First — talk for 2 minHave you noticed vaping getting more or less common in the last few years? What changed it — flavor bans, school stuff, family stuff? Pass freely if you'd rather just listen.
How fast is teen vaping really falling?
Each year the CDC runs the National Youth Tobacco Survey — a big, school-based survey that asks high schoolers across the country whether they have used an e-cigarette in the past 30 days. The answers, year by year, tell a story.
In 2019, vaping among high schoolers hit 27.5% — about one in four. Then it started falling. By 2024 it was down to 7.8% — fewer than one in twelve. In between: a federal flavor ban on cartridges (2020), Tobacco 21 (the legal sales age moved from 18 to 21), Truth Initiative quit-texts, state bans, FDA letters to disposable importers. The curve did not fall by itself. Real interventions, with measured effects, moved it.
The math today gets us a model that fits those five years. Once we have it, we can ask the questions that matter. What rate is the curve actually falling at — and what would it take to push it the rest of the way to zero?
Worked Example #1 — Fit the post-2019 decay (find the model)
ProblemLet \(P(t)\) be the % of US high schoolers who vaped in the past 30 days, where \(t\) = years since 2019. Find a continuous-decay model in the form \(P_0 \cdot b^{t}\).
DataNYTS reports 27.5% in 2019 (\(t = 0\)) and 7.8% in 2024 (\(t = 5\)). Because vaping is decreasing, expect \(0 < b < 1\).
Word Bank
starting value
growth factor
decay rate
divide
subtract
substitute
root
equation
Number Bank
27.5
7.8
0.2836
0.777
5
0.252
2024
(plus 2 distractors: 1.286 · 19.7)
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Identify the \(P_0\) (the value when \(t = 0\), from the 2019 reading). |
\(P_0 = \) |
| 2 |
the second data point into \(P_0 \cdot b^{t}\). |
\(27.5 \cdot b^{\,5} = \) |
| 3 |
both sides by 27.5 to isolate \(b^{5}\). |
\(b^{5} = \dfrac{7.8}{27.5} \approx \) |
| 4 |
Take the 5th of both sides to find \(b\) (raise to the \(\tfrac{1}{5}\) power). |
\(b = (0.2836)^{1/5} \approx \) |
| 5 |
Write the . The per year is \(1 - b \approx 0.223\) (about 22% of vapers stop each year, net). |
\(P(t) = \) \(\cdot\) \(^{\,t}\) |
Page 1 of 3 · Reading the Curve on Teen Vaping · Algebra 2 · Unit 5 · Lesson 15
Worked Example #2 — Use \(\ln\) to find when \(P(t)\) reaches a target (and a half-life)
ProblemUsing the continuous-rate form \(P(t) = 27.5 \cdot e^{-0.252\,t}\), find the year the model predicts vaping reaches 5%. Then find the model's \(\log_b(0.5)\) half-life.
DataThe same fit, written with base \(e\): \(P(t) = 27.5 \cdot e^{-0.252\,t}\). The continuous rate \(k = -0.252\) per year. The natural log \(\ln\) is the inverse of \(e^{x}\), so \(\ln(e^{kt}) = kt\).
Word Bank
isolate
divide
natural log
exponent
target
half-life
add
solve
Number Bank
5
0.1818
−1.704
−0.252
6.76
2.75
−0.693
2025
(plus 2 distractors: 1.704 · 5.50)
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Set \(P(t)\) equal to the (5%). |
\(27.5 \cdot e^{-0.252\,t} = \) |
| 2 |
both sides by 27.5 to the exponential. |
\(e^{-0.252\,t} = \dfrac{5}{27.5} \approx \) |
| 3 |
Take the of both sides — this pulls the out. |
\(-0.252\,t = \ln(0.1818) \approx \) |
| 4 |
Divide by to for \(t\). |
\(t = \dfrac{-1.704}{-0.252} \approx \) \(\Rightarrow\) year |
| 5 |
: set \(P(t) = k\) with \(k = 13.75\) (half of 27.5). Solve. |
\(t = \dfrac{\ln(0.5)}{-0.252} = \dfrac{-0.693}{-0.252} \approx \) yr |
Activity 2 — graph the intersection (fill in as you work with your partner)
Step A (set up the system): Plot \(y = 27.5 \cdot e^{-0.252\,t}\) and \(y = \) . Their intersection point gives the time when vaping rate equals the target.
