Lesson 4: Representing Exponential Decay

Representing Exponential Decay

5 min

Narrative

This Warm-up asks students not only to identify that a quantity is changing linearly or exponentially, but also to identify a term when the preceding value is not given.

Student Task

Use the patterns you notice to complete the tables. Show your reasoning.

Table A

$x$	0	1	2	3	4	25
$y$	2.5	10	17.5	25

Table B

$x$	0	1	2	3	4	25
$y$	2.5	10	40	160

Sample Response

Table A: 32.5 when $x$ is 4, and 190 (or $2.5 + 25 \boldcdot 7.5$ ) when $x$ is 25.

Table B: 640 when $x$ is 4, and $2.5 \boldcdot 4^{25}$ when $x$ is 25.

Activity Synthesis (Teacher Notes)

Focus the discussion on how students used the patterns in the tables to generate a $y$ -value for a non-consecutive value of $x$ . Make sure students are able to express the pattern of repeated addition in the first table, using an expression like $2.5 + 7.5x$ , and that they can express the pattern of repeated multiplication in the second table using an exponential expression like $(2.5) \boldcdot 4^x$ . Highlight the words “linear“ and “exponential“ to describe how the quantities are changing suggested by the tables.

Anticipated Misconceptions

For the last entry in each table, students may struggle to find the actual value to write in the table. Explain that an expression representing the value is all that’s needed.

Standards

Building Toward

F-BF.A·Build a function that models a relationship between two quantities
F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
HSF-BF.A·Build a function that models a relationship between two quantities.
HSF-LE.A.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

15 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Representing Exponential Decay

5 min

Narrative

This Warm-up asks students not only to identify that a quantity is changing linearly or exponentially, but also to identify a term when the preceding value is not given.

Student Task

Use the patterns you notice to complete the tables. Show your reasoning.

Table A

$x$	0	1	2	3	4	25
$y$	2.5	10	17.5	25

Table B

$x$	0	1	2	3	4	25
$y$	2.5	10	40	160

Sample Response

Table A: 32.5 when $x$ is 4, and 190 (or $2.5 + 25 \boldcdot 7.5$ ) when $x$ is 25.

Table B: 640 when $x$ is 4, and $2.5 \boldcdot 4^{25}$ when $x$ is 25.

Activity Synthesis (Teacher Notes)

Anticipated Misconceptions

For the last entry in each table, students may struggle to find the actual value to write in the table. Explain that an expression representing the value is all that’s needed.

Standards

Building Toward

F-BF.A·Build a function that models a relationship between two quantities
F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
HSF-BF.A·Build a function that models a relationship between two quantities.
HSF-LE.A.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Representing Exponential Decay

Warm-upTwo Other Tables5 minKCs

Narrative

Student Task

Sample Response

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Knowledge Components

Representing Exponential Decay

Warm-upTwo Other Tables5 minKCs

Narrative

Student Task

Sample Response

ActivityThe Algae Bloom15 minKCs

ActivityGlow Stick Luminescence15 minKCs

5 min

15 min

15 min

5 min

15 min

15 min