Montana Math MT.PLUS.ALG.EXP.3.a
The Standard
- MT.PLUS.ALG.EXP.3
- Understand that exponential and logarithmic functions can be represented using multiple forms by: Expressing exponential functions in the form f(x) = a · b^x and f(x) = Pe^(r·t), and Expressing logarithmic functions in base 10 and base e.
Understand that exponential and logarithmic functions can be represented using multiple forms by: Expressing exponential functions in the form f(x) = a · b^x and f(x) = Pe^(r·t), and
2026 Montana Mathematics Content Standards · Algebraic and Functional Reasoning
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students write the same growth or decay function as a·b^t and as P·e^(rt). They keep the starting value and time unit the same, using r = ln(b) or b = e^r to switch forms.
What Mastery Looks Like
- Given 120(1.08)^t, a student writes 120e^(ln(1.08)t) and finds a continuous rate of about 0.077 per year. They check that both forms give the same value at t = 0 and t = 1.
Common Misconceptions
- Students may use 0.08 as the continuous rate for a multiplier of 1.08, instead of ln(1.08). They may change the starting amount or use a positive rate for a multiplier below 1.
How to Assess It
- Exit ticket: Rewrite 60(0.9)^t in the form P·e^(rt). Give P, an approximate value of r, and say whether the function grows or decays.
Lesson moves
Ways to Teach It
Start with three counters, double them for three rounds, then write the count as 3·2^t and 3·e^(ln(2)t).
Ask students: Why do 5(1.1)^t and 5e^(ln(1.1)t) give the same result at t = 1?
Give pairs of cards showing forms such as 50(0.8)^t and 50e^(ln(0.8)t); have students match and check them with a calculator.
Model a $100 account growing 3% each year as 100(1.03)^t, then find the equivalent continuous rate with ln(1.03).
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Related Standards
- MT.PLUS.ALG.EXP.3
Understand that exponential and logarithmic functions can be represented using multiple forms by: Expressing exponential functions in the form f(x) = a · b^x an...
- MT.PLUS.NUM.REAS.1
Extend the properties of exponents to rational exponents, including converting between exponential and radical forms.
- MT.CORE.ALG.EXP.2
Understand a as the initial value and b as the growth/decay factor for an exponential function written in standard form, y = a · b^x.
- MT.PLUS.ALG.EXP.3.b
Expressing logarithmic functions in base 10 and base e.
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