If bacteria double every hour, how many are there after 90 minutes — and is the answer what you expect?
Learning Targets
I can determine the value of exponential functions at non-whole number inputs.
I can use equations and graphs to interpret rational inputs to exponential functions in context.
This lesson examines rational inputs to exponential functions. Students use equations and graphs to make sense of inputs like \(\frac{1}{2}\) and \(1.8\) in population growth and chemical decay contexts. They connect fractional exponents from the warm-up to function evaluation throughout. MP2 (reason abstractly and quantitatively) is the primary practice standard.
Standards: HSN-RN.A.1, HSF-LE.A.1.a, HSF-LE.A.2
Prep Checklist
Graphing technology (one per student)
Warm-Up ⏱ 5 min
Keeping Equations True
Review of rational exponents. Students identify which expressions are solutions to equations involving \(\frac{1}{2}\) and \(\frac{1}{3}\) exponents. This prepares them for evaluating exponential functions at non-integer inputs. Give 2 minutes of silent individual work, then compare with a partner before whole-class discussion.
Determine which expressions are solutions to each equation. Be prepared to explain your reasoning.
On your own, then share with a partner
Select all solutions to \(x^2 = 9\).
A. \(3\) B. \(-3\) C. \(\sqrt{(9)}\) D. \(-\sqrt{(9)}\) E. \(9^{\frac{1}{2}}\) F. \(-9^{\frac{1}{2}}\)
Select all solutions to \(x^3 = 8\).
A. \(2\) B. \(-2\) C. \(\sqrt[3]{(8)}\) D. \(-\sqrt[3]{(8)}\) E. \(8^{\frac{1}{3}}\) F. \(-8^{\frac{1}{3}}\)
⏱ 2 min individual think time
Warm-Up Synthesis ⏱ 3 min
What does a fractional exponent mean?
Survey the class on each expression. If there is disagreement, ask contrasting students to share. Make sure students recall: \(b^{\frac{1}{2}} = \sqrt{(b)}\) and \(b^{\frac{1}{3}} = \sqrt[3]{(b)}\). Highlight that squaring \(3\) and \(-3\) both give \(9\), and cubing \(2\) gives \(8\) but cubing \(-2\) gives \(-8\).
The purpose of this discussion is to recall that fractional exponents represent roots, and to verify solutions by reversing the operation.
Try Saying
“I know ___ is a solution because when I ___, I get ___.”
“___ is not a solution because ___.”
click to advance discussion ▶
“Which expressions are solutions to \(x^2 = 9\)?”A, B, C, D, E, F — all six are solutions. Both \(3\) and \(-3\) satisfy the equation. \(\sqrt{(9)} = 3\), \(-\sqrt{(9)} = -3\), \(9^{\frac{1}{2}} = 3\), and \(-9^{\frac{1}{2}} = -3\).
“Which expressions are solutions to \(x^3 = 8\)?”A, C, E only. \(2^3 = 8\), \(\sqrt[3]{(8)} = 2\), and \(8^{\frac{1}{3}} = 2\). \(-2\) is not a solution because \((-2)^3 = -8 \neq 8\).
Key takeaway: A number raised to the exponent \(\frac{1}{2}\) equals the square root of that number. A number raised to the exponent \(\frac{1}{3}\) equals the cube root. These rules will help us evaluate exponential functions at non-whole-number inputs today.
Sample Responsesclick to reveal ▶
\(x^2 = 9\): solutions are \(3\), \(-3\), \(\sqrt{(9)}\), \(-\sqrt{(9)}\), \(9^{\frac{1}{2}}\), and \(-9^{\frac{1}{2}}\).
\(x^3 = 8\): solutions are \(2\), \(\sqrt[3]{(8)}\), and \(8^{\frac{1}{3}}\).
\(b^{\frac{1}{2}} = \sqrt{(b)}\) and \(b^{\frac{1}{3}} = \sqrt[3]{(b)}\). Squaring reverses a \(\frac{1}{2}\) exponent; cubing reverses a \(\frac{1}{3}\) exponent.
Math As A Second Language
Fractional Exponent: One-Half
Math · We Say · Meaning
★ Math (given)
\(b^{\frac{1}{2}}\)
▲ We Say
“b to the one-half”
or
“b to the power of one-half”
● Meaning
A base raised to the \(\frac{1}{2}\) power equals the square root of that base. Example: \(9^{\frac{1}{2}} = 3\).
Read the Math card aloud. Ask a student to read the We Say card. Ask another student to explain the Meaning card in their own words. Emphasize both pronunciations are correct.
