First — talk for 2 minHave you ever seen rooftop solar — on a house, a school, a parking lot? Anyone lived somewhere besides PA where solar is more common? If no one, Mr. Bennett will share.
Why does Texas install rooftop solar so much faster than Pennsylvania?
The sun shines on every state. The same federal tax credit (30% of the install cost, claimed by 752,300 households in 2023) is available everywhere. The same panels ship from the same factories. Yet Texas added 11.6 gigawatts of new solar in 2024, while Pennsylvania’s cumulative total is only about 955 megawatts — less than one-tenth of what Texas added in a single year.
The difference is not weather. It is state policy: how long permits take, who pays soft costs, what utilities owe a homeowner who feeds extra power back into the grid. Pennsylvania’s SREC market is essentially closed. Texas auto-approves many permits. New Jersey has full net metering. California cut its export-credit by 70–80% in 2023 — and the next year, California’s installations dropped 41% and 17,000 solar jobs disappeared.
So each state grows on a different base \(b\). Texas’s rooftop solar grows at about \(b = 1.30\) per year. Pennsylvania’s grows at about \(b = 1.05\). Those two numbers look almost the same on paper. But how different do you think their futures actually look?
Worked Example #1 — Texas residential solar (write the model two ways)
Problem\(N(d)\) is the number of Texas residential solar systems (in thousands), where \(d\) = number of doublings. Write the model as \(N_{0} \cdot 2^{d}\), then rewrite the “3 doublings” relationship in logarithmic form.
DataTexas had about 25 thousand residential systems in 2019. Counts doubled every 2 years. After 3 doublings, the count is \(25 \cdot 2^{3}\) thousand systems.
Word Bank
argument
starting value
multiply
exponent
base
logarithm
divide
growth factor
Number Bank
2
3
8
25
200
5
16
| # | Reasoning (fill in the word) | Math (fill in the number) |
|---|
| 1 |
The when \(d = 0\) is the count in 2019. |
\(N_{0} = \) |
| 2 |
The for one doubling is 2. Each doubling takes 2 years, so the count after \(d\) doublings multiplies by \(2^{d}\). |
\(b = \) (per doubling) |
| 3 |
Write the count after \(d\) doublings using the -form \(N_{0} \cdot 2^{d}\). |
\(N(d) = 25 \cdot 2^{d}\) |
| 4 |
Plug in \(d = 3\) and the powers of 2: \(2^{3} = \) . |
\(N(3) = 25 \cdot \) \( = \) thousand |
| 5 |
Now rewrite \(2^{3} = 8\) in ic form. The base 2 stays the base; the 3 moves out front; 8 becomes the . |
\(\log_{2}(\) \() = \) |
Page 1 of 3 · Reading the Solar Curves · Algebra 2 · Unit 5 · Lesson 10
Worked Example #2 — Texas vs. Pennsylvania (years to double)
ProblemEach state’s rooftop solar follows \(\text{capacity}(t) = P_{0} \cdot b^{t}\). For Texas, set up the doubling equation \(1.30^{t} = 2\) and find \(t\). Repeat for Pennsylvania at \(b = 1.05\).
DataTexas grows at \(b = 1.30\) per year (when \(b > 1\), the curve grows). Pennsylvania grows at \(b = 1.05\) per year. (If a state had \(0 < b < 1\), the curve would decay instead.)
Word Bank
double
base
exponent
logarithm
subtract
argument
substitute
Number Bank
2
2.6
14.2
1.30
1.05
5.3
7.3
| # | Reasoning (fill in the word) | Math (fill in the number) |
|---|
| 1 |
For Texas, the per-year is given. |
\(b = \) |
| 2 |
To “” means to grow by a factor of 2. Write that as an exponential equation. |
\(1.30^{t} = \) |
| 3 |
Rewrite in log form: the 2 becomes the input; \(t\) becomes the answer. |
\(t = \log_{1.30}(2)\) |
| 4 |
Use a calculator to compute \(t\) (years for Texas to double). |
\(t \approx \) years |
| 5 |
Now \(b = 1.05\) for Pennsylvania and repeat steps 2–4. |
\(t = \log_{1.05}(2) \approx \) yrs |
Activity 2 — Same relationship, two ways
(a) TX, 3 doublings: \(2^{3} = 8\) becomes \(\log_{2}(\)\() = \).
