Name _____________________________________ Date ____________ Period ___________
Shading Grids
Directions: Below are four different patterns for shading squares in a 10 x 10 grid. For
each of the grids below, do the following:
 Study each pattern and assuming it continues, color in the correct number of
squares (or partial squares) for grids 4 and 5,
 Complete the table for each pattern
 Graph the ordered pairs on the graphs provided, using a different color for each
function. (Pay attention to which Grids to graph on each graph provided!)
Grid A
Grid B
Grid C
Grid D
1.
1.
1.
1.
2.
2.
2.
2.
3.
3.
3.
3.
Linear vs. Exponential Growth: Shading Grids
1
4.
4.
4.
4.
5.
5.
5.
5.
Tables
Grid A
Grid
#
1
2
3
4
5
6
10
100
x
# of
squares
shaded
Grid B
Grid
#
1
2
3
4
5
6
10
100
x
# of
squares
shaded
Grid C
Grid
#
1
2
3
4
5
6
10
100
x
Linear vs. Exponential Growth: Shading Grids
# of
squares
shaded
Grid D
Grid # of
#
squares
shaded
1
2
3
4
5
6
10
100
x
2
Graph # 1: Grid A and Grid B
Color to represent Grid A: ____________ Color to represent Grid B: ____________
Analysis Questions Graph #1
1. What is the same about the shading you did for Grids A and B?
In both grids, I ______________________________________________________
2. What is different about the shading you did for Grids A and B?
In grid A, I __________________________________________, but in grid B, I
_____________________________________________.
3. What is similar and what is different about the graphs of the functions represented by
the Grid A and B patterns?
Similar: _______________________________________________________
Different: _____________________________________________________
4. Let’s call the function represented by Grid A’s pattern A(x) and the function
represented by Grid B’s pattern B(x).
a. Do A(x) and B(x) have the same rate of change?
Yes or no: ___________. How do you know? _________________________________
b. How does A(x) grow in terms of the shading? How does B(x) grow in terms of
shading?
A(x) grows by _____________________, while B(x) grows by ______________.
Linear vs. Exponential Growth: Shading Grids
3
c. How does A(x) grow when looking at the graph? How does B(x) grow when looking
at the graph?
A(x) grows by _____________________, while B(x) grows by ______________.
d. How does A(x) grow in terms of the Table or rate of change (show the differences)?
How does B(x) grow in terms of the Table or rate of change (show the differences)?
A(x) grows by _____________________, while B(x) grows by ______________.
e. Predict: Which grid would get to 1,000 shaded squares the fastest? ___________
How do you know? __________________________________________________
5. A(x) represents a linear function and B(x) represents an exponential function.
a. How does a linear function grow? (What can you say about its rate of change?)
A linear function ___________________________________________________
b. How does an exponential function grow? (What can you say about its rate of change)?
An exponential function ___________________________________________________
6. Predict: Are the functions represented by grids C and D linear or exponential? Why?
I predict grid C will be represented by a ___________________ function because
_________________________________________________________________.
I predict grid D will be represented by a ___________________ function because
_________________________________________________________________.
Linear vs. Exponential Growth: Shading Grids
4
Graph #2: Grids B, C & D
Grid B Color: ____________ Grid C Color: __________ Grid D Color: ________
Graph #2 Analysis Questions
1, a. Do the patterns represented by shading grids B, C and D represent functions? Why
or why not?
_______________________________________________________________________
b. What type of function do they represent? How do you know?
_______________________________________________________________________
2. What is different about the shading, table and graph of Grid C as compared with Grids
B and D?
Shading: ________________________________________________________________
Table (rate of change): ____________________________________________________
Graph: ________________________________________________________________
3. What is different about the shading, table and graph of grid D as compared with Grid
B?
Shading:________________________________________________________________
Table (rate of change):____________________________________________________
Graph: ________________________________________________________________
4. In this activity, you have represented the following types of functions: Linear,
Exponential Growth and Exponential Decay. Which type of function do you think each
Grid pattern represented and why?
Grid A: _________________________
Grid B: _________________________
Grid C: _________________________
Grid D: _________________________
Linear vs. Exponential Growth: Shading Grids
5
Teacher Directions
Materials:
Rulers (1 per student)
Colored Pencils (6 colors per student)
Objective:
By extending and analyzing 4 shading patterns, students will come to understand the
similarities and differences between the rate of change in linear and exponential functions
as seen in the pattern, the table and the graph.
