Summer Math Trail Questions
In the summer workshop, the participants participated in a math trail in the Bellingham Ferry
Terminal, then worked in small groups trying to generate appropriate math trail questions for
a local middle school. The results (unedited) are as follows:
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Find the entryway to the main courtyard. Locate the two brick pillars. How many
bricks were used to construct the pillars? What assumptions did you make?
(In the courtyard . . .)Estimate the height of the roof.
In the front of the school, locate the wheelchair ramp and the front stairs. How much
steeper is the front entry when compared to the wheel chair ramp? How did you
calculate this? What idea did you use to find “steepest”?
Find the bike racks in the front of the school. If you straightened out the pipe used to
make the bike racks, how long would the pipe be?
Find the fire zone map at the front of the school. Which zone is the smallest? Which
zone is the largest? How did you determine this?
Find the flag pole and look at the American flag. If two more states were added to the
union (DC & Puerto Rico), how would you redesign the stars on the flag?
How many different size squares are there in the top wire frame of the door?
What is the amount of runoff if one inch of rain falls on the courtyard?
How many mirrors are on the disco ball?
How many bricks are in the square?
Pick out the tree in the courtyard you think is the most pleasing to your eye. Measure
it and see if it fits the golden rectangle.
Maximum occupancy of the auditorium is 999. If that number of people are in the
auditorium, how much space would each person have?
The auditorium doors open on the east side of the room. Describe, using a
measurement of your choosing, the directions from the stage to room 207. Give this
to another student and see if they can follow your directions.
On the grid below, carefully plot each point for the Braille letters which spell the word
“boss.” The letter b has been plotted for you. Rules: Skip one space between letters,
and put a number on each point and give the coordinates for each point. *Answers: b
= (0,1) (0,2); o = (2,0) (3,1) (2,2); s = (5,0) (5,1) (6,2); s = (8,0) (8,1) (9,2)
In front of the school is a monument with a plack titled Fairhaven Middle School.
Notice the two large cylinders. List the following measurements in cm: height,
diameter, radius. Using your measures, calculate the total surface area and volume
of the cylinder. If each cylinder were a container, how many 2 litter pop bottles would
it take to fill one?
Look at the two sets of stairs in front of your school. Measure the height of a step (in
inches). There are eight steps in each set. How much do you go up in elevation when
you climb one set? How much do you climb when you go up both sets? Assume it is
level between the two sets. Mt. Baker is 10,972ft tall. If you start from Bellingham
Bay, how many steps would it take to reach the top?
An ant moving at three feet/minute is crossing the diagonal distance of the courtyard,
so is a beetle who travels at 10 inches/minute. How long will it take each to cross the
courtyard?
a)What’s the maximum number of round cafeteria tables that you could put in the
cafeteria? Explain how you got your answer.
19. b) How many table if the radius of the table was doubled?
20. c) If the radius was halved?
21. In the music hall there’s a banner with FMS. Redraw each letter so that . . . (1) when
it’s reflected it makes a new letter not F, M, or S. (2) it has a line of symmetry (3) it
has a perimeter of 13.
22. The soil in the planters needs to be replaced. How many cubic yards will be needed?
Leave 3 inches from the top. How much gravel will be needed to put a 2 inch layer on
top of all the planters?
23. Compare the slope of the main entrance ramp to the slope of the main entrance
stairs.
24. Find: Parallel Lines, perpendicular lines, 90 degree angles, angles less than 90
degrees, angles greater than 90 degrees, 45 degree angles, angles less than 45
degrees, and tessellation not using just a square.
25. Terra Cotta Headpiece: Two cylinders can be found in the original Headpiece of
FHMS. Find the radius, diameter, circumference, height, volume, and surface area.
26. Examine the patterns in the front windows and draw the next couple of iterations
(teachers: have the first two drawn). Where is another place where this pattern is
repeated? *Answer: the Cafeteria
27. In the courtyard: 1 square on the ground = one coordinate point. Find the coordinates
of each tree in the courtyard. Define your axes.
28. Using the circular window in the cafeteria: How many squares can you create?
Estimate the area of it’s inside diameter.
29. Find two objects that contain a Fibonacci number. *Answer: Dandelion.
30. Ramp on 2nd floor: estimate the angle of the ramps slope. List the tools you used.
31. Find the following shapes on the school’s property: Circle, square, triangle, rectangle.
32. Find a flag with the following lines of symmetry: (a) 0 lines, (b) 1 line,(c) 2 lines.
33. Estimate the following by comparing the mural and it’s legend: (a) ratio of mural’s
length to legend length, (b) ratio of mural’s width to legend’s width, (c) ratio of mural’s
area to legend’s area.
