The OISE Math Trail for
Intermediate/Senior
Teacher Candidates
Ron Lancaster
35 Oak Knoll Drive
Hamilton, ON
L8S 4C2
Senior Lecturer
Mathematics Education
Ontario Institute for Studies in
Education of the University of Toronto
[email protected]
[email protected]
© Ron Lancaster - [email protected]
Introduction
Welcome to the OISE Intermediate/Senior Math Trail! The location for this Trail is downtown
Toronto, a perfect place to visit while wearing a pair of mathematical glasses.
The idea behind a Math Trail as it is now commonly called is quite simple: students follow a planned
route and answer or create mathematical questions related to what they encounter along the path.
Through this experience students are given the chance to connect the mathematics curriculum to many
subjects including art, design, architecture, science, geography and history.
You are probably going to be incredibly busy during your practicum and the thought of taking your
students on a Math Trail might end up being just that - a thought. The same might be true in your first
few years of teaching. My hope is that at some point you will be able to take your students on a Math
Trail. When you do, I suggest you start with a short Trail right on the school property. This will
allow you to avoid the complications of taking your students off campus and it will provide you with a
relatively simple way of experimenting with the idea of taking your students outside.
In the meantime, please consider using some of the questions from this Trail with your students now.
Take photographs (where permitted) along the Trail and show them to your students when you assign
the questions. You might even want to arrange the photographs with the accompanying questions
around your classroom and have students move from one station to the next. Participating in a
simulated Trail is not as good as actually being there, but it will provide your students with an
opportunity to view the world through a mathematical lens.
At the end of this document you will find a bibliography and information about my work with Math
Trails. Happy Trails.
Ron Lancaster, Senior Lecturer in Mathematics Education
Ontario Institute for Studies in Education of the University of Toronto (OISE/UT)
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The Trail
Our meeting point will in the courtyard behind the Commerce Court North building at 25 King
Street West. This building is located just west of Yonge Street. To get to the courtyard, enter the
building and walk through to the opposite end and then exit the building.
Question 1
Walk to the large, circular area (during the summer there is a fountain here) in the middle of the
courtyard. Look closely around the outside of this area and you should be able to find three huge
concentric circles that look like the ones shown in Figure 1.
Figure 1
You are going to use this outdoor setting to answer the following few questions.
When the radius of a circle is repeatedly increased by the same amount each time, what happens to
the circumference of the circle? How will it grow in size? Will it go up by the same amount each
time or will it grow in some other manner?
(a)
Use a tape measure to find the length of the line segments shown in Figure 2. These two
lengths should be approximately the same and by measuring them you will have shown that
this is indeed an appropriate setting for the above questions.
Figure 2
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(b)
Use the tape measure to find the circumference of each of the three circles. Record your
answers in Table 1.
While it might appear that your tape measure is not long enough to measure such a large
circumference, there is in fact a way!
Table 1
circumference of the inner circle
circumference of the middle circle
circumference of the outer circle
(c)
Use your results in part (b) to answer the questions posed at the beginning of this activity.
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Question 2
While I was planning this Trail I encountered a mother with her daughter in the area near the fountain.
The mother was sitting near the fountain reading a book and her daughter was running around and
around the fountain. As I watched the daughter go around the fountain I began to think about how the
distance between the mother and daughter changed over time.
Have two people in your group (let's call them Mahima and Deepti) play the role of the mother and
her daughter. The person playing the role of the mother gets to sit at point M while the other person
should start at point P and then move at a constant rate around the fountain in a counter-clockwise
direction (see Figure 3).
Deepti
d
P
M ahima
Figure 3
(a)
Let d represent the distance (in metres) between Mahima and Deepti (the length of line
segment MD) and let t represent the time (in seconds). Graph the distance d versus the time t.
Since you have no actual data to work with, a precise graph is not required.
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(b)
How would the graph in part (a) change if Deepti ran at a faster rate?
