Problem Solving: Tips For TeachersAuthor(s): Phares G. O'Daffer, Carole E. Greenes and George E. ImmerzeelSource: The Arithmetic Teacher, Vol. 34, No. 7 (March 1987), pp. 34-35Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193124 .

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| Strategy Spotlight N Decomposition; or

A Piece at a Time

Sometimes one can simplify a "big" problem by de- composing it into smaller "bite size" pieces, analyz- ing the pieces, then recombining the pieces to solve the big problem. Often other problem-solving strate- gies, such as making lists, observing patterns, check- ing patterns for consistency, and forming generaliza- tions, are part of the decomposition process and subsequent analysis. Consider this "timely problem":

Problem

Interviewer for Up-to-the-Minute News: Today we / have with us Mr. Al Arm, President of Time Out, In- ( I corporated. Tell me, Mr. Arm, how many hours a day U do you work? '

Al Arm: Well, to be exact, each day I work as many minutes as a 9 (at least one) is displayed on a digital clock in a 24-hour period.

Interviewer: And just how long is that? Al Arm: To find out, you'll have to look at a digital /

clock face to face! 1 1

Solution: We can first decompose the problem by IV examining a 12-hour period. Once we find the num- ' ber of minutes that at least one 9 is displayed in 12 hours, we can double that number of minutes for 24 у hours. (

Edited by Phares G. O'Daffer Illinois State University Normal, IL 6176I

Prepared by Carole E. Greenes Boston University Boston, MA 02215

George E. Immer zeel New Impressions, Inc. Lexington, MA 02173

Let's consider the places in a clock display in which we would see the digit 9. A 9 can appear in the hour position (e.g., 9:00) or in the minutes position (e.g., 8:49).

Looking at the hour position first, we see that at least one 9 is displayed for 60 minutes from 9:00 through 9:59. For the other 1 1 hours, a 9 does not appear in the hour position.

Examine the other 1 1 hours. Decompose this part of the problem and consider only one of those hours, say from 1 :00 through 1 :59. Make a list of the times when at least one 9 is displayed.

gi 1 1:09

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1 1:19

lf"^Ç™J55^H ■■■■■m mi- DQD 1 a 1:29 д éej§ set «■» řt 1 1 "T?*! I 3

In this hour, a 9 is displayed for 6 minutes. Thus, in the 1 1 hours, a 9 will be displayed for 1 1 x 6, or 66 minutes. Combining the 66 minutes with the 60 min-

i utes for the 9 o-clock hour, we find that a 9 is dis- ' played for 126 minutes in 12 hours or 252 minutes in I 24 hours. Mr. AI Arm works 6 hours and 12 minutes I a day.

The Interview Continues

Interviewer: My, my. That is quite a short work day, Mr. Arm.

Al Arm: Yes, that's true. But my work day last year ' was quite a bit longer. Last year I worked as many

minutes each day as a 2 (at least one) is displayed on a digital clock in a 24-hour period!

I Have students discuss how this new problem can ' be decomposed and analyzed. Encourage them to follow the procedures used in the first problem, con- sidering a 12-hour period and the positions that a 2

i can occupy in the clock's display.

34 Arithmetic Teacher

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Solution 1 . At least one 2 will be displayed in the hour posi-

tion during 2 hours:

12:00-12:59 2:00-2:59

2. In 10 other hours a 2 is not displayed in the hours position. During those 10 hours, a 2 may ap- pear in either the tens place or the ones place of the minutes.

Example: From 3:00-3:59, at least one 2 occurs at 3:02, 3:12, 3:20, 3:21, 3:22, 3:23, 3:24, 3:25, 3:26, 3:27, 3:28, 3:29, 3:32, 3:42, 3:52. During each of those 10 hours, at least one 2 is displayed for 15 minutes. In 10 hours, that is 10 x 15, or 150, min- utes (21/2 hours).

3. In a 12-hour period, at least one 2 is displayed for 41/2 hours; in a 24-hour period, 9 hours. So Mr. Arm worked 9 hours each day last year.

Extensions

Some students may be interested in determining which of the digits 0-9 is displayed most often on a digital clock. (At least one 1 is displayed for 6 hours in a 12-hour period of time!) Other students may want to consider the more complex listing problem of de- termining the total number of minutes in 24 hours in which a particular digit is displayed. For example, to determine the total number of minutes 1's are dis- played, the time 1:11 would be counted as 3 minutes, since each 1 is displayed for 1 minute.

•- ' П Classroom Climate- A '^^y i - | "Wonder-Full" Atmosphere -

Encourage students to wonder about things in I I their environment, things that they see every

day. Once their curiosity is piqued, then a math- | I ematical investigation can take place. Try ask-

ing questions such as these: ' - ' • How long a piece of wire is needed to make

| ' - i

' all the paper clips in a new box of paper clips?

' - ' • How many reams of paper are used by your class in a year?

• How many times can you write your name with a pencil before you have to throw it away?

• During which half-hour period in a school day do the most cars pass by the front of your school?

• Can you roll a marble in the chalk tray as fast as a car can drive past your school at the

^=^ legal speed limit?

LJ Problem Corner

Students in grades 3 and 4 might like to solve these: • At what time is the sum of the digits on a digital clock the greatest? (Answer: 9:59)

• In a 12-hour period, for how many minutes is the sum of the digits displayed on a digital clock equal to 10? (Answer: 62 minutes)

Students in grades 5-8 might like this problem: • The following departure times contain each of the digits 0-9 exactly once and span a 7-hour and 51- minute period: ^ | |

12:08 p.m. ^^pt^^^^

^M

What three departure times that contain the digits 0-9 exactly once, span the shortest period of time? (Answer: 8:57, 9:46, and 10:23 span 1 hour and 26 minutes.)

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