Unit 3 Topic 5 · week 9 · appeared in 3 of the last 4 externals · both formulas are printed in the book
The concept in one sentence: degrees convert to distance (111.2 km per degree along a great circle) and to time (15° of longitude per hour), and every question reduces to two decisions: subtract or add the angles, and cos or no cos.
The thing that makes it click: the two-question decision tree. Same side of the equator? Subtract. Opposite sides? Add. Travelling along a meridian (north-south)? No cos. Along a parallel (east-west)? Multiply by cos θ, because circles of latitude shrink toward the poles.
1 · Prerequisite check (30 seconds)
"Point north-south on a globe sketch. Which lines are latitude, which are longitude?" and "Find the cos button, what's cos 30°?"
If lat/long is shaky: the ladder mnemonic (LATitude lines lie FLAT, rungs of a ladder). Don't proceed until he can label a sketch, everything hangs on reading positions correctly. Calculator must be in DEGREES mode, check it now, not mid-question.
2 · The teaching path
Concrete: a real orange or a drawn globe. Meridians = full-size circles through the poles. Parallels = rings that shrink as you go poleward. Squeeze the top of the orange: THAT is why cos θ exists, the ring is smaller so each degree of longitude is worth fewer kilometres
The one number: 111.2 km per degree on any great circle (all meridians, plus the equator). Where from? Earth's circumference ÷ 360. It's printed in the formula book, he never memorises it
Meridian distances first (no cos): 40°N to 15°N = 25° apart → 111.2 × 25 = 2 780 km. Same side subtract. Then 40°N to 10°S = 50° apart (add across the equator)
Parallel distances second: two cities on the 30°N ring, 20° of longitude apart → 111.2 × cos 30° × 20 ≈ 1 926 km. The cos shrinks it because the ring is smaller than the equator
Time zones: 360° ÷ 24 h = 15° per hour. East is ahead (the sun gets there first). 150°E vs 90°E = 60° = 4 hours, and the westerly one is behind: 3 pm in P is 11 am in Q. Do ONE mixed-hemisphere example too: 150°E vs 30°W = 150 + 30 = 180° apart, the same add-across rule as latitude, the guides' examples otherwise never cross the prime meridian
D = 111.2 × angular distance · D = 111.2 × cos θ × angular distance · 15° = 1 hour (head-carried: 360 ÷ 24, NOT printed)
Same side subtract, opposite sides add.
North-south, no cos. East-west around a ring, cos.
East is ahead, the sun gets there first.
3 · Misconception catalogue
Looks like
Why brains do it
The fix script
Uses cos θ on a meridian (north-south) question
The cos formula looks more "proper", so it must be the real one
The orange: "meridians are all FULL-size circles, nothing shrank, nothing to fix. cos only pays for the shrunken rings"
Subtracts latitudes across the equator (40°N to 10°S = 30)
Subtracting two numbers is the habit
"Walk it: 40 down to the equator, then 10 more. You walked ALL of both." Same side subtract, opposite add, draw the two cases once
Goes the wrong way on time ("west is ahead")
No anchor for direction
Anchor it at home: "Brisbane sees the sun before Perth. East ahead." Then every answer gets the sense-check: which city meets the morning first?
Uses the LATITUDE angle in the cos for a parallel question, or the longitude difference
Two angles in one question, roles unclear
"cos eats the LATITUDE of the ring you're standing on. The angular distance is the LONGITUDE gap you travel." Label both before computing
Calculator in radians
Set-and-forget settings
cos 30° should be about 0.87. If it says 0.15, the calculator is in radians, the wrong universe. Check the mode at the start of every session this week
Reads 23°45′ as 23.45°
The minute marks look like a decimal point
"Minutes are SIXTIETHS: 45′ = 0.75°, so 23°45′ = 23.75°." Two minutes on the calculator's °′″ button, once. Degrees and minutes are named in the syllabus, expect them
Ignores the date line or daylight saving fine print
Feels like trivia, not maths
"Both are in the syllabus. 'Daylight saving applies' shifts the answer an hour; crossing the date line changes the DAY. Read the question's fine print out loud before computing"
4 · Questioning ladder
Rung
Ask
Listen for
Recall
"15° of longitude is how much time?"
1 hour, instantly
Do
"Same meridian, 50°N and 20°N. Distance?"
Subtracts (30°), × 111.2 = 3 336 km, no cos
Explain back
"Why does the parallel formula need cos but the meridian one doesn't?"
The shrinking-rings story in his own words
Transfer
"P at 150°E, Q at 90°E. 3 pm Friday in P, what time in Q, and how do you know which way?"
4 hours, Q behind (west), 11 am Friday, with the sun-first reasoning
5 · How QCAA examines it
Frequency: in the external 2022, 2023 and 2024, absent 2025, so treat it as expected. Usually SF marks: position reading, one distance, one time question
The two DISTANCE formulas are printed in the book (15° per hour is not, he carries that one: 360 ÷ 24), so the assessable skill is the two decisions (add/subtract, cos/no cos), not recall. Train with the book open
Combined questions: flight/call questions chain time zones with elapsed time ("departs 9 am, flies 7 hours, lands at what local time?"). Method: convert everything to ONE city's clock, do the arithmetic, convert back at the end
Rounding: distances to the nearest km unless told otherwise, and keep cos θ's full value on the calculator (round last, same rule as finance)
6 · Stuck scripts
Same side or opposite sides?
Meridian or parallel, which line are you moving along?
Which city sees the sun first?
One clock. Move everything to one city's time first.
7 · ADHD delivery notes
The two-question decision tree is the whole topic: put it on a sticky note, let him run it out loud each question. Decisions-as-checklist again
The orange is worth the mess, squeeze it once and cos θ never needs re-explaining
This is the LAST new content before the Season Final: frame the week as "one small topic, then we see the whole board", the finish line helps
Time-zone questions about places he cares about (game release times, UFC start times in Vegas) convert practice to genuine curiosity