8.NS.A.1 and 2

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    I’m troubled by the interpretations I’m seeing of these standards and would love some clarification, correction and/or support. It seems that folks are interpreting the second of these standards as a call to lead students through memorizing a procedure for finding approximations of irrational numbers to the tenths and then hundredths place by essentially pretending that the growth between, say, sqrt(16) and sqrt(25) is linear. I see no use in 8th graders muddying their mathematical waters with such procedures as opposed to, say, reasoning that sqrt(22) must be closer to 5 than 4. I suppose that it makes sense to then choose 5.6 and then 5.7, … , square them and see which is closer, but it makes no sense at all to pretend that sqrt(22) is 6/9 from sqrt(16). This is corrupting the sense of what a square root IS. Yes?
    Secondly, most people seem to be ‘treating’ this standard early in an 8th grade text / course, and so are teaching how to rewrite a repeating decimal as a fraction. This process is a clever application of solving by elimination, which we haven’t learned yet (I assume) until the end of our 8th grade year. Using mathematical techniques that have not been derived, explored and established seems epically contrary to everything I love and admire about CCSS-M.
    As always, advice and feedback are most appreciated.
    My Best,
    Joanna Burt-Kinderman


    Bill McCallum

    In answer to your first paragraph: Yes! The standard says “Use rational approximations …” not “Find rational approximations …” so it is a mystery to me how people could misinterpret it. Of course, as you say, this might sometimes involving finding them using simple methods of trial and error, as you suggest, but how anybody could interpret this as requiring a systematic method for finding approximations is beyond me.

    In answer to your second paragraph, I’m not sure what you mean by “a clever application of solving by elimination.” 8.NS.1 does in fact say “convert a decimal expansion which repeats eventually into a rational number,” so you have to have some way of doing this. One way is to solve a linear equation: if $x = 0.171717 …$ then I can write the equation $100x-x = 17.171717 … – 0.171717 … = 17$ and solve the equation $100x – x = 17$ to write $x$ as a fraction. You are right that there is some cleverness involved in thinking of multiplying by 100 and subtracting the original number, but this in itself is a nice application of MP7, Look for and use of structure.

    Students have been solving equations like this since Grade 7:

    7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

    Converting repeating decimals to fractions strikes me as a nice “mathematical problem” that falls under this standard.




    I guess the issue is that I am seeing such huge payoff within my district (I am a K-12 math coach for a rural district) in leading with the message that students should build on steps they can justify. We are avoiding cross-multiplying, FOILing, and all sorts of shortcuts that are previously accepted as if they were ideas unto themselves… I am a strong believer in keeping the arrows of this change all pointing in the same direction, as the change is just enormous and is so very important.

    If we are to unpack the thinking of this particular repeating decimal to fraction process, and have students replicate it, as I see it, the following is what is going on:

    x = 17.171717…

    100x = 1717.1717…
    (here student should be able to answer why we choose 100)
    (student should also be able to answer why the second equation is true, assuming the first / why the equations say the same thing)

    Now, to move forward, student needs a reason subtracting the first from the second results in an equally true equation. If we are to regularly say:
    “well, then, why would it be really cool to know what 99x is? why would this be a better idea that 101 x?”

    then I suppose we are teaching students to replicate a teacher seeing a cool structure and making use of it, but I don’t think we can reasonably call this the students looking for the structure. In the classrooms that I’m observing and coaching, this task as a standard does not build meaning, but rather a notion that math is magic…

    If we teach this as a most simple application of systems, I don’t think you encounter the same disconnect.

    As an underlying issue, I’m not even sure that I see the real relevance here… Perhaps you can illuminate the importance of the idea of changing repeating decimals (we would only see this with a calculator) to fractions? If this is truly just a nice mathematical problem, should it not be a resource for MP 7 or a specific suggested treatment of 7.EE.4? Can you give me another example of a nice mathematical problem that is important enough to be turned into a standard?

    I so appreciate the feedback and debate, because the process is so clarifying and cleansing. Further, as someone doing my work without a peer group, This site is a gem. It is beyond wonderful to have this level of feedback. If I’m missing something here, I’d enjoy seeing it from a new angle.

    So Many Thanks,



    Bill McCallum

    Joanna, to address your last question on the importance of converting repeating decimals to fractions, I would say that the important thing here is not so much actually doing the converting as understanding why it can always be done. Repeated reasoning with the conversion can lead to this understanding. So, to achieve the understanding, you need to do a few conversions, just to see how it works. But it is the seeing how it works that is important point.

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