Chris Shore passes on the above, and he thinks that things just aren’t clicking for this kid. I’m inclined to agree. What would you all recommend? What’s the next step for this kid?

I just want to mention that I really don’t like the first question. If you’re going to use the same diagram with numerous lengths and widths, I think the relationship between length and width should be proportional.

Yes! What is given is impossible. This is the sort of thing that can make students think math makes no sense!

Yes, this question is terrible. I am a working mathematician and it was completely unclear to me that the question was implying a relation between length and width. It looks to me like a table of some arbitrarily chosen rectangles. I would have answered the way the kid did (except that I know to look back and second-guess what the test writer is asking for).

The question is completely at fault here, not the kid.

Is that a 560, or a very sloppily-written 500? And I would have preferred to see “wL” instead of “a” in that bottom-right corner. Still, I think this student needs more of a “look for a pattern” kind of advice — see what you can find in the question to make sense of, or how you might find enough information to get to a unique answer in a case where it seems at first like there’s not enough information to make a decision.

Like the previous commenters, I would have preferred to see more in the question talking about a sequence of rectangles following a pattern.

It seems fair to critique the question since the author (follow the link) does some consulting and sales.

As others pointed out we are asked to believe that a single rectangle can magically have several different areas. If you want to go this route, you need to make it clear that we are dealing with more than one rectangle – maybe call them special rectangles and show a few diagrams. Also might be nice to add a little variety by having a row where the area & length are given, but not the width, etc. You have asked the student to fill out a lengthy table with a lot of structure. Why not ask a question related to the structure. For these special rectangles, if the width doubles, does the area double? How about the length? Would there be two special rectangles like these where the area of one was twice the other…

I suppose the intent is to have the student notice that the length is five more than the width for each rectangle, and to generalize the length as w + 5. Using “w” as the variable in the table is probably a poor choice. Most folks will see that “w” has been associated with width, so like the student, will think to use “L” for length & “A” for area. You would have a better shot using “k” or something like that. But, really, what is the point of this? You have artificially inserted a pattern that serves no purpose and isn’t used in a meaningful way. You also have not provided enough rows to give a student a chance to notice what goes on the the area column, nor asked them to draw enough figures to see why the area column produces the pattern that it does. Either make use of the pattern, or don’t include it in the question.

I think we probably need some official ground rules for the comments on this site. The first rule would almost certainly be, “We don’t comment on the quality of the question.”

There are a lot of reasons why I think that this is an important rule, but most importantly, the submitter is the absolute best judge of whether the question was a good one or not. Without the fuller context of class (or even the rest of the assignment) it’s really hard to get a sense of the quality of a question.

Here’s another reason: it’s the easy way out. Even if you think this question isn’t up to par, the student’s work still reveals much about her understanding of rectangles and area. There’s plenty that we could talk about in the above submission: the way the student chose to represent width and length and area with separate variables, the sort of rectangle that they chose to draw, and the next steps that would help this student understand how to solve problems of this sort. Talking about the quality of the question is the easy way out.

Finally, the teachers here are opening themselves up with their submissions. If you want more submissions, you’ve got to play nice, and part of playing nice in this case is ignoring stuff that you think is bad, and commenting more productively on aspects of the student work that you think are meaningful.

This isn’t official site policy yet, but it’s awfully close.

Hi Michael, I thought about that, but I assumed the teacher was offering questions from a textbook or a curriculum they had little say about.

However, I hadn’t looked at who the teacher was. I totally trust that Chris Shore knows what he’s doing. Chris, can you tell us more about the context of this question? I am intrigued. (And I hope my comment did not feel harsh.)

Well then, leaving the question alone, I wonder why the student made a checkerboard in the second response? Where have they seen such a thing that they’re picking it up?

Leaving aside 560 or 500 (sloppy 0s….we’ve all done it) the student abstracts the length to a variable but not the area. Clearly there is some confusion of what is meant by a general rectangle, or generality in general (sorry).

Hey Guys,

It’s all good here. Michael, thanks for having my back, but I actually appreciate the critique of the question (though I agree that we should all keep a professional tone). I have my own criticism of the question, but not anything that the others have pointed out here. I felt that the directions should have explicitly asked to write the length and area in terms of w. This was a common assessment, and that issue was not caught by any of us teachers before hand. We have been doing pattern charts similar to this in class, so I think we assumed too much. With that said, for the vast majority of the students, they must have also been used to these types of charts because they appeared to understand what we were asking. I did not think about the proportionality of the rectangle, though; I appreciate that comment. It was there strictly to define the terms. This question was for a remedial algebra class (our lowest performers on campus), who come to high school not knowing how to calculate the area of a rectangle, and who get hung up on things like which side is the width. With all that said, from looking at the other students’ work, the question was not the issue, so if I may, I would like to comment on the student error and why I submitted it to this site.

