Unit 4, Lesson 12
The Number $e$
Integrated Math 3
12.1
Warm-up
Standards Alignment
Building On
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).
Addressing
HSF-LE.B.5
Interpret the parameters in a linear or exponential function in terms of a context.
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-up, students match equations with descriptions representing exponential functions in familiar contexts. When written in the form of , all functions have the same value for the parameter and for the exponent. The goal is to prompt students to examine and interpret the different bases, setting the stage for thinking about as a base.
Launch
NoneActivity
NoneMatch each equation to a situation it represents. Be prepared to explain how you know. Not all equations have a match.
- A scientist begins an experiment with 400 bacteria in a petri dish. The population doubles every 10 hours. The function gives the number of bacteria hours since the experiment began.
- An erosion model begins with 400 kilograms of sand. The amount of sand decreases by 25% every 10 days after the experiment begins. The function gives the amount of sand left days after the experiment begins.
- The half-life of a radioactive element is 10 years. There are 400 g of the element in a sample when it is first studied. The function gives the amount of the element remaining years later.
- In a lake, the population of a species of fish is 400. The population is expected to grow by 25% in the next decade. The function gives the number of fish in the lake years after it was 400.
Activity Synthesis
Select students to share their responses and explanations for each description. If not brought up by a student, point out that all four examples of exponential equations show the same initial values and exponents but different bases and, as such, different growth factors. The numbers 0.5, 1.25, 0.75, and 2 were used. Tell students that, in this lesson, we will encounter another number that is used as a base in many contexts.
If time allows, invite students to write a situation to match , and then select 2–3 students to share.
12.2
Activity
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this activity is to elicit the idea that , while being a new symbol, is a number and when that number is used as a base in an exponential expression, the function behaves just like those with other bases that represent exponential growth. This will be useful when students use as a base in a later activity.
The Notice and Wonder routine is used here to give students an opportunity to notice several familiar things—a description, table, graph, and equation—and to draw parallels between the familiar bases used in the Warm-up matching activity and the new base value, . While students may notice and wonder many things about these representations, a curiosity about as a number describing the growth factor is the important discussion point.
12.3
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
Activity Narrative
This activity serves as a brief and light introduction to how is visible in the behaviors of certain functions. The standards addressed in this course don’t require a deep understanding of the meaning of , so the observations from this activity should not be assessed.
has a special connection to the expression . Jumping to this expression would likely be an opaque process for most students, so they first investigate the range of behavior of its component pieces, for very small and very large positive values of .
Entering increasingly larger or smaller values of in the functions and making observations about them is a chance to look for and express regularity through repeated reasoning (MP8).
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Lesson Synthesis
To help students build a better sense for the constant , consider discussing these questions:
- “How is like ?” (They are both irrational numbers and cannot be represented as a fraction. If written as a decimal, the decimal numbers don’t repeat or terminate. They are both important in math.)
- “How is different from ?” ( is about 2.718 and is about 3.14. is the ratio of the circumference and the radius of a circle. It shows up a lot in geometry. We’re not quite sure yet how came about, but we saw it used in an exponential function.)
- “In the moldy wall problem, we saw the mold growth represented with , where is time in weeks and is the mold-covered area in square centimeters. How would you describe how fast the mold was growing to someone who doesn’t know about ?” (Each week, the area it covers is growing by a factor of about 2.7. It more than doubles, but it doesn’t quite triple.)
Student Lesson Summary
Scientists, economists, engineers, and others often use the number in their mathematical models. What is ?
is an important constant in mathematics, just like the constant , which is important in geometry. The value of is approximately 2.718. Like , the number is irrational, so it can’t be represented as a fraction, and its decimal representation never repeats or terminates. The number is named after the 18th-century mathematician Leonhard Euler and is sometimes called Euler’s number.
has many useful properties and it arises in situations involving exponential growth or decay, so often appears in exponential functions.