34. Estimate the height of the ceiling in the main hallway.
35. Estimate the number of rectangular ceiling tiles in the main hallway.
36. Estimate the number of bricks on the courtyard ground.
37. Locate at least two golden rectangles on school property (sketch them and state their
location).
38. Skylight on the 2nd floor: There is a large “plus” sign dividing the window. Estimate its
volume.
39. In the mural in the corridor: Consider 5 yellow flowers, 5 pink-purple flowers, and 4
blue flowers. How many combinations and permutations can be selected if 4 flowers
are picked at random of any color.
40. Determine how much rope is needed to replace the rope on the flagpole in front of
the school.
41. Estimate the width of the hallway leading from the cafeteria doors to the stairs going
up to the second level.
42. Estimate the volume of the concrete used to construct the handicap ramp.
43. Looking at the doors: How much wire is needed to fit in the door windows.
44. What is the area and perimeter of the large window above the front doors? How does
the perimeter (size) of the window compare to the planters and benches in the
entryway?
45. At the front of the school: In the brick column to the right of the front door, how many
half size bricks are there per row and per column. (recessed rows are half size.
46. Looking at the handicap ramp at the front of the school: How fast would a ball be
moving at the bottom of the ramp if you let it roll down from the top? Do four time
trials.
47. In the hall: Immediately to the left when entering the building, compare the area of
the gray carpet to the maroon carpet. Estimate which is greater and by how much?
What methods did you use for your estimation?
48. Estimate the length of large and small tubing needed to construct railings along the
drive of the bus drop in front of the school.
49. At the front entrance, estimate the surface area and volume of one concrete
rectangular prism. Explain your process.
50. At the NE corner of the building, compute the slope of the stairway and the handrail.
Describe your process. Are they the same?
51. At the front entrance, how many total rectangles exist out of the glass? Remember,
squares are rectangles.
52. In the cafeteria: How many times bigger is the mural than the legend?
53. How many squares can you create using the circular window? Estimate the area of
the window.
54. On the second floor walkway above the main entrance, find how many feet the
hallway rises/falls and explain your method.
55. Suppose that the stairs in the front hall contained a storage closet below. If the
volume of a vending machine could be redistributed, how many would fit under the
stairs?
56. Estimate the number of bricks used in the construction of the two posts at the
entrance of the courtyard. If the bricks cost 69 cents each, what would the total cost
be? If a concrete pillar is $790, would you still build it from concrete?
57. If you stand at the entrance to the cafeteria on the first dark gray square and start
counting from the lower right corner, where should you end up if you walk . . . (a) 2 up
and 2 left? (b) 1 up and 2 left? (c) 7 back and 1 left?
58. In the hall that has the men’s bathroom and the art room: What is the area of the
beige acoustic tile on the walls only? What is the total area of the acoustic tile on the
walls? What is the area of the white acoustic tile?
59. Estimate the number of florescent bulbs needed to maximize the light in the
commons.
60. The flags in the hallway are 6 feet apart. Estimate the length of the hallway from the
stairs to the end of the last set of red doors.
61. Push the water fountain button until you get a nice arc. What type of function
represents this? Vary the pressure on the button. How does this change the
function? Build a model representing your observation.
62. How many students could fit inside the pop machine?
63. Estimate the distance between floors.
64. How high is the flagpole?
65. For students you hate. How many squares in the top pane of the door to the
commons? Rectangles?
66. Take the 1st hallway to the left of the main entrance. Go to the lockers. How many
different three number combinations could you have? Why do you suppose most
lockers (+ locks for that matter) are only three numbers?
67. How many people does it take to go around the blue pillar outside the commons
fingertip to fingertip? Shoulder to shoulder?
68. 25 bottles of each selection fit inside the Dasani water machine. How much change
should the machine be able to hold if it only took quarters? Nickels?
69. There is an interesting design of white, gray, and dark gray tiles. How many dark gray
tiles would it take to cover the light gray? How many dark grays to cover the white?
How many light grays to cover the whites?
70. In the main entrance hallway there are 78 lockers. All of them are closed. If you open
multiples of two, then change (open or close) multiples of 4 and so on, up to 78, what
lockers will be left open?
71. In the front entrance of the school there is a rectangular pattern. Calculate how many
white squares are in it.
72. middle school In the main office door window there is safety wire embedded in the
glass. Would you need more, less, or the same amount of wire if the wire goes
horizontal and vertical, but creates the same size squares. Why or why not?
73. 8th If you put your hand over the drinking fountain drain in the entrance or it became
clogged, and it was stuck running, how much time would pass before it overflows?