The student obviously understood the pattern (length is 5 more units than the width). For this particular student to have recognized that and been able to complete the next two rows of the start was no small victory on the year. I was very excited about that. The fact that he did not generate an algebraic expression, I believe is based somewhat on the difficulty he had with the diagram. That difficulty was shocking during the teaching of the unit.

There are plenty of teachers out there using an area model to teach multiplying of polynomials. Something as simple as distributing a variable, x(x+5), is often represented by a rectangle with x representing the width, and x + 5 representing the length. Or the students may be posed with a prompt like: “The length of a rectangle is 5 units more than the width. Write an expression that represents its area.” For those teachers who have students who are struggling with concepts like this, I challenge you to ask those students to draw, on grid paper, a rectangle that is 5 units longer than it is wide, and see what you get.

I was floored at how many of my high-needs kids in the class could not generate that kind of rectangle. I asked for others like it: “The width is 3, and the length is twice as long.” Their drawings bordered on randomness. My point is that if they cannot generate the diagram, then how are they suppose to understand the abstract polynomial? If you want to see how I developed most of the students to a level of competency and understanding, check out my 180Blog: http://bit.ly/X0Q1FM . Polynomials is the second of the three units shown so scroll to the middle where you see pics of the student work. Thanks again.

hmmm… first time reply’r person. but i like this dialogue! i looked long at hard at the problem before i read any comments and the only thing that i was wondering about was why couldn’t he draw a 4 by 9 rectangle? he obviously picked up on the pattern. the l, w, and a… didn’t even notice.

sorry about my capitalization. or the lack thereof…

The only thing I can tell the young student might be missing is desire to communicate an abstraction. He clearly found a (good!) pattern, and knows how to use it; but there’s nothing but the word “complete” to prompt him to explain his chosen abstraction in the last line. To me it looks like an exam about rectangles, which are described by width and length, and have area. I suppose the same question *could* arise in a section about describing patterns, but (without criticism) that exam would still look different.

Without having set this question (without expectations to be disappointed) I think I’d award full marks; why do you think there’s more to “click”?

## 11 replies on “Area, Rectangles, Variables”

I just want to mention that I really don’t like the first question. If you’re going to use the same diagram with numerous lengths and widths, I think the relationship between length and width should be proportional.

Yes! What is given is impossible. This is the sort of thing that can make students think math makes no sense!

Yes, this question is terrible. I am a working mathematician and it was completely unclear to me that the question was implying a relation between length and width. It looks to me like a table of some arbitrarily chosen rectangles. I would have answered the way the kid did (except that I know to look back and second-guess what the test writer is asking for).

The question is completely at fault here, not the kid.

Is that a 560, or a very sloppily-written 500? And I would have preferred to see “wL” instead of “a” in that bottom-right corner. Still, I think this student needs more of a “look for a pattern” kind of advice — see what you can find in the question to make sense of, or how you might find enough information to get to a unique answer in a case where it seems at first like there’s not enough information to make a decision.

Like the previous commenters, I would have preferred to see more in the question talking about a sequence of rectangles following a pattern.

It seems fair to critique the question since the author (follow the link) does some consulting and sales.

As others pointed out we are asked to believe that a single rectangle can magically have several different areas. If you want to go this route, you need to make it clear that we are dealing with more than one rectangle – maybe call them special rectangles and show a few diagrams. Also might be nice to add a little variety by having a row where the area & length are given, but not the width, etc. You have asked the student to fill out a lengthy table with a lot of structure. Why not ask a question related to the structure. For these special rectangles, if the width doubles, does the area double? How about the length? Would there be two special rectangles like these where the area of one was twice the other…

I suppose the intent is to have the student notice that the length is five more than the width for each rectangle, and to generalize the length as w + 5. Using “w” as the variable in the table is probably a poor choice. Most folks will see that “w” has been associated with width, so like the student, will think to use “L” for length & “A” for area. You would have a better shot using “k” or something like that. But, really, what is the point of this? You have artificially inserted a pattern that serves no purpose and isn’t used in a meaningful way. You also have not provided enough rows to give a student a chance to notice what goes on the the area column, nor asked them to draw enough figures to see why the area column produces the pattern that it does. Either make use of the pattern, or don’t include it in the question.

