options: Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
problem_activity: Optional activity
This activity is optional. Use this activity to allow students to practice analyzing graphs of exponential functions involving
\(e\)
and setting an appropriate graphing window on their graphing technology in order to produce useful graphs.
Students have graphed exponential functions and analyzed the graphs to solve problems prior to this point. Though the functions here use base
\(e\)
, the analytical work is fundamentally the same as that when the base is another number. Students will also have opportunities to graph exponential functions with base
\(e\)
later in the unit.
Launch
Provide access to devices that can run Desmos or other graphing technology.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses to the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How did you get . . . ?”, “Why did you . . . ?”, and “How do you know . . . ?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for the population of bacteria 10 hours after the scientist put the initial bacteria in the dish.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
Use graphing technology to graph the function defined by
\(f(t) = 50 \boldcdot e^{(0.25t)}\)
. Adjust the graphing window as needed to answer these questions:
The function
\(f\)
models the population of bacteria in
\(t\)
hours after it was initially measured. About how many bacteria were in the dish 10 hours after the scientist put the initial 50 bacteria in the dish?
About how many hours did it take for the number of bacteria in the dish to double? Explain or show your reasoning.
Use graphing technology to graph the function defined by
\(p(t)=192 \boldcdot e^{(0.014t)}\)
. Adjust the graphing window as needed to answer these questions:
The equation models the population, in millions, in the U.S.
\(t\)
years after 1964. What does the model predict for the population of the U.S. in 1974?
In which year does the model predict the population will reach 300 million?
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Student Facing
Are you ready for more?
Research what the population of the U.S. was in the year the model predicted 300 million people. How far off was the model? What factors do you think account for the actual population in that year being different from the prediction of the model? In what year did the U.S. actually reach 300 million people?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Focus the discussion on how students set the graphing window and how they used the graphs to answer the questions.
Depending on the graphing technology used, students may use the following strategies:
Visually estimate the value of the output given a certain input, or the input given an output.
Use a tracing tool to trace the graph and identify the coordinates when the input or output has a certain value.
To find an unknown input: graph a horizontal line at a given
\(y\)
-value (for example,
\(y=100\)
, a doubled bacteria population), find the intersection of that line and the graph of the exponential function, and identify the
\(x\)
-value at that intersection.
To find an unknown output: use a table or type something like
\(b(10)\)
after
\(b\)
was defined.
Some students may also notice that they could calculate the output value given an exponential equation and an input value, but had to rely on the graph to find an unknown input value because they were unsure how to solve for
\(x\)
when the equation has
\(e\)
as a base. Tell students that in a future lesson they will learn how to use logarithms to solve this type of equation algebraically similar to how they have used logarithms previously.


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