How many cups of sugar do I need for these 5th grade problems?

Question

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?)

Answer 1

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.

Answer 2

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.

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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.