CoUrSe ModUle SUMMary and UnpaCkIng of StandardS | 85Students discover Euler’s number, e, by applying repeated reasoning (MP.8) and numerically
approximating the base b for which the constant c is equal to 1. Euler’s number is used
extensively in the future and occurs in many different applications.
Focus Standards: N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending
the properties of integer exponents to those values, allowing for a notation for radicals
in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5
because we want (51/3)^3 = 5(1/3)3 to hold, so (51/3)^3 must equal 5.
N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.★
F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically
or as a table) over a specified interval. Estimate the rate of change from a graph.★
F-BF.A.1a Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
F-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).★
Instructional Days: 6Student Outcomes
Lesson 1: Integer Exponents
● (^) Students review and practice applying the properties of exponents for integer
exponents.
● (^) Students model a real-world scenario involving exponential growth and decay.
Lesson 2: Base 10 and Scientific Notation
● (^) Students review place value and scientific notation.
● (^) Students use scientific notation to compute with large numbers.
Lesson 3: Rational Exponents—What are 2
21
and 2
(^13)
?
● (^) Students calculate quantities that involve positive and negative rational exponents.
Lesson 4: Properties of Exponents and Radicals
● (^) Students rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Lesson 5: Irrational Exponents—What are 22 and 2π?
● (^) Students approximate the value of quantities that involve positive irrational exponents.
● (^) Students extend the domain of the function fx()=bx for positive real numbers b to all
real numbers.
Lesson 6: Euler’s Number, e
● (^) Students write an exponential function that represents the amount of water in a tank
after t seconds if the height of the water doubles every 10 seconds.
● (^) Students discover Euler’s number e by numerically approaching the constant for which
the height of water in a tank equals the rate of change of the height of the water in the
tank.
● (^) Students calculate the average rate of change of a function.