LessonTitle:
Stacking Cubes and Rates of Change
Alg 5.3
Utah State Core Standard and Indicators Algebra Content Standard 2 Process Standards 1-5
Summary
In this lesson, students explore patterns in stacking cubes, find a shortcut to predict the number of cubes in the 10th
stage, and find equations and rates of change in the growth pattern.
Enduring Understanding
Essential Questions
The slope in a linear equation tells the story of a unit rate How do you represent the growth rates and equations for
such as miles per hour, students per teacher, pay per
the patterns found in the stacking cubes growth?
hour, cost per bagel , sit-ups per minute, words read per
minute, miles per gallon, commercials per hour of TV
watching, cost per oz., cost per lb., cost per candy kiss,
etc.
Skill Focus
Vocabulary Focus
 Tracking data
 Unit rates and slope
 Graphing linear equations
Assessment
Materials: Drawings (see below the activity sheets)
Launch
Explore
Summarize
Apply
Directions:
Students should keep track of their methods or shortcuts and be prepared to describe the secret to finding the equations and
growth rates.
1
Alg 5.3
Stacking Cubes
I. Use the stacking towers drawings (separate handout) to answer the questions below.
II. Discuss this question in your small group. Be prepared to share your ideas in a large group
discussion.
What is the secret to finding the growth rates and equations for the stacking cube patterns?
Problem 1
a)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? _____________________
b)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ______________________
c)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? _____________________
d)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? _____________________
Problem 2
a)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ___________________
b)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ____________________
2
c)
d)




Find the number of cubes in the tenth tower (find a shortcut). ________
How many cubes would be in the Zero-th building? ________
What is the rate of change? _________
How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ____________________




Find the number of cubes in the tenth tower (find a shortcut). ________
How many cubes would be in the Zero-th building? ________
What is the rate of change? _________
How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ______________________
Problem 3
a)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? _____________________
b)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ______________________
c)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? _____________________
d)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? _____________________
e)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? _____________________
3
f)




Find the number of cubes in the tenth tower (find a shortcut). ________
How many cubes would be in the Zero-th building? ________
What is the rate of change? _________
How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? _____________________
Problem 4
a)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ___________________
b)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ____________________
c)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ____________________
d)
 Find the number of cubes in the tenth tower (find a shortcut). ________
 How many cubes would be in the Zero-th building? ________
 What is the rate of change? _________
 How could the pattern be described symbolically? Or, if you know the pattern number, can you
write a formula that will give you the height of any tower? ______________________
4
Activity One: Stacking Towers
Problem 1a.
Problem 1b.
#1 #2 #3 #4
#1 #2 #3 #4
Problem 1c.
Problem 1d.
#1 #2 #3 #4
#1 #2 #3 #4
Note: Problem 1c and 1d present the zeroth tower as 'underground' or in the 'basement'.
5
ACTIVITY 2: Representing Ordered Pairs on the Cartesian Plane
Problem 2a.
Problem 2b.
24
26
20
29
23
16
20
12
#1
#2
#3
#4
#1
Problem 2c.
#2
#3
#4
Problem 2d.
47
42
37
21
23
25
27
32
#1
#2
#3
#4
#1
6
#2
#3
#4
ACTIVITY 3: Building Toward A Rate Of Change
Problem 3a.
Problem 3b.
64
56
40
20
6 12
3 12
Problem 3c.
Problem 3d.
56
56
30
20
3 12
3
7
7
Problem 3e.
Problem 3f.
66
52
41
31
8 12
3 13
8
ACTIVITY 4: Points On The Cartesian Plane
Problem 4a.
80
70
60
50
40
30
20
10
0
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
Problem 4b.
80
70
60
50
40
30
20
10
0
9
Problem 4c.
160
120
(32, 100)
80
(8, 40)
40
0
0
8
16
24
32
40
48 52
Problem 4d.
(48, 152)
160
120
(40, 128)
80
40
0
0
8
16
24
10
32
40
48 52