Math As A Second Language
Fractional Exponent: One-nth
Math · We Say · Meaning
★ Math (given)
\(b^{\frac{1}{n}}\)
▲ We Say
“b to the one over n”
or
“b to the power of one-nth”
● Meaning
A base raised to the \(\frac{1}{n}\) power equals the nth root of that base. Example: \(8^{\frac{1}{3}} = 2\).
Connect this to the previous card. Ask: "How is this card a generalization of the previous one?" (When \(n = 2\), we get the square root; when \(n = 3\), the cube root; this generalizes to any root.)
Activity 1 · Launch ⏱ 5 min
Moon Base Population
Arrange students in groups of 2. Read the context aloud. Use Co-Craft Questions: display only the problem stem (the context paragraph) without the numbered tasks. Give students 1–2 minutes to write mathematical questions that could be asked about this situation. Partners compare questions. Cold-call several pairs to share. Ask: "What do these questions have in common? How are they different?" Listen for and amplify language related to "predict," "each year," and "growth factor." Then reveal the full task. Every student needs graphing technology.
In a video game, Jada is building a moon base to support a growing population. Her base has a population of 54,500 in the year 2240. Between 2240 and 2270, the population grows exponentially by about 60% per decade.
“What mathematical questions could you ask about this situation?”
⏱ 1 min individual think time before partner discussion
Activity 1 · Work Time ⏱ 15 min
Moon Base Population
🧑🏫 Circulate and look for: students who express the population as an exponential expression (like \(54500 \cdot 1.6^{0.5}\)) vs. those who use a calculator for a numerical estimate. Both approaches are important — select one of each for synthesis. Common misconception: students may think 60% growth in half a decade means 30% growth (additive thinking instead of multiplicative). If you see this, ask: "If the population grows by 60% per decade, does that mean it grows by 30% in half a decade? How could you check?"
If Students Are Stuck
"How did you choose the horizontal and vertical dimensions for your graph?"
"How could adjusting the horizontal and vertical dimensions help you see how the population grows between 2240 and 2270?"
Amplify · Activity Builder · Type-in
Setup (do once):
1. Add a Text component → include ALL student-facing text from the slide.
2. Add a Math Response component → name it answer
3. Add a Note component below it
4. Click </> on the Note → paste CL below
5. Straight quotes only — curly quotes break CL
Problem 1: Population in 2250 and 2260
content:
when answer.submitted and answer.latex = "87200"
"Correct for 2250! Now find 2260."
when answer.submitted and answer.latex = "87,200"
"Correct for 2250! Now find 2260."
when answer.submitted and answer.latex = "139520"
"Correct for 2260! Now find 2250."
when answer.submitted and answer.latex = "139,520"
"Correct for 2260! Now find 2250."
when answer.submitted
"The population grows by 60% each decade. What is 54,500 times 1.6?"
otherwise ""
Problem 2: Equation for P(d)
content:
when answer.submitted and answer.latex = "54500(1.6)^d"
"Correct!"
when answer.submitted and answer.latex = "54500 \\cdot 1.6^d"
"Correct!"
when answer.submitted and answer.latex = "54500(1.6)^{d}"
"Correct!"
when answer.submitted and answer.latex = "54500*1.6^d"
"Correct!"
when answer.submitted
"The initial population is 54,500. Each decade multiplies by 1.6. How do you write that as an exponential equation?"
otherwise ""
Work with your partner
Find the population of Jada's moon base in 2250 and in 2260 according to this model.
The population is a function \(P\) of the number of decades \(d\) after 2240. Write an equation for \(P(d)\).
Explain what \(P(0.5)\) means in this situation.
Graph your function using graphing technology. Estimate the value of \(P(0.5)\).
Explain why we can find the value of \(P(0.5)\) by multiplying 54,500 by \(\sqrt{(1.6)}\). Find that value.
Based on the model, what was the population of Jada's base in 2258? Show your reasoning.
Try Saying
“\(P(0.5)\) means ___ because ___.”
“I can find \(P(0.5)\) by ___ because ___.”
Activity 1 · Synthesis ⏱ 5 min
Making Sense of Rational Inputs
Focus on the meaning of \(P(0.5)\) and \(P(1.8)\). Display a table to fill in during discussion. Select previously identified students — one who used an exponential expression and one who estimated numerically. Help students see: we can think of \(P(2)\) as \(54500 \cdot 1.6^2\), but for \(P(0.5)\) we use exponent rules. Three ways to write the growth factor for 5 years after 2240: \(1.6^{0.5}\), \(1.6^{\frac{1}{2}}\), or \(\sqrt{(1.6)}\). Conclude by asking students to write three ways for 18 years after 2240: \(1.6^{1.8}\), \(1.6^{\frac{9}{5}}\), or \(\left(\sqrt[5]{(1.6)}\right)^9\).
