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What feedback would you give to this student? Some considerations…

  • Would you ask a question or make a statement?
  • What written feedback would be most helpful?
  • If you were able to have a conversation with this student, how would you start it?
  • What would the student’s job be once you handed the paper back to him/her?
  • Some “Good news, bad news” here. The good news, surely, is that the student gets that comparing the concentrate to the total might be a useful way to go. The bad news seems to be that regardless whether s/he compares the concentrate to the water or the concentrate to the total, not knowing how to compare fractions is going to ruin any potential insight to be gained thereby.

    I have always liked the shortcut of comparing a/b to c/d by taking bc and placing it over the right-hand fraction and ad placed over the left hand fraction. This is a sneaky but mathematically valid way of getting numerators properly placed over the corresponding fraction with IMPLIED like denominators (never specified, since the specific denominator doesn’t matter as long as it is the SAME for both fractions. This allows student to just compare integer numerators to integer numerators and then draw the simpler conclusion about the relative size of the rational numbers by just asking which integer is larger (that’s generally not a problem for all but the most numerically-challenged students).

    Of course, alternative strategies include finding equivalent fractions with like denominators, and finding decimal equivalents. There are likely others, though with inconvenient numbers like 11 and 17 as denominators, lots of luck trying diagrams. 🙂

    I think it’s worth starting with asking the student to explain how s/he determine which was the larger fraction, and then push at that (mis-)understanding with questions about friendlier fractions using the same strategy so that the student has a better sense of it not working. After that, if there was time to really not just TELL the student ‘the’ right way to solve this sort of thing, I’d try to help him/her drawn on prior knowledge of rational numbers which, if correct, should be the basis for one or more strategies that would work here. If the student understands that the job entails knowing which of two numbers (neither of which is an integer, let alone a natural number) is larger, you’re off to the races. If not, well, you’ve got a long, nasty slog ahead of you, at least potentially. Students who “solve” problems in mathematical domains in which they are essentially clueless can come up with rather remarkably creative buggy algorithms. I wonder if those sorts of buggy algorithms are harder to dispel than those of students with a greater degree of relevant numeracy.

  • Tough one. I’m not totally happy using this as feedback, but the argumentative East Coaster in me always likes the strategy Michael Paul Goldenberg described above as “push at that (mis-)understanding with questions about friendlier fractions using the same strategy so that the student has a better sense of it not working.” “What if you had started with 1/6 and tried to compare it to 3/12? The numerator increases by 2 and the denominator increases by 6, but is 1/6 really bigger than 3/12?” Unfortunately this may just give the kid the take-home message that the teacher is a smartass.

    I kind of like the idea of just converting to decimals. After all, the main insights about ratios are already there. The missing part is comparing fraction size, which is valuable but not essential to this problem.

    Maybe decimals-on-the-calculator could be the spoonful of sugar that helps the medicine of you-can’t-do-that-with-fractions go down.

  • As Michael said, I’d ask the student to explain what they did and why, and take it from there. I think the student was not thinking much about “comparing fractions” in the abstract, but about comparing how much water and juice would be added to turn Mix A into Mix B — and that is a good, instinctive way to think about this problem. But they had two misconceptions:

    (1) The “+2” tells how much juice would be added, but the “+6” is NOT the amount of water added. It includes both the juice and the water, so they’ve counted the juice twice without realizing it.

    (2) The student seems to be under the impression that, if more water was added than juice, then the mixture will get weaker. But what they need to decide is whether the additional fluid is stronger or weaker than the original mixture.

    If their numbers had been right, adding 2 cups of juice and 6 cups of water, it would indeed have made a weaker Mix B. But since the true addition was 2c juice and 4c water — which is the equivalent of 4c juice in 8c water, so it’s “orangier” than Mix A — the mix got stronger.

  • The student clearly knows what he is doing and gets an answer to the original problem. in order to let him find out that something is wrong somewhere I would ask “For A, how many cups of concentrate for 16 cups of water, for 24 cups of water, for 32 cups of water…?” and “For B, how many cups of concentrate for 24 cups of water, for 36 cups of water, for 48 cups of water…?” and sit back for a short while.

  • justin

    i would like the person to get some order in their work and show more of what they have done and what they are doing its all over they place but when i got a good understanding of the problem it looks good until this person got to the ratio parts

  • monique

    always remember to divide by the numerator for the first and divide by the denominator for the second and then compare and i think u can think of this as a real life situation if that would help more water can take the place of an orangey flavor.