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I'm working with a middle school student (grade 8) who recently displayed a misunderstanding of place value in decimal numbers. The student believes, for example, that $0.125$ is bigger than $0.12$ in the same proportion that $125$ is bigger than $12$.

What are some lines of thinking / questioning that might undo this confusion? I have an idea or two of my own, but I'm not totally satisfied with them so I'd like some input.

The student suffers from no specific cognitive impairments, but has a history of poor performance in math and would be a poster-child for math anxiety.

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    $\begingroup$ Do they believe $0.115$ is bigger than $0.12$? $\endgroup$ – quid Feb 18 '16 at 20:24
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    $\begingroup$ Are you sure that this is a confusion about place value, rather than a confusion about proportion? $\endgroup$ – mweiss Feb 18 '16 at 22:24
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    $\begingroup$ @quid Without having the student in front of me to check, I believe that they would. Or, at least, that they would some of the time, depending on what they had recently been considering. $\endgroup$ – NiloCK Feb 19 '16 at 13:54
  • $\begingroup$ @mweiss I think that it's about place value. The specific bit about proportion is my own description of what's going on - the student didn't use those words. $\endgroup$ – NiloCK Feb 19 '16 at 13:55
  • $\begingroup$ Can the student articulate any kind of "scale" relationship between $125$ and $12$, like "$125$ is more than ten times larger than $12$"? $\endgroup$ – mweiss Feb 19 '16 at 18:02
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I always emphasize comparing decimals with the same number of places to my students. Therefore before comparing 0.12 and 0.125, I would first teach them that 0.12 = 0.120. Students can than grasp that 0.120>0.012. This is further brought home when students read it as 12 hundredths is greater than 12 thousandths.

I suggest you work on converting decimals to different numbers of places with this student to make it easier for him to think about decimals.

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  • $\begingroup$ Agreed on the read aloud method; this leads 0.12 to be read as "twelve hundredths" and 0.125 to be read as "one hundred twenty five thousandths," and now it can be viewed as a question about fractions. I think that addending a 0 (e.g., 0.12 becomes 0.120) can seem a bit too much like a trick divorced from sense-making unless it is turned into an idea that's focused on (which seems to me a reasonable idea!). $\endgroup$ – Benjamin Dickman Feb 22 '16 at 18:56
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    $\begingroup$ @BenjaminDickman I agree that It adding the zero can become a trick with no meaning attached to it. I try to introduce the idea that just as we wouldn't compare numerators unless the denominators were equal, we don't compare decimals unless they have the same number of places and therefore equal denominators. $\endgroup$ – Amy B Feb 22 '16 at 22:34
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I think that the overriding principle would be to model comparing like terms, one term at a time. Fundamentally that's the key to any comparisons, additions or subtractions -- not just in decimals, but more generally for fractions, mixed units, etc., etc. So I'd go through and model looking at those digits one at a time, and make sure that the student was doing the same thing.

@AmyB's answer is also useful to give additional context. Chiefly the student needs to be disabused of the idea that just looking at string-length can tell which decimal is larger.

This is a common enough problem that there's a speed-quiz for this skill for my incoming community-college statistics students on my site Automatic-Algebra. Consider having your student drill there for a few days to prove they've mastered it:

http://www.automatic-algebra.org/decimalrelations.htm

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Decimal notation is an ingeniously efficient tool for working with rational quantities, but too often students and teachers forget just why/where it came from. There are a bunch of ways to say what a decimal like "0.125" means, but, assuming this student has a "classical" background in fractions the only way to bring concrete meaning to a notation like "0.125" is to relate it to dividing a whole into parts.

In this case "0.125" is a way of writing the fraction 125/1000 or the ratio 125:1000 and "0.12" is a way of writing the fraction 12/100 or the ratio 12:100. If the student is unable to identify which of these fractions is greater than the other, then you must review those concepts first as they are more fundamental than the mechanics of decimal notation.

In this case, the computational method is to produce a common denominator and use the numerators of the fractions with common denominator to decide whether one is greater than the other. The appeal here is to the physical origin of these notations: 125/1000 means 125 parts out of the 1000 equal parts we've broken a whole into. Thus 12/100 is equal in quantity to 120/1000 and from here it is easily seen that 125 of 1000 parts is greater than 120 of 1000 parts of the same whole.

With regards to the place value: terminating decimals are sums of fractions. So 0.123 is an abbreviation of the following expression

$$ \frac{1}{10} + \frac{2}{100} + \frac{3}{1000} $$

and it is clear that here, to calculate or "know" the sum is to find a common denominator and add them all together, BUT THIS IS EXACTLY WHY WE INVENTED POSITIONAL NUMERAL NOTATION, so that we can read off from 0.123 the corresponding fraction 123/1000.

I hope that helps.

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