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WASHINGTON STATE MATHEMATICS COUNCIL
2006 MIDDLE SCHOOL MATH OLYMPIAD
5
Session I: FIFTH GRADE PROBLEM SOLVING
Trailblazer Problem – Sample Solution
Sample calculation for the simplest route. The calculations for one route are sufficient to
establish understanding, and to communicate strategy.
Fox and Kinnikinik Trails (FK) :Tree points = 50 + 160 = 210
Length = 3 + 1 = 4 km
Route time = (4km x 8min) = 32 min
(since 1 km per 8 min)
Route time points = 400
from second row of table 2 because 30 < 32 < 40.
Total points for the Route = 210 + 400 = 610
The subsequent routes will be designated by the order of the marked trees on the route (or by first
letter in the trail name which is the same thing).
The following calculations need not be shown, a table or list summarizing the results for their
routes is sufficient as long as a sample calculation such as the one above is show.
ABC (meaning Acorn-Bobcat-Crow Trails) - A three trail route
Tree points = 100 + 200 + 120 = 420
Length (L) = 2 + 2 + 2.6 = 6.6 km
Route time (T) = 6.6 * 8 = 52.8 min
Route time points = 200
Total points = 620
ADGK – a short route containing both the Acorn and Goose Trails as requested by Louise
Tree points = 100 + 125 + 180 + 160 = 565
L = 2 + 1 + 2 + 1 = 6km
T = 6 * 8 = 48 min
Route time points = 300
Total points = 865
ADHJGEFK – a route containing the most number of trails
Tree points = 100 + 125 + 75 + 80 + 180 + 0 + 50 + 160 = 770
L = 2 + 1 + 1.7 + 2.1 + 2 + 1.7 + 3 + 1 = 14.5km
T = 14.5 * 8 = 116 min
Route time points = 0
Total points = 770
Items in italics are what I would expect in a response that goes beyond the requirements of the
problem. There will be a few teams that list more than the 4 routes required and draw inferences
and make recommendations that are clearly beyond the minimum asked for.
ABHGK - a long route that contains both the Acorn and Goose trails one that leaves out the low
point Eagle and Fox trails.
Tree points = 100 + 200 + 75 + 180 + 160 = 715
L = 2 + 2 + 1.7 + 2 + 1 = 8.7 km
T = 8.7 * 8 = 69.6 min
Route time points = 100
WSMC 2006 Middle School Math Olympiad
Session 1 Sample Solution, Grade 5, Page 1 of 2
Total points = 815
Route
FK
ABC
ADGK
ADHJGEFK
ABHGK
Tree points
210
420
565
770
715
Route length
in km
4
6.6
6
14.5
8.7
Route time in
minutes
32
52.8
48
116
69.6
Route time
points
400
200
300
0
100
Total points
for Route
610
620
865
770
815
We notice the following when designing the different routes:1. A short route earns more time points but possibly fewer tree points.
2. If we make the routes long in order to get as many tree points as possible from the marked tree,
the travel time for the route becomes longer and Louise will not get as many points for the travel
time as she would with a shorter route. (There is also the chance that that she will get too tired on
the long route and be unable keep up the estimated pace of 1 kilometer in 8 minutes over the
entire route.)
3. Even if we wanted to, we can not design a route that will get her all the points from the marked
trees (i.e. Tree points) because then she would have to go on some of the trails twice and will be
disqualified.
4. A route that has a travel time close to the upper time limit of the route time point time interval
is risky in case Louise stops for a short while along the trail to look at something or slows down a
little.
Our recommendation is that she take the Acorn-Dogwood-Goose-Kinnikinik Trail route because,
a. It has the highest points among the routes that we considered.
b. It has 565 sure tree points that do not depend on Louise being able to keep up the one kilometer
in 8 minute pace (she has some slack time).
c. It is one of the shortest routes, it is the second shortest among the routes that we considered (and
even though we did not show it in the table, there are only two routes, FK and EGK, that are
shorter). So we think that Louise will be able to do this route in the estimated time and earn the
Total points (865) that we have calculated.
d. Fortunately, this route has both her favorite trails, but we are warning her not to slow down too
much at her favorite trails because she has only 2 minutes to spare before she drops down into a
route time interval that is 100 points lower.
We see that our choice is reasonable because we have chosen a short route that earns Louise a lot of
time points, yet includes trails that have trees that are marked with high tree points. Justification b
and c make the same point. So this statement is not necessary to earn the reasonableness
points. In the absence of justification b and c or some thing similar (observation 3 or 4) and
appropriate for their solution, then something along the lines of statements 1 and 2 will earn 3
reasonableness points. If the solution does not contain anything similar to 1, 2, b, or c that is
appropriate for their solution, a statement that the calculations were double checked will earn
one reasonableness point and the fact (either stated or unstated) that they choose the route
with the highest total points will earn one reasoning point for a maximum of 2 in the
reasonableness/reasoning category.
WSMC 2006 Middle School Math Olympiad
Session 1 Sample Solution, Grade 5, Page 2 of 2