Exchange between Volunteers

HallsofIvy said:
Isn't it wonderful that rectangles only have integer length and width.
Click to expand...

Yes, HallsofIvy, the point (by way of sarcasm in this case) you were bringing up is correct.

1) Students should not be limited in this type of problem by considering only integer divisors.

2) A beginning student presented with this problem should never assume the values would
have to be integers. They should have sufficient training that allows for non-integer
values.

Now, they might try to anticipate integer values.

Suggestion:

Trust (that there could be other than integers) but verify (it).


Another point: In a more generalized area/perimeter problem,
the existence of integer solutions does not preclude
additional solutions for the problem which are non-integers.


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Edit:

Here is a problem where the values cannot be assumed to be integers:


There is a rectangular box. If the height's dimension is 3 units longer than the width's,
and the length's dimension is eight times the width's dimension, then what are the
dimensions of the box if it is known that its volume is 81 cubic units?


Let x = the measure of the width of the box

Let x + 3 = the measure of the height of the box

Let 8x = the measure of the length of the box

x(x + 3)(8x) = 81 **


Without regard to order, can one assume that the sides
must be one of the following scenarios where each
dimension of the box has an integer number of units?

1*1*81

1*3*27

1*9*9

3*3*9



So, after all, what is/are the solution/s for (positive) x in the equation ** above ?
 
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lookagain said:
Yes, HallsofIvy, the point (by way of sarcasm in this case) you were bringing up is correct.

1) Students should not be limited in this type of problem by considering only integer divisors.

2) A beginning student presented with this problem should never assume the values would
have to be integers. They should have sufficient training that allows for non-integer
values.

Now, they might try to anticipate integer values.

Suggestion:

Trust (that there could be other than integers) but verify (it).


Another point: In a more generalized area/perimeter problem,
the existence of integer solutions does not preclude
additional solutions for the problem which are non-integers.


---------------------------------------------------------------------------


Edit:

Here is a problem where the values cannot be assumed to be integers:


There is a rectangular box. If the height's dimension is 3 units longer than the width's,
and the length's dimension is eight times the width's dimension, then what are the
dimensions of the box if it is known that its volume is 81 cubic units?


Let x = the measure of the width of the box

Let x + 3 = the measure of the height of the box

Let 8x = the measure of the length of the box

x(x + 3)(8x) = 81 **


Without regard to order, can one assume that the sides
must be one of the following scenarios where each
dimension of the box has an integer number of units?

1*1*81

1*3*27

1*9*9

3*3*9



So, after all, what is/are the solution/s for (positive) x in the equation ** above ?
Click to expand...

Lookagain,

I did not get the "point" of your cubic equation.

I am fully aware of the fact that most of the solutions of quadratic equations are complex numbers (with non-zero imaginary part).

However, quadratic equations with rational coefficients - are amnable to "observations" and quick guesses.

It is also an easy vehicle to teach use of quadratic in middle school and talk about practical application (my opinion).
 
Last edited by a moderator:
lookagain said:
1) Students should not be limited in this type of problem by considering only integer divisors.
Click to expand...

I view the problem type in this exercise as 'Easy' (as in "easy for pre-algebra").



2) A beginning student presented with this problem should never assume the values would
have to be integers. They should have sufficient training that allows for non-integer
values.
Click to expand...

I'm not sure that I can agree with this viewpoint. The reason for my uncertainty is that I don't really know what schools are doing, anymore. Yet, I still think that all beginning students tend to grasp for Whole-numbered solutions as soon as they're presented with something abstract.

Tunnel vision will surely become an issue later on, but, for pre-algebra exercises, I don't have an issue with keeping assignments "simple" (i.e., sticking with the Integers).

I remember that my first algebra teacher gave every student in the class optional handouts from time to time; these handouts presented some of the lecture material in greater depth, and they always included exercises or brain teasers that were more challenging or thought-provoking than the material presented in class. These handouts were never mandatory, but they were eagerly received (and sometimes requested) by curious students.

Again, I'm not really sure what-all pre-algebra covers (and I've never worked as a math instructor), but I've come to realize that not much beyond 2+2 is manageable for some students faced with abstract arithmetic or concepts for the first time.

Cheers :cool:
 
Denis said:
No comments...except
Click to expand...

except for your comments :p

Yes -- many threads end up with side debates about math education, wording, and the like. Can't be helped.

My memory recently reminds me of the old woman in Garp's world (Robin Williams character) who described life as a merry-go-round. Some people like a fast ride, others not so much. Some like the ups-and-downs, while others like to get off. Can't really remember the exact dialogue, but the point was made that many different kinds of people make the world go 'round.
 
Denis said:
This sho is a strange thread, sho is...
Now the OP flew the coop, and bravely replaced by H.O.F. Ivy.

Subhotosh, will you teach me cricket? S'gotta be less confusing than this....
Click to expand...

Sorry, that was my doing. I just cut the thread into two - after the OP flew the coop.

The original thread is still there - without these trailing part.

Sorry for the confusion - I thought changing title would eliminate confusion - it didn't, at least not for Denis....
 
lookagain said:
Yes, HallsofIvy, the point (by way of sarcasm in this case) you were bringing up is correct.
Click to expand...
Actually, there was no intended sarcasm! I was just amused at how often checking integer values works.
(Well, maybe a little subconscious sarcasm. While writing this, I suddenly remembered how one teacher we had just hired actually taught students in Calculus to find max and min of a function by finding its values over a range of integers and picking out the largest and smallest. When I pointed out that max and min were not necessarily at integer values, his response was that in Calculus problems they always are! He only lasted the one year of his contract- not solely for that reason.)

1) Students should not be limited in this type of problem by considering only integer divisors.

2) A beginning student presented with this problem should never assume the values would
have to be integers. They should have sufficient training that allows for non-integer
values.

Now, they might try to anticipate integer values.

Suggestion:

Trust (that there could be other than integers) but verify (it).


Another point: In a more generalized area/perimeter problem,
the existence of integer solutions does not preclude
additional solutions for the problem which are non-integers.


---------------------------------------------------------------------------


Edit:

Here is a problem where the values cannot be assumed to be integers:


There is a rectangular box. If the height's dimension is 3 units longer than the width's,
and the length's dimension is eight times the width's dimension, then what are the
dimensions of the box if it is known that its volume is 81 cubic units?


Let x = the measure of the width of the box

Let x + 3 = the measure of the height of the box

Let 8x = the measure of the length of the box

x(x + 3)(8x) = 81 **


Without regard to order, can one assume that the sides
must be one of the following scenarios where each
dimension of the box has an integer number of units?

1*1*81

1*3*27

1*9*9

3*3*9



So, after all, what is/are the solution/s for (positive) x in the equation ** above ?
Click to expand...
 
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