those problems.Â 5 Quote Link to comment Share on other sites More sharing options...

Sherry in OH Posted October 24, 2016 Share Posted October 24, 2016 Division is repeated subtraction. Context does matter though. Â a) Grandma knit 18 hats. If she gives each grandchild 3 hats, how many grandchildren does she have? Â b) Grandma knit 18 hats. If she gives an equal number to each of her 3 grandchildren, how many hats will each child receive? Â a)Â Â Â Â She has 6 grandchildren 18 hats Ă¢â‚¬â€œ 3 hats for grandchild 1 = 15 hats remaining 15 hats Ă¢â‚¬â€œ 3 hats for gc2 = 12 hats remaining 12 hats Ă¢â‚¬â€œ 3 hats for gc3 = 9 hats remaining 9 hats Ă¢â‚¬â€œ 3 hats for gc 4= 6 hats remaining 6 hats Ă¢â‚¬â€œ 3 hats for gc5 = 3 hats remaining 3 hats Ă¢â‚¬â€œ 3 hats for gc6 = 0 hats remaining Â a) is easiest to illustrate by drawing 18 hats and then circling groups of 3 hats.Â There will be 6 groups of 3. Â b) Each child will receive 6 hats. 18 hats - 3 hats (one to each grandchild) = 15 hats remaining, each child has 1 hat 15 hats Ă¢â‚¬â€œ 3 hats = 12 hats remaining, each child now has 2 hats 12 hats Ă¢â‚¬â€œ 3 hats = 9 hats remaining, each child now has 3 hats 9 hats Ă¢â‚¬â€œ 3 hats = 6 hats remaining, each child now has 4 hats 6 hats Ă¢â‚¬â€œ 3 hats = 3 hats remaining, each child now has 5 hats 3 hats Ă¢â‚¬â€œ 3 hats = 0 hats remaining, each child now has 6 hats. Â b) is easiest to illustrate by drawing 18 hats then coloring them in 3 colors (for example red for child 1, blue for child 2, and green for child 3).Â Color one hat for each child, then repeat until you run out of hats.Â There will be 6 hats of each color, or 3 groups of 6. Â Both a) and b) are solved by the equation 18/3=6. 1 Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 Another link that highlights the usefulness of understanding more than one way of modeling division:http://corwin-connect.com/2014/07/rethinking-fraction-division-one-hardest-topics-teach-learn/ Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 (edited) Division of fractions is the context in which different models came up in Liping Ma's book. Teachers were asked to come up with and solve a word problem that would accurately depict something like 1 3/4 ĂƒÂ· 1/2 and while some of the American elementary teachers could provide an accurate answer to the problem, not one in her study was able to provide a conceptually correct word problem model. Edited October 24, 2016 by maize Quote Link to comment Share on other sites More sharing options...

regentrude Posted October 24, 2016 Share Posted October 24, 2016 Division of fractions is the context in which different models came up in Liping Ma's book. Teachers were asked to come up with and solve a word problem that would accurately depict something like 1 3/4 ĂƒÂ· 1/2 and while some of the American elemtary teachers cforould provide an accurate answer to the problem, not one in her study was able to provide a conceptually correct word problem model. Â Ouch. But I am not surprised. 1 Quote Link to comment Share on other sites More sharing options...

heartlikealion Posted October 24, 2016 Share Posted October 24, 2016 I'm just impressed you are able to say you've done all the stuff. I have gotten two of those books and we never cover everything inside. Well, we might cover the basic math concepts or something, but we often fail to read all the literature section or we learned about something else in our history, etc. Â The way I interpret it is we have 18 hats that need to be divided for the three children. How many hats does each child get? Six. Â So I guess I think like you. Â I made it through my math classes fine. And my father has a Master's in Math. I think if there was a big gaping problem it would have been pointed out to me long ago. I said I think lol. 1 Quote Link to comment Share on other sites More sharing options...

