"Boulder is full of math."      Meagan Chandler

Requirements: Write up a total of three investigations from the Math Trail in downtown Boulder. These are to be selected from the starred (**) investigations in this booklet. You may work with a friend on the investigations and submit five that you worked on together.
Each investigation must be written in the following format:

I. PROBLEM STATEMENT - Write a concise statement of the problem. Write clearly enough so that someone picking up your paper could understand exactly what you plan to solve.

II. WORK - Explain in detail what you did to solve the problem. Tell about the information that you gathered and how you used it. Draw diagrams and/or sketches to help clarify the explanation.

III. ANSWER - State your answer to the problem. What makes you think that this is the correct answer?

Investigations and completed Trail booklet are due Tuesday, May 26.
All summaries of investigations must be written or typed clearly with correct grammar and spelling.

MAP OF THE AREA AROUND DOWNTOWN BOULDER

TIME LINE

As you walk along the BMT be on the lookout for the year that buildings were erected. Each building contains this date in a special location. Usually it can be found on the cornerstone, but sometimes you will have to look elsewhere. As you find this information, enter the name and/or location of the building on the timeline below. Find a good place for this entry. What/Where are the oldest and newest buildings that you can find?

Look for the following geometric shapes and mark down the address or location (street or corner) where you find them. Draw a small sketch of each shape as you find it.
Triangles
Isosceles __________________________________________________________________
Right _____________________________________________________________________
Equilateral ________________________________________________________________
Scalene ____________________________________________________________________
Quadrilaterals______________________________________________________________
Rectangle __________________________________________________________________
Square _____________________________________________________________________
Parallelogram ______________________________________________________________
Rhombus ____________________________________________________________________
Trapezoid __________________________________________________________________
Kite _______________________________________________________________________
Other polygons
Pentagon ___________________________________________________________________
Hexagon ____________________________________________________________________
Octagon ____________________________________________________________________
others _____________________________________________________________________

Solids
Sphere _____________________________________________________________________
Pyramid (type) _____________________________________________________________
Cone _______________________________________________________________________
Rectangular Prism __________________________________________________________
Cylinder ___________________________________________________________________
others _____________________________________________________________________

THE ROUTE OF THE BMT NE corner of Spruce and 14th Streets

1. First Methodist Church
• In what year was this church erected? ________
      Hint: The information can be found on the west side of the church in two locations.

• There is an addition to the church on the east side. How many years after the church was erected, was the addition completed? _______

• You should be able to find many solid and plane geometric shapes. Some are decorative and others have a practical purpose. Think about this as you locate the shapes.

Walk north on 14th Street to Pine Street.

2. St. John's Episcopal Church
• Is this church on the NE, NW, SE, or SW corner of Pine and 14th Streets?________

• Locate the cornerstone of the church. Notice that it contains two different dates. See if you can find out why this is so.

• Each year is preceded by the letters "AD." What does this mean?_______________________

• There is a fire hydrant on the corner. The bolts on the hydrant are two different shapes.
  What are these shapes? ______________________________________________________
  What is the reason for this difference? Is it decorative or practical?

Continue north on 14th Street to Mapleton Avenue.

3. Street signs
• Since there is a school near here, there are many traffic signs. Locate these signs and determine the shape of each and its purpose.

  Purpose ________________________________________ Shape _____________

  Purpose ________________________________________ Shape _____________

  Purpose ________________________________________ Shape _____________

• As you continue on the BMT look for additional traffic signs. You will notice that there is a connection between the shape of each sign and its purpose. Keep a record of this.

Before crossing 14th Street, look at the steeple on top of the Catholic church. There are some angels circling the steeple.
     How many degrees are in a circle? _______
     How many angels are there? ____________
Determine how many degrees apart the angels are from one another.

• What shapes are in the amber glass in the steeple? --__________________

• When you cross 14th Street, find the date that this church was erected. A different number system is used to write this date. Record the date and the system being used in the timeline.

5. Boulder First Church
• This church is on the __--___ corner of Mapleton Ave. and 13th St.

• Identify the type of triangle that is above the main entrance to the church.

** There is a STOP sign on the corner. Can you find the area of this sign?
     It may help to think of the number of congruent triangles that make up this shape.
     What is the formula for the area of a triangle?
     What measurements do you need to take in order to find the area?
NOTE: You may want to take the measurements now and determine the area at a later time.
Walk south on 13th St. to Pine St. Turn right on Pine in front of the First Baptist Church. Continue west on Pine to Broadway.

