Algebra 2 · U5 L8 · Unknown Exponents (Philly Rent vs. Income)

First — talk for 2 minHas your family or someone you know had to move because rent got too expensive? Take on a roommate or a second job? If no one, Mr. Bennett will share what he's seen in the neighborhood.

When did Philly cross the 30% line?

About 52% of Philadelphia households rent. The U.S. Department of Housing (HUD) calls a household "cost-burdened" when rent eats up 30% or more of monthly income. That is the line where families start choosing between rent and groceries, rent and SEPTA, rent and a doctor visit.

Between 2010 and 2024, two numbers in Philly grew at different speeds. Median rent climbed from about $870/month to about $1,500/month — roughly 4% per year. Median household income climbed from about $51,000 to about $60,500 — roughly 1.25% per year. Pennsylvania's minimum wage has not moved off $7.25 since 2009.

So the rent curve and the income curve are pulling apart. Somewhere along the way, the typical Philly renter crossed the 30% line. In what year did that happen — and what would it take to push the line back?

Worked Example #1 — When does Philly rent first reach $2,000/month? (integer exponent)

Problem$R(t)$ is the median monthly rent in Philly (dollars), where $t$ counts five-year steps after 2010. Find the first integer step where rent reaches \$2,000.

DataMedian rent was about \$800 in 2010 and grew by a factor of 1.25 every five years (roughly 4% per year). Use the form $R(t) = a \cdot b^{t}$.

Word Bank starting value growth factor exponent multiply guess-and-check divide subtract smallest

Number Bank 800 1.25 1,000 1,562 1,953 2,441 5 2035 2,000 (plus 2 distractors: 1,250 · 8.00)

#	Reasoning (fill in)	Math
1	Identify the at $t = 0$ (the rent in 2010).	$a = $
2	Identify the per-step (every 5 years).	$b = $
3	Write the equation. We want the smallest $t$ so the rent is at least \$2,000. The unknown is the .	$800 \cdot 1.25^{\,t} = $
4	Try $t = 4$. 800 by 1.25 four times.	$R(4) = 800 \cdot 1.25^{4} = $ \$ — under \$2,000.
5	Try $t = 5$. The integer that crosses the target wins.	$R(5) = $ \$. So $t = $ , year .

Page 1 of 3 · Unknown Exponents — Philly Rent vs. Income · Algebra 2 · Unit 5 · Lesson 8

Worked Example #2 — Bracket and squeeze the 30% crossing year (fractional exponent)

ProblemLet $I(t)$ be Philly median annual income, with $t$ = years since 2010. Find the first $t$ where rent eats 30% of income — that is, where $R(t) = 0.30 \cdot I(t)$.

DataAnnual rent: $R(t) = 10{,}440 \cdot 1.04^{t}$. Annual income: $I(t) = 51{,}000 \cdot 1.0125^{t}$. The ratio of two exponentials is itself exponential. Use guess-and-check — bracket between integers, then squeeze.

Word Bank divide ratio isolate bracket squeeze decimal cost-burdened crossing

Number Bank 0.205 1.027 1.466 1.455 1.494 14 15 14.3 2024 (plus 2 distractors: 0.502 · 1.072)

#	Reasoning (fill in)	Math
1	Build the rent-to-income . the two equations.	$\dfrac{R(t)}{I(t)} = \dfrac{10{,}440}{51{,}000} \cdot \left(\dfrac{1.04}{1.0125}\right)^{t} \approx $ $\cdot $ $^{\,t}$
2	Set the ratio equal to 0.30 (the threshold). the exponential by dividing both sides by 0.205.	$0.205 \cdot 1.027^{\,t} = 0.30 \;\Rightarrow\; 1.027^{\,t} = $
3	the answer between two integers. Try $t = 14$ and $t = 15$.	$1.027^{14} \approx $ (under). $1.027^{15} \approx $ (over). So $t$ is between and .
4	to one place.	$1.027^{14.3} \approx 1.467$. So $t \approx $ .
5	Convert $t$ to a year. The year is the answer.	14.3 years after 2010 $\Rightarrow$ mid-.

Activity 2 — fill in as you work with your partner

Step A (set up the ratio): $\dfrac{R(t)}{I(t)} \approx $ $\cdot $ $^{\,t}$. The combined base $\tfrac{1.04}{1.0125} \approx $ .

Step B (the equation to solve): Set the ratio equal to . After dividing, we need the $t$ that makes $1.027^{\,t} = $ .

Step C (bracket then squeeze): $t$ lives between and . Squeezed to one decimal, $t \approx $ , which is the year .

Step D (model vs. reality check): Pew reports about % of Philly renters were cost-burdened in 2024. Our model says the typical renter hit cost-burden right around the same year. Does that match?

Agency Move · Pick a Lever

If you got to set the rate — what would you push for?