The purpose of this discussion is to connect fractional exponents to the meaning of non-whole-number inputs in the population model.
Try Saying
“An input of ___ means ___ because ___.”
“I can write the growth factor as ___ or ___ because ___.”
click to advance discussion ▶
“What does \(P(0.5)\) mean in this situation, and how did you find its value?”\(P(0.5)\) is the population half a decade (5 years) after 2240 — the population in 2245. Its value is \(54500 \cdot 1.6^{0.5} \approx 68{,}938\).
“Why can we write \(1.6^{0.5}\) as \(\sqrt{(1.6)}\)?”Because raising a number to the \(\frac{1}{2}\) power is the same as taking the square root. So \(1.6^{0.5} = 1.6^{\frac{1}{2}} = \sqrt{(1.6)}\).
“What does \(P(1.8)\) mean, and what is its value?”\(P(1.8)\) is the population 1.8 decades (18 years) after 2240 — the population in 2258. Its value is \(54500 \cdot 1.6^{1.8} \approx 127{,}003\).
Write three ways to express the growth factor for 18 years after 2240.\(1.6^{1.8}\), \(\;1.6^{\frac{9}{5}}\), \(\;\left(\sqrt[5]{(1.6)}\right)^9\)
Sample Responsesclick to reveal ▶
Population in 2250: \(54500 \cdot 1.6 = 87{,}200\). Population in 2260: \(54500 \cdot 1.6^2 = 139{,}520\).
\(P(d) = 54500 \cdot 1.6^d\)
\(P(0.5)\) means the population in 2245 (half a decade after 2240). \(P(0.5) \approx 68{,}938\).
Population in 2258: \(P(1.8) = 54500 \cdot 1.6^{1.8} \approx 127{,}003\).
Math As A Second Language
Rational Exponent: m over n
Math · We Say · Meaning
★ Math (given)
\(b^{\frac{m}{n}}\)
▲ We Say
“b to the m over n”
or
“b to the power of m-nth”
● Meaning
A rational exponent \(\frac{m}{n}\) means take the nth root of b, then raise it to the mth power. Example: \(8^{\frac{2}{3}} = 4\).
This generalizes both previous MASL cards. Ask: "How does this connect to what we just did with the moon base?" (Writing \(1.6^{1.8} = 1.6^{\frac{9}{5}}\) is an example of \(b^{\frac{m}{n}}\) where \(b = 1.6\), \(m = 9\), \(n = 5\).)
Math As A Second Language
Non-Whole Number Inputs
Math · We Say · Meaning
★ Math (given)
\(f(0.5)\)
▲ We Say
“f of zero point five”
or
“f of one-half”
● Meaning
A non-whole number input to an exponential function represents a fractional time period. Example: \(f(0.5)\) is the value halfway through one time unit.
Ask students to connect this to the moon base context: "\(P(0.5)\) is the population half a decade — 5 years — after 2240." Emphasize that non-integer inputs are meaningful and common in real-world exponential models.
Activity 2 · Launch ⏱ 2 min
Cleaning Up a Spill
Read the context aloud. Point to the graph and ask students to identify key features before they begin working: "What are the coordinates of the point where the curve starts?" (0, 12). "Is the amount of chemical increasing or decreasing?" (Decreasing — this is exponential decay.) Then release students to work in pairs.
A chemical is accidentally spilled into a lake. The cleaning process decreases the amount of the chemical roughly exponentially. The graph represents \(f\), an exponential function that models the amount of chemical left in the lake, in liters, \(t\) hours after cleaning begins.
⏱ 1 min individual think time before partner work
Activity 2 · Work Time ⏱ 8 min
Cleaning Up a Spill
🧑🏫 Circulate and look for: how students estimate \(f\!\left(\frac{1}{3}\right)\) from the graph (reading the y-value at \(t = \frac{1}{3}\), which is about 20 minutes). Watch for students who confuse \(\frac{1}{3}\) of an hour with 30 minutes — ask: "How many minutes is one-third of an hour?" Common error on part 2: students may set the decay factor to \(\frac{0.75}{12}\) instead of recognizing it as the multiplicative factor. Ask: "What fraction of the original 12 liters is 0.75 liters?"