I think we probably need some official ground rules for the comments on this site. The first rule would almost certainly be,

“We don’t comment on the quality of the question.”There are a lot of reasons why I think that this is an important rule, but most importantly, the submitter is the absolute best judge of whether the question was a good one or not. Without the fuller context of class (or even the rest of the assignment) it’s really hard to get a sense of the quality of a question.

Here’s another reason: it’s the easy way out. Even if you think this question isn’t up to par, the student’s work still reveals much about her understanding of rectangles and area. There’s plenty that we could talk about in the above submission: the way the student chose to represent width and length and area with separate variables, the sort of rectangle that they chose to draw, and the next steps that would help this student understand how to solve problems of this sort. Talking about the quality of the question is the easy way out.

Finally, the teachers here are opening themselves up with their submissions. If you want more submissions, you’ve got to play nice, and part of playing nice in this case is ignoring stuff that you think is bad, and commenting more productively on aspects of the student work that you think are meaningful.

This isn’t official site policy yet, but it’s awfully close.

Hi Michael, I thought about that, but I assumed the teacher was offering questions from a textbook or a curriculum they had little say about.

However, I hadn’t looked at who the teacher was. I totally trust that Chris Shore knows what he’s doing. Chris, can you tell us more about the context of this question? I am intrigued. (And I hope my comment did not feel harsh.)

Well then, leaving the question alone, I wonder why the student made a checkerboard in the second response? Where have they seen such a thing that they’re picking it up?

Leaving aside 560 or 500 (sloppy 0s….we’ve all done it) the student abstracts the length to a variable but not the area. Clearly there is some confusion of what is meant by a general rectangle, or generality in general (sorry).

Hey Guys,

It’s all good here. Michael, thanks for having my back, but I actually appreciate the critique of the question (though I agree that we should all keep a professional tone). I have my own criticism of the question, but not anything that the others have pointed out here. I felt that the directions should have explicitly asked to write the length and area in terms of w. This was a common assessment, and that issue was not caught by any of us teachers before hand. We have been doing pattern charts similar to this in class, so I think we assumed too much. With that said, for the vast majority of the students, they must have also been used to these types of charts because they appeared to understand what we were asking. I did not think about the proportionality of the rectangle, though; I appreciate that comment. It was there strictly to define the terms. This question was for a remedial algebra class (our lowest performers on campus), who come to high school not knowing how to calculate the area of a rectangle, and who get hung up on things like which side is the width. With all that said, from looking at the other students’ work, the question was not the issue, so if I may, I would like to comment on the student error and why I submitted it to this site.

The student obviously understood the pattern (length is 5 more units than the width). For this particular student to have recognized that and been able to complete the next two rows of the start was no small victory on the year. I was very excited about that. The fact that he did not generate an algebraic expression, I believe is based somewhat on the difficulty he had with the diagram. That difficulty was shocking during the teaching of the unit.

There are plenty of teachers out there using an area model to teach multiplying of polynomials. Something as simple as distributing a variable, x(x+5), is often represented by a rectangle with x representing the width, and x + 5 representing the length. Or the students may be posed with a prompt like: “The length of a rectangle is 5 units more than the width. Write an expression that represents its area.” For those teachers who have students who are struggling with concepts like this, I challenge you to ask those students to draw, on grid paper, a rectangle that is 5 units longer than it is wide, and see what you get.

I was floored at how many of my high-needs kids in the class could not generate that kind of rectangle. I asked for others like it: “The width is 3, and the length is twice as long.” Their drawings bordered on randomness. My point is that if they cannot generate the diagram, then how are they suppose to understand the abstract polynomial? If you want to see how I developed most of the students to a level of competency and understanding, check out my 180Blog: http://bit.ly/X0Q1FM . Polynomials is the second of the three units shown so scroll to the middle where you see pics of the student work. Thanks again.

hmmm… first time reply’r person. but i like this dialogue! i looked long at hard at the problem before i read any comments and the only thing that i was wondering about was why couldn’t he draw a 4 by 9 rectangle? he obviously picked up on the pattern. the l, w, and a… didn’t even notice.

sorry about my capitalization. or the lack thereof…

The only thing I can tell the young student might be missing is desire to communicate an abstraction. He clearly found a (good!) pattern, and knows how to use it; but there’s nothing but the word “complete” to prompt him to explain his chosen abstraction in the last line. To me it looks like an exam about rectangles, which are described by width and length, and have area. I suppose the same question *could* arise in a section about describing patterns, but (without criticism) that exam would still look different.

Without having set this question (without expectations to be disappointed) I think I’d award full marks; why do you think there’s more to “click”?