"Never make a calculation until you know the answer. Make an estimate before every calculation, try a simple physical argument (symmetry! invariance! conservation!) before every derivation, guess the answer to every paradox and puzzle. Courage: No one else needs to know what the guess is. Therefore make it quickly, by instinct. A right guess reinforces this instinct. A wrong guess brings the refreshment of surprise. In either case life as a spacetime expert, however long, is more fun!"
Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics,
W. H. Freeman (1992), p. 20.
Preparation
Due 12:00 PM before start of lab
Prelab assignment 1 (*.html), 1 point
Equipment
M&M's® "fun pack" (1 per group)
M&M's® big box (Costco™ bulk size, for display)
rulers/metersticks (12", 1 m, 2 m)
whiteboards, markers
Big Ideas
Complex calculations can be approached by first formulating a systematic stepbystep procedure. Measured and expected values can be compared by calculating percent errors.
Goals
Students work in (selfassigned?) groups to compare the coverage ratios of different tiling schemes, and apply this to make estimates and verify claims.
Students formulate their own systematic stepbystep procedure to determine the data to be collected and calculations to be made.
Students are introduced to basic crystallography concepts (primitive cells), and calculations of percent errors.
Students practice rules for handling proper significant figures and decimal place rules for arithmetic operations.
Tasks
(Record your lab partners' names on your worksheet for tasks 12, to be turned in at the end of today's lab for randomly selected grading for your group.)
1. Coverage Ratios of Penny Tiles Pennies have been used as tiles to cover floors, walls, and countertops. Given that the diameter of a penny is 1.905 cm, determine the (circular) area of a penny, in cm^{2}. (Show your work and steps, and then truncate your raw result to the proper number of significant figures using scientific notation, if necessary.)
Area of a penny = __________ cm^{2}.
Area of a penny = __________ cm^{2} (proper sig figs).
 A twodimensional primitive cell is the smallest section of a tile pattern that can be repeated to cover any arbitrary area. For two possible methods for tiling pennies such that they all touch each other, calculate the total areas contained within the boundaries of the square primitive cell and the rhomboid primitive cell, in cm^{2}. (Show your work and steps, and then truncate your raw result to the proper number of significant figures using scientific notation, if necessary.)
Area of square primitive cell = __________ cm^{2}.
Area of square primitive cell = __________ cm^{2} (proper sig figs).
Area of rhomboid primitive cell = __________ cm^{2}.
Area of rhomboid primitive cell = __________ cm^{2} (proper sig figs).
 Note that the square primitive cell contains exactly onequarter the area of four separate pennies, or equivalently encloses exactly one penny. Discuss mathematically (using geometry and/or logical reasoning) why the rhomboid primitive cell also encloses exactly one penny. Show your work and explain your reasoning,
Brief discussion/explanation:
 Calculate the nearest whole number of pennies required to tile over one square meter of area. (Hint: use unit analysis and cancellation; the number of pennies required is also the number of primitive cells to fill this area. Also don't worry about the "edges" of this square meter, as this is a general result that for simplicity will not depend on the shape of the boundaries, just the area covered. Show your work and steps.)
Number of squaretiled pennies = __________.
Number of rhomboidtiled pennies = __________.
 Write a concluding statement discussing which tiling method (square or rhomboid) would require more pennies to cover a given area, and specifically why. Include the specific relevant numbers in this statement, such that it can be read (and cited) on its own without referring to the above calculations and numbers.
Brief concluding statement:
2. The Penny Bar The Penny Bar in the McKittrick Hotel is a room tiled with pennies.
(i) "over 800,000 pennies" (Maxim, issue 54 (June 2002)).
(ii) "over 1 million pennies" (Penny Bar & Cafe menu).
