WASHINGTON STATE MATHEMATICS COUNCIL

#### 2001 MIDDLE SCHOOL MATH OLYMPIAD

Session I: 5th ¾ 8th Grade Problem Solving

PHONE TREE

5-8

Sample Solution & Scorer Notes

(Adjust for grade level.)

1.  Table or Diagram

Example of Table or Diagram for 3-Call Plan

 Round Number Time of Round Total Minutes at End of Round Total # of Calls Made During Round Cumulative # of Calls at End of Round 1 10:21  ¾  10:24 3 3 3 2 10:24  ¾  10:27 6 3x3 or 32  = 9S 12 3 10:27¾  10:30 9 3x3x3 or 33 = 27 39 4 10:30  ¾  10:33 12 34 = 81 120 5 10:33  ¾  10:36 15 35 = 243 363 6 10:36  ¾  10:39 18 36 = 729 1,092 7 10:39  ¾  10:42 21 37 = 2,187 3,279 8 10:42  ¾  10:45 24 38 = 6,561 9,840 9 10:45  ¾  10:48 27 39 = 19,683 29,523 10 10:48  ¾  10:51 30 310 = 59,049 88,572

S We multiplied 3x3 because each of the first 3 people call 3 more people.  That makes 3x3 people called at the end of this round.

Example of Table or Diagram for 5-Call Plan

 Round Number Time of Round Total Minutes at End of Round Total # of Calls Made During Round Cumulative # of Calls at End of Round 1 10:21  ¾  10:26 5 5 5 2 10:26  ¾  10:31 10 5x5 or 52 = 25S 30 3 10:31  ¾  10:36 15 5x5x5 or 53 = 25 155 4 10:36  ¾  10:41 20 54 = 625 780 5 10:41  ¾  10:46 25 55 = 3,125 3,905 6 10:46  ¾  10:51 30 56 = 15,625 19,530 7 10:51  ¾  10:56 35 57 = 78,125 97,655

S We multiplied 5x5 because each of the first 5 people call 5 more people.  That makes 5x5 people called at the end of this round.

2.  Exact Time Needed for Each Plan:

 3-Call Plan 10,000 ®  27 min ®  by 10:48 50,000 ®  30 min. ® by 10:51 5-Call Plan 10,000 ®  30 min ®  by 10:51 50,000 ®  35 min. ® by 10:56

3.  Report for Community Leaders:

Statement of Understanding    (Sample)

® All Grades:  Our assignment was to investigate the two proposed plans for this community.  One proposed plan called for the superintendent to call three households, who would then call 3 more households, and this would continue until the current community of 10,000 households and future community of 50,000 households were all contacted.  The other plan was the same, except for the 3 households being 5 households each time.  The calls all last 1 minute in the plan, and each round lasts 3 minutes or 5 minutes depending on the plan.  Our job was to find out which plan would work the best for the 10,000 and 50,000 households, and to find out if the plans would notify the whole community within 15 minutes.  Then we were to prepare our report re-stating our assignment, summarizing our findings about the two different plans, and making recommendations to the community leaders about notifying their community within 15 minutes in the case of an emergency.

® 7/8 only:  In addition to this, we were asked to try a third phone tree plan, and to make our final recommendation based on all three phone tree plans.  We must show our reasoning for our recommendation.

Summary of Findings   (Sample:  Note that findings for 5/6 are based on only 2 plans, 7/8 are based on 3 plans.)

® All Grades:  Our findings were that the two plans ended up reaching the whole community less than 3 minutes apart, and that neither of the plans would reach the community of 10,000 households or the community of 50,000 households in less than 25 minutes.  Counting the time until the end of the last round, the 3-call plan would reach the community of 10,000 within 27 minutes, and the 5-call plan would take 3 minutes longer.  The 3-call plan would notify a community of 50,000 within 30 minutes, and the 5-call plan would take 5 minutes longer.  Although 3 or 5 minutes could make a big difference to people in terms of their families’ lives, there is not a tremendously huge advantage of one plan over the other.  Also, the whole last round is not needed in any of the four cases because the 10,000 and 50,000 households would be contacted before the whole round was up.  What is more important is the fact that neither plan meets the 15 minute requirement.   A severe emergency would leave some households not being notified for from 27 to 35 minutes under these phone tree plans.

® 7/8 only:   Accept summaries based on the third phone tree of the team’s choosing.

Note:   7/8 students who have tried a third tree may observe some of the following ¾

(Note that they will probably not have time to, but for the record these are some possible observations!)

ä   Times for the 3-person and 5-person trees do not differ as much as expected.  This is because even though more people are being called each round, these additional people do not start calling until the end of the round.

ä   Times for trees of more than 6 people begin losing too much time because of callers waiting for a new round to start calling.  The round times are longer.  The times to reach 10,000 and 50,000 people are longer.

ä   Times in the 2­-5 minute range are the best times.  See the tables below, in rank order of best plan:

Top 5 Plans Ranked by  End-of-Round Times                             Top 5 Plans Ranked by  Within-Round  Times

 10,000 Households 50,000 Households 10,000 Households 50,000 Households 2-Call Plan 26 minutes 30 minutes 3-Call Plan 25 minutes 29 minutes 3-Call Plan 27 minutes 30 minutes 2-Call Plan 25 minutes 30 minutes 5-Call Plan 27 minutes 32 minutes 4-Call Plan 26 minutes 30 minutes 4-Call Plan 28 minutes 32 minutes 5-Call Plan 27 minutes 32 minutes 6-Call Plan 36 minutes 36 minutes 6-Call Plan 31 minutes 35 minutes

Recommendation    (Sample:  Note that 5/6 is based on only 2 plans, 7/8 is based on 3 plans.)

Our recommendation is that if the community leaders are wanting to use one of these two phone tree plans, they choose the 3-call plan.  However, if they are not wanting to limit themselves to one of these two plans, they should do one or more of the following:

1)     Investigate other phone tree plans, such as 2-call, 4-call, or even an 8-call or 10-call plan.  Our data is limited, although it does not suggest that there will be major differences.  It should be noted that the more calls a household has to make, though, the less time the family has to prepare for the emergency.  (Hopefully one member of the family could make the calls while other family members prepare for the emergency.  Households with only 1 member could be given fewer calls to make, which would change the plan a little.)

2)     Investigate other methods of notifying the community, such as:

§        Radio or television (which are accessible even in the remote parts of the country outside the town),

§        Loud sirens (which would need to be hear-able in the remote parts of the countryside,

§        Combination of siren with phone tree (phone tree part for people outside of hearing of the sirens), etc.

Our reasons for recommending an alternative to either the 3-call or the 5-call plan is that 25 to 35 minutes is a very long to spend reaching the last households in a community.  Should the phone system in the community system go down, or not be strong enough to bear all of the phone calls going on at the time, that would leave households unable to be contacted too.  All of these reasons would support trying the other methods listed in number 2 above.