(c)
How would the graph in part (a) change if, before Deepti began to run, Mahima moved to a
point further away from P to stay in the shade?
Question 3
Suppose that a concert is to be held in this courtyard and that you are in charge of determining how
many people could stand comfortably within this area.
(a)
Decide by estimating, which of the following numbers is closest to the number of people who
could gather in this area for the concert.
(i)
(ii)
(iii)
(iv)
(b)
100 people
1,000 people
10,000 people
100,000 people
Develop a plan for determining an accurate estimate of the number of people who could stand
in this area.
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Question 4
Enter Commerce Court North through the back door and stand under one of the beautiful
chandeliers.
(a)
Identify by name all the geometrical shapes that appear on the ceiling and on the floor.
(b)
Stand underneath one of the chandeliers hanging from the ceiling. Count the number of lights
on each of the three levels and record your results in Table 2.
Table 2
number of lights on the lower level
number of lights on the middle level
number of lights on the upper level
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(c)
Calculate the two ratios in Table 3. Are they the same? Did you expect them to be the same?
Why?
Table 3
ratio of the number of lights on the upper level to
the number of lights on the middle level
ratio of the number of lights on the middle level to
the number of lights on the lower level
(d)
Use your results from parts (b) and (c) to predict how many lights might be used if an extra
level of lights were added above the existing upper level.
(e)
Suppose that a multi-leveled chandelier is to be made and that the number of lights per level is
to follow the same pattern exhibited by the chandelier in this building (the number of lights on
the lowest level is 10 and after that the number per level is always double the previous level).
Explain whether it is possible to make a chandelier for which the total number of lights is
equal to
(i)
(ii)
(iii)
99 lights?
1,000 lights?
1,000,000 lights?
for Wayne Gretzky’s home
not a light light for sure
for a hotel in Las Vegas
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(f)
Suppose that a chandelier has n levels. The lowest level has 10 lights and after that the
number per level is always double the previous level. Let tn represent the number of lights on
the nth level and let sn represent the total number of lights on all levels from the first to the
last. Mark each of the following statements as true or false.
Table 4
statement
tn 102n1
tn 20n1
tn 52n
tn
2tn1 t1
1
0
n
sn
1
02

1
0
sn102n1
n
sn
1
0
2

1

sn
2
02
n1
1
0
sn  tn
sn2tn10
T or F?
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(g)
What is a watt? What is a kilowatt-hour (kWh)?
(h)
Assume that this chandelier uses 40-watt bulbs and that businesses in Toronto are currently
paying on average 5 cents/kWh for electricity. Calculate the average cost of leaving this
chandelier turned on for a 24-hour period.
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Return to the courtyard. Walk to the southwest corner and stop when you reach the family of
elephants. These sculptures were created by the artist Derrick Stephen Hudson.
Question 5
(a)
Decide by estimating, which ratio is closest to the ratio of the height of the mother elephant to
the height of one of the babies.
1:1, 2:1, 3:1, 4:1, 5:1, 6:1
(b)
Visit the http://www.citywindsor.ca/000356.asp?artist=hudson1#hudson1 and use the
information to determine the ratio of the height of the mother elephant to the height of one of
the babies. Compare this answer with your estimate from part (a).
A building to visit next time you are in this area ...
Look in a westerly direction at the Ernst and Young Tower located at 222 Bay Street. The next time
you are in this area, be sure to visit the lobby of this building. Arlene Stamp, a Canadian artist and
former mathematics teacher, created all of the artwork that is on permanent display in the lobby of the
building. The pattern of black and white squares that appears high up on the walls was created using
the binary system (Base 2)! You can see another major commission (Sliding Pi) by Ms. Stamp at the
Downsview subway station. Throughout the cavernous station, she created a non-repeating design
made up of thousands of small coloured titles using the decimal expansion of π.
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Walk to the southeast corner of the courtyard. Enter the building through the doors near the Running
Room store. You should see the JUMP Cafe and Bar to your left once you are inside the building.