The math says: at current rates, severe cost-burden (50%) hits the typical Philly renter around 2043. Pick one intervention. Each rate change below has a measured source.

Option A · Rent stabilization (NJ-style 2% cap) Hard cap on rent increases: rate $\to$ ~1.02/yr. Crossing pushed past ?
Trade-off: Diamond et al. (2019) found a % supply drop in SF.

Option B · Raise PA minimum wage + index Income rate $\to$ ~1.04/yr (matches rent). New ratio: $0.205 \cdot \left(\tfrac{1.04}{1.04}\right)^{t} = $ .
Effect: 30% line is crossed.

Defend your pick: I would pick option because it changes the rate of by , which would push the crossing year to .

Page 2 of 3 · Unknown Exponents — Philly Rent vs. Income · Algebra 2 · Unit 5 · Lesson 8

Tier 2 · General Academic Words

approximate — close to the right answer, but not exact.
bracket — to pin a value between two numbers you already know.
cap — a top limit on how high something can go.
cross — to pass over a line or threshold.
crossing — the point or year when one curve passes another.
decade — a stretch of ten years.
estimate — a careful guess based on what you know.
exceed — to go past a limit.
index — to tie a number to another number so it grows along with it.
intersect — when two lines or curves meet at a point.
median — the middle value — half are above, half are below.
nominal — the dollar amount on a price tag, before adjusting for inflation.
ratio — one quantity divided by another (like rent divided by income).
squeeze — to narrow the gap between two guesses until they almost meet.
threshold — a line where things start to count differently (like 30% of income).
trajectory — the path a number is on if nothing changes.

Tier 3 · Housing & Math Words

median rent — the middle rent in a city — half the rentals cost more, half cost less.
median income — the middle household income — half earn more, half earn less.
rent burden — the share of a household's income that goes to rent.
cost-burdened — HUD's name for a household paying 30% or more of income on rent.
severely cost-burdened — paying 50% or more of income on rent.
gentrification — when a neighborhood's rents and home prices climb so fast that long-time residents are priced out.
displacement — when families have to move out because they can no longer afford to stay.
AMI (Area Median Income) — the middle income for a city or region; affordable-housing rules use it as a yardstick.
inclusionary zoning — a city law that says new buildings have to set aside some units as affordable.
rent stabilization — a law that caps how much a landlord can raise rent each year.
housing voucher — a government coupon that helps a family pay rent on the open market.
minimum wage — the lowest hourly pay an employer is allowed to pay by law.
unknown exponent — the variable $t$ in an equation like $a \cdot b^{t} = c$, where we know $a$, $b$, and $c$ but not $t$.
guess-and-check — trying values, seeing if they work, and adjusting until you land close enough.
logarithm — the exact tool that "undoes" an exponent (you'll meet it tomorrow).
growth factor (b) — the number you multiply by each step in exponential growth.
starting value (a) — the amount at $t = 0$, before any growth has happened.
ratio of exponentials — $\frac{a \cdot b^{t}}{c \cdot d^{t}} = \frac{a}{c} \cdot \left(\frac{b}{d}\right)^{t}$ — still one exponential.

Page 3 of 3 · Unknown Exponents — Philly Rent vs. Income · Algebra 2 · Unit 5 · Lesson 8

#	Reasoning (fill in)	Math
1	Identify the at \(t = 0\) (the rent in 2010).	\(a = \)
2	Identify the per-step (every 5 years).	\(b = \)
3	Write the equation. We want the smallest \(t\) so the rent is at least \$2,000. The unknown is the .	\(800 \cdot 1.25^{\,t} = \)
4	Try \(t = 4\). 800 by 1.25 four times.	\(R(4) = 800 \cdot 1.25^{4} = \) \$ — under \$2,000.
5	Try \(t = 5\). The integer that crosses the target wins.	\(R(5) = \) \$. So \(t = \) , year .

#	Reasoning (fill in)	Math
1	Build the rent-to-income . the two equations.	\(\dfrac{R(t)}{I(t)} = \dfrac{10{,}440}{51{,}000} \cdot \left(\dfrac{1.04}{1.0125}\right)^{t} \approx \) \(\cdot \) \(^{\,t}\)
2	Set the ratio equal to 0.30 (the threshold). the exponential by dividing both sides by 0.205.	\(0.205 \cdot 1.027^{\,t} = 0.30 \;\Rightarrow\; 1.027^{\,t} = \)
3	the answer between two integers. Try \(t = 14\) and \(t = 15\).	\(1.027^{14} \approx \) (under). \(1.027^{15} \approx \) (over). So \(t\) is between and .
4	to one place.	\(1.027^{14.3} \approx 1.467\). So \(t \approx \) .
5	Convert \(t\) to a year. The year is the answer.	14.3 years after 2010 \(\Rightarrow\) mid-.