Amplify · Activity Builder · Type-in
Setup (do once):
1. Add a Text component → include ALL student-facing text.
2. Add a Math Response → name it answer
3. Add a Note → click </> → paste CL below
Problem 2: Equation for f(t)
content:
when answer.submitted and answer.latex = "12(0.0625)^t"
"Correct!"
when answer.submitted and answer.latex = "12 \\cdot 0.0625^t"
"Correct!"
when answer.submitted and answer.latex = "12(\\frac{1}{16})^t"
"Correct!"
when answer.submitted and answer.latex = "12 \\cdot (\\frac{1}{16})^t"
"Correct!"
when answer.submitted
"Good start - what is the initial amount? What fraction of 12 is 0.75?"
otherwise ""
Work with your partner
Use the graph to estimate \(f\!\left(\frac{1}{3}\right)\). Explain what it tells us in this situation.
After one hour, 0.75 liters of the chemical remains. Find an equation that defines \(f(t)\).
Try Saying
“\(f\!\left(\frac{1}{3}\right)\) means ___ because ___.”
“The decay factor is ___ because ___.”
Activity 2 · Synthesis ⏱ 5 min
Interpreting Decay at Rational Inputs
Invite students to share how they derived the equation for \(f(t)\). Discuss the meaning of 12 (initial amount) and 0.0625 (or \(\frac{1}{16}\)) as the decay factor. Focus on interpreting \(f\!\left(\frac{1}{3}\right)\) — both graphically and algebraically. Ask all four discussion questions below in order. For the fish question, give 30 seconds of think time before cold-calling.
The purpose of this discussion is to interpret the decay function at rational inputs and connect the graphical estimate to the algebraic calculation.
Try Saying
“I found the decay factor by ___ because ___.”
“The point at \(t = \frac{1}{3}\) means ___ because ___.”
click to advance discussion ▶
“How did you find the growth (or decay) factor for this situation?”The amount decayed from 12 liters to 0.75 liters after 1 hour. The decay factor is \(\frac{0.75}{12} = 0.0625\), or \(\frac{1}{16}\).
“What does a point at \(t = \frac{1}{3}\) mean in this situation?”20 minutes after cleaning begins, there are about 4.8 liters of chemical still in the lake.
“What percentage of the chemical is still in the lake after 20 minutes? What are some ways to calculate this value?”About 40% is left after 20 minutes. This can be found using \(\frac{4.8}{12}\) or \(0.0625^{\frac{1}{3}}\).
“The fish will be safe if the chemical is reduced to less than \(\frac{1}{2}\) of a liter after 80 minutes of cleaning. Will the fish be safe?”Yes, because \(f\!\left(\frac{4}{3}\right) = 12 \cdot 0.0625^{\frac{4}{3}} \approx 0.18\) liters, which is less than \(\frac{1}{2}\).
Sample Responsesclick to reveal ▶
\(f\!\left(\frac{1}{3}\right) \approx 4.8\) liters — the amount of chemical 20 minutes after cleaning begins.
Display a graph of \(p(t) = 1000 \cdot 2^t\) modeling bacteria population (in thousands) \(t\) hours after being counted. Walk through the two discussion questions below. If students notice that \(p(1.25)\) is exactly twice \(p(0.25)\), invite them to reason about why — the interval is 1 hour, so outputs differ by a factor of 2. Deeper reasoning about this property is a focus of a future lesson, so it is fine to note the observation without fully explaining it. End by emphasizing the distinction between exact values (like \(1000 \cdot 2^{\frac{1}{4}}\)) and approximate values (like 1,189). Use \(\approx\) notation when rounding.
The purpose of this discussion is to summarize how fractional exponents and exponential functions come together in this lesson, and to distinguish between exact and approximate values.
Try Saying
“The exact value is ___, and the approximate value is ___.”
“The exact value is useful for ___, but the approximate value is useful for ___.”
click to advance discussion ▶
Consider the function \(p(t) = 1000 \cdot 2^t\), which models the population of bacteria (in thousands) \(t\) hours after being counted.
“What are the coordinates of \(p(0.25)\), and what do they mean in this context?”\(p(0.25) = 1000 \cdot 2^{0.25} = 1000 \cdot 2^{\frac{1}{4}} \approx 1{,}189\). Fifteen minutes after being counted, the population is about 1,189 thousand.
“What are the coordinates of \(p(1.25)\), and what do they mean in this context?”\(p(1.25) = 1000 \cdot 2^{1.25} = 1000 \cdot 2^{\frac{5}{4}} \approx 2{,}378\). One hour and fifteen minutes after being counted, the population is about 2,378 thousand.
