options: 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.

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.

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

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

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.

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.

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.

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.

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

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
problem_activity: In Class Launch
Use after Unit 4, Lesson 16
Ask students to give examples of situations where there is exponential change (population growth, compounding interest, radioactive decay). Write down students’ ideas for all to see.
Then ask students for an example of an exponential function (
\(y=2e^{\text-t}\)
). Write it for all to see. Here are some possible questions for discussion:
“What situation could this function model?” (Since the exponent is negative, this models decay. The coefficient of the exponential part is 2, so the initial value is 2. The amount is divided by
\(e\)
every time
\(t\)
increases by 1. So it’s a small amount of something and it’s decreasing. How fast it decreases depends on what
\(t\)
represents.)
“What questions about a situation could this function help you answer?” (I could find out what
\(t\)
is when the value of the function is a specific number, like 0.01. If
\(t\)
represents time, I could find out how often the value of the function doubles.)
Tell students that in this task they will ask and answer their own questions about a situation that can be modeled with an exponential function.
Teacher Instructions
After students have finished the task, they should see what questions were asked about each situation. One way to do this would be to have groups that looked at the same situation get together to compare their questions and the assumptions they made about the situation. Groups that made up their own situations can get together and share what they did. Then one or two representatives from each large group can summarize the discussion for the whole class.
Student-Facing Statement
Choose one of these situations to think about, or write a similar one of your own:
There are currently 61 million cars in a certain country, increasing by 6% annually.
A vampire enters a town. He looks around for about an hour, and then chooses someone to bite. The person he bit is now also a vampire. Both of them take an hour to choose new victims, and then they each bite someone. All the vampires continue choosing victims in the same way.
The population in millions,
\(W\)
, of the country of West Mathlandia is modeled by
\(W(x) = 17.9e^{0.0015x}\)
, where
\(x\)
is the number of years since September 1998. The population in millions,
\(E\)
, of East Mathlandia is modeled by
\(E(x) = 13.5e^{0.0128x}\)
.
There are 10,000 bacteria in a petri dish. The population of bacteria declines to 4,000 in 4 hours.
An artifact is discovered. It has 57% of the carbon-14 it originally contained. Carbon-14 decays at the rate of 0.0125% per year.
The number of books,
\(B\)
, in a small library increases according to the function
\(B = 5,\!300e^{0.02t}\)
, where
\(t\)
is measured in years.
The growth in the population,
\(P\)
, of a certain species of rodent at a landfill is modeled by the exponential function
\(P(t)= 648e^{0.016t}\)
, where
\(t\)
is the number of years since 1987.
The initial mass of a certain compound is 448 g, and every 27 hours it decreases by half.
Write down questions you could ask about the situation you chose. Write as many questions as you can think of. As a group, you will choose some interesting questions to answer. If you would need to do calculations, make a sketch, or draw a graph in order to answer a question, then it’s probably an interesting question.
You should answer 2 questions for every person in your group. Multiple people can work on a question together—each person in the group does not have to answer 2 questions by themselves.
Write a report to explain your questions and answers. The report should include:
A context for your situation. Write a story that includes all the relevant information. If the description of the situation is missing some details, fill them in however you want.
An equation to describe the situation, with an explanation of what each part of the equation represents.
A labeled graph of the situation.
The questions you chose and your answers. Include your process for finding the answers. Say what the answers mean in the context of this situation.
Teacher Instructions
After students have finished the task, they should see what questions were asked about each situation. One way to do this would be to have groups that looked at the same situation get together to compare their questions and the assumptions they made about the situation. Groups that made up their own situations can get together and share what they did. Then one or two representatives from each large group can summarize the discussion for the whole class.
Student-Facing Statement
Choose one of these situations to think about, or write a similar one of your own:
A vampire enters a town. He looks around for about an hour, and then chooses someone to bite. The person he bit is now also a vampire. Both of them take an hour to choose new victims, and then they each bite someone. All the vampires continue choosing victims in the same way.
The population in millions,
\(W\)
, of the country of West Mathlandia is modeled by
\(W(x) = 17.9e^{0.0015x}\)
, where
\(x\)
is the number of years since September 1998. The population in millions,
\(E\)
, of East Mathlandia is modeled by
\(E(x) = 13.5e^{0.0128x}\)
.
There are 10,000 bacteria in a petri dish. The population of bacteria declines to 4,000 in 4 hours.
The number of books,
\(B\)
, in a small library increases according to the function
\(B = 5,\!300e^{0.02t}\)
, where
\(t\)
is measured in years.
The growth in the population,
\(P\)
, of a certain species of rodent at a landfill is modeled by the exponential function
\(P(t)= 648e^{0.016t}\)
, where
\(t\)
is the number of years since 1987.
Write down questions you could ask about the situation you chose. Write as many questions as you can think of. As a group, you will choose some interesting questions to answer. If you would need to do calculations, make a sketch, or draw a graph in order to answer a question, then it's probably an interesting question. To get you started, here are some example questions you could ask about each situation:
If there are 200 people in the town, how long will it be before all of them are vampires?
Will the populations of East and West Mathlandia ever be the same, and if so, when?
How much does the population of bacteria decline each hour?
How long will it be before the library has 1 million books?
How many rodents were born between 1987 and now?
You should answer 2 questions for every person in your group. Multiple people can work on a question together—each person in the group does not have to answer 2 questions by themselves.
Write a report to explain your questions and answers. The report should include:
A context for your situation. Write a story that includes all the relevant information. If the description of the situation is missing some details, fill them in however you want.
An equation to describe the situation, with an explanation of what each part of the equation represents.
A labeled graph of the situation.
The questions you chose and your answers. Include your process for finding the answers. Say what the answers mean in the context of this situation.


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