To your right there are two flights of stairs leading to street level. Walk to the stairway on the left
side and walk down to street level.
Question 6
(a)
How are these steps different from most other steps? Describe how you would you find the
slope of these steps. Describe how you would you find the slope of the stairway? Calculate
these slopes?
(b)
At the bottom of the stairs, turn right and stop at the top of the stairs that lead to the stores
underground and the PATH. Measure the slope of these stairs and then the slope of the stairs
on the left side that lead back up one level. Compare these slopes.
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Question 7
At the bottom of the stairs locate the large sign for Reitman's on your left side. The sign appears to
consists of a number of small squares and four rectangles. Are the small "squares" actually square?
How do you know? For this exercise assume that they are!
(a)
I recently showed students a photo of this sign and asked them to find several different ways of
counting the number of small squares. Three different strategies emerged and the calculations
are shown below. Explain the strategies that lead to these calculations.
(i)
(ii)
(iii)
(b)
3 8
27 10
81078
Let the area of each of the small squares be equal to one square unit. Determine the exact area
of the four rectangles (combined) in terms of the area of a small square. Determine the area of
each rectangle in terms of the area of a small square.
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Question 8
Walk in the direction of the Commerce Court Food and Shops. After a few steps, Rexall Pharma
Plus will be on your right side and Smart Set will be to your left. Stop when you reach the
mmmuffins store.
(a)
Using exponents, rewrite the name of this store. Does this alternative version require fewer
characters?
(b)
Fill in the cost for each item in Table 5 and then calculate the cost per ounce for each size.
Table 5
size of coffee
small (12 ounces)
medium (16 ounces)
large (20 ounces)
(c)
(d)
cost
cost per ounce
How many more ounces of coffee does a customer get by buying
(i)
a medium coffee instead of a small?
(ii)
a large coffee instead of a medium?
Suppose that mmmuffins were to add a new size (see Table 6) to the existing sizes for their
cups of coffee. Based on your results from part (c), suggest the number of ounces for this new
size.
size
extra large
Table 6
amount of coffee (ounces)
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Question 9
(a)
Use the data from Table 5 to sketch the graph of the cost of a cup of coffee versus the size of
the cup. Find the line of best fit for this data. How well does this line fit the data?
cost of a cup of coffee
3
2
1
size of cup
5
(b)
10
15
20
Use the line of the best fit that you determined in part (a) to predict what mmmuffins might
charge for the hypothetical size discussed in Question 8, part (d).
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Walk along the hallway that leads from the mmmuffins store to the Herbal Magic store. Stop when
you reach the escalators that go up to ground level.
Question 10
Have one member of your group ride up the escalator to the floor above and then stand in front of the
down escalator. Have another member of your group stand still in front of the up escalator. At
exactly the same time, have both people get on their escalators. Everyone else should watch these two
people as they go for a ride. Which graph describes how the distance between the two people changed
from the time that they got on the escalators to the time that they got off?
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
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Question 11
If the two people decided not to get on the escalators, then the left graph in the middle row would be
the one that describes how the distance between them changes with time. Setting aside this graph and
the one you selected in Question 10, describe what the two people could do in order to “make” each of
the four remaining graphs (in other words so that a particular graph shows how the distance between
the two people changes with time).
Table 7
Figure
Description of how to make the graph
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Question 12
(a)
How would the graph you selected in Question 10 change if the speed of both escalators were
doubled? Redraw the graph in the space given below.
(b)
How would the graph you selected in Question 10 change if the speed of one of the escalators
were twice the speed of the other? Redraw the graph in the space given below.
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Question 13
Let P and Q represent the two people on the escalators (Figure 10). Let x represent the distance
between P and A and also the distance between B and Q. Let y represent the distance between P and
Q.
(-1, 1)
A(1, 1)
P
O(0, 0)
Q
B(-1, -1)
(1, -1)
Figure 10
(a)
Develop a formula for y in terms of x.