6. Trinity Lutheran Church
• Look at the leaded glass windows. What shapes are the pieces of glass? ________________ The shapes form a tessellation.
       Look for other tessellations as you continue along the BMT.

• Find out how many years elapsed between the organization of the church and its building.
       ______________________________________

Cross Broadway.

7. Masonic Lodge
• There is an interesting marble stone on the NW corner of Pine and Broadway.
       Read the inscription on the stone.
       Make an hypothesis about what was deposited here and why.
       ____________________________________________________________________
       In how many years will it be opened? _______________________
• Notice the triangle laid out around the stone.
       Record the type of triangle this is?

       What is its perimeter?_________________________________

Continue south across Pine St.

8. First Congregational Church
• How many years elapsed between the organization of the church, its first building and its completion?______________________________

• Find the leaded glass windows on the east side of the First Congregational Church. What shapes are tessellated in this window?
       _________________________________________________________

• A tessellation can be regular or irregular. A regular tessellation is composed of congruent shapes. Is this tessellation regular or irregular? ______________________

Continue south on Broadway.

9. Notice the speed limit sign.
• What shape is this sign? _________________________

• What is the speed limit in downtown Boulder?_________
       Is it the same on all streets? ________ Why is this so?_________________________

Cross Spruce St. and turn right. Walk west on Spruce St.

10. Parking garage
• There are two different green and white signs that indicate PARKING. Identify the shape of each sign. _______________________________________________________

• The solid shape has two congruent faces. What is their shape? _______________________

Continue walking west on Spruce St. Turn left at 11th St. and walk south to Pearl St.

11. DAILY CAMERA building

**There is a large Daily Camera sign near the top of the building. About how high is the letter "D" in meters?

Walk west on Pearl Street.

12. Look at the address numbers on the north and south sides of the street.
• Which numbers are even and which are odd? _____________________________________

• Is this true of all streets? ________________

13. Armory Building (west of the Daily Camera building)
• In what year was this building erected? ___________

• What was this building used for when it was built?
     _________________________________

• What is this building used for now? __________________________________________

14. Look across Pearl Street. You will see an office building with a large clock.
• What is the number for the address of this building? __________
       Notice that this number is the same forwards and backwards.
       This is called a palindromic number.
       See if you can find other palindromic numbers as you complete the BMT.

** Look at the large clock. Imagine that someone as small as Tinkerbell were sitting on the tip of the minute hand.
     How far would this person travel in one hour?
     Use what you have found out about circumference and about using bricks to estimate measurements.

How many digits are used on the face of a clock? _______
     **Part A - If you were building a clock and numerals cost $4.95 each, plus tax, how much would you need to pay for the digits on the clock?
You will need to know the rate of sales tax in Boulder.

     Part B - The hands of a clock make various angles as they move during the day.
     Name two times when the hands form a right angle.
     Name two times when the hands form an acute angle.
     Name two times when the hands form an obtuse angle.

Continue west on Pearl Street, across 9th Street and walk south on 9th St. on the west side.

15. Office building on 9th Street between Walnut St. and Canyon Blvd.
• What is the number in the address of this building? __________

       What kind of number is this?___________________

• Look at the parking meters in front of the building.

       If there were cars parked at a meter for the entire day, what is the most money that the city of Boulder could legally collect from a meter? ________________________

Continue south on 9th Street, crossing Canyon Blvd. and stopping on the island.

16. YIELD sign

** YIELD signs are red and white. Approximate the percent of the area of the sign that is red.

Continue walking south.

** Determine the length of this bridge in meters. (NOTE: Do this with investigation #21)

Walk south to Arapahoe and 9th. There is an office building on the northwest corner.

18. Highland Office Building
• In what year was this building erected? _______________

• What was the building used for when it was first built? _____________________________

Walk east on Arapahoe, crossing 9th Street.

19. Bus Shelter

** Imagine that is has just started to rain. You notice a bus shelter ahead.
     How many people could fit under the bus shelter in order to remain dry?
     NOTE: All people must be standing.
     You will need to determine how much area on the ground a person needs in order to be able to stand comfortably. You will need to measure the dimensions of the shelter.
     Use this information and the area of the floor of the bus shelter to solve the problem.

• What two shapes are the bricks around the bus shelter?
        ______________________________
• What type of tessellation do they create? ____________________________________

Continue walking east.

