Lessons / Logical reasoning
Logical reasoning · Day 9

Directions

Draw it. Put a plus sign on paper and track every turn; right turn = clockwise. Distances are Pythagoras at the end.

Practise (6 questions) Present as slides

Why this topic matters

Draw it. Every direction question is easy on paper and hard in your head.

Directions questions ask where someone ends up, how far from the start, or which way they face. A compass, a few arrows and Pythagoras solve all of them.

Turns and the compass

North is up, East is right. Facing north, a right turn faces east, a left turn faces west. Right turn = clockwise.

Keep track of which way the person faces after each turn, then draw that leg.

NSEWNENWSESW4 km N3 km E
4 km north, then right 3 km: 5 km from the start, north-east
Example 1Ravi walks 4 km north, turns right and walks 3 km. How far is he from the start?
  1. After the right turn he faces east.Right from north is east.
  2. He is 4 km north and 3 km east of the start.Draw it.
  3. √(16 + 9) = 5 km.
Example 2A man walks 10 m south, turns left and walks 6 m, turns left and walks 10 m. Where is he from the start?
  1. South 10, then left from south = east 6, then left from east = north 10.Track the facing direction.
  2. The south and north legs cancel: 6 m east of the start.

Shadows

In the morning the sun is in the east, so shadows fall to the west. In the evening shadows fall to the east. At noon there is almost no shadow.

If a person's shadow falls to his right in the morning, the west is on his right, so he is facing south.

Example 1In the morning, Asha's shadow falls exactly to her left. Which way is she facing?
  1. Morning shadows point west.The sun rises in the east.
  2. West is on her left.
  3. She faces north.

Formula summary

Right turn
N → E → S → W → N (clockwise).
Left turn
N → W → S → E → N.
Shortest distance
√(net east2 + net north2).
Shadow model
Morning sun in the east → shadow to the WEST. Evening → shadow to the EAST.
Angles
Clockwise 90 = right, 180 = about-turn. 45 lands on NE/SE/SW/NW.

Shortcuts and tricks

Net displacement

Use it when: Several legs, then 'how far from start'.

  1. Add all N-S legs, add all E-W legs separately.
  2. Then one Pythagoras.
ExampleWalk 3 km N, 4 km E. Distance from start?
  1. 3-4-5.

Before you move on, you should be able to...

  • draw every walk
  • turn right = clockwise
  • use Pythagoras for the straight-line distance
  • use sunrise-east for shadows

Practice questions (6)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy From home, walk 5 km east, 12 km north. Straight-line distance?
  1. 5-12-13.

Answer: 13 km

medium Ravi walks 10 m north, turns right and walks 6 m, turns right and walks 18 m. How far (m) and which way from start?
  1. Net: 6 east, 8 south.
  2. √(36 + 64) = 10, towards SE.

Answer: 10 m, south-east

medium In the morning, Sita stands facing a pole; the pole's shadow falls to her right. Which way is she facing?
  1. Morning shadow points west.
  2. West on her right → she faces south.

Answer: South

medium A man faces north, turns 90 clockwise, then 180 anticlockwise, then 45 clockwise. Which way does he face?
  1. N → E → W → NW.

Answer: North-west

hard A goes 4 km south, turns left and walks 3 km, turns left again and walks 8 km. Distance from start?
  1. South 4, left = east 3, left = north 8.
  2. Net: 3 east, 4 north.
  3. 3-4-5 → 5 km, north-east.

Answer: 5 km

hard From home, walk 7 km north, turn right and walk 4 km, turn right and walk 1 km, turn left and walk 4 km. Distance from home?
  1. N7, E4, S1, E4: net 8 east, 6 north.
  2. 6-8-10 triangle → 10 km, north-east.

Answer: 10 km

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
30 minCompass and turns: draw 5 walks.
20 minDistance with Pythagoras; final direction.
15 minShadows.
40 minPractice set.
5 minRecap.
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