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(b)
Use the TI-84 (or your favourite graphing utility) to graph y versus x for 0  x  2 . Draw the
graph in the space given below. Compare this graph to the one you selected in Question 10.
Question 14
(a)
Decide by estimating, which number in Table 8 is closest to the slope of the escalator.
(b)
Decide by estimating, which angle in Table 9 is closest to the angle between the escalator and
the ground.
Table 8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Table 9
00
100
200
300
400
500
600
700
800
900
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Question 15
Photograph 1 shows a triangle that appears on the left side of the escalator that goes down. After the
Trail use this photograph to answer the following questions. Compare your answers with your
estimates from Question 14.
(a)
Find the slope of the escalator.
(b)
Determine the angle between the escalator and the ground.
Photograph 1
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Question 16
Write a question of your own for any grade level from 7 to 12 related to the geometric design on the
ceiling above the escalators (on the lower level).
Go up the escalators to the main floor of the building, turn left and leave the building through the
nearby doors. As soon as you are outside, look to your right and locate the window on the side of
the building that resembles the shape shown in Figure 11. A window such as this one that consists
of a rectangle surmounted by a semi-circle is called a Norman window.
semi-circle
radius
height
width
Figure 11
Figure 12
Question 17
(a)
Suppose the perimeter of a Norman window is 6 metres. If the width (Figure 12) of the
window is 1 metre, find
(i)
the radius of the semi-circle
(ii)
the circumference of the semi-circle
(iii)
the height of the rectangle
(iv)
the area of the rectangle
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(b)
Suppose the perimeter of a Norman window is 6 metres. Let the width of the window be x
metres. Find the area (represented by y) of the window in terms of x.
(c)
Referring to part (b), find the value of x for which y is as large as possible.
Question 18
(a)
From where you are standing, locate each of the buildings (Table 10). Place a check mark
beside each name once you have located it. To get a better view of all these buildings you
might want to walk towards the southeast corner of King and Bay Streets.
Table 10
Name of building
Name of architectural firm
Commerce Court West
Toronto Dominion Centre
First Canadian Place
Scotia Plaza (historic component)
Scotia Plaza (modern component)
(b)
After the Trail (or during if you have internet access on a smartphone) visit the wikipedia.org
website and look up the name of the architectural firm that designed each of the buildings
listed in Table 10.
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(c)
Study the design of the buildings listed in Table 10 and discuss how certain designs allow for a
higher number of corner offices. Draw a diagram that shows a design that allows for more
corner offices.
Cross King Street and walk to the northeast corner of King and Bay Streets. Turn right, walk in an
easterly direction and stop when you reach the entrance to 44 King Street West. Go inside the
building. Walk forward towards the old clock, turn right and enter the adjacent hall. Walk around the
rotunda in a counter clockwise fashion until you locate a large collection of beautiful flags from all
over the world on a nearby wall.
Question 19
(a)
How many of these flags can you identify? Express your answer as a percentage.
(b)
How many of these flags have a line of symmetry?
Use the nearby escalators to go down one level, turn left and stop at the nearby Investors Update
area.
Question 20
(a)
Watch the electronic sign as it cycles through the current financial numbers and record the
numbers that appear on the screen for the exchange rate between US and Canadian dollars
(Table 11).
Table 11
country
CDN$
US$
BID
ASK
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(b)
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See if there is anyone within your group who knows the meaning of the words BID and ASK.
If no one knows, determine the meaning after the Trail and record what you found out in the
space given below and the source you used.
Question 21
On Friday, September 17 at 10:30 PM the following information was posted at the website
www.oanda.com/convert/classic
100 US Dollar = 103.272 Canadian Dollar
100 Canadian Dollar = 96.7380 US Dollar
(a)
How much Canadian money would you need to buy $100 US dollars?