20. Boulder Public Library

• Notice that the library window on Arapahoe is composed of a tessellation of rectangles.
       About how many panes make up this tessellation? ____________________

• Sketch the shape of one of the handrails on the steps on Arapahoe. Identify the acute and obtuse angles in your sketch. What type of angle does the handrail make with the top of the step?________________

• There are many benches outside the library in front and to the north.
       How many benches are there ? _______________
       If approximately four persons could comfortably sit on a bench at a time, how many people in total could be seated on benches outside the library at one time? _______
  

** There are two different types of trash cans outside the library. One is black metal and the other has wooden slats.
     What is the volume of each trash can?
     Which trash can has the greater volume?
     How much greater is this than the other trash can?

Walk north toward Boulder Creek.

21. Library bridge across Boulder Creek

** What is the length of this bridge in meters?
     You will notice that it is not the same length at the 9th Street bridge that spans the Creek.
     Which bridge is longer?
     How much longer?
     Why do you think this is so?

• The side railings of the bridge are composed of many little squares. About how many squares are there in all? __________

Continue north to Canyon Blvd.

22. Parking lot on Canyon Blvd.

• About how many parking spaces are in this lot? ______________

** If all the spaces were occupied all day on a weekday, how much money could the city earn from the meters in this lot?

Walk east on Canyon Blvd. to 11th Street

23. Pedestrian Crossing sign
• What shape is this sign? ______________

** Find the area of this sign.      What measurements would you need?

Cross Canyon Blvd. in the crosswalk and then go back across 11th Street.

24. Oasis Restaurant

** There is a large window on the east side of the restaurant.
     The window contains many, many rectangles. How many rectangles are there in all?
     Be sure to count the rectangles of various sizes.

** Behind the window are two beer vats. One vat is about 120 cm high and has a radius of approximately 51cm.
     The vat is cylindrical in shape.
     You can use the formula for the volume of a cylinder to find out how many liters of beer the vat holds.
     When you find the volume of the cylinder, your answer will be in cubic cm.
     There are 1000 cubic cm in one liter.
     Often been is sold in pints. How many pints does the vat contain? Be sure to check with the conversion charts in your text.

Walk north on 11th Street to Walnut Street. Turn right and continue to Broadway. Turn left and walk north on Broadway.

25. Walnut Brewery

** There is a replica of a bottle cap hanging on the wall outside the restaurant.
     What is the scale factor of the cap compared with the real bottle cap?
     What would the dimensions of a bottle need to be if it were capped by the enlarged bottle cap?
     Be sure to take measurements of the large bottle cap in order to complete this investigation.

26. Look at the address numbers on the east and west sides of the street.
• What is the pattern of these address numbers?____________________________________

27. Pearl Street and Broadway

• There are four interesting buildings at this intersection. Try to find out the dates that the buildings were erected and some historical information about each. You will need to look carefully to find the dates on each building. Be creative in your search!

** There are dual trash cans suspended on posts. What is the volume of each trash can?
     How does the volume of the two trash cans compare with the volume of one of the cans at the library?

28. Ice cream store

** An ice cream cone has a diameter of 6 cm and a height of 11.5 cm. Find the volume of the cone.
     A spherical scoop of ice cream has a diameter of 6 cm.
     If it is placed in the cone and left there to melt into the cone, will the melted ice cream overflow the cone?
     You need to find the volume of the sphere in order to solve this problem.

Cross Broadway.

29. Lady on the swing statue

** Imagine that the artist who created this statue wishes to ship it to another location. The statue, without the base, will be packed in a crate that is the shape of a rectangular prism.
     What will be the dimensions, in meters, of such a crate if you want to be as economical as possible?
     What would be the volume of the crate?

30. Kiosk with notices

** There are many notices of varying sizes on the kiosk. IF all notices had to measure 28 cm by 22 cm and none could overlap, what is the maximum number of notices that could fit on the kiosk at one time? (HINT: The answer is NOT the lateral surface area!)

Walk north on Broadway to Pine Street.

31. Boulderado Hotel

** Can you estimate the height of the Boulderado Hotel in the front of the hotel from street level to the peak?
     It will help if you used the width and height of the bricks and mortar to assist you with this problem.

• The original hotel is on 13th Street.

       Notice the similarities and differences in design between the old and new portions of the hotel.

Continue south on 13th Street to Spruce . Turn on Spruce and walk east to 14th Street. Continue to NE corner of Spruce and 14th Streets. You may contintue with activity #1 or you have come to the end of the Trail.

Click here for Student Problems and Solutions

Back to Colorado menu