stupidest question ever

[kiana]

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?

 

Do you mean the example word problems of partitive vs. measurement examples? (If you meant something else, sorry). 

 

The point that they're trying to make is that they're asking for different things based on the language, and both of these model division. So 24/4 could be modeled either by (to steal examples from their page) "I have 24 cookies. I want to put them on 4 plates, so that I have the same number of cookies on each plate. How many cookies should I put on each plate?" or "I have 24 cookies. I want to put 4 cookies on each plate. How many plates do I need to hold my cookies?"

 

Edit: To follow your edit -- yes, they are not saying something fundamentally different where 18/3 is not equal to 6. 

Edited October 24, 2016 by kiana

[eternalsummer]

Arctic mama, I don't think it was trying to tell me that 18/3 and 18/6 were the same, but it had an abstract problem like 18/3 and then presented a pictoral solution.

 

I expected the pictoral solution to represent the conception of 18/3 as 18 hats divided into 3 sets, so 6 hats in each set, because that is how I've always visualized the abstraction.

 

But the pictoral solution they had represented the conception of 18/3 as 18 hats divided into sets of 3, so 3 hats in each set with 6 sets total.

 

 

I can see how both are valid, I'd just never thought of the second one as a possibility!

 

Except now I am remembering that when I do long division, of course I say to myself how many times does 4 go into 29, or whatever, which is the same as the second conception, so that makes sense.  

 

 

 

Thanks all, I can't believe this is something I never considered! 

[maize]

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.

 

I was already having an awful day and now I feel totally incompetent in something I thought I enjoyed teaching. Screw this day :(

There are two common and equally valid conceptualizations of division. Both yield the same numerical result.

 

In the equation 18÷3=6, the Partitive model of division would have us dividing 18 into 3 groups. The 6 on the other side of the equation then represents the number of items in each group.

 

The quotative or measurement model has us dividing 18 into groups of 3. The 6 on the other side of the equation in this case represents the number of groups formed.

 

Both models are valid representations of the equation 18÷3=6.

 

I think some of the confusion that is arising comes from the fact that many people are used to using only one model and not the other. If, for example, you always think of division as partitive--that is, the divisor represents the number of groups formed, then you think that dividing 18 into six groups of three must be a different equation: 18÷6=3. In the quotative model, however, 18÷6 would mean forming three groups of six, and it is 18÷3 that gives you six groups of three.

 

The point of this discussion is that both quotative and partitive models of division are equally valid. Both yield the same numerical result for the same problem. 18÷3=6 no matter which model you use.

 

It's just that in one, 3 represents the number of groups and six the items or units in the group; in the other, 3 is the number of items in a group and six the number of groups.

Edited October 24, 2016 by maize

[LMD]

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. 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.

 

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. Add to that any other specifics from a word problem or deeper problem they are solving. They should also know they could use completing the square, etc. if they preferred.

 

What I'm finding IRL with many students is that they need a calculator and have memorized the "/" key. 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. Most of the time they will just shove their book or homework aside and decide they don't care when it gets that deep. 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. When one actually understands math, it's not that difficult because one can break it down mentally from building blocks. That understanding is helpful if it comes from an early age rather than trying to fill in the gaps later on.

 

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. Hang on, it's on our formula sheet somewhere." If the problem isn't already set up where y=0, I can count on numerous mistakes.

I wish I'd had a high school maths teacher like you!

[MaeFlowers]

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.

 

Furthermore, I'm not implying that you can't switch between equivalent forms of the same problem.

 

But I do believe that it's the same problem.

 

 

 

 

It may not have been snark. The past few days have been craptastic. I may be taking things personally. I sincerely apologize.

 

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.

I was already having an awful day and now I feel totally incompetent in something I thought I enjoyed teaching. Screw this day :(

Yep. The only thing I've ever been good at in life is math. Now I'm finding out that I am, in fact, not good at math. Now I got nothin'. Screw this day. Edited October 24, 2016 by MaeFlowers

[Guest]

There are two common and equally valid conceptualizations of division. Both yield the same numerical result.

 

In the equation 18÷3=6, the Partitive model of division would have us dividing 18 into 3 groups. The 6 on the other side of the equation then represents the number of items in each group.

 

The quotative or measurement model has us dividing 18 into groups of 3. The 6 on the other side of the equation in this case represents the number of groups formed.

 

Both models are valid representations of the equation 18÷3=6.