El... Posted October 24, 2016 Share Posted October 24, 2016 There was a thread about this, wasn't there?Â And it is NOT a dumb question. Â A programmer had written an article, explaining that computers view division problems in a defined way (one number meant columns, and the other meant rows, if you thought of the problem occurring in a rectangle).Â Â I can't say I understood his explanation.Â :tongue_smilie: Quote Link to comment Share on other sites More sharing options...

nd293 Posted October 24, 2016 Share Posted October 24, 2016 I tend to conceptualise it as the OP does although I recognise both. I can't say I've run into a problem because of that, nor have my kids (both now at school). Quote Link to comment Share on other sites More sharing options...

Guest Posted October 24, 2016 Share Posted October 24, 2016 What is happening right now? I don't see any difference whatsoever between Â 18 divided by 3 means that you've got 18 hats and you are dividing them into sets of 3, so 18 divided by 3 equals 6.Â and Â 18 divided by 3 meant you were dividing 18 things into 3 *sets* - so each set has 6 things.Â Â ^^Those say the exact same thing! Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 Ouch. But I am not surprised. Conceptual understanding among elementary math teachers in the US is rather depressingly bad in general. If I could fix just one thing about the American education system it would be to hire specialist teachers--people who understand and like math--to teach at all levels. Â Even high school. I was talking with a high school math teacher some time ago about teaching multiplication, I said something about teaching that 3 x 7 means 3 groups of 7 and she told me that was all wrong and I had to teach it as 7 groups of 3 BECAUSE the 3 comes first so you have to start with that because in addition you start with the first number and the kids would be too confused if you didn't do the same with multiplication. ?!? 2 Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 What is happening right now? I don't see any difference whatsoever between Â and Â Â ^^Those say the exact same thing! No they don't. In the first you get 6 groups of 3 hats, in the second 3 groups of six hats. Â Quantitive vs. Partitive models. 1 Quote Link to comment Share on other sites More sharing options...

SKL Posted October 24, 2016 Share Posted October 24, 2016 IMO it does not matter.Â :) Quote Link to comment Share on other sites More sharing options...

poppy Posted October 24, 2016 Share Posted October 24, 2016 I recently read the stupidest question ever.Â It was: who would win in a battle of 30,000 lions vs the sun? 1 Quote Link to comment Share on other sites More sharing options...

Guest Posted October 24, 2016 Share Posted October 24, 2016 That there are different names for different ways of finding the same answer is interesting trivia (and necessary I am sure for some non-fourth grade people), but doesn't change the fact that Â 18 divided by three equals 6 and 18 divided by 6 equals 3 and 6 multiplied by 3 equals 18 and 3 multiplied by 6 equals 18 Â SO you can divvy up your 18 hats into two groups of nine; nine groups of two; 6 groups of three; one group of nine and three groups of three, etc etc etc but you still ultimately have 18 hats. Those things are all the exact same thing: 18Â hats. Â Isn't the idea to recognize that immediately upon seeing the problem, at a certain point? Quote Link to comment Share on other sites More sharing options...

Guest Posted October 24, 2016 Share Posted October 24, 2016 I had to teach it as 7 groups of 3 BECAUSE the 3 comes first so you have to start with that because in addition you start with the first number and the kids would be too confused if you didn't do the same with multiplication. ?!? Â wut :cursing: Quote Link to comment Share on other sites More sharing options...

kiana Posted October 24, 2016 Share Posted October 24, 2016 Even high school. I was talking with a high school math teacher some time ago about teaching multiplication, I said something about teaching that 3 x 7 means 3 groups of 7 and she told me that was all wrong and I had to teach it as 7 groups of 3 BECAUSE the 3 comes first so you have to start with that because in addition you start with the first number and the kids would be too confused if you didn't do the same with multiplication. ?!? Â WUT. Â Well, no wonder my students have issues with understanding that 2+x is the same damn thing as x+2.Â Â Sadly, they finally think they get that and then immediately conclude that 2-x is the same thing as x-2.Â 6 Quote Link to comment Share on other sites More sharing options...

