33rd Middle School Math Bowl

General Information

The thirty-third Armstrong Middle School Math Bowl will be held
in University Hall, on Saturday morning, October 28, 2017.
The competition is open to public and private middle schools in
the greater Savannah area.

For questions not answered on this web page, contact Dr. Jim Brawner, Dept. of Mathematics, Armstrong State University, 11935 Abercorn Street, Savannah, GA 31419, or call 344-3186, or send email to James.Brawner@armstrong.edu . In particular, for a middle school to participate in this year's Math Bowl, the teacher serving as math team coach should do the following by Friday, October 20:

- Let Dr. Brawner know how many students will be coming. [A team consists of four students, but can also include up to four additional alternates.]
- Let Dr. Brawner know if you can bring a quiz-bowl type buzzer
set for the oral rounds.

- Send a check made out to Armstrong State University to help cover the cost of refreshments and awards. [The charge is $10 for a team of 4-6 students, $11 for a team of 7 students, and $12 for a team of 8 students.] Mail the check to Dr. Brawner at the address given above.
- (Optional) Send 5 pairs of toss-up/bonus questions for the oral rounds. [There are examples of such questions from past exams on the web page. The toss-up questions should be reasonable for one person to work mentally in about 20 seconds or less; the bonus questions can be more challenging since teams can discuss, use paper and calculators, and have 2 minutes. Each toss-up can have only one part, but a bonus can have several parts. Please do not include any questions that have only two or three possible answers. Topics from which we encourage questions include: whole numbers, fractions, decimals, percent, integers, geometry, measurement, estimation, logical thinking, problem solving, probability, vocabulary. Strive for variety in your five pairs to help us develop a well balanced set of questions. One of the strengths of our contest is the inclusion of questions directly from the teachers who know what their students are studying.]

**Note:** It is especially important that we know the number
of schools coming, because this affects the number of necessary
rooms and helpers, and can even affect the number of rounds and
questions. It would also be helpful to have your best guess about
how many students you will be sending, to be sure we have enough
copies of the multiple-choice exam and enough food, but we can
allow last-minute adjustments on the number of alternates.
Similarly, the question sheets are made up in advance, and any
questions submitted at the last minute cannot be included among
those used in the oral rounds.

Awards are given for the top three teams and for the top three
individuals. Over the years, eight different schools have won the
competition, thirteen different schools have finished among the
top three, and five additional schools have had individual
winners. The students can be any combination of sixth, seventh, or
eighth graders. We expect a teacher to accompany your
team---several, if you like. It is up to your mathematics
department or your school to decide how to select your team
members. If you have more than 8 students who want to come, you
may want to hold a qualifying contest at your school to decide
which students will participate. Unfortunately, we cannot
accommodate more than eight students participating from any one
school.

Any parents wishing to watch the competition are invited to come at about 11:30am to University Hall Room 156 to watch the final two rounds. The earlier rounds are held in smaller classrooms, where space is limited.

Calculators will be allowed on the multiple choice test and on the team part of the oral competition. For the oral competition, each room will have two or three teams competing. One player from each team will be designated for the first question, called a "toss-up." No calculators will be allowed on the toss-up portion. If the first student to answer is correct, his/her team is awarded 10 points. If the first to answer is incorrect, the representative of the other team(s) will have ten seconds to answer, after the question has been repeated. All teams will be given two minutes to work together to answer the 12-point follow-up team question. Answers to the team question will be written on paper instead of given orally. Calculators will be allowed on the team questions.

The process then continues with a second representative from each team, then a third, and so on, for about fifteen minutes. Each team will be directed to the proper room through the preliminary rounds. Cumulative scores will be kept for each team in order to determine the top four teams, who will then compete in the auditorium for first, second, and third place.

Students designated as alternates may take the multiple choice test and compete for top individual honors. Teachers may choose to substitute an alternate for a regular team member in the oral competition, as long as the substitution is for an entire round.

**Directions to University Hall**

To get to University Hall when driving on Abercorn Street: Turn
onto campus at the Science Drive traffic light. Then turn right
onto University Drive and continue on University Drive behind Fine
Arts Hall, bearing left until you reach the parking lot in front
of University Hall (on your left). See
http://about.armstrong.edu/Maps/index.html for a campus map.
There is also a detailed schedule
below of the morning's events.

SATURDAY, October 28, 2017

8:15 Registration (lobby of University Hall, then gather in Univ. Hall 156)

8:30 - 9:00 Written Exam

9:00 - 9:20 Refreshment Break

9:20 - 9:45 Orientation (Univ. Hall 156)

9:50 - 10:10 Round 1 (rooms in Univ. Hall)

10:15 - 10:35 Round 2 (rooms in Univ. Hall)

10:40 - 11:00 Round 3 (rooms in Univ. Hall)

11:05 - 11:25 Round 4 (rooms in Univ. Hall)

11:35 - 11:55 Round 5 (Univ. Hall 156)

12:00 - 12:20 Round 6 (Univ. Hall 156)

12:25- Awards Presentation (Univ. Hall 156)

We do not recommend bringing expensive graphing or programmable calculators. There will be no opportunity to use the fancier features of such calculators in our competition. Also, we would hate for anything to happen to an expensive calculator, with students moving from room to room. A simple 4-function calculator with a square root key and a single memory will be quite sufficient. Many of these cost less than $10. If your students are using scientific calculators, that is fine, but of course, there will be no need for the trigonometric or logarithmic functions on such calculators. In general, we recommend solar calculators to reduce the problem of batteries going bad, but we certainly will not require solar calculators.

