How to teach exponential growth and decay

Why this matters

A student can copy y = a · b^x from the board and still not know what the exponent counts. Asked how long music at 100 decibels is safe, that student sees 5 halvings and divides 8 hours by 5. On a worksheet the wrong answer looks like any other number, and the teacher has to wait for quiz day to find out.

Exponential growth and decay

An exponential function has the form y = a · b^x, where a is the starting value and b is the factor the value is multiplied by at each step. When b is greater than 1, the function shows exponential growth, and when b is between 0 and 1, it shows exponential decay (OpenStax College Algebra 2e). Exponential change multiplies by the same factor over equal steps, and linear change adds the same amount. To solve for the exponent, students need a logarithm.

In the Flint activity, students turn a real noise limit into an exponential decay rule. NIOSH recommends a limit of 85 decibels averaged over an 8-hour workday, and halves the time for every 3 decibels louder (NIOSH), so b is 1/2 and each step is 3 decibels. A worksheet can only mark 1.6 hours wrong. In the chat, a student who divides by the number of halvings is asked for the safe time after one halving, then two, and a student who gets 15 minutes solves for their own commute with a logarithm.

Jayden's chat: 100 decibels on a 40-minute bus ride

Jayden is an 11th grader who listens to hip hop on earbuds during a 40-minute ride to school. He knows exponents and can halve a number in his head. He hasn't used logarithms yet.

The whole class gets the same activity: the same greeting, the same noise rule and the same 100-decibel question. Each student's chat is their own. Sparky asks about their music and their listening time, and its next question depends on what they just wrote.

Sparky uses the student's music and commuteSparky sends the student back to a step they already didA right answer gets a harder question

J

Jayden

Interest: Hip hop on a 40-minute bus rideStarting point: Knows exponents, new to logarithms

Sparky

Welcome to a quick math challenge about music and safe listening. What kind of music are you into, do you usually use earbuds or headphones, and how long is a typical listening session for you, like a bus ride?

Jayden

mostly hip hop. earbuds on the bus to school, the ride is like 40 minutes

Sparky

Hip hop on the bus is a solid way to start the day. Suppose you're playing your music at 100 decibels. NIOSH says 85 dBA is safe for 8 hours, and every 3 dBA louder, the safe time is cut in half. So at 88 dBA it's 4 hours, at 91 dBA 2 hours, and so on. Can you write a formula for the safe listening time t (in hours) at any sound level L?

Jayden

ok so from 85 to 100 is 15, and 15/3 = 5 halvings. so t = 8 / ((L-85)/3)? at 100 that's 8/5 = 1.6 hours

Sparky

You're right that it's 5 halvings. Let's slow down on what halving 5 times means. What is the safe time at 88 decibels, just one halving from 85?

Jayden

4 hours

Sparky

Exactly. Now what about 91?

Jayden

oh ok. 91 is 2 hours, 94 is 1 hour, 97 is half an hour, 100 is 15 minutes. so it's not 8 divided by 5, it's 8 halved 5 times. t = 8 * (1/2)^((L-85)/3)

Understanding check

That's it. You caught your own mistake and landed on the right formula. Plug in L = 100 and confirm you get 15 minutes.

Jayden

8 * (1/2)^5 = 8/32 = 0.25 hours = 15 minutes

Sparky

That's exactly right. You said your bus ride is 40 minutes. What's the loudest level you can play your hip hop for the full ride? Set up the equation and see how far you can get.

Jayden

40 min is 2/3 hour. 8 * (1/2)^((L-85)/3) = 2/3, so (1/2)^((L-85)/3) = 1/12. so 2^((L-85)/3) = 12. 2^3 is 8 and 2^4 is 16 so it's somewhere between 3 and 4 halvings? idk how to get it exactly

Sparky

Your setup is correct, and you know the exponent is between 3 and 4. To get the exact value, you need a logarithm. If 2^x = 12, how do you solve for x exactly using a log?

Jayden

x = log base 2 of 12 = log(12)/log(2) = 3.58. so (L-85)/3 = 3.58, L = 85 + 3*3.58 = 95.75. so like 95 decibels max for the whole ride

Sparky

That's exactly right: about 95.75 dB. For your 40-minute hip hop commute on earbuds, the safe max is just under 96 dB.

Fixed his own rule, then solved for 95.75 decibels with a logarithm

Jayden's first rule divided 8 hours by the number of halvings. Sparky didn't tell him the rule was wrong. It asked for the safe time after one halving, then two, and Jayden listed the times until he reached 15 minutes and rewrote the rule himself. The 40-minute question came only after he got 15 minutes. His own guess, between 3 and 4 halvings, is what Sparky built the logarithm question on.

Jayden is an example student, and Sparky's replies are shortened.

What Flint did for Jayden

• Sparky searched the web for the NIOSH noise limit and cited the page, so Jayden could open the source of the rule he turned into a formula.
• When Jayden divided 8 hours by 5, Sparky asked for the safe time after one halving instead of giving him the answer.
• Sparky checked Jayden's 15 minutes, his equation for the bus ride and his 95.75 decibels with its calculator before replying.
• After Jayden got 15 minutes, Sparky gave him a harder question built on his own 40-minute bus ride.

A teacher can't do this by hand for 30 students, and Sparky does it in every student's chat.

Who divided instead of halving, and who reached the logarithm

When students finish, Sparky grades each chat against the activity's rubric: how the student built the rule, how they used the logarithm, and how they worked through hints. The activity page shows who needs help right now, what the class does well, and what to practice next.

Set up the activity for free

The safe listening activity is in Flint's public library, with Sparky's guidelines, greeting and a rubric already written. Open it, select Use activity to copy it into a free Flint account, and move it to a class. In the activity's Preview tab, select Try session myself to test the activity as a student before you assign it.

For more math activities, see Flint for math.