 

I think some of the confusion that is arising comes from the fact that many people are used to using only one model and not the other. If, for example, you always think of division as partitive--that is, the divisor represents the number of groups formed, then you think that dividing 18 into six groups of three must be a different equation: 18÷6=3. In the quantitive model, however, 18÷6 would mean forming three groups of six, and it is 18÷3 that gives you six groups of three.

 

The point of this discussion is that both quantitive and partitive models of division are equally valid. Both yield the same numerical result for the same problem. 18÷3=6 no matter which model you use.

 

It's just that in one, 3 represents the number of groups and six the items or units in the group; in the other, 3 is the number of items in a group and six the number of groups.

The order of numerals in a problem means something when translating a word problem or image to the number sentence/equation. I am fully aware of the model but I STILL cannot see how that translates into what you're explaining. If we are saying then that there are two equally correct answers for each iteration, how would the student know which one to pick on a multiple choice test? I can accept two different ways of *thinking* about division, but are there not still rules and best practices governing which is used and when? It's like the date format - much confusion arises when someone uses an atypical format for the culture and location they are in, especially day/month/year and month/day/year in digital form. This seems like the same sort of issue?

 

If you're giving the numbers 18, 3, and 6 and asking students to demonstrate their relationships to one another in the form of number sentences or pictures, you'll 3*6=18, 6*3=18, 18/6=3, and 18/3=6. These can all be represented by a direct, physical model in a different way, but they are not interchangeable. How then would there be any certainty about how to translate a word problem into a number sentence?

 

How would a student know the difference between how they're representing each iteration of the problem when they're backwards from one another, depending on the method used?

Edited October 24, 2016 by Arctic Mama

[Guest]

It may not have been snark. The past few days have been craptastic. I may be taking things personally. I sincerely apologize.

 

Yep. The only thing I've ever been good at in life is math. Now I'm finding out that I am, in fact, not good at math. Now I got nothin'. Screw this day.

I'll drink some strong coffee to that one. We are in the middle of a do over from the awful morning, and some of that was over math. This whole thing makes me want to cry after I've spent so much time and energy working on pedagogy to help my kids avoid the sort of math by mysticism and discovery nonsense that had *me* thinking I was idiotic by the time I hit high school. It can be useful to know there are three or five ways to think about a problem and come to the correct answer, but it can also be absolutely intellectually paralyzing to never be quite sure what is expected of you when solving a stupid flipping problem.

 

:grouphug:

[eternalsummer]

Well if someone gave me the word problem:

 

You have 18 hats, and you hand them out in sets of 3.  How many people get a set of hats?  

 

or something like that,

 

I'd automatically know to write:

 

18/3=6

 

And someone gave me the word problem:

 

You have 18 hats, and you split them between 3 people.  How many hats does each person get?

 

I'd know to write: 

18/3=6

 

So going from the word problem to the abstract equation, there is no confusion and all is well.

 

 

But if you ask me to go from the abstract 18/3=6 and put it into a word problem, every time I am going to say, you've got 18 hats and you're splitting them between 3 people, because that is the way I think about it.

 

What PP is saying is that it is equally valid (if basically foreign to me) to also say (when given the abstract problem 18/3) that you've got 18 hats and you're splitting them into groups of 3 hats.

 

 

[eternalsummer]

Arctic mama, no worries, I didn't know this part of math (well, not overtly, anyway - obviously I can do long division so I must have intuited it at some point).  But anyway, no one taught it to me, and I tested out of Calc. in college.  So I don't know if it's one of those things you have to know to do well in math :)

 

 

I am glad I posted this thread, though, because I was going to write to the publishers of that book and inform them of their errata, hah.

[Guest]

Well if someone gave me the word problem:

 

You have 18 hats, and you hand them out in sets of 3. How many people get a set of hats?

 

or something like that,

 

I'd automatically know to write:

 

18/3=6

 

And someone gave me the word problem:

 

You have 18 hats, and you split them between 3 people. How many hats does each person get?

 

I'd know to write:

18/3=6

 

So going from the word problem to the abstract equation, there is no confusion and all is well.

 

 

But if you ask me to go from the abstract 18/3=6 and put it into a word problem, every time I am going to say, you've got 18 hats and you're splitting them between 3 people, because that is the way I think about it.

 

What PP is saying is that it is equally valid (if basically foreign to me) to also say (when given the abstract problem 18/3) that you've got 18 hats and you're splitting them into groups of 3 hats.

