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Background:

I am working with 10-13 year olds.

I saw a problem which requires the students to convert $6/27$ into decimal notation. I can think of some methods, but I have not tested using any of the following methods in a classroom setting:

Method 1: One can do the calculation

\begin{array}{c c c c c c c c} 6 \hspace{5mm} 0 & 0 & : & 27 & = & 0 & 2 \\ 0 \hspace{5mm} 0 & \\ \overline{6 \hspace{5mm} 0} &\\ 5 \hspace{5mm} 4 &\\ \overline{\hspace{7mm} 6} \\ \end{array}

with appropriately placed commas, to arrive at

$ 6/27 = 0.\overline{2} $

Method 2: If one recalls that $1/9 = 0.\overline{1}$, then one can compute

$ \frac{6}{27} = \frac{2}{9} = 2 \cdot \frac{1}{9} = 2 \cdot 0.\overline{1} = 0.\overline{2} $

Method 3: If one recalls that $1/3 = 0.\overline{3}$, then one can compute

$ \frac{6}{27} = \frac{2}{9} = \frac{2}{3} \cdot \frac{1}{3} = 0.\overline{6} \cdot 0.\overline{3} $

and then one can perhaps proceed with a multiplication strategy.

Question:

What are strategies that a 10-13 year old could use to convert $6/27$ into decimal notation?

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    $\begingroup$ Method 4: Simplify first (always!). Then do the division by hand. You repeatedly get 9 into 20 is 2 remainder 2. $\endgroup$
    – Sue VanHattum
    May 31 at 18:54
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I prefer method 2, which has the student simplifying first. Even if the student doesn't know that $ \frac{1}{9} = 0.\overline{1} $, the student can still divide 2.00 by 9.

If the student uses method 1, s/he misses the chance to see the pattern in ninths and is more likely to make a mistake because s/he is dividing by 27 (a 2 digit number) instead of a one digit number. On the other hand, it is a good method because it works for all fractions.

I would never use method 3, since the student is multiplying 2 repeating decimals and they have no understanding of how that will work.

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Simplify first. Then do the problem by long division. With a number of such problems, the student will quickly realize that 2/9 gives .222... and 4/9 gives .4444 and the like. It's actually a good introduction to such stuff. But make them grind the long division first, several times, to see where it all comes from.

The "times .111..." seems cute and appeals to the mathy sorts here, but I don't think it is as foundation building (for a neophyte) as cranking the long division a few times and then realizing all the x/9 (x from 1 to 8), end up being .xxx. (This is a pedagical, psychological issue, not a math concept.)

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If the decimal expansion of $\frac13$ is known (as in Method 2), then $$\frac6{27}=\frac29=\left(\frac23\right)/3=(0.\overline{66})/3=0.\overline{22}.$$

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