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

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?

• Method 4: Simplify first (always!). Then do the division by hand. You repeatedly get 9 into 20 is 2 remainder 2. May 31 at 18:54

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