Directions:
Pass out the activity sheet and colored pencils to each student. Direct the class’ attention
to Grid A. Ask them to study how the shading is changing between each grid and then
shade in what grids 4 and 5 would be if the pattern continued. Select students to explain
how they see the pattern growing and how they knew how to shade the next grid.
Encourage all different methods of seeing the growth. Next, ask the students to record
the information from Grid A on the table on page 2, listing the coordinates. Give them a
few minutes to try to complete the table by looking for a pattern or drawing if needed.
For students stuck on 100 or x, ask some questions, such as “Describe how to build figure
10” or “What is changing each time? How many times will you have added ____ by
figure 10?” Note: it is okay if students cannot write the general rule for this lesson for
Grids B-D, but they should be able to for Grid A.
Once students understand this, have them follow the same process for Grids B, C and D.
Circulate to ensure students see each shading pattern. Grid B is doubling the number of
squares shaded each time; Grid C is shading half of the number of squares previously
shaded; Grid D is tripling the number of squares previously shaded. Once all tables are
complete, pass out colored pencils and have students draw a graph to represent each
function. When all students have completed the tables, stop to have students share their
tables and explain how they saw the pattern growing and how they came up with the rule
for x (if they were able to). Again, encourage different methods of seeing the growth or
coming up with an equation, as these methods will help students understand what makes
a function linear vs. exponential.
Graph #1: Make sure the students notice that they will only graph the data from Grids A
and B onto graph #1. The goal here is to see the differences and similarities between
linear and exponential functions. Give the students 10 minutes to answer the analysis
questions from Graph #1, having them think silently and record their ideas for a few
minutes before working with a partner to discuss ideas. Come together as a class and ask
students, at random, to share ideas to each of the questions. When you get to question
#4d, record the differences on your table for Grids A and B (see table below). Allow the
class to try to look for a common difference (first difference, second difference, third
difference, etc) until they see there will never be a common difference. Then ask them to
consider (or have someone share) what type of common change they do see between the
# of squares shaded in Grid B. Guide them to see that this function has a common ratio
or a consistent multiplicative change (as opposed to additive change in Grid A and all
Linear vs. Exponential Growth: Shading Grids
6
linear functions). Complete this section by giving the class 5 minutes to complete
questions 5 and 6 independently and consider collecting this as formative assessment.
Grid A
Grid B
Grid #
1
2
3
4
5
6
10
100
x
# of
squares
shaded
2
4
6
8
10
12
20
200
2x
+2
+2
+2
+2
Grid # of
#
squares
shaded
1
2
2
4
3
8
4
16
5
32
6
64
10
1024
100 2100
x
2x
First
Differences
+2
+4
+8
+16
Grid B: Common Ratio or Multiplicative
Grid # of
#
squares
Multiplier
shaded
1
2
•2
2
4
•2
3
8
•2
4
16
•2
5
32
6
64
10
1024
100 2100
x
2x
Next have the students graph the data from Grids B, C and D onto Graph #2. The goal of
this section is for students to compare rates of change of different exponential functions
(this topic will be explored further later in this unit). After students have completed the
graph, give them 10 minutes to work through the analysis questions for graph #2. Give
pairs or groups 10 minutes to discuss their answers and then use numbered heads to call
on students to share their answers to each question. It is okay if they are not crystal clear
on exponential growth and decay at this point!
Answer Key
Linear vs. Exponential Growth: Shading Grids
7
Grid A
Grid
#
1
2
3
4
5
6
10
100
x
# of
squares
shaded
2
4
6
8
10
12
20
200
2x
Grid B
Grid
#
1
2
3
4
5
6
10
100
x
# of
squares
shaded
2
4
8
16
32
64
1024
2100
2x
Grid C
Grid
#
1
2
3
4
5
6
10
100
x
Linear vs. Exponential Growth: Shading Grids
# of
squares
shaded
64
32
16
8
4
2
1/8
128(1/2)100
128(1/2)x
or
64(1/2)x-1
Grid D
Grid # of
#
squares
shaded
1
3
2
9
3
27
4
81
5
243
6
729
10
59,049
100 3100
x
3x
8