creekland Posted October 24, 2016 Share Posted October 24, 2016 I disagree with you on the usefulness of understanding different models. Â Being able to consider a problem from a different but equally valid perspective can be extremely useful. Â I might not be wording my thoughts correctly, but I don't think we're all that different in thought. Â Conceptual understanding among elementary math teachers in the US is rather depressingly bad in general. If I could fix just one thing about the American education system it would be to hire specialist teachers--people who understand and like math--to teach at all levels. Â :iagree:Â but right with it I would eliminate dependence on calculators in the early years (for typical students). Â Â I have far more problems with kids and their dependence on calculators - not being able to do 2x100 without one - or recognizing a mistake like forgetting parentheses around negative numbers when squaring them (and thinking (-2)^2 is -4 because they plugged it in as -2^2) than I do with kids not understanding that 18/3 can be broken into 6 sets of 3 or 3 sets of 6 - and in word problems knowing which way it should go. Â We've now graduated a whole set of teachers who were brought up dependent upon calculators and teach the only thing they know. Â I've been in Alg classes where the teacher is emphasizing being careful with the calculator when doing quadratic formula equations. Â She seemed mystified when I told her I found it easier to teach kids to do the section under the radical in their head rather then trying to plug it all in correctly. Â She refused to teach the kids that as an option. Â Being typical "me" I did it as soon as I came in for her - and encouraged it. Â :coolgleamA:Â Â It was tough for some kids to be able to do it, but even more made mistakes by forgetting parentheses - and didn't recognize that there had been a mistake. Â This was 8th grade. Â If they had been taught more without a calculator earlier it would have been easier for them. 2 Quote Link to comment Share on other sites More sharing options...

MaeFlowers Posted October 24, 2016 Share Posted October 24, 2016 (edited) I don't understand how both answers are correct. I get it conceptually but...how does 18/3=18/6? If you asked a child to divide 18 pencils among 3 classmates and they gave you 6 sets of 3 pencils, would that not be wrong? I mean, yes, 3*6=18, but isn't understanding that 18/3 is not the same as 18/6 important? What about as you've into higher math? Edited October 24, 2016 by MaeFlowers Quote Link to comment Share on other sites More sharing options...

kiana Posted October 24, 2016 Share Posted October 24, 2016 I don't understand how both answers are correct. I get it conceptually but...how does 18/3=18/6? If you asked a child to divide 18 pencils among 3 classmates and they gave you 6 sets of 3 pencils, would that not be wrong? I mean, yes, 3*6=18, but isn't understanding that 18/3 is not the same as 18/6 important? What about as you've into higher math? Â That's actually not what I'm saying. Â "Divide 18 pencils among 3 classmates" has only one answer, and of course 18/3 is modeled by that. But if I asked a student to model 18/3 and come up with a word problem, they could equally well ask "I divide 18 pencils among my friends, and each friend gets 3 pencils. How many friends do I have?" to which the answer is "I have 6 friends, 18/3".Â 1 Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 (edited) I don't understand how both answers are correct. I get it conceptually but...how does 18/3=18/6? If you asked a child to divide 18 pencils among 3 classmates and they gave you 6 sets of 3 pencils, would that not be wrong? I mean, yes, 3*6=18, but isn't understanding that 18/3 is not the same as 18/6 important? What about as you've into higher math?No-one is saying that 18ĂƒÂ·3=18ĂƒÂ·6.Â What we are saying is that 18ĂƒÂ·3 can mean either 18 divided into three groups (in which case the answer of 6 is the number of items in each group) or 18 divided into groups of three (in which case the answer, still 6, is the number of groups created.) Â It is different ways of modeling 18ĂƒÂ·3, both ways are correct. Of course a given word problem will tend to require one type of modeling and not the other. Edited October 24, 2016 by maize 2 Quote Link to comment Share on other sites More sharing options...

