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Numbers at the Airport

Four short lessons on the numbers a family meets on a trip by air: adding minutes on a departure board, why each end of a runway wears a different number, how a seat map is a grid, and why the shortest route between two cities bends toward the pole.

Spark & Whetstone · sparkwhetstone.org/learn/numbers-at-the-airport · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. The departure board

A departure board is a table of clock times, and the arithmetic it asks for is elapsed time: a flight that leaves at 10:40 and takes 95 minutes lands at 12:15. Ninety-five minutes is one hour and thirty-five; 40 minutes and 35 minutes make 75, which is one more hour and 15 minutes.

The trap is that a clock carries at 60, not at 100. Children who add the numbers as if they were ordinary write 10:40 + 1:35 = 11:75, a time no clock shows. On MeasurePals they read a small board of invented flights and pick the landing time from four, one of which is exactly that trap.

Every time on the board is local. A flight that leaves at 9:00, flies 5 hours and crosses into a time zone three hours behind lands at 11:00 by the clock on the arrivals wall, though everyone aboard has lived through five hours.

Question. A flight leaves at 1:30 and takes 2 hours 5 minutes. What time is it at the departure city when it lands?

  1. 3:35
  2. 4:05
  3. 3:75

Say it back. In your own words: why does adding times on a departure board need a carry at 60, and what does a time zone change?

Lesson 2. The number on the runway

The big number painted at the end of a runway is its compass direction, in degrees, divided by ten and rounded. A runway pointing 268° wears 27; a runway pointing due north, 360°, wears 36, never 0.

The two ends of one strip of concrete point opposite ways, and opposite directions differ by 180°. So the far end is the near number plus or minus 18: runway 27 is runway 09 from the other end. Every runway number pair differs by 18.

The directions are magnetic, and the magnetic pole drifts. When the drift moves a runway’s heading past a rounding line, the airport repaints it: a runway can be renamed without anyone moving it. On GeometryForge older children turn a compass needle, find the painted number, and then find the other end.

Question. A runway points to 044° on the compass. What numbers are painted on its two ends?

  1. 04 and 22
  2. 04 and 40
  3. 44 and 22

Say it back. In your own words: where does a runway’s number come from, and why do the two ends always differ by 18?

Lesson 3. The seat map is a grid

A seat number like 14C is a coordinate: the row, then a letter for the place across. In a cabin with six seats a row, A to F with the aisle between C and D, the letters at the ends (A and F) are windows and the letters beside the aisle (C and D) are aisles. So 14C is an aisle seat on the left side, fourteen rows back.

Because it is a grid, counting is multiplication. Thirty rows of six is 180 seats, and every row has two windows, so thirty rows hold 60 window seats. Children on ArrayPals find a seat on a small cabin, then count a whole cabin without counting every seat.

Question. A cabin has 25 rows, each with seats A to F and the aisle between C and D. How many window seats does it have?

  1. 50
  2. 25
  3. 150

Say it back. In your own words: what do the number and the letter of a seat each tell you, and how does that make the seats easy to count?

Lesson 4. Why the route bends north

On a flat map a long flight between two northern cities looks like a curve bowing toward the pole. On a globe it is the straight one. The shortest way between two points on a sphere runs along a great circle, a circle whose centre is the centre of the Earth, and for two cities on the same northern latitude that circle swings toward the pole.

Take two cities at 40° north on opposite sides of the world, 180° of longitude apart. Following the line of latitude covers about 138° of arc (180 × cos 40°). Going straight over the North Pole covers 50° up to the pole and 50° down: 100°. At about 60 nautical miles a degree, that is roughly 6,000 miles instead of 8,300, a bit more than a quarter shorter.

The flat map draws the straight route as a curve because no flat map can keep every distance true. The Globe Table on this site shows the same pair of cities both ways.

Question. Two cities lie on the same line of latitude in the far north. Which way does the shortest route between them go?

  1. Bowed toward the pole
  2. Along the line of latitude
  3. Bowed toward the equator

Say it back. In your own words: why does the shortest flight between two northern cities look curved on a flat map?

At the table: The kitchen-table departure board

Players: Two or more, ages 7 and up. You'll need: Paper ruled into a table, a pen, a clock with hands or a phone clock face.

  1. Make up five flights: a place, a departure time and how many minutes each one takes.
  2. Each player works out one landing time in their head and writes it in. A time like 11:75 is the trap: say why no clock shows it.
  3. Swap boards and check each other’s times on the clock face, moving the minute hand all the way round.
  4. Now give one flight a time zone two hours behind and ask: what does the arrivals clock say, and how long were we really flying?

Answers

  1. 3:35 — 1:30 plus two hours is 3:30; plus five minutes is 3:35. No carry is needed, because 30 + 5 stays under 60.
  2. 04 and 22 — 44 ÷ 10 is 4.4, which rounds to 04. The other end points 44 + 180 = 224°, which is 22.4 and rounds to 22. And 22 − 4 = 18, as every pair must.
  3. 50 — Each row has two windows, A and F. 25 rows × 2 = 50.
  4. Bowed toward the pole — The great circle through two northern cities passes nearer the pole than their shared latitude does, so the shortest route bows poleward. On a flat map that straight route looks like a curve.