Problem centered math

Developed by LEARN NC with Grayson Wheatley

A photograph that is 6" on the base and 8" high is to be enlarged so that the new base is 15". What will the height of the enlargement be?

Two rectangles, one 6 by 8 inches and the other 15 inches by unknown.

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Transcript

Wheatley
Here is a problem. If you would just read this aloud.
Student
A photograph that is 6 inches on the base and 8 inches high is to be enlarged so that the new base is to be 15 inches. What will the height of the enlargement be?
Wheatley
And so here we have…You take this into the store, this photograph, and you say “I want you to blow it up and my new picture frame is 15 across the bottom.” How high would it be?
Student
I would do a proportion.
Wheatley
All right, you want to go ahead and solve it?
Student
I multiply the 8 and the 15 and divide by 6.
Wheatley
OK, and what would you get?
Student
And, I would divide that by 6.
Wheatley
So what answer did you get?
Student
X equals 25.

Wheatley
OK, so that would be your solution to the problem. So now, on here, when you get your photograph back it would be 2.5.
Student
No, that is wrong.
Wheatley
Why doesn’t that work?
Student
Because it is smaller.
Wheatley
About what would you expect that to be? Just sort of glancing at it.
Student
18 or 20.
Wheatley
OK, so let’s go back and think about it. So, you set up 8 over 6 is equal to x over 15. And your next step is you multiplied 8 times 5 and divided by 6. So this 8 times 15, Lets check out the multiplication.
Student
45, no 40. That’s wrong.
Wheatley
40. Why don’t you move it over here?
Student
Then 8 times 1 equals 8 plus 4. 120. And divide that by 6 goes into 12 twice, subtract 0. 20.
Wheatley
20. Now does that seem reasonable?
Student
Yes.
Wheatley
OK. You like that better than 2.5?
Student
Yes
Wheatley
I was using this problem with another student and they said the answer was 17. Can you think how they might have gotten 17 as the answer to that problem?
Student
They could have just estimated.
Wheatley
Actually what they did when they explained it to me is they said “Well, this 15 is 9 more than 6 so I need a number that is 9 more than 8. So that is how I get 17.” So, what do you think about that reasoning?
Student
I think it could work….but, I don’t think it could work on any problem.
Wheatley
You had an answer of 20. So do you think your 20 is correct or the 17 that this person gave? Which would you go with?
Student
I would go with the 20.
Wheatley
All right. Instead of 15, suppose this number was 24. Now would, without going through your procedure here, would there be a way of reasoning what this height would be?
Student
Like if you just stuck in a 24, just like estimated it?
Wheatley
OK, let’s see. This is 24 so this 6 now has instead of a 6 a 24 base. How high would it be? There’s some way we could just think about that rather than using the proportion, which works very well.
Student
We could do 6 times 4 and 8 times 4.
Wheatley
And that would give 32. So that would be another way you could think about that problem. Could you use that line of reasoning with 15?
Student
Well, 6 doesn’t go into 15.
Wheatley
OK, What if you do divide 15 by 6, what do you get?
Student
2, 3, 4
Wheatley
OK, 2 times 6 is 12
Student
Right

Wheatley
And what is left over?
Student
Like 3. So it would be 2.3?
Wheatley
So it would be 2.3. OK, so you have seen several ways of doing it. 20 makes a lot of sense to me. And I really appreciate the way you explain your thinking so clearly for us.

The student’s work

student's written work

Analysis

It is significant that the student recognized this as task for which setting up a proportion (a procedure she had learned) would be appropriate. She makes a computational error and arrives at 2.5 as an answer. She rejected this number after I asked to reflect on it. She provides a reasonable estimate of the height of the new photograph and seems happy with 20 as the computed height. She is unsure about the additive method I propose, not rejecting it. However, she likes her computed answer better. Her proportional reasoning in not well developed.