by Soo Jin Lee and Jaehong Shin
Primary Learning Outcomes
Help students appreciate the value of 0.999... as the same one of 1.
Grade : 8
Objective : Understading infinite circulating decimals or Preparation for the idea of 'limit' in precalculus context
Procedures/Activities
The purpose of following activities is to help student's appreciation of a infinite decimal 0.999... by interpretating it in three diffenent ways using calculator. The activities also make students to think that it could be possible to represent infinite series as a finite bounded value and they naturally start to learn the concept of limits.
DISCUSSION
Let students define the meaning of 0.999.... and discuss each other about their own expectation value.
Interpretation 1 : Converting Fractions to Decimals
Students can easily get the value of 1/3 by using a calculator, that is, 1/3 = 0.3333.... and by sum of three 1/3 students can get the values of 1 on both sides like above table.
Interpretation 2 : Using equation.
Let x=0.999....,then 10x= 9.999....
10x-x = 9x = 9.999.... - 0.999.... = 9
Therefore, x = 1 = 0.999....
By above two different interpretations, students will easily understand why 1.0 is equal to 0.999..... Actually, above contents would be taught at middle school levels, However, those concepts can play a good role in introducing the notion of 'limit' by suggesting third interpretation about 0.9999.....
Interpretation 3 : The sum of infinite series (Expansion)
0.999.... = 9/10 + 9/100 + 9/1000 + ..... = ???
Students already know that the answer is 1 by previous two interpretations. That makes students to think that it could be possible to represent infinite series as a finite bounded value and they naturally start to learn the concept of limits.
