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Problem 1a: If 4 glasses of a mixture needs 1 cup of sugar how many cups of sugar are needed for 5 glasses?

This one is easy and makes sense. It's just simply $\frac{1}{4}*5$ Now taking it a notch higher:

Problem 1b: If 4 glasses of a mixture needs 1 cup of sugar and 6 glasses need 2 cups, how cups of sugar are needed for 7 glasses?

Geometrically you can solve the problem by plotting a line between (4,1) and (6,2); calculate the slope/intercept and get the answer. But how can I do this (algebraically) from a 5th grader's perspective?

Problem 1c: (A variation) If 4 glasses need 1 cup sugar, 6 glasses need 2 and 8 glasses need 4 then how many cups of sugar do I need for 7 glasses?

Geometrically, you get a triangular region and there are two solutions to the problem! How could a 5th grader solve this variation algebraically? Geometrically it's easy, but since there are two answers, how can a 5th grader interpret the solution since there isn't a unique solution?

(Background: my nephew came to me with #2 and wasn't aware of slope/intercepts but was struggling with solving it using simple algebra. Maybe I'm overlooking something but it seems to be a bit tricky for a 5th grader. What are some good ways of approaching these sets of problems?)

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These are strange questions. There isn't enough information to define a unique solution. Why should we necessarily expect that the amount of sugar needed is modelled by a straight line, when for 1b that would mean we need no sugar for 2 glasses of mixture? –  Tara B Feb 23 '13 at 0:26
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Too much sugar and the kids will get hyperactive. You don't want that. –  Asaf Karagila Feb 23 '13 at 1:23
    
@TaraB - good point on zero cups for 2 glasses. Let's assume that the 'line passes through the origin' so you always have some sugar for some mixture. As for them being ill defined, that's the part I'm struggling with myself and hence the question :) –  PhD Feb 23 '13 at 1:27
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I'm not sure there is a good answer to 1b. Probably the answer you give is the best available. For a 5th grader, the last 2 glasses of water needed 1 cup, so one more glass will need 1/2 cup more for a total of 2 1/2. For an algebra student you could fit a quadratic through (0,0), (4,1), and (6,2), which gives approximately (7,2.6264).

For 1c I would do the same and say 7 is halfway from 6 to 8, so I need halfway between 2 and 4 cups of sugar, getting 3. But I'm not sure I believe that answer either.

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Problem 1b: If 4 glasses of a mixture needs 1 cup of sugar and 6 glasses need 2 cups, how cups of sugar are needed for 7 glasses?

One possible solution might be to argue that if 4 glasses of a mixture needs 1 cup of sugar, then $4 + 4 = 8$ glasses of mixture needs $1 + 1 = 2$ cups of sugar. And we know that $6$ glasses needs $2$ cups of sugar.

So $4 + 4 + 6 = 14$ glasses require $1 + 1 + 2 = 4$ cups of sugar. And so for $1/2(14) = 7$ glasses, we would then need $1/2(4) = 2$ cups of sugar.

But as noted, this question is too ill-defined (missing too much information) to arrive at a unique correct solution.

For example, the reasoning I used above would lead to two possible solutions to (c). The first would be, using the reasoning above, 2 cups of sugar. (Based on the information about 4 glasses and 6 glasses. But one could argue similarly that $6 + 8 = 14$ glasses require $2 + 4 = 6$ cups sugar, which would seem to mean that $1/2(14) = 7$ glasses would require $1/2(6)$ = 3 cups of sugar.


Perhaps these questions are posed in order to challenge students to explore ways of approaching the problem, which may result in different solutions. I've seed this pedagogical strategy before: more of a task in problem solving and understanding why one needs more information to arrive at one unique, correct solution, than a task of rote application of algebra manipulations. Certainly, at any rate, one could hardly expect 5th graders to have the sophistication and knowledge necessary to "fit a curve" other than that of a straight line.

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