Montana Math MT.PLUS.ALG.EXP.2

MathGrades 9–12Exponential and Logarithmic Functions

The Standard

Understand why e is defined as the natural base.

2026 Montana Mathematics Content Standards · Algebraic and Functional Reasoning

Teacher's field guide

What Students Need to Do: Students calculate what happens when one unit of growth is split into more and more compounding periods. They see the results approach a fixed number, e, and connect that number to continuous growth.
What Mastery Looks Like: A student can use a table of (1 + 1/n)^n values to estimate e and explain why the values do not grow without bound. They can explain that the growth rate of e^x at any point equals its value there.
Common Misconceptions: Students may think e is a variable or that it equals exactly 2.718. Some expect more frequent compounding to make the balance grow without limit.
How to Assess It: Exit ticket: Calculate (1 + 1/n)^n for n = 1, 12, and 365, then name the number these values approach and what is special about the growth rate of e^x.

Lesson moves

Have students use calculators to make a table of (1 + 1/n)^n for n = 1, 2, 4, 12, and 365.
Ask students to write why compounding more often raises the balance but does not make it grow without limit.
Give students cards showing n = 1, 2, 12, and 365 and cards showing their calculated results to match.
Compare $1,000 at 12% interest compounded monthly for one year with $1,000e^0.12 using a calculator.

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