Repeating Decimals Exploration

My students have been working on rational number conversions.  We started with fraction to decimal conversions.  I blogged about the notes I used and question stack my students practiced with here.  Next we reviewed how to convert terminating decimals to fractions.  Students put this review sheet in their notebooks.

After a brief review of terminating decimals, I had my students complete a repeating decimals exploration.  I asked them to convert special fractions to decimals using long division or a calculator, and then look for patterns.

 

Students noticed that all of the decimals were repeating.  They noticed that the denominators were all 9s.  They noticed that the numerators showed the digits that repeat when written as a decimal.  They noticed that the number of digits in the numerator matched the number of 9s in the denominator.

Then I asked students to predict what the fraction would be if I gave them the repeating decimal.  After making their predictions, students checked their answers with a calculator.

Things I Like About This Investigation

• Students get practice with converting fractions to decimals - whether that is by long division or entering it correctly in a calculator.
• Students do some notice/wonder while looking for patterns and making predictions.
• Students discover the 9s trick for repeating decimals themselves. 

Limitations and Room for Improvement

• Students didn't learn why the 9s trick works.
• Students only discovered how to convert repeating decimals to fractions if all the digits after the decimal repeat. 
• There was not a big emphasis on simplifying fractions, so students may not realize that simplified fractions with denominators other than 9 can still repeat (e.g., students may not recognize that 7/11 repeats and that it is the same as 63/99).

We summarized the 9s trick in their notebooks with this notes sheet.

View/Download: Converting Decimals to Fractions Notes and Repeating Decimals Exploration