(please estimate and calculate, don’t do it).
74. How much lumber is needed to construct the planter box out of the 2x6 material?
75. 8th What percent of the rain water in the courtyard drains to trees instead of the
drains?
76. 8th-10th In the main entry, there is a plastic garbage can with a dome lid. What is the
volume of the largest object that can be put in the can without lifting the lid?
77. 8th There is a tree in the center of the main entry. Above the tree is a skylight. At
what sun angles will the tree receive direct sunlight?
78. What is the volume of cement used to create the design around the plaque on the
front lawn?
79. Divide the stage into a 3x3 grid. For each of the nine areas, determine how many
lights are hitting that part of the stage given their current position. Will they hit at eye
level?
80. Find and then draw three sets of concentric shapes.
81. On the window above the Fairhaven Middle School sign: What percentage of the
rectangles that can be made are squares?
82. At what angle would you have to stand from the front center for the building for the
window pattern to appear symmetrical when viewing the school?
83. How many isosceles triangles can you find in the railings surrounding the original
Fairhaven building-front structure? What if the railing went all the way around? Then
how many isosceles triangles would be found?
84. Estimate how much concrete was needed to create the rectangular prisms flanking
the front stairway.
85. There are 10 median railings in front of the school. About how many total railings
would you need to extend all the way to both street intersections? What is the
approximate ratio of railed to not railed distance of the road passing in front of the
building?
86. If a bus is ____ feet long, how many buses can fit in the bus lane?
87. If picnic tables have to have 3 feet on each side, how many can fit in the courtyard?
88. Find the ratio of gray bricks to red bricks in the courtyard.
89. Assuming the bricks go all the way around the pillar, about how many bricks does it
take to build each pillar between the parking lot and the courtyard?
90. How many more loading zones can be made along the yellow line west of the current
loading zones?
91. At what angle is the wood trim mitered at along the stairs?
92. On the Einstein Award Plaque at the top of the stairs (above the window near the
clock): What percentage of the names are filled in? At the end of next year what will
the percentage be?
93. Above the cafeteria, there are panels. Assuming the room is 82 ft long, how many
more panels would you need if the room was 289 ft long?
94. The cafeteria floor: What is the total area? How many gray tiles are there? What area
do they take up? How can you estimate this quickly? How many dark tiles are there?
What area do they take up? How can you estimate this quickly? Determine the
percentage of each color tile to the total floor.
95. Find the locker number(s) that sum to 11 and multiply to 40. (looking at the three digit
locker numbers)
96. Measure the slope of the grassy field/hill from the parking lot to the basketball court.
What resources do you think you should have? Should you get the same answer as
another group? Why/why not?
97. How much wire would be needed to go from the bottom to the top of a cyclone fence.
(Notice all the loops that each wire makes around another wire – does that make a
difference?)
98. Middle School If the school replaces 3 fluorescent bulbs for every 26 hours, how
many light bulbs will it need to replace over the school year (180 days)?
99. Middle School How many mirror tiles do you think cover the “disco-ball” in the
cafeteria? How did you get your answer?
100. Middle School Look at all the lockers in the 7th grade hall. What is the probability of
getting a locker that has a 4 as part of the #? Write your answer as a fraction,
decimal, and percent.
101. Middle School Look at the maroon and gold sides of the flags in the foyer. How many
have exactly 1 line of symmetry? 2? 3? 4? Illustrate here.
102. Middle School In the cafeteria, if one round table seat 8 people and one long table
seats 16 people, what is the best arrangement of tables in the cafeteria to
comfortably seat the most students?
103. How many squares do you see on the circular frame photo of Bellingham in the
cafeteria?
104. In the Cafeteria: How many different rectangles do you see?
105. Find the slope of every six steps and compare.
106. Approximate how many of the bulletin boards would cover the hallway.
107. If you toss a penny on the floor of the cafeteria, what is the probability that is will fall
on a (a) light blue tile? (b) dark blue tile? (c) beige tile?
108. Using the column at the entrance to the cafeteria, find: (a) area of base, (b)
circumference of base, (c) explain your method, (d) how accurate do you think you
are?
109. In the courtyard find: a square, a rectangle, a circle, a trapezoid, a semicircle. Can
you find more than one of each (not the same size).
110. Looking at the circular Fairhaven photo: How many circles are there? How many
squares are there?
111. Name all of the polygon shapes within the flags above the cafeteria entrance.
112. Give an example of two shapes that are similar but not congruent.
113. How many isosceles triangles can you find in the railing around the old headpiece of
the original Fairhaven HS.
