Learning Fractions with Understanding - Repeated Decimals
I had the following conversation with Andy, who was in the last month of 6th grade. We had this conversation without and paper and pencil. I have added some notation in green to help the reader follow his ideas. (Please note that I am writing divided by rather than the division sign because I can’t get the division sign to appear in this blog.)
Andy: I was just thinking that point 9 9 9 9 and on and on forever must be the same thing as one.
.99999…. = 1
Ms L: Why were you thinking that?
Andy: Well, you know with 1/3. If you wanted to write 1/3 as a decimal you would think about how many tenths it was the same as. One is the same as 10 tenths. One third would be 10 tenths divided by 3. Ten tenths divided by 3 is 3 tenths with one tenths left over.
1/3 equals 10/10 divided by 3
1/3 = 3/10 R1/10
(R stands for Remainder)
You could take that one tenth left over and again divide it by 3. One tenth is ten hundredths. Ten hundredths divided by 3 is 3 hundredths with a hundredth left over. So 1/3 is 3/10 plus 3/100 with 1/100 left over.
1/3 = 3/10 + 3/100 R1/100
You could do it again with the left over 1/100. 1/100 is the same as 10/1000. 10/1000 divided by 3 is 3/1000 with 1/1000 left over.
1/3 = 3/10 + 3/100 + 3/1000 R1/1000
It would just go on like that forever. There would always be one piece left over and you would divide that again by 3. You get three with again one piece left over. I think that is what it means when they write .33333…..
1/3 = .33333….
Ms L: What got you thinking about this?
Andy: My teacher told me that one third as a decimal was point 3 3 3 3 and on and on forever and I wanted to know why. So I started thinking about how many tenths 1/3 was, and how many hundredths it was. Is my idea what it means when you write .33333….?
Ms L: Makes sense to me. But you said something about point 9 9 9 9 9 and on and on being the same as 1.
Andy: Well, I already told you that.
Ms L: You did?
Andy: Yes, 1/3 + 1/3 + 1/3 = 1. Of course. Point 3 3 3 3 and on forever plus point 3 3 3 3 3 and on forever plus point 3 3 3 3 and on forever is point 9 9 9 9 and on forever which means that point 9 9 9 9 and on forever must be the same as 1.
1/3 + 1/3 + 1/3 = 1
1/3 = .3333
.3333…. + .3333…. + .3333…. = .9999….
.9999….. = 1
Ms L: Are you sure of that?
Andy: Yes, I just proved it.
When I taught high school more than 20 years ago, I taught a proof for .9999…. = 1. My daughter’s babysitter, Mary, is taking Pre-Calc and was taught this same proof a few weeks ago. The proof typically taught in high school is very different from the one generated by Andy. The typical high school proof is a series of equations that proceed logically. Even though many high school students can understand how one equation logically follows from the previous one, the proof seldom convinces students that .9999…. = 1. When I asked Mary whether or not .9999…. equaled one, she said, “We proved it in class a few weeks ago. I understand the logic of the proof, but I’m still not convinced that .9999…. equals 1. I think it is very close but just a little bit less than 1.”
In my professional development work with teachers, I have been asked to talk about .9999… = 1. Before talking with Andy, I hadn’t thought of a way to talk about the relationship between .999…. and 1 other than the typical formal proof. Since I didn’t know what else to do, I presented the formal proof to the teachers. After the presentations, teachers were left feeling pretty much like Mary. There were no “aha’s!” coming from the group.
I explained Andy’s proof to Mary she immediately said, “Oh, I get it! Now I am convinced, .9999… is the same as 1” and added that she was going to tell her friends about the proof when they went bowling that evening. (She later reported that her friends were now convinced as well.)
Before I proceed, I want to state that I am not advocating we abandon logical sequence of equations as proofs in high school. The typical proof of .9999…. = 1 probably should remain in the high school curriculum. However, wouldn’t it be nice if students had an informal understanding of .9999…. before working with a formal proof that involved .9999….? To state the question more generally, wouldn’t it be nice if students had an informal understanding of repeated decimals and their relationship with fractions before working abstractly with repeated decimals?
How did Andy gain an understanding of .9999….? Andy had been in a CGI class for 1st – 5th grade. I had often worked in his class when he was in elementary school. Andy wasn’t an unusual student. Unlike some of his classmates, he wasn’t put in a math class with older kids. He tended to master concepts a little earlier than some of his classmates, but he always had peers in his classroom who were working at the same level that he was.
Andy’s CGI teachers had engaged with Susan Empson’s work on learning fractions with understanding. Susan is a member of the CGI research and development team who specializes in the learning and teaching of fractions. She has developed a framework of problems and strategies that link learning fractions to children’s understanding of whole number operations. I can’t provide a full explanation of her framework in this blog, but I encourage you to check out her blog at http://www.edb.utexas.edu/empson/.
In elementary school, Andy’s learning of fractions was integrated into his learning of whole number operations in many different ways. He developed an understanding of many relationships between fractions and whole number operations. In this proof, we see a strong understanding of the relationship between fractions and whole number division in that he has a deep understanding of one third as one divided by three.
Throughout elementary school, Andy’s CGI teachers posed problems to help their students to develop an understanding of the relationship between fractions and whole number division. For example, in first grade, he solved problems like this,
4 kids were sharing 13 cookies, how many cookies would each kid get?
In third grade, his teacher used problems like the following to develop an understanding of the relationship between fractions and whole number division:
4 kids were sharing 3 small cakes. How much cake would each kid get?
The problems in fifth grade were more sophisticated:
8 kids were sharing 21 sandwiches. How many sandwiches would each kid get?
Andy was not taught fraction in isolation from whole number operations until he got to sixth grade. He was not shown pictures of circles and asked to color in ¾. He did not make fraction strips or use commercially produced fraction manipulatives. His teachers used their students’ understanding of whole number operations as a foundation to teach fraction concepts. Fractions were never an isolated unit. They were never something brand new.
Elementary and middle school teachers have the potential to teach a great deal of important mathematics when they teach fractions. Fraction instruction can help students understand the relationships between fractions and whole number operation. Fraction instruction can help students develop a foundation that allows them to learn later mathematics with understanding. I believe the best way to do this is to integrate fraction instruction with whole number instruction and to use what children understand about how number operation as a foundation for learning fractions.
posted by Linda Levi @ 8:53 AM
