Tiling a Patio
Mr. Gomez designs square patios for backyards. His designs are unique. In the center of each patio is a square space for a garden. Below are his three smallest patios. The dark tiles represent the garden and the white are tiles.
Mr. Gomez has hired you to as his assistant. He wants you to figure out the number of white tiles needed for the various sizes of patios.
Mr. Gomez’s 3 Smallest Patios
Patio 1 / Patio 2 / Patio 3a. Draw patio 4 and 5. How many white ties are in patio 4? Patio 5”
b. Make some observations about the patios that could help you describe e larger patios.
c. Describe a method for finding the total number of white tiles needed for patio 50 (without constructing it.)
d. Describe a method for finding the total number of white tiles needed for any patio. Explain how your rule relates to the visual representation of the patio.
e. Write a different rule that could be used to determine the number of white tiles needed for any patio. Explain how your rule relates to the visual representation of the patio.
f. Make a table of (patio #,#tiles).
g. On a coordinate plane, plot the points from your table of values. Be sure to label the axes appropriately.
h. After comparing the symbolic rule(e), the table of values(f), and the graph(g), what conclusions can you make?
Tiling a Patio and Garden Space
Mr. Gomez designs square patios for backyards. His designs are unique. In the center of each patio is a square space for a garden. Below are his three smallest patios. The dark tiles represent the garden and the white are tiles.
Mr. Gomez has hired you to as his assistant. He wants you to figure out the number of black tiles needed to represent the garden size.
Mr. Gomez’s 3 Smallest Patios
Patio 1 / Patio 2 / Patio 3i. Draw patio 4 and 5. How many dark tiles are in patio 4? Patio 5?
j. Make some observations about the patios that could help you describe the larger gardens.
k. Describe a method for finding the total number of dark tiles needed for patio 50 (without constructing it.)
l. Describe a method for finding the total number of dark tiles needed for any patio. Explain how your rule relates to the visual representation of the patio.
m. Write a different rule that could be used to determine the number of dark tiles needed for any patio. Explain how your rule relates to the visual representation of the patio.
n. Make a table of (patio#,#tiles(garden size)).
o. On a coordinate plane, plot the points from your table of values. Be sure to label the axes appropriately.
p. After comparing the symbolic rule(e), the table of values(f), and the graph(g), what conclusions can you make?
Tiling a Patio, Garden Space, Salary
Mr. Gomez designs square patios for backyards. His designs are unique. In the center of each patio is a square space for a garden. Below are his three smallest patios. The dark tiles represent the garden and the white are tiles.
Mr. Gomez has hired you as his assistant. He will pay you for the work that you do. If you build Patio 1 he will pay you $200. If you build Patio 2 he will pay you $400. If you build Patio 3 he will pay you $800. The salary amounts continue to increase in this manner.
Mr. Gomez’s 3 Smallest Patios
Patio 1 / Patio 2 / Patio 3q. Draw patio 4 and 5. What is the salary for each patio?
r. Make a table of (patio#, salary amt)
s. On a coordinate plane, plot the points from your table of values. Be sure to label the axes appropriately.
t. Describe a method for finding the salary earned for Patio 50.
u. Describe a method for finding the salary earned for any patio.
v. Write a different symbolic rule that could be used to determine the salary earned for any patio.
w. After comparing the symbolic rule(g), the table of values(c), and the graph(d), what conclusions can you make?
Adapted from NCTM’s Lessons and Resources