MaeFlowers Posted October 24, 2016 Share Posted October 24, 2016 That's actually not what I'm saying. Â "Divide 18 pencils among 3 classmates" has only one answer, and of course 18/3 is modeled by that. But if I asked a student to model 18/3 and come up with a word problem, they could equally well ask "I divide 18 pencils among my friends, and each friend gets 3 pencils. How many friends do I have?" to which the answer is "I have 6 friends, 18/3". I would say the problem would be 18/x=3. The question is asking you what the divisor is. Yes, 18/3 would work give you the right answer but would not generate the right problem. Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 I would say the problem would be 18/x=3. The question is asking you what the divisor is. Yes, 18/3 would work give you the right answer but would not generate the right problem. It is well accepted in math education circles that both the partitive (18ĂƒÂ·3 seen as dividing 18 into three groups)and quantitive or measurement (18ĂƒÂ·3 seen as 18 divided into groups of 3) models of division are valid. Â Your insistence that only one model can be valid is not the professional standard here. 5 Quote Link to comment Share on other sites More sharing options...

MaeFlowers Posted October 24, 2016 Share Posted October 24, 2016 (edited) Well, I'm not a professional. LOL. But, I think the argument that both are the same is invalid. While 3*6 and 6*3 are the same, 18/3 and 18/6 are not. That's what you are arguing when you say 18 by 3 and 18 divided into groups of three are the same thing. They are two different equations and my lowly non-professional self thinks that a student who doesn't understand that is going to have major issues in math down the line. Edited October 24, 2016 by MaeFlowers Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 Well, I'm not a professional. LOL. But, I think the argument that both are the same is invalid. While 3*6 and 6*3 are the same, 18/3 and 18/6 are not. That's what you are arguing when you say 18 by 3 and 18 divided into groups of three are the same thing. They are two different equations and my lowly non-professional self thinks that a student who doesn't understand that is going to have major issues in math down the line. You might take a look at some of the links I posted up thread. Quote Link to comment Share on other sites More sharing options...

creekland Posted October 24, 2016 Share Posted October 24, 2016 Going back to overall concepts, personally, I want a student to know how to make up a word problem fitting 18/3 (being able to do either way) and to be able to know to use division if given the word problem first. Â Then it would be nice if they remember division as a shortcut for subtraction, but many forget that detail as they aren't reminded of it often. Â With my Alg students, I want them to be able to use the quadratic formula to solve quadratic equations, but I also want them to know how to derive it if they "forget" the exacts and to know that they are really finding spots where y=0 on a parabolic graph - if it exists. Â Add to that any other specifics from a word problem or deeper problem they are solving. Â They should also know they could use completing the square, etc. if they preferred. Â What I'm finding IRL with many students is that they need a calculator and have memorized the "/" key. Â They have to think to know what it means if asked - and yes - if dividing by fractions all they really remember is to put it in parentheses or maybe to multiply by the reciprocal. Â Most of the time they will just shove their book or homework aside and decide they don't care when it gets that deep. Â If one has to memorize math, it can really be tough to remember all the specifics, though top students can do it - to a point. Â When one actually understands math, it's not that difficult because one can break it down mentally from building blocks. Â That understanding is helpful if it comes from an early age rather than trying to fill in the gaps later on. Â The best I usually get with the quadratic is that it's "that weird formula where you plug things in and have to be careful. Â Hang on, it's on our formula sheet somewhere." Â If the problem isn't already set up where y=0, I can count on numerous mistakes. 2 Quote Link to comment Share on other sites More sharing options...

heartlikealion Posted October 24, 2016 Share Posted October 24, 2016 (edited) Well, I'm not a professional. LOL. But, I think the argument that both are the same is invalid. While 3*6 and 6*3 are the same, 18/3 and 18/6 are not. That's what you are arguing when you say 18 by 3 and 18 divided into groups of three are the same thing. They are two different equations and my lowly non-professional self thinks that a student who doesn't understand that is going to have major issues in math down the line. Â They are two different things, but the wording makes it hard to understand which question is really being asked. That's my take, anyway. Â Is the book asking them to divide the items into three groups or display groups that each hold three objects? Â I interpreted it to be divide these 18 hats among three people. Their drawing would confuse me because it is not how I interpreted it. Â Edited October 24, 2016 by heartlikealion Quote Link to comment Share on other sites More sharing options...