Step B (read the graph): The curves cross near \(t \approx \) , \(y = \) . That \(t\)-coordinate is the year-count since 2019, so the year is about .
Step C (check with \(\ln\)): Solving algebraically, \(t = \dfrac{\ln(5/27.5)}{-0.252} \approx \) . Graph and \(\ln\) agree to within of a year.
Step D (push it further): What \(t\) makes \(P(t) = 1\)? \(t \approx \) , which is roughly the year .
Agency Move · Pick a Bundle
What rate would halve teen vaping again by 2028 — and what would that take?
The current model halves vaping every 2.75 years. Halving from 7.8% (2024) by 2028 (4 years away) needs roughly the same rate as today — but every later halving is harder, because there are fewer marginal users left. Pick two interventions to combine. Each card below has a real, measured effect.
Bundle A · "This is Quitting" + Disposable enforcement
Quit-texts: +9.8 pp absolute abstinence (RCT). Elf Bar share among HS vapers fell % → 36.1% (one year). Together, plausible new \(k \approx \) ?
Bundle B · State flavor ban + 75% excise tax
MA HS vaping fell % → 18.3% (2019→23). CT (no ban) fell similarly — effect partly confounded. Plausible new \(k \approx \) ?
Defend your pick: I would push for bundle because it changes through , shifting \(k\) from \(-0.252\) toward .
Page 2 of 3 · Reading the Curve on Teen Vaping · Algebra 2 · Unit 5 · Lesson 15
Tier 2 · General Academic Words
- survey — a set of questions asked to many people, used to learn about a whole group.
- prevalence — the share of a group doing or having something right now (like the % of HS students vaping this month).
- decline — a drop; a steady decrease over time.
- halve — to cut something in half (so it ends up half its previous size).
- intervention — a planned action meant to change a behavior or outcome.
- enforcement — making sure a rule is followed, by checking and giving consequences (e.g., FDA letters to disposable importers).
- peak — the highest point a number reached before turning around.
- baseline — the starting point or trend that you compare other things against.
- extrapolate — to extend a pattern past the data you have, into the future.
- trajectory — the path a number is on if nothing changes.
- target — the value you are trying to reach (like 5%).
- bundle — a set of separate things picked together to act as one package.
- plausible — believable; could reasonably be true.
- policy — a written rule a government, company, or school chooses to follow.
- population — the whole group being studied (here, US high schoolers).
Tier 3 · Vaping & Math Words
- vaping — using an e-cigarette to breathe in a heated mist that usually contains nicotine.
- e-cigarette — a battery-powered device that heats a liquid into a mist the user inhales.
- nicotine — the addictive chemical in tobacco and most vape liquids; affects a teenage brain harder than an adult one.
- CDC — the Centers for Disease Control and Prevention, the US government agency that tracks public-health data.
- NYTS — the National Youth Tobacco Survey, the CDC’s yearly school-based survey of middle and high schoolers.
- flavor ban — a law that stops companies from selling flavored vape products (like fruit or candy flavors).
- cartridge — a small refill pod that snaps into a vape device (e.g., JUUL pods).
- disposable vape — a one-use vape device that is thrown away when it runs out (e.g., Elf Bar).
- Tobacco 21 (T21) — the federal law (2019) raising the legal age to buy tobacco or vape products from 18 to 21.
- intervention bundle — a planned combination of policies that work on different parts of the problem at once.
- exponential decay — a shrinking pattern where a quantity is multiplied by the same factor (less than 1) each step.
- decay rate — how fast a quantity is shrinking; in \(P_0 \cdot e^{kt}\) the rate is the negative number \(k\).
- continuous rate (\(k\)) — the rate used in base-\(e\) models; for our fit, \(k \approx -0.252\) per year.
- half-life — the time it takes for the modeled value to fall to half of any earlier value; here, about 2.75 years.
- logarithm (log) — the operation that “undoes” an exponent; for any base \(b\), \(\log_b(b^{x}) = x\).
- natural log (\(\ln\)) — the logarithm with base \(e\); the exact inverse of \(e^{x}\).
- intersection — the point where two graphs cross; its \(t\)-coordinate solves the equation \(f(t) = c\).
Page 3 of 3 · Reading the Curve on Teen Vaping · Algebra 2 · Unit 5 · Lesson 15