For simplicity, assume that the Penny Bar is a simple bare room with a floor space of 20 ft × 50 ft with walls 10 ft tall; ignore furniture, doors and cabinets; and that only the floor and walls are covered with pennies (using rhomboid tiling). Calculate how many rhomboid primitive cells are required to cover the floors and walls (1 ft = 30.48 cm). (Show your work and steps, and then truncate your raw result to the proper number of significant figures using scientific notation, if necessary.)
Number of rhomboid primitive cells = __________ (raw result).
Number of rhomboid primitive cells = __________ (proper sig figs).
 Determine the number of pennies to cover this simplified model of the Penny Bar. (Show your work and steps, and then truncate your raw result to the proper number of significant figures using scientific notation, if necessary.)
Number of pennies = __________ (raw result).
Number of pennies = __________ (proper sig figs).
 The percent error (or "percent approximation error") is used to compare the discrepancy between an experimental value (here, your calculation) with a theoretical, known, established, or accepted value (here, the values cited by various media sources):
% error = 100 ×  experimental − theoretical  / theoretical.
(Note that the percent error is a positive definite value, because of the absolute value operation.) Determine the percent error of your estimate for the number of pennies compared to each of the published values (i)(ii) above. (Both results are expected to be somewhat low, as we are ignoring furniture and other items that are also covered with pennies. Express your answer with the proper number of significant figures using scientific notation, if necessary.)
(i) % error = __________.
(ii) % error = __________.
 Write a concluding statement regarding which of the published values (i)(ii) above is closer to your estimate. Include the specific relevant numbers in this statement, such that it can be read (and cited) on its own without referring to the above calculations and numbers.
Brief concluding statement:
3. The M&M's®Covered Classroom
(Done on whiteboards only, to be worked on and presented as a group.) Consider if M&M's® were used to completely tile the floor and walls of this classroom. Ignore the presence of furniture, doors and cabinets on the floors and walls, and that only the bare floor and walls are covered with M&M's® (using rhomboid tiling). You will ultimately decide how many whole bulk boxes of M&M's® need to purchased in order to do this.
For your procedure, write out on a whiteboard a detailed, stepbystep description of evidence that needs to be collected, and calculations to be donenot just "divide the area of the room by the primitive cell area," but exactly what would someone need to do, stepbystep, to accomplish this (e.g., "measure the length and width of the room; calculate the area of the room; measure the diameter of M&M's®; unit conversion(s)? etc.).
 Record your data on another whiteboard, and briefly show your work for each calculation (clearly labeled).
 Write out a concluding statement on this whiteboard (or another one if needed), regarding the number of whole bulksized boxes of M&M's® needed to cover the floor and walls of a simple model of this classroom. Include the specific relevant numbers in this statement, such that it can be read (and cited) on its own without referring to the above calculations and numbers.
 Bring up your whiteboards to the front of the class, to be presented to the instructor, which should include:
 A descriptive abstract: a brief summary of the purpose and methods used, but not of the data nor the conclusions. (Use the sample abstract below for today's lab.)
"We determined the number of bulksize boxes of M&M's® to be purchased in order to completely tile the floor and walls of a simplified model of a physics laboratory room, using the concept of a rhomboid primitive cell area for hexagonally closepacked circles. As a result, an estimated _________ bulk boxes are needed to do so."  Stepbystep procedure.
 Data table, calculations and/or results.
 Evidencebased conclusion statement.
 Documentation Rubric (tasks 12)
(Graded from randomly selected group member)
Score  Description 
3  Explanations complete and calculations correct, or very nearly so. 
2  Essentially complete; few explanations/calculations missing or incorrect. 
1  Substandard effort; substantive amount of explanations/calculations missing or incorrect. 
0  Unacceptable or no significant effort. 
 Whiteboard Rubric (task 3)
(Graded as a group, evaluated by instructor during debrief session)
Score  Description 
3  (Only graded for completion this week.) 
FollowupFollowup
Complete this week's postlab assignment, next next week's prelab assignment, and review lab instructions.
Due 12:00 PM before start of next lab
Postlab assignment 1 (*.html), 1 point
