G8M7 Lesson 6: Finite and Infinite Decimals

Lesson Transcript

As part of the opening exercises the students were directed to write the decimal expansion*using long division for the following rational numbers:

Solutions:

* The posters show belowhang on the wall

While the students were working on the opening activity, Mrs. S walked around the room checking student work.

1. / Mrs. S / Now that you have determined the decimal expansion of these rational numbers, turn to your partner and compare your answers. Talk to your partner about any similarities or differences you observe in the decimal expansions.
2. / Mrs. S / [Mrs. S gives the students about forty seconds to discuss and when the conversations start to dwindle, she redirects the students]
Raise your hand if you or your partner noticed any similarities in the decimal expansions for any of the rational numbers.
[All of the students raise their hands. Mrs. S uses her equity sticks to choose a student to share his/her observation] (equity sticks are popsicle sticks, each with a student’s name written on it. As students enter the room, they pick up their equity stick and place it in a container. Mrs. S takes roll by marking students whose equity stick still remain on the table at the start of class absent)
3. / Mrs. S / Juan, will you share an observation that you or your partner made?
4. / Juan / We noticed that two of the decimals stopped and two didn’t.
5. / Mrs. S / Excellent observation. Can anyone restate Juan’s observation using one of this unit’s vocabulary words?
[Several students raise their hands]
Juan, please pick someone to restate your observation.
[Juan chooses a student whose hand is raised]
6. / Juan / Erica
7. / Erica / Terminate. Some of the decimals terminate and some repeat.
8. / Mrs. S / Very nice! Terminate and repeat. Did we notice anything about how the decimals repeat?
[Several students raise their hands]
If your hand is not raised, turn to someone who did raise their hand and ask them what they noticed.
9. / Mrs. S / [The students share their observations for approximately 15 seconds then Mrs. S chooses another equity stick from the cup]
Franklin, what did you or your partner notices about the decimal expansions that resulted in repeating decimals?
10. / Franklin / The number repeat.
11. / Mrs. S / Can you be more specific? Which numbers repeat? All of the numbers?
12. / Franklin / With the fraction five elevenths, forty-five kept repeating. With the fraction eight ninths just the eight repeated.
13. / Mrs. S / Does everyone agree with Franklin’s observation? [The students indicate they do agree]
Are we sure the pattern will continue? How do we know that if we were to keep using the long division algorithm we wouldn’t eventually get different numbers? Can you convince yourself that this won’t happen? [Some student immediately raise their hands while others look somewhat confused]
14. / Mrs. S / [Mrs. S calls on a students whose work she had noticed as being particularly neatly written and well organized when she was observing the students during the opening activity]
Desiree’, do you think the repeating pattern will continue?
15. / Desiree’ / Yes.
16. / Mrs. S / Can you convince the class that this is the case?
17. / Desiree’ / When you divide you keep getting the same number left over.
18. / Mrs. S / Can you use a different term for “number left over”?
19. / Desiree’ / Remainder
20. / Mrs. S / So, after each step of the long division algorithm we get the same remainder?
21. / Desiree’ / Sort of. I did when I was doing eight ninths. After each time I had one left over, I mean remaining. Each time I had 8 left over. It was a little different for the one with five elevenths.
22. / Mrs. S / It might be easier for the class to understand what you mean if you show us your work under the Elmo.
[Desiree’ walks to the front of the room and places her paper under document camera]
23. / Desiree’ / With the eight-ninths each time I subtracted I got eight because eighty minus seventy-two is eight, see. [Desire’ points to the remainders at each of the iterations] After a few times I stopped because I knew it was just going to keep going like that.
24. / Mrs. S / Does anyone have any questions they would like to ask Desiree’ about how she found the decimal expansion of eight ninths?
[Several students raise their hands. Mrs. S calls on Sean]
25. / Sean / Why did you write your answer as ?
25. / Desiree’ / Because I stopped after three tries so I had eight three times.
26. / Mrs. S / That was an excellent question Sean. Desire’s notation indicated that the string of numbers 888 repeats indefinitely. Does Desiree’s notation give us exactly enough information about the decimal expansion, or has she given us more information than we actually need?
[Students call out, most of them saying “too much information”]
27. / Mrs. S / While Desire is correct that the numbers 888 do repeat indefinitely, is there a more efficient way in which we can indicate this same pattern? Desiree’, what do you think?
28. / Desiree’ / I could write it with just one eight with a bar over it, like this. [Desiree’ writes on her paper]
29. / Mrs. S / Very nice. Now, if Desire’ doesn’t mind, I would like someone else to explain how we determined that the decimal expansion of five elevenths repeats using Desiree’s work.
30. / Desiree’ / I don’t mind. [Desiree’ returns to her seat]
31. / Mrs. S / Who would like to volunteer? [Mrs. S selects Richard from the students whose hands are raised]
32. / Richard / If you look you can see that the first time we subtract we get six. The second time we get five, then we get six and then we get five. It keeps going like this. [Richard points to the remainders each time he refers to spthem]
