Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In middle grades, students solved problems involving percentages, including percent increases
and decreases. In Algebra I, students identify and describe exponential functions in terms of
growth or decay rates. In later courses, students will further develop their understanding of
exponential functions and how they are characterized by having a constant percent of change per
unit interval.
- Provide opportunities to reference MA.912.AR.1.1 as students identify and interpret parts
of an exponential equation or expression as growth or decay.
- Instruction includes the connection to growth or decay of a function as a key feature
(constant percent rate of change) of an exponential function and being useful in
understanding the relationships between two quantities.
- Instruction includes the use of graphing technology to explore exponential functions.
- For example, students can explore the function () = and how the -value
and -value are affected. Ask questions like “What impact does changing the
value of have on the graph? What about ? What values for cause the function
to increase? Which values cause it to decrease?” As students explore, formalize
the terms exponential growth and decay when appropriate.
- As students explore the graph, have students choose values of and to
complete a table of values. Once completed ask students what causes the value of to increase or decrease as the value of increases. Guide students to see that
it’s because > 1 or < 1. Have students adjust the graph and repeat this
exercise.
- Once students have an understanding of what causes exponential growth and
decay, both graphically and algebraically, ask students if they think the curve ever
passes = 0. Have students extend their function table for exponential decay to
include more extreme values for to explore if it ever does. As student arrive at
an understanding that it does not cross = 0, guide them to understand why it
doesn’t algebraically. Once students arrive at this understanding, define this
boundary as an asymptote.
- As students explore the provided graph, they will move the slider for to have
negative values. The resulting graphs provide an interesting discussion point.
Have students complete a function table using negative values. Students should
quickly see the connection between the two “curves” and why neither is
continuous. Let students know that for this reason, most contexts for exponential
functions restrict to be greater than 0 and not equal to 1.
- As students solidify their understanding of () = , use graphing technology again to
have them explore the form () = (1 ± ) . Guide students to use the sliders for and to visualize that only determines whether the function represents exponential
growth or exponential decay.
- Have students discuss which values for cause exponential growth or decay.
They should observe that negative values cause exponential decay while positive
values cause exponential growth.
Common Misconceptions or Errors
- Students may not understand exponential function values will eventually get larger than
those of any other polynomial functions because they do not fully understand the impact
of exponents on a value.
- Students may not understand that growth factors have one constraint ( > 1) while decay
factors have a compound constraint (0 < < 1). Some students may think that as long as < 1, the function will represent exponential decay.
- Students may think that if is negative and > 0 or > 1, the function represents an
exponential decay. To address this misconception, help students understand that the
negative values are growing at an exponential rate.
Strategies to Support Tiered Instruction
- Teacher provides instruction to identify exponential functions in all methods (i.e., graphs,
equations and tables).
- For example, instruction may include providing a comparison of the two forms of
exponential functions. Having a side-by-side comparison of both as an equation,
graph and a table of values will provide a visual aid.
- Teacher provides student with examples and non-examples of exponential functions in
tables.
- For example, teacher can provide the following tables for students to determine
which ones represent an exponential function.
Instructional Tasks
Instructional Task 1 (MTR.4.1, MTR.7.1) - After a person takes medicine, the amount of drug left in the person’s body changes over
time. When testing a new drug, a pharmaceutical company develops a mathematical model to
quantify this relationship. To find such a model, suppose a dose of 1000 mg of a certain drug
is absorbed by a person’s bloodstream. Blood samples are taken every five hours, and the
amount of drug remaining in the body is calculated. The data collected from a particular
sample is recorded below.
- Part A. Does this data represent an exponential growth or decay function? Justify your
answer.
- Part B. Create an exponential function that describes the data in the table above.
Instructional Items
Instructional Item 1 - Given the function () = 125(1 − 0.26)x , does it represent an exponential growth or
decay function?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.