QUANTIFYING EXPONENTIAL GROWTH:
THE CASE OF THE JACTUS
Abstract
The Landscape of Exponential Functions in Common School Treatments
-
-
-
93
QUANTIFYING EXPONENTIAL GROWTH
The Repeated Multiplication Approach
-
-
-
-
-
Alternate Approaches to Exponential Growth
-
94
QUANTIFYING EXPONENTIAL GROWTH
3
a as
a
-
y-
The Rate of Change Approach to Exponential Functions:
Covariation and Continuous Variation
y=
y
y
y
to y
95
QUANTIFYING EXPONENTIAL GROWTH
y
-
Covariation for Exponential Growth
y
y
@
@.
-
Continuous Variation
-
96
QUANTIFYING EXPONENTIAL GROWTH
-
-
-
The Jactus: Building Exponential Growth by
Reasoning with Continuously Covarying Quantities
-
Figure 1:
QUANTIFYING EXPONENTIAL GROWTH
-
Encouraging Coordination
-
-
-
QUANTIFYING EXPONENTIAL GROWTH
-
y
) to
y
Figure 2:
Encouraging Smooth Reasoning
) to
a
99
-
QUANTIFYING EXPONENTIAL GROWTH
th
The Teaching Experiment
-
Data Analysis
-
QUANTIFYING EXPONENTIAL GROWTH
From Repeated Multiplication to Constant Multiplicative Rates of Change:
Three Conceptual Shifts
-
y
y
) to
y
)
to
.
Figure 3:
-
Shift 1: From Repeated Multiplication to Coordinating x and y
-
QUANTIFYING EXPONENTIAL GROWTH
Figure 4:
-
Figure 5:
QUANTIFYING EXPONENTIAL GROWTH
Figure 6:
3
Shift 2: From Coordinating x and y to Coordinated Constant Ratios
QUANTIFYING EXPONENTIAL GROWTH
Figure 7:
Figure 8: J
-
QUANTIFYING EXPONENTIAL GROWTH
Figure 9:
-
3
-
Figure 10:
QUANTIFYING EXPONENTIAL GROWTH
Shift 3: Generalizing to Non-Natural Exponents
y
a
a
Estimation and Reversion to Linear Thinking
-
Reliance on Equations to Assist with Interpolation
QUANTIFYING EXPONENTIAL GROWTH
Figure 11:
-
x with b
-
Figure 12:
QUANTIFYING EXPONENTIAL GROWTH
y
.
th
-
-
th
QUANTIFYING EXPONENTIAL GROWTH
Figure 13:
Generalizing to Fractional Exponents
) to
Figure 14:
5
th
QUANTIFYING EXPONENTIAL GROWTH
Discussion
.
5
y
-
-
-
5
%
-
-
QUANTIFYING EXPONENTIAL GROWTH
a
as a
a
-
References
-
.
.
-
-
QUANTIFYING EXPONENTIAL GROWTH
-
-
-