Well see this makes absolute sense and I agree with your conclusions. Maybe there is something about the way it's being stated in some other posts that totally isn't clicking for me? ARGH!

[regentrude]

. It can be useful to know there are three or five ways to think about a problem and come to the correct answer, but it can also be absolutely intellectually paralyzing to never be quite sure what is expected of you when solving a stupid flipping problem.

 

But isn't that rather the issue with a poorly designed problem?

 

A student who understands division conceptually and has been taught to think rather than apply algorithms should have no trouble solving a well designed division, unambiguously phrased, problem.

Edited October 24, 2016 by regentrude

[regentrude]

If you're giving the numbers 18, 3, and 6 and asking students to demonstrate their relationships to one another in the form of number sentences or pictures, you'll 3*6=18, 6*3=18, 18/6=3, and 18/3=6. These can all be represented by a direct, physical model in a different way, but they are not interchangeable. How then would there be any certainty about how to translate a word problem into a number sentence?

 

That direction has a unique answer. The word problem has a unique mathematical solution expressed by a particular equation.

 

The issue was the other direction: phrasing an equation as a word problem. That is not uniquely defined; there are different word problems that lead to the same equation, and asking a person to phrase a particular equation as a word problem may lead to different answers.

[kiana]

Ananemone is right on.

 

The only way this would ever come up as in any kind of dispute in a MC test is if a student were asked "what is an appropriate model for 18/3?" and both were presented as alternatives. I would consider such a test to be incredibly poorly written mathematically although I do acknowledge such tests exist. Similarly, I could see a student being marked wrong because they were told to present a word problem for division and presented the wrong model -- but I would also consider the teacher to have erred. I'd mark it wrong if they were told to write a word problem for 18/3 and came up with "Johnny had 18 apples and ate 3" or something. It's similar to (and I've heard of it happening) marking a kid wrong because they were told to come up with a word problem for 2+1 and wrote "Johnny had an apple and found two more" because the teacher considered that to be 1+2.

 

I have noticed that many of my algebra students struggle with translating things such as "2 is subtracted from 4"and tend to write "2-4" and now I am wondering if they had somehow intuited "The number written first needs to come first" or something similar? I will have to investigate this further. 

[regentrude]

The only way this would ever come up as in any kind of dispute in a MC test is if a student were asked "what is an appropriate model for 18/3?" and both were presented as alternatives. I would consider such a test to be incredibly poorly written mathematically although I do acknowledge such tests exist. Similarly, I could see a student being marked wrong because they were told to present a word problem for division and presented the wrong model -- but I would also consider the teacher to have erred.

 

That is what I meant by "poorly designed problem".

I did not teach my kids so they could do well on stupidly written multiple choice tests. I preferred to teach them so they understand math.

 

 

I have noticed that many of my algebra students struggle with translating things such as "2 is subtracted from 4"and tend to write "2-4" and now I am wondering if they had somehow intuited "The number written first needs to come first" or something similar? I will have to investigate this further.

 

I suspect this may stem from them drilling formulaic ways of "solving" word problems without understanding. Such as looking for trigger words and indiscriminately memorizing "subtracted means put a minus sign". Sigh

 

[Paige]

The order of numerals in a problem means something when translating a word problem or image to the number sentence/equation. I am fully aware of the model but I STILL cannot see how that translates into what you're explaining. If we are saying then that there are two equally correct answers for each iteration, how would the student know which one to pick on a multiple choice test? I can accept two different ways of *thinking* about division, but are there not still rules and best practices governing which is used and when? 

 

Nobody is talking about different valid numerical answers. The answer to 18/6 is always 3. The word problem can change what the 3 means for that particular time, kwim?

 

I get annoyed when teachers insist that 18/6 means I have to make 6 groups; always and forever that it means that's the number of groups I'm making and I can only have 3 in each group. Because sometimes I want to think that 18/6 means I want 6 in a group because that's the useful info I need, and I don't care how many groups I get as long as 6 are in the group, but the answer will still be 3. 

 

I think it's similar to multiplication. Some people insist that 6x3 means 6 groups of 3, but other people think it can mean 6 three times. I say it can mean whatever you need it to. 

[Guest]

I think it's a thinking problem and not an algorithmic one, in the case of mixing up the order of a subtraction word problem. I'm probably a crap math teacher according to this forum, but I always remind them to think through what the words are actually asking and model it in their heads or on paper *before* constructing the equation for the problem. Not thinking through what is being asked seems to be 90% of the issue with word problems.