33. / Mrs. S / When we look at Desiree’s work we see that she stopped using the long division algorithm after four steps. How can we know for sure that we won’t eventually get a different remainder if we continued the process?
34. / Richard / Because once we get back to 5 the whole things starts again.
35. / Mrs. S / Very interesting. Does everyone see what Richard is saying? [The students respond in the affirmative] Thank you Richard. [Richard returns to his seat].
35. / Mrs. S / Show me using your white boards how we write the decimal expansion of five elevenths using conventional notation. [The students write on their white boards and hold them up]
36. / Mrs. S / We have looked at examples of rational numbers with infinite decimal expansions and observed that, when using the long division algorithm, eventually the remainder repeats. What did you notice when you used the long division algorithm with rational numbers whose with decimal expansions that terminate? Take a few seconds to look over your work and to formulate an answer. [Mrs. S waits about fifteen seconds and then selects a student using the equity sticks (the cup is divided into two sections using masking tape. Once a stick has been used it is placed on the other side of the cup)]
Peter, what did you notice?
37. / Peter / I knew to stop when I got a remainder of zero.
38. / Mrs. S / Does everyone agree with Peter? [Students indicate using thumbs up that they do]
38. / Mrs. S / Do you think you can tell by looking at a rational number whether the decimal expansion will terminate or repeat? Take a minute to write a conjecture about when the decimal expansion will terminate and when it will repeat infinitely.
39. / Mrs. S / We are now going to tests your conjectures. Using your calculator, divide to find the decimal expansion for a variety of rational numbers. Classify each as having a terminating decimal expansion or a repeating decimal expansion. Find at least three examples of each and write your examples on the corresponding piece of poster paper at the front of the room. You have five minutes to complete this activity.
[The two pieces of chart paper are labeled “Group 1 – Terminating Decimal Expansion” and “Group 2 – Repeating Infinite Decimal Expansion”
40. / Mrs. S / [After 5 minutes has elapsed]
Finish writing your examples on the chart paper.
In your table groups, identify any similarities within each group and identify any differences between the two groups.
[After about a minute Mrs. S chooses an equity stick]
Joey, please share with the class some of the observations the members of your group made.
41. / Joey / We noticed that a lot the denominators in the repeating group were prime numbers and that a lot of the numbers in the other group had denominators that were even.
42. / Mrs. S / Excellent observations. Did any of the other groups make similar observations?
I think your observation about the second group having denominators that were prime numbers is an interesting one. Did any of the numbers in the first group have denominators that were prime? [Mrs. S selects a students whose hand is raised]
43. / Jessica / Some of them have two in the denominator and two is prime.
44. / Mrs. S / Great observation. Were any of the denominators in the second group even?
45. / Class / Yes
46. / Mrs. S / Take a closer look at the two groups. Can you identify any characteristics that will let you know for sure that the decimal expansion terminates? What do all of the rational numbers you listed in Group 1 have in common?
47. / Mrs. S / [After about a minute, Mrs. S used the equity sticks to select another student]
Dorrell, what does your group think?
48. / Dorrell / We think that if the denominator is even or a multiple of five then it will stop.
49. / Mrs. S / Six is and even number, but I notice that one sixth is listed in Group 2. Does your group want to make any adjustments to your statement?
50. / Dorrell / We mean, the denominator has to be either two or a multiple of five.
51. / Mrs. S / Do we agree with this groups’ statement? All rational numbers with terminating decimals have denominators that are either two or a multiple of five?
[Students use thumbs up, down, sideways to agree, disagree, or remain undecided. The majority of the students indicate that they disagree. Mrs. S selects one of these students]
Calli, can you tell us why you disagree?
52. / Calli / Fifty-three over seventy-five is in group two and seventy-five is a multiple of five.
53. / Mrs. S / Well put Calli. What did Calli just use to disprove this conjecture? [The majority of the class responds “a counter example”]
54. / Mrs. S / It looks like we are getting close but appears that we need to do a little more refining of our conjecture, as not all? rational numbers with denominators that are multiples of five have decimal approximations that terminate.
55. / Mrs. S / In your groups, write the prime factorization of the rational numbers your group used as examples for Group 1 and Group 2. You have three minutes to complete this.
56. / Mrs. S / [after three minutes]
Looking at the prime factorizations, what do you think determines whether a rational number’s decimal expansion will terminate?
[Mrs. S selects an equity stick]
Chaslyn
57. / Chaslyn / All of the denominators only have factors of two or five or both.
58. / Mrs. S / Very interesting. I wonder why that is.

During the remainder of the class period the students worked on writing rational numbers whose decimal expansions terminate as equivalent fractions with powers of ten in the denominator. (refer to lesson)