(b) FL, 2 yrs at \(b = 1.20\): \(1.2^{2} = 1.44\) becomes \(\log_{1.2}(\)\() = \).
(c) PA, 1 yr at \(b = 1.05\): \(\log_{1.05}(1.05) = 1\) becomes \(= 1.05\).
(g) TX, \(t\) years to 4×: \(\log_{1.30}(4) = t\) becomes \(1.30^{t} = \). Solving: \(t \approx \) years.
(h) PA, \(t\) years to double: \(1.05^{t} = 2\) becomes \(t = \log_{1.05}(\)\() \approx \) years.
Conclude · Agency Move
What policy could move Pennsylvania’s base?
Texas doubles its rooftop solar in about 2.6 years. Pennsylvania takes about 14.2 years — more than 5× longer for the same outcome. Same panels, same federal tax credit. The difference is the base \(b\), and the base is moved by policy.
Pick one intervention. Estimate the new base \(b'\) it could push PA toward. Then write PA’s new doubling time in both forms.
Intervention 1 — SolarAPP+ permitting
Auto-approves residential permits in hours. Cuts project times ~12 days. Targets ~25% cost reduction. Used in 160+ jurisdictions.
Intervention 2 — SREC market reform
Massachusetts SMART program: predictable per-kWh incentives. Built MA into top-5 per-capita rooftop. PA’s SREC market is closed.
Estimate: If PA adopted , I think \(b\) would rise from 1.05 to about . New equation: \(b'^{\,t} = 2\) becomes \(t = \log_{b'}(2) \approx\) years.
Defend it: I picked this because
Page 2 of 3 · Reading the Solar Curves · Algebra 2 · Unit 5 · Lesson 10
Tier 2 · General Academic Words
- adoption — how widely something new is being picked up and used.
- cumulative — the total added up over all the years so far.
- decade — ten years.
- distributed — spread out across many small places (like rooftops) instead of one big plant.
- exceed — go past; be more than.
- infrastructure — the basic systems (wires, pipes, roads) a community needs to work.
- install — to put a piece of equipment in place so it is ready to use.
- permit — an official paper a city gives that says you are allowed to build something.
- policy — a rule or plan that a government or company decides to follow.
- rate — how fast something happens or grows.
- residential — having to do with homes where people live.
- rooftop — on top of a building.
Tier 3 · Solar & Math Words
- asymptote — a line that a curve gets closer and closer to but never touches.
- base of an exponential (\(b\)) — the number being raised to a power. In \(P_{0} \cdot b^{t}\), \(b\) is what you multiply by each year.
- capacity — the most power a system can produce, measured in watts, kilowatts, or megawatts.
- decay factor — the value of \(b\) when \(0 < b < 1\); the curve shrinks each year.
- doubling time — the number of years it takes for an amount to grow to twice its original size.
- exponential function — a function in the shape \(P_{0} \cdot b^{t}\), where the variable is in the exponent.
- growth factor — the value of \(b\) when \(b > 1\); the curve grows each year.
- interconnection — the rules and paperwork for hooking a rooftop solar system to the local power grid.
- kilowatt (kW) — one thousand watts of power; a typical home solar array is 5–10 kW.
- logarithmic form — the equation \(\log_{b}(x) = e\), which is the same relationship as \(b^{e} = x\) written so the exponent is the answer.
- megawatt (MW) — one million watts; one MW can power roughly 200 average homes.
- net metering — a rule that pays a homeowner for extra solar power they send back to the grid.
- permit timeline — how many days or weeks a city takes to approve a solar install.
- photovoltaic (PV) — the technology that turns sunlight directly into electricity using panels.
- rooftop solar — solar panels mounted on a home or building’s roof, owned by the building owner.
- soft costs — the parts of a solar install that are not the panels: permits, labor, paperwork, sales.
- SREC (Solar Renewable Energy Credit) — a credit a homeowner earns for each megawatt-hour of solar electricity, then sells.
- y-intercept — the point where a graph crosses the y-axis (when \(t = 0\)). It equals the starting value \(P_{0}\).
Page 3 of 3 · Reading the Solar Curves · Algebra 2 · Unit 5 · Lesson 10