114. Estimate the measure of the central arc for the arc in the Fairhaven sign on the front
steps.
115. Estimate the number of red floor bricks in the courtyard.
116. Estimate the height of the building on the north side, (the highest building).
117. Give an example of perpendicular and parallel lines.
118. How many people holding hands would it take to go around the perimeter of the
courtyard? What would the diagonal length be in people?
119. The handicap ramp: how does it’s slope compare to the mass of two human arms
and how does the slope of the stairs compare to the mass of a human leg?
120. Discrete math Amount of time taken to get pop as a function of button pressed on
pop machine.
121. What proportion of the flag pole is visible as a function of distance from the front
window?
122. Develop a strategy to find the height of the building, and use it.
123. Write your own math trail question. Be creative.
124. Find the lines of symmetry in the fence around the bench under the tree.
125. Estimate the number of branches on the labeled tree.
126. Examine the symmetry of leaves on different trees.
127. Estimate how many bikes can be locked on the bike rack. Does position matter?
Why/why not?
128. Calculate the angles of the triangles formed by the three trees in the courtyard.
129. Estimate the perimeter of the planter box/bench in the main hall. How did you find it?
Now, give the area of the same planter/bench in square feet.
130. Stand in the middle of the commons. Look around. Make a list of all the different size
rectangles you see. How are these rectangles used in this room?
131. Using students standing shoulder to shoulder, what is the length of the 300 hallway
from the end of the carpet to the far door? What is that length in inches? In feet? In
yards?
132. Calculate the slope and distance of the diagonal of the retaining wall in back.
133. Find the ratio of glass area to door area.
134. How many shapes are in the square ball court? Find the area of each shape.
135. Estimate the number of rock holes in the back wall.
136. Using the ratio of small bricks to large bricks on the back wall, how many bricks do
you need to cover ____ square feet of wall.
137. How many lockers numbers are prime? Which ones are they?
138. How many people would it take with hands joined at arms length to encircle the
archway along the inner edge of the sidewalk? Find the area of the sidewalk.
139. Write three dates you find on some of the engraved plaques and give the significance
of each.
140. Using the same spacing as the east outside wall of the school, how many plants
could be planted along the north side on the wall assuming no plants are placed
along the entrances.
141. Find the vents on the North wall. How many isosceles triangles are in each vent?
142. Find something that is in the shape of a cylinder and calculate its volume.
143. How many rectangles are in the lights at the ceiling of the auditorium?
144. In the mural in the corridor, I see 2 bears, 3 moose, 1 eagle, and 1 raccoon. How
many arrangements can be made if you choose one animal from each group? A
moose has 13 ends on his two horns. If each moose has 13 ends on his two horns,
how many moose would there be if 117 ends were counted?
145. What is the volume of one of the drink machines? Assuming all the bottles are
standard 20 oz. Pop bottles, how many could fit inside? State your assumptions.
146. Maximum capacity for the auditorium is 999 people. Using the floor tiles can you
figure out what formula is used?
147. Draw and identify all the geometric shapes on the flags in the hall.
148. At what angle has the “historical” window in the cafeteria been rotated off the x/y
grid?
149. Find the pattern between the number 999 and the Braille representation of it.
150. In the middle of the courtyard gate there is a circle. Find the area of the upper region,
find the area of the circle, find the area of the lower region, use that information to
find the total area. Now find the area of the whole rectangular panel. How do these
areas compare? Explain your observations.
151. Above the entry way there is a “welcome to Fairhaven” banner. What fraction of the
banner is yellow? What percent is yellow?
152. The entry way hall: To replicate a terribly interesting ancient sea battle we need to
flood the first floor. Close the doors to the commons, office, and all the classrooms,
and fill up the space with water until it reaches the top of the stairs in the hall. How
many gallons of water would you need?
153. Entryway: You need to reach the top rail (above the banner) by making a human
pyramid of kids. How many kids would you need to reach the rail? What assumptions
did you make?
154. Cafeteria: the door has two windows that are rectangles. How do the rectangles
compare?
155. Cafeteria: Ms. O’Neill has decided that all students should get to sit in chairs during
assemblies. She is not sure if there is adequate space. So, get off the floor! Figure
out how many chairs could be set up, allowing space for aisles, exits, and room in
front of the stage.
156. Handicap ramp: Rolling different sizes of balls down the ramp, how long would it take
for each ball to get down the ramp? Would a small or large ball be faster?
157. Locker: How much larger will a locker need to be (height, length, width) in order to fit
an average 7th grader inside?