(b)
Suppose that after purchasing the $100 US dollars you realized that you actually had enough
US money for an upcoming trip and so you went and exchanged the $100 US dollars back to
Canadian money. How much Canadian money would you get back if you made the exchange
almost immediately after the first transaction? Would you end up with the exact same amount
of money that you started with? If not, would you have more or less?
(c)
Determine what happens if you continued to exchange your money back and forth from
Canadian dollars to US dollars for four more times. In the space given below, graph how
much money you would have from the start to the end.
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Walk a short distance in the direction of the nearby Batteries and Gadgets store. Stop when you
reach the Shoe Company store on your left side.
Question 22
Have someone in your group enter the store and locate one of the Women's Conversion Charts that
displays shoe sizes in Canada and Europe, Last week they were posted at the rear of the store, near
the stockroom door.
(a)
Record the data displayed in this chart in Table 12.
Table 12
Women's shoe size in Canada
Women's shoe size in Europe
x
y
(b)
In the space given below create a graph of the shoe size for women in Europe (represented by
y) versus the shoe size for women in Canada (represented by x). Find the line of best fit for
this data. How well does this line fit the data?
(c)
Use the line of best fit that you determined in part (b) to predict the shoe size in Europe for a
size 12 women's shoe in Canada.
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Locate the Teaopia store opposite the Shoe Company store.
Have someone in your group enter the store and locate the stand that contains five levels of tea cans,
each arranged along the perimeter of a circle.
Question 23
(a)
Starting from the top level and working your way down, count the number of cans per level
and record your results in Table 13.
Table 13
level
1
2
3
4
5
number of cans
(b)
Teaopia's Tea List brochure contains 100 varieties of premium loose leaf teas. Is there enough
room on the stand to hold all these varieties?
(c)
Let the circumference of the circle for the top level of the stand be 1 unit. Use your results
from part (a) to determine the circumference of the circles on the other four levels in terms of
this unit.
I hope you have enjoyed the Trail and that you get the opportunity to do one with your own students
some day soon. Here is one more problem for you!.
Question 24
In the puzzle given below, each letter represents a digit from 0 to 9. Any given letter stands for just
one digit (in other words the value of E cannot be say 3 and 9 at the same time) and two different
letters cannot stand for the same digit (so N and D can’t both be say 7). Also the first letter of any
word cannot be 0. There are 144 solutions to this puzzle. All you have to do is to find one of them!
E
M
P
A
A
N
O
T
T
D
F
H
H
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Bibliography
Websites
Math for America Math Trails in Manhattan
http://www.mathforamerica.org/mathtrails
The National Math Trail
www.nationalmathtrail.org
The Canadian Math Trail
http://www.brocku.ca/cmt/English/index.htm
Vancouver 2010 Olympic Math Trail
http://www.brocku.ca/cmt/upload/1070004370.8764/index.html
Math Trails
http://www.cmste.uregina.ca/MathTrails/
Math Trails
http://www.saskschools.ca/~otl_el/grassroots/mathtrail/index.html
Welcome to the Welland Canal Math Trail
http://spartan.ac.brocku.ca/~emuller/mathtrail/wcmt/wcmtstart.html
NCTM Journal Articles
"Designing Math Trails for the Elementary School". Kim Margaret Richardson, Teaching Children
Mathematics, August 2004
"Making Mathematics Real: The Boston Math Trail". M Rosenthal and Clement K Ampadu,
Mathematics Teaching in the Middle School, November 1999
"A Mathematics Trail at Phillips Exeter Academy". Vince Delisi and Ron Lancaster, The Mathematics
Teacher, March 1997
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Tales about my Trails by Ron Lancaster
I created my first Math Trail on Toronto's Centre Island in the summer of 1985 for teachers who were
enrolled in an Additional Qualifications (AQ) course I taught at the Faculty of Education of the
University of Toronto. The suggestion to go to Centre Island came from Al Fleming, a well known
mathematics teacher, who had developed a tradition of going to the island for the last class of his AQ
courses. In the early 1990s I began to create Trails for teachers who attended the Phillips Exeter
Academy Mathematics, Science and Technology Conference in Exeter, New Hampshire. I developed
these Trails with Vince Delisi, a good friend of mine, who now teaches at the Bishop Strachan School
in Toronto. Vince and I created Trails on the Exeter campus, in the downtown area and in stores
including Whirlygigs Toys and Exeter Candy.