[regentrude]

 Not thinking through what is being asked seems to be 90% of the issue with word problems.

 

This, precisely. It is astounding how many people attempt to "do math" without thinking.

[Guest]

Nobody is talking about different valid numerical answers. The answer to 18/6 is always 3. The word problem can change what the 3 means for that particular time, kwim?

 

I get annoyed when teachers insist that 18/6 means I have to make 6 groups; always and forever that it means that's the number of groups I'm making and I can only have 3 in each group. Because sometimes I want to think that 18/6 means I want 6 in a group because that's the useful info I need, and I don't care how many groups I get as long as 6 are in the group, but the answer will still be 3.

 

I think it's similar to multiplication. Some people insist that 6x3 means 6 groups of 3, but other people think it can mean 6 three times. I say it can mean whatever you need it to.

I agree this changes depending on the word problem and what it is asking to solve. That's why I'm confused as to the argument! But with the open ended pictorial modeling there are two ways you could represent division in the fact family, and whichever one you'd choose would and should be dictated by the problem at hand. Talking about this in the abstract is what is confusing me, I think. It always seemed pretty straightforward to me until this thread!

 

I *think* I am getting the quibble with the OP now, so I will just quit while I'm ahead :lol:

 

This day still sucks so bad :(

Edited October 24, 2016 by Arctic Mama

[maize]

So in summary:

 

Given the word problem: You have 18 apples and want to divide them equally between three baskets, how many apples will be in each basket? The appropriate equation would be 18/3=6. Given the problem: you have 18 apples and want to put them in baskets with three in each basket, how many baskets will you need? The equation would be, again, 18/3=6. The equation is the same, but what is represented (the apples or the baskets) by the 3 and the 6 is reversed.

 

Given the problem: You have 18 apples and want to divide them equally between six baskets, how many apples will be in each basket? The equation would be 18/6=3. For the problem: you have 18 apples and want to put them into baskets with six in each basket, how many baskets will you need? The equation would be again 18/6=3.

 

A student asked to make up a word problem for the equation 18/3=6 could use either of the first two (or any other appropriate valid response), if asked to make up an word problem for 18/6=3 either of the last two problems listed above would  be valid.

Edited October 24, 2016 by maize

[maize]

Pictorially: 18/3=6 can be represented with equal validity as either *** *** *** *** *** *** or ****** ****** ******

[Guest]

So in summary:

 

Given the word problem: You have 18 apples and want to divide them equally between three baskets, how many apples will be in each basket? The appropriate equation would be 18/3=6. Given the problem: you have 18 apples and want to put them in baskets with three in each basket, how many baskets will you need? The equation would be, again, 18/3=6. The equation is the same, but what is represented (the apples or the baskets) by the 3 and the 6 is reversed.

 

Given the problem: You have 18 apples and want to divide them equally between six baskets, how many apples will be in each basket? The equation would be 18/6=3. For the problem: you have 18 apples and want to put them into baskets with six in each basket, how many baskets will you need? The equation would be again 18/6=3.

 

A student asked to make up a word problem for the equation 18/3=6 could use either of the first two (or any other appropriate valid response), if asked to make up an word problem for 18/6=3 either of the last two problems listed above would be valid.

Yes, I agree with all this. I think I wasn't understanding the OP as being asked in the abstract to model this, but I'd teach it the same was above.

[HomeschoolingHearts&Minds]

So what would be a good word problem for 1 3/4 divided by 1/2?

 

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.

 

You are sewing pillows for Christmas gifts and have 1-3/4 yards of fabric.  If each pillow requires 1/2 yard, how many pillows can you make?

Edited October 24, 2016 by HomeschoolingHearts&Minds

[maize]

You are sewing pillows for Christmas gifts and have 1-3/4 yards of fabric.  If each pillow requires 1/2 yard, how many pillows can you make?

 

I like this one!

[EmseB]

Pictorially: 18/3=6 can be represented with equal validity as either *** *** *** *** *** *** or ****** ****** ******

I think a key in this conversation to add is that 18/6=3 can also be represented those two ways. At least, it helped me to clarify my thoughts on everything.

[maize]

I think a key in this conversation to add is that 18/6=3 can also be represented those two ways. At least, it helped me to clarify my thoughts on everything.

Absolutely.