creekland Posted October 24, 2016 Share Posted October 24, 2016 That's what you are arguing when you say 18 by 3 and 18 divided into groups of three are the same thing.Â Â No one is saying this. Â What folks are saying is the simple math equation written out as 18/3 = x can be interpreted IRL two different ways with groups of three or creating 3 groups from the 18. Â A student should understand that. Â Â The way the OP's book described it is with groups of three. Â It's the one most easily modeled by subtracting 3 each time. Â 4 Quote Link to comment Share on other sites More sharing options...

MaeFlowers Posted October 24, 2016 Share Posted October 24, 2016 No one is saying this. What folks are saying is the simple math equation written out as 18/3 = x can be interpreted IRL two different ways with groups of three or creating 3 groups from the 18. A student should understand that. Â The way the OP's book described it is with groups of three. It's the one most easily modeled by subtracting 3 each time. To me, if you ask for 18/3, you should get 3 groups of six somethings every time. I see it as 18 * 1/3. If you get 6 groups of somethings, you are not answering the right equation. If you get 6 groups, you're switching the divisor and quotient. While you could do that all day long and prove the associative property of multiplication, it doesn't work for division. Switching those creates two distinct problems with two different answers. That matters in math, does it not? Â I feel like we are going around on circles. I will look at the links maize provided a little later and see if I can better understand what you are trying to say. Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 (edited) To me, if you ask for 18/3, you should get 3 groups of six somethings every time. I see it as 18 * 1/3. If you get 6 groups of somethings, you are not answering the right equation. If you get 6 groups, you're switching the divisor and quotient. While you could do that all day long and prove the associative property of multiplication, it doesn't work for division. Switching those creates two distinct problems with two different answers. That matters in math, does it not? Â I feel like we are going around on circles. I will look at the links maize provided a little later and see if I can better understand what you are trying to say. We are going around and around because you are sure that your model of division--the partitive model--is the one and only correct model and that anyone who views the quantitive or measurement model as equally correct and valid is simply wrong.Â You've offered no evidence for your point of view, you basically keep saying "my way is right therefor the other way is wrong." (logic folks, is that an example of "begging the question"?) Edited October 24, 2016 by maize 4 Quote Link to comment Share on other sites More sharing options...

MaeFlowers Posted October 24, 2016 Share Posted October 24, 2016 Another link that highlights the usefulness of understanding more than one way of modeling division:http://corwin-connect.com/2014/07/rethinking-fraction-division-one-hardest-topics-teach-learn/ I don't understand his argument. Â In lesson three he divides by 1/2 when he should be multiplying by 1/2 (or dividing by two). This wouldn't be the right equation to begin with... Â That's as far as I've gotten right now. I keep getting interrupted. Quote Link to comment Share on other sites More sharing options...

Guest Posted October 24, 2016 Share Posted October 24, 2016 To me, if you ask for 18/3, you should get 3 groups of six somethings every time. I see it as 18 * 1/3. If you get 6 groups of somethings, you are not answering the right equation. If you get 6 groups, you're switching the divisor and quotient. While you could do that all day long and prove the associative property of multiplication, it doesn't work for division. Switching those creates two distinct problems with two different answers. That matters in math, does it not? Â I feel like we are going around on circles. I will look at the links maize provided a little later and see if I can better understand what you are trying to say. Â I think you actually do understand BUT this is the hardship of using words and symbols to represent mathematics, which exists outside the words and symbols we use to describe and manipulate it.Â Â If the question is 18 divided by 3 equals ? then YES there is one correct numeral to write down. Any other numeral would be incorrect. <<--- You are talking about this. Â But you can hold that question in your mind in two different ways to arrive at that correct numeral. <<--- Others are referring to this. Quote Link to comment Share on other sites More sharing options...