In 1996 I began to develop Trails for my own students. At the time I was teaching at St. Mildred’s, an
all-girls school in Oakville, and I was looking for new ways of making the study of mathematics more
attractive to the young women in my classes. The first Trail that I created was set in the Financial
District of Toronto and the day was a major success. The students loved the active nature of the
project, they enjoyed being able to work collaboratively and they were stunned at how much
mathematics there is in the world around us. The following year my grades 7 and 8 students
participated in three Trails. The first Trail took place in Toronto and I used it as a way of getting to
know my students and as a way for my students to be exposed to the big ideas of the course. The
second Trail was on the campus of the school and I used it to help students practice what they had
learned to date. The third Trail was also in Toronto and this one was used to help students review
everything we had done during the year. The third Trail also contained questions from the curriculum
for the next year in an effort to give students a sense for what was to come.
Prior to joining the faculty of OISE I worked as an independent mathematics consultant for six years
and through my work I ended up creating Trails in many locations throughout North America and
Asia. The settings for my Trails have literally been all over the map and have included shopping
malls, parks, art museums, schools, farms, downtown areas of big and small cities and even a church
in Kilgore Texas. Recently I have been creating Math Trails in Manhattan for Math for America.
Locations of these Trails include the Museum of Modern Art, The American Museum of Natural
History, Ellis Island, NYU, Columbia University, Madison Square Park and the Bronx Zoo. These
Trails can be downloaded for free at the Math for America website.
Many of the teachers I have worked with have created their own Trails and several have used the idea
of a Trail as the basis of their Masters thesis. By now I have seen a tremendous amount of evidence
that the idea works well for students in all grades, at all levels of learning and in all settings (co-ed,
all-boys, all-girls schools).
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I would be delighted to discuss any issues with you regarding the logistics of organizing a Trail along
with the assessment of student work. With regard to logistics, I have a few points to share with you.
Set up the Trail so that it forms a large circuit with the end of the Trail near the start. Put your
students into very small groups (three or four students per group) and have the groups start the Trail in
different locations. If they all move in the same direction around the circuit, there will never be a
point when too many groups are in the same place at the same time. For supervision, ask parents and
other teachers to join you. The best idea of all is to involve senior students as the guides and mentors.
If you are designing a Trail for students in say grade 9, have students from grade 12 help out. Be sure
to check with officials at every site on the Trail. Ask for permission to be on the property and make
sure that everyone knows what is going on.
There have been other people around the world who have developed their own Trails. Kay Toliver is
one of the best-known Trail blazers and information about her work can be found at
www.nationalmathtrail.com. Several years ago I worked with Kay in Los Angeles and New Orleans
and she was a pleasure to be with. Geoff Kavanagh, an Instructor at OISE/UT, was a real pioneer in
the development of Math Trails and he was out with his students in downtown Toronto many years
ago. Eric Muller, a Mathematics Professor at Brock University in Canada was another early Trail
blazer. Many years ago he developed Math Trails in Niagara Falls and at the Welland Canal.
Other trail developers include close friends of mine such as Brigitte Bentele, who has developed
Trails in Manhattan; Larry Ottman, Lisa Ledwith and Marcia Wexler who have developed Trails in
Philadelphia; Diane Devine who has developed Trails in Boston and Carly Ziniuk who has developed
Trails in Toronto and in Ottawa at the National Gallery of Canada.
Ron Lancaster, Senior Lecturer in Mathematics Education
Ontario Institute for Studies in Education of the University of Toronto (OISE/UT)
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