We recommend that you check that all your calculators are in working order before you come. If your team includes alternates, they should have calculators, too, since they will also have the opportunity to take the multiple choice test. (Also, if you have more than 4 calculators, then there will be one to fall back on in case one fails during the oral competition, when only 4 students can compete at a time. You may want to bring an extra calculator, just in case.) Another potential problem is a student leaving his calculator somewhere. You may want to gather all of them when they are not in use, or at least check that each has put her own in a pocket or purse. Once everyone has arrived, the crucial times would be: (1) after the multiple choice test when students get refreshments, (2) between each oral round when teams often move to a different room, and (3) at the end of the competition when everyone leaves campus.

You will want to be sure that each student is familiar with his or her own calculator. For instance, different calculators have different names for the buttons used to store a number in memory, and some do it by adding to whatever is already in memory, while others simply replace previous memory contents with the new number. For a more complicated calculator, you may need a manual. The simpler ones work pretty much the same way. I strongly recommend that you let your students use calculators to work problems when they prepare for this competition (except the toss-up type of problems). Then if there are things they do not understand about the calculator's use, there will still be time to find out what to do.

Some of the following problems illustrate the fact that the brain should be used first, that the calculator may not be necessary at all, or may simply be used as a tool to do the dirty work, once the brain has determined what must be done.

__Example 1__ Which of the following fractions has the
greatest value?

(A) 3/7 (B) 4/9 (C) 50/100 (D) 151/301 (E) 201/401

[Of course, all five fractions can be converted to decimals by dividing on the calculator, but the student who notices that only D and E are greater than 1/2 will save time by considering only these two fractions.] (Correct answer is D.)

__Example 2__ If 2/7 is expressed as a repeating decimal, the
1989th digit to the right of the decimal point is

(A) 1 (B) 2 (C) 5 (D) 7 (E) 8

[Doing the division on the calculator furnishes enough digits to see a repeating pattern. The student would still have to determine what the 1989th digit in that pattern would be, but the calculator saves the time it would take to do the division] (Correct answer is C.)

__Example 3__ Find the value of the following sum:

1/2 + (1/2 + 1/3) + (2/3 + 1/4) + (3/4 + 1/5) + (4/5 + 1/6) + (5/6
+ 1/7)

+ (6/7 + 1/8) + (7/8 + 1/9) + (8/9 + 1/10).

(A) 3/4 (B) 8 (C) 8 1/2 (D) 8 1/10 (E) 9 1/2

[Adding all these fractions in order, even with a calculator, would be a pain, and would likely involve an error somewhere along the line. It is much better to regroup the fractions in pairs whose sums are 1 (legal since addition is associative)]. (Correct answer is D.)

__Example 1__ The sum of all of the digits in the numbers 34,
35, and 36 is 24 because 3 + 4 + 3 + 5 + 3 + 6 = 24. Find the sum
of all of the digits in the first fifty natural numbers 1, 2, 3,
...,49,50.

[Here students may be tempted to actually add all the digits involved on their calculator. Much better would be to notice a pattern, such as adding 1 + 2 + 3 + ... + 9, noticing that that group occurs five times in the units place, and that in the tens place there will be ten 1's, ten 2's, ten 3's, ten 4's and one 5.] (Correct answer is 330.)

__Example 2 (harder)__ One year, a girl started to work on
Labor Day (Monday, September 4) and continued to work every
weekday (Monday through Friday) through Columbus Day (Monday,
October 9) the same year. She agreed to be paid 1 cent for her
first day's work, 2 cents for her second day's work, 4 cents for
her third day's work, etc. That is, her pay for each day was
double the amount she was paid the previous work day. How much was
she paid, in all?

[The students would have to determine that there are 26 work
days, then compute

1 + 2 + 2^{2} + 2^{3} + 2^{4} + ... + 2^{24}
+ 2^{25} (cents). As they add the first few, they might
notice that the sum from the beginning to any point is simply one
less than the next number. For example,

1 + 2 + 4 = 7 = 8 - 1, 1 + 2 + 4 + 8 = 15 = 16 - 1, etc. With that
insight, they could take a shortcut to the whole sum by using the
calculator to compute 2^{26} - 1 = 67108864 - 1 =
67,108,863¢ = $671,088.63.]

The past tests available from this page are **PDF files (**portable
document format**)**. These files can be read using the Adobe
Acrobat Reader software. When properly installed, the software
will automatically start-up and display the PDF document when you
attempt to download it over the web. You may already have the
Acrobat Reader software installed, if not, you should have little
difficulty in downloading and printing these files. It is easy to
download the FREE software by clicking
http://www.adobe.com/prodindex/acrobat/readstep.html
and following the directions for your type of computer.

- Oral (questions only) 1990
- Oral (answers only) 1990
- Oral 1995
- Multiple-choice 1995
- Oral 1998
- Multiple-choice 1998
- Oral 2003
- Multiple-choice 2003
- Multiple-choice 2011
- Oral
2012