MaeFlowers Posted October 24, 2016 Share Posted October 24, 2016 Okay, so I think I understand what the argument is. I think my confusion was a wording issue. Or I'm am idiot. (more likely) Â I was thinking that you were saying that the same problem could result in two different answers. Although the math is the same, these are actually two different problems that result in the same answer. Â 18/3 will yield 3 groups of 6 only when you ask for things to be divided into groups. If I ask for 3 items in each group, then 18/3 will yield 6 groups. 4 Quote Link to comment Share on other sites More sharing options...

Guest Posted October 24, 2016 Share Posted October 24, 2016 Okay, so I think I understand what the argument is. I think my confusion was a wording issue. Or I'm am idiot. (more likely) Â I was thinking that you were saying that the same problem could result in two different answers. Although the math is the same, these are actually two different problems that result in the same answer. Â 18/3 will yield 3 groups of 6 only when you ask for things to be divided into groups. If I ask for 3 items in each group, then 18/3 will yield 6 groups. Â You have shown no evidence of idiocy :laugh:Â here. Â And, yep! Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 Okay, so I think I understand what the argument is. I think my confusion was a wording issue. Or I'm am idiot. (more likely) Â I was thinking that you were saying that the same problem could result in two different answers. Although the math is the same, these are actually two different problems that result in the same answer. Â 18/3 will yield 3 groups of 6 only when you ask for things to be divided into groups. If I ask for 3 items in each group, then 18/3 will yield 6 groups. Bingo! Quote Link to comment Share on other sites More sharing options...

Paige Posted October 24, 2016 Share Posted October 24, 2016 Â With my Alg students, I want them to be able to use the quadratic formula to solve quadratic equations, but I also want them to know how to derive it if they "forget" the exacts and to know that they are really finding spots where y=0 on a parabolic graph - if it exists. Â Add to that any other specifics from a word problem or deeper problem they are solving. Â They should also know they could use completing the square, etc. if they preferred. ... Â The best I usually get with the quadratic is that it's "that weird formula where you plug things in and have to be careful. Â Hang on, it's on our formula sheet somewhere." Â If the problem isn't already set up where y=0, I can count on numerous mistakes. Â This makes me feel better about DS. He says he doesn't need to bother memorizing the quadratic formula because deriving it is just as easy as memorizing it!Â Â He's in HS now and driving his teachers crazy with his lack of calculator usage. I think they don't trust him.Â 3 Quote Link to comment Share on other sites More sharing options...

MaeFlowers Posted October 24, 2016 Share Posted October 24, 2016 Sorry about being so dense. I was truly just misunderstanding what you were trying to explain to me. 1 Quote Link to comment Share on other sites More sharing options...

kiana Posted October 24, 2016 Share Posted October 24, 2016 So what would be a good word problem for 1 3/4 divided by 1/2? Â The one I came up with was if you have 1 3/4 lbs of m&ms and they fill one box halfway how much would a full box hold? I think it would only work as a demonstration if you first did the same problem with whole numbers - if 6lbs of m&ms fills 2 boxes each box holds 3 lbs because 6/2= 3, if 10 lbs of m&ms fills 4 boxes each box holds 2.5 lbs, etc. Â That's a reasonable example although I'd modify it to "how many 1/2 pound boxes can I make?" Â Yes, this only works if the students actually understand word problems with numbers. One of the big issues that we have with math education in this country is that we attempt to run before we can walk. For example, we try to have students who cannot compose a coherent sentence write a research paper.Â Â Students should not be attempting dividing fractions until they are fully competent with dividing numbers, including both solving word problems for given values and constructing word problems that fit given problems.Â 1 Quote Link to comment Share on other sites More sharing options...

