Put numbers on the sun's height

Before the lesson — what this assumes

• Assumes: the tilt model of the seasons, argued and believed — this lesson quantifies it, it doesn't re-litigate it.
• Assumes: sine-level trig and reading residuals off a table.
• Pre-work: find your town's latitude on its sun page and compute 90° − latitude + 23.4° and 90° − latitude − 23.4°. Bring both numbers; class tests them against the sky.

The plan — every step carries its minutes

The minutes are there for the teacher's pacing — and so that any single step can be handed to a student as their own five-minute lesson. See students teach the class.

What they should find — the teacher's key

• Formula vs page agrees to well under a degree once read at true solar noon; reading at clock noon injects the equation-of-time error deliberately.
• Day-length change: ~2–3 min/day at equinox (mid-latitudes), ≲ seconds/day at solstice.
• Earliest sunset precedes the winter solstice by ~1–2 weeks at mid-northern latitudes; latest sunrise follows it.

Standards: HS-ESS1-4 practice; error analysis on a real model.

What the picture fakes: Every sunrise is computed in the browser from the date and the place, never looked up, so any date works. The solver assumes a flat horizon — a mountain to the east makes the real sunrise later than the page says, and the methodology pages state the accuracy bounds.

Questions to chase on your own

For the student who wants more — each answerable with the tools, no teacher required:

• The tropics sit at 23.4° latitude and the Arctic Circle at 66.6°. Show that both numbers are the tilt wearing different hats.
• Using any city's page: find a latitude where the formula predicts a midnight sun on June 21, then verify the page agrees.

Questions the curious actually ask

Real questions, mostly from real kids — the kind that sound simple and open trapdoors. Worth raising in class before someone raises them for you:

Why isn't the hottest day the longest day?
The longest day is around June 21 — check yours — but the hottest weeks come in July and August. Why the lag? Because the land and the oceans are still filling up with heat, like an oven that keeps warming after you turn the dial. As long as each day brings in more heat than the night lets out, temperatures keep climbing — even as the days start shrinking. The sea does the same thing harder: beach water is warmest in September, months after the sun's peak.

Do people at the equator have seasons?
Not ours. Day length barely moves — check Singapore on any date, it's near 12 hours year-round — and there's no warm-and-cold cycle to hang "summer" on. Instead the year is carved into wet and dry seasons as the planet's rain belt migrates north and south, chasing the overhead sun. And there's a bonus strangeness: at the equator the sun passes straight overhead twice a year, and on those days at noon, you have almost no shadow.

If we're closer to the sun in January, shouldn't January be warmer?
It is — in Australia: look at Sydney's daylight today against your own town's. That's the tell that unravels the whole "closer = summer" idea: Earth really is 3 million miles closer to the sun in early January, and the southern hemisphere really is in summer then. The distance change is real but small (about 3%); the tilt's effect on sun-angle and day length is enormous. The 7–8 lesson turns this exact trap into a full period of hypothesis-testing.

Go further — beyond this site

• NOAA Solar Calculator — the professional reference — includes the equation-of-time drift the class just uncovered.
• NASA: Earth — context reading on orbit, tilt and climate patterns.

Same question, other grades

Each grade band re-asks this topic's question one level deeper — observe it, describe the pattern, measure it, explain the mechanism, quantify it and question the model. This page is the quantify it — and question the model rung.