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Tiny Nines

What do you notice about these families of recurring decimals?

Repetitiously

Age 14 to 16
Challenge Level

Why do this problem?

This problem offers carefully chosen examples intended to give students the opportunity to prove that any recurring decimal can be written as a fraction. Together with the problems Terminating or Not and Tiny Nines, this problem offers valuable insights into the relationship between fractional and decimal representations.

Possible approach

Before starting on the main task, make sure students are happy with the notation $0.\dot{2}$ to mean $0.222222\ldots$ Perhaps the video from Tiny Nines could be used to open up the conversation about recurring decimals. It's important that students understand that the 2s are repeated forever; common misconceptions about recurring decimals arise from people worrying about what might be on the end of it after all those 2s! 

 
Define $x = 0.\dot{2}$ and invite students to discuss with a partner what they need to do to $x$ to get to $2.\dot{2}$. If they need a hint, try the following questions:  
"What could you add to $x$ to get $2.\dot{2}$?"
"What could you multiply $x$ by to get $2.\dot{2}$?"
 
In the discussion that follows, make sure to draw out the idea that as the decimals recur (repeat forever), adding 2 and multiplying by 10 both give the same answer. Then discuss with the class the process of setting up and solving an equation using the two expressions for $2.\dot{2}$ together. You can find another worked example here.

Once students are happy with the method for converting a recurring decimal such as $0.\dot{2}$ into a fraction, invite them to consider the other decimals in the problem:

  • $0.\dot{2}\dot{5}$
  • $0.\dot{4}0\dot{5}$
  • $0.8\dot{3}$
  • $0.002\dot{7}$

Students could check their answers by performing the division calculation by hand or with a calculator.

Those who finish could be invited to choose a fraction, convert it to a decimal, and then challenge their partner to convert it back into a fraction. This might offer a good opportunity to discuss the limitation of calculators, which have to round numbers after a number of decimal places.

Finally, in a plenary discussion, draw together students' thinking to develop a general method for writing any recurring decimal as a fraction. 

Key questions

What could you add to $x$ to get $2.\dot{2}$?
What could you multiply $x$ by to get $2.\dot{2}$?
What happens when you multiply a recurring decimal by 10? or 100? or 1000?

 

Possible support

For some students, the stumbling block with this activity is uncertainty about what it means for a decimal to recur. Spend plenty of time discussing what happens when you multiply terminating decimals by powers of ten, gradually increasing the number of decimal places (such as $0.2, 0.22, 0.222, 0.2222$ and so on). Then invite them to imagine that the decimal places continue forever, to help them understand that even after multiplying by 10, there are still infinitely many digits after the decimal point.
 

Possible extension

Invite students to consider recurring decimals such as $0.\dot{9}$ and $0.4\dot{9}$, and to construct a proof that they are equivalent to 1 and $\frac12$ respectively.