6.G.2

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  • #2221
    kgartland
    Member

    Hi,
    I have a question about the reference to “unit cubes” and fractional side lengths in the 6.G.2 standard:
    It states “Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas v=lwh and V=Bh to find the volume of right rectangular prisms with fractional edge lengths in the context of real-world problems”.

    As it says to use “unit cubes” and yet we are to have students work with fractional edge lengths, I’m wondering if you could elaborate on what kind of unit cubes you could suggest be used in which fractional side lengths could then be created?
    Is there such a manipulative? Or would you suggest conducting this activity using technology?
    Thanks for any help you can provide.

    #2222
    Anonymous
    Inactive
    #2236
    Bill McCallum
    Keymaster

    Alexei is right, I think it makes sense to replace “unit cubes” with “rectangular prisms with unit fraction side lengths” here. One of these days I will publish my glitch file!

    Although I would point out that you could use unit cubes. For example, you can pack a $\frac13$ by $\frac15$ by $\frac17$ rectangular prism with unit cubes with side length $\frac1{3\times5\times7}$, forming a $5 \times 7$ by $3 \times 7$ by $3 \times 5$ array. But in the end you still have to find the volume of the cube, and the natural way to do that is by seeing how many of them fit into a cube with side length 1. Since that’s also the way you would find the volume of a rectangular prism with unit fraction side lengths, I think it makes more sense to do the latter directly.

    Probably way more answer than you wanted!

    #2237
    kgartland
    Member

    Thanks so much for responding so quickly. I agree with the proposed language suggestion of “unit fraction side lengths.” Your added bonus of describing how you could use unit cubes is intriguing, however, it would certainly be a very small cube 🙂
    Perhaps an activity could be suggested where students make unit cubes that are 1.5 by 1.5 out of grid paper (or something like that) and then use it to measure the volume as you suggest in your post.
    So important to keep it concrete while they are first learning the concept rather than just multiplying l x w x h without reason.

    #2254
    Anonymous
    Inactive

    Bill,

    Your reply above seems to suggest that 6.G.2 is calling for some sort of hands-on manipulative activity, which seems wildly impractical to me. Your reply above is also at odds with the answer you gave us in July 2010 when we asked essentially the same question. Your answer then was, “Certainly the intention was not to mandate any manipulative activity,
    but rather to describe how to conceive of the volume of a prism with
    fractional side lengths. This could be aided by manipulatives,
    drawings, computer animation, verbal description … how to bring
    about the conceptual understanding is not mandated.” Jason Zimba also responded to us at that time, saying, “I myself at least always thought of this as a pencil and paper diagrammatic argument, not a manipulative activity.”
    So, which is it? Are you proposing that teachers try to obtain actual “unit cubes of the appropriate unit fraction edge lengths”? Or even, as you now seem to be proposing, “rectangular prisms with unit fraction side lengths”? Or is this a thought experiment, as we thought you and Jason were saying three years ago?

    Andy

    #2296
    Bill McCallum
    Keymaster

    Andy, you are talking about my August 23 reply, right? I didn’t mean it to be taken any differently from my previous comments on this. To me the unit cubes or the rectangular prisms with unit fraction sides are mathematical objects, and when I talk about packing them this way or that I am talking about a mathematical activity. So, this could be represented by “manipulatives, drawings, computer animation, verbal descriptions …” as in my previous answer. Sorry for the miscommunication.

    • This reply was modified 10 years, 6 months ago by Bill McCallum.
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