regentrude Posted October 24, 2016 Share Posted October 24, 2016 So what would be a good word problem for 1 3/4 divided by 1/2? Â I have 1 3/4 pizza. A serving size is half a pizza. How many servings are in my 1 3/4 pizza? Answer: 3 and a half. Â For fractions, anything with pizza is easy to visualize for the student and can be demonstrated with cut up paper plates. 5 Quote Link to comment Share on other sites More sharing options...

maize Posted October 24, 2016 Share Posted October 24, 2016 (edited) So what would be a good word problem for 1 3/4 divided by 1/2? Â The one I came up with was if you have 1 3/4 lbs of m&ms and they fill one box halfway how much would a full box hold? I think it would only work as a demonstration if you first did the same problem with whole numbers - if 6lbs of m&ms fills 2 boxes each box holds 3 lbs because 6/2= 3, if 10 lbs of m&ms fills 4 boxes each box holds 2.5 lbs, etc. Your story works.Â The one I came up with is if I am wallpapering a wall that is 1 3/4 meters wide and one strip of wallpaper is 1/2 meter wide, how many strips of wall paper will it take? Â Another option is to ask an area question-- if the area of a rectangle is 1 3/4 square feet and one side is 1/2 ft, what is the length of the other side? Edited October 24, 2016 by maize 1 Quote Link to comment Share on other sites More sharing options...

kiana Posted October 24, 2016 Share Posted October 24, 2016 I would say the problem would be 18/x=3. The question is asking you what the divisor is. Yes, 18/3 would work give you the right answer but would not generate the right problem. Â 18/x = 3 is exactly the same problem as 18/3 = x. Being able to casually flip between them is something that I strive for as a level of deep mathematical understanding.Â 1 Quote Link to comment Share on other sites More sharing options...

MaeFlowers Posted October 24, 2016 Share Posted October 24, 2016 18/x = 3 is exactly the same problem as 18/3 = x. Being able to casually flip between them is something that I strive for as a level of deep mathematical understanding. It's not deep mathematical understanding and I don't appreciate the snark. If you read my responses later on, I said you could switch those all day long but that doesn't mean they are the same problem. For her particular example, x would be the divisor. Sometimes the placement matters. Â If I still completely wrong, I will quit teaching math. Quote Link to comment Share on other sites More sharing options...

kiana Posted October 24, 2016 Share Posted October 24, 2016 It's not deep mathematical understanding and I don't appreciate the snark. If you read my responses later on, I said you could switch those all day long but that doesn't mean they are the same problem. For her particular example, x would be the divisor. Sometimes the placement matters. Â If I still completely wrong, I will quit teaching math. Â I am actually not intending that as snark. I went through several times attempting to delete stuff that could be taken as snark, but clearly I was unsuccessful. Â Furthermore, I'm not implying that you can't switch between equivalent forms of the same problem.Â Â But I do believe that it's the same problem.Â Quote Link to comment Share on other sites More sharing options...

Guest Posted October 24, 2016 Share Posted October 24, 2016 I'm totally confused here too. It seems like a badly worded problem if 18/3=6 and 18/6=3, as these are inherently different groupings. I can see the discussion of eighteen hats being divided among three people versus eighteen hats being split into groups of threes, but I don't get being asked for one but seeing a picture drawn of the other. Â I was already having an awful day and now I feel totally incompetent in something I thought I enjoyed teaching. Screw this day :( Quote Link to comment Share on other sites More sharing options...

Guest Posted October 24, 2016 Share Posted October 24, 2016 (edited) This talks about both models: http://langfordmath.com/ECEMath/Multiplication/DivModelsText.html Okay, I'm reading the link but these problems are NOT worded the same way. They're asking for different things based on the language. What am I missing? They're demonstrating different properties of the same number families and different laws by which we can solve them, but they're not saying something fundamentally different whereby a problem that is asking for 18/3 doesn't equal 6? Edited October 24, 2016 by Arctic Mama Quote Link to comment Share on other sites More sharing options...

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