158. Cafeteria: If the radius of each round table is approx. 31 inches, find the surface area
of one table. How many of the round tables could be used to fill the room (sides
touching, not stacked)?
159. Outside: Estimate the number of small bricks along the front of the building.
160. If the tree in the foyer grows 3 inches per year, how many years will it take for the
tree to reach the skylight?
161. Basketball court: (a) determine if the painted square for four square is a perfect
square, and if the center is accurately painted. (b) Predict the number of tilted
squares per fence panel. How many would there be in all? What if the fence
surrounded the entire court? What is the surface area of the court? What is your unit
of measure? (c) How many parallelograms can you find in the “out of bounds” strip?
162. Looking at the field by the basketball court: using string and tape measure, create a
grid, count dandelions within a small area, and use that to predict the number of
dandelions in the entire field.
163. Lockers: How many Mountain Dew cans will fit in a locker? How much air space is
left when the locker is filled with full cans?
164. Using one round table in the commons, create a cylinder with butcher paper (going
from the ground to the top of the table). How many average sized 7th graders would
fit under the table? What is the volume of the cylinder?
165. Outside: measure the angles of the angled parking spaces in front of the school.
What is their sum?
166. Basketball court: What is the diameter of the hoop? How can you determine if the
hoop is a true 10 feet above the ground? What is the radius of curvature of the bend
in the pole?
167. Basketball court: Does each square in the foursquare “court” have the same area? If
the foursquare “courts” were allowed to boarder each other, how many would fit into
both basketball courts?
168. Lockers: In the locker bay at the top of the main stairs, express the total number of
lockers in terms of the high and low locker numbers.
169. Outside the room whose number is closest to 100 times the approximation of pi:
Estimate how many 1x1 wire squares exist within the windows outside the room?
170. How many ways can you get from locker number 209 to locker 234, just going down
or right? Is the sequence of numbers an arithmetic sequence, geometric sequence,
or neither?
171. Find a sky light that “sums it up the smallest,” take the number of outlets you see,
multiply by the number of dark gray tiles you can see, subtract Baskin Robins, add
the number of can lights, what perfect square do you have?
172. What is the ratio of tile flooring to carpet for just the hallways and foyer? (of the whole
building)?
173. What are the dimensions of the courtyard in terms of the concrete benches? What
assumptions do you make?
174. At the bottom of trees, there is a cement sunburst, find the area of surface covered
by concrete shape.
175. Find the number of lockers in the school that are palindromes.
176. Find the number of lockers in the school that are: (a) a perfect square, (b) triangular
numbers, (c) prime numbers.
177. The molding on the wall by the stairwell is sometimes parallel to the floor and
sometimes at an angle. Write a function to approximate this shape.
178. Use a toy car or ball. Allow it to roll down the ramp above the foyer. Calculate velocity
and acceleration. If the ramp was 3 times as long, estimate the velocity and
acceleration.
179. Design an experiment to find the probability that a ping pong ball would hit a person
during lunch if dropped from the balcony.
180. At what rate does a shadow change length? Is this a constant? Design an
experiment and collect the data, model with a function.
181. How many animals are in the large mural? If each dimention was increased by 3,
how many animals would there be? What assumptions did you make?
182. There is a red curtain hanging on the stage. Estimate the surface area and the
volume of the curtain.
183. Lunch table are folded and placed against the wall. Looking from left to right you see
bench, tabletop, bench, bench, tabletop, bench, . . . what will the 100th item be? A
tabletop or a bench?
184. How many people could comfortably stand on the ramp above the foyer? What are
your assumptions?
185. What are the most common shapes in the structure of FMS? Speculate - why were
those shapes used?
186. Describe three patterns within the banners over the cafeteria.
187. If the trophy display case were halved in volume, what size would the glass panels
that cover it be?
188. What is the area of the sheet rock portion of the wall surrounding the exit to the gym?
189. What is the maximum capacity of 6th graders who could stand on the bricks of the
courtyard? What assumptions did you make?
190. What is the probability that a flag outside the commons area contains a convex
polygon?
191. Find the maximum occupancy of the cafeteria. Add pi^0. Multiply by the number of
flags above in the foyer. Square root this result. Would you be able to fit this number
of 6th graders comfortably into the seats of a standard school bus?
192. If it takes 25 seconds to wash the outside of the larger window. How long would it
take to wash all of the windows of the courtyard that are surrounded by red paint?
193. Find a boy’s restroom sign. Copy the brail symbols. Identify all lines and points of
symmetry that exist.
194. Determine the number of tiles in the main entrance and hallway to the base of the
steps. If you used tiles with double the side length, how many tiles would you need to
cover the same area?