Exact values such as \(1000 \cdot 2^{\frac{1}{4}}\) reveal mathematical structure but can be difficult to interpret in real situations. Approximate values such as 1,189 are useful for understanding and measuring, but can hide the structure. When a value is approximate, use \(\approx\) to show that the given value is not exact.
By the end of this lesson, you can:
Understand the meaning of a rational input, particularly positive non-whole number inputs, to an exponential function in context.
Use equations and graphs to identify the value of an exponential function at a rational input.
Determine the value of exponential functions at non-whole number inputs.
Reference · How ToCool-Down ⏱ 5 min
🧑🏫 Leave this slide on screen during the cool-down. The reference table supports students evaluating the exponential function at rational inputs. Look for: students who confuse "1 day" with \(t = 1\) instead of \(t = \frac{1}{7}\). Also watch for students who multiply by \(\frac{1}{2}\) for problem 2 instead of raising \(\frac{1}{2}\) to the \(\frac{1}{7}\) power. For problem 3, students should recognize \(\frac{4}{7}\) as 4 days out of a 7-day week.
Amplify · Activity Builder · Type-in
Setup (do once):
1. Add a Text component → include ALL student-facing text.
2. Add a Math Response → name it answer
3. Add a Note → click </> → paste CL below
Input format example: Example: 25
Problem 1: Sand after 1, 2, 3 weeks
content:
when answer.submitted and answer.latex = "25, 12.5, 6.25"
"Correct! 25 kg, 12.5 kg, 6.25 kg."
when answer.submitted and answer.latex = "25,12.5,6.25"
"Correct! 25 kg, 12.5 kg, 6.25 kg."
when answer.submitted
"Each week, half the sand remains. Start with 50: what is 50 times 1/2?"
otherwise ""
Problem 2: Expression for 1 day
content:
when answer.submitted and answer.latex = "50(\\frac{1}{2})^{\\frac{1}{7}}"
"Correct!"
when answer.submitted and answer.latex = "50 \\cdot (\\frac{1}{2})^{\\frac{1}{7}}"
"Correct!"
when answer.submitted and answer.latex = "50(0.5)^{1/7}"
"Correct!"
when answer.submitted and answer.latex = "50 \\cdot 0.5^{1/7}"
"Correct!"
when answer.submitted
"Not quite - 1 day is what fraction of a week? Use that fraction as the exponent."
otherwise ""
Problem 3: Meaning of f(4/7)
content:
when answer.submitted and isBlank(answer.latex)
"Write your explanation of what f(4/7) means in context."
when answer.submitted
"Does your answer mention the number of days and what the output represents?"
otherwise ""
A laboratory sets up a model river so that half of the sand is washed away every week. The model starts with 50 kilograms of sand. The function \(f(t) = 50 \cdot \left(\frac{1}{2}\right)^t\) gives the weight of sand left \(t\) weeks after the water begins flowing.
On your own
Find the amount of sand left 1 week, 2 weeks, and 3 weeks after the water begins flowing.
Write an expression to represent the amount of sand 1 day after the water begins flowing.
Explain what \(f\!\left(\frac{4}{7}\right)\) means in this context.
Lesson Summary
Key Takeaway: Rational Inputs to Exponential Functions
This summary slide formalizes the lesson's key ideas. Read it aloud or have a student read it. Reference the algae example as a new context that mirrors the moon base and chemical spill. Students should see the same structure across all three contexts.
Some exponential functions can have inputs that are any numbers on the number line, not just integers.
Suppose the area of a pond covered by algae \(A\), in square meters, is modeled by \(A(t) = 200 \cdot \left(\frac{1}{2}\right)^t\), where \(t\) is the number of weeks since a treatment was applied.
Since \(t\) is one week and each week has 7 days, \(\frac{1}{7}\) is 1 day. So after 1 day, the algae covers \(200 \cdot \left(\frac{1}{2}\right)^{\frac{1}{7}} \approx 181\) square meters.
The expression \(\left(\frac{1}{2}\right)^{\frac{1}{7}}\), which is equivalent to \(\sqrt[7]{\left(\frac{1}{2}\right)}\), is about 0.906. This means that after 1 day, only about 91% of the algae from the previous day remains.
Key Principle
Exact values like \(1000 \cdot 2^{\frac{1}{4}}\) reveal mathematical structure.
Approximate values like \(1{,}189\) are useful for measuring in real situations.
Use \(\approx\) when a value is not exact.
