Direction and Distance Sense

What does the question ask?

A person walks along a path with several turns. You must find where they end up.

Usually you find the direction of the end point from the start, or the straight-line distance.

Why do students slip?

Your mind cannot hold five turns at once.

A quick sketch holds them for you.

What do you need to know first?

The four main directions and the four in between.

And what a left turn or a right turn does to the way you face.

How do you set up the map?

Draw a plus sign. Put North at the top and South at the bottom.

Put East on the right and West on the left.

What are the in-between directions?

North-East lies between North and East. North-West lies between North and West.

South-East and South-West work the same way.

Is there a trick to remember the order?

Going clockwise from the top, the order is North, East, South, West.

A right turn moves you one step clockwise. A left turn moves you one step back.

Can you see the in-between directions used?

A point lies 3 km East and 3 km North of you. It is exactly North-East.

A point lies 5 km West and 5 km South of you. It is exactly South-West.

Why this works: a point up and to the right sits between North and East.

What does a right turn do?

It turns you a quarter circle clockwise.

Facing North, a right turn makes you face East. Facing East, it makes you face South.

Facing South, it makes you face West. Facing West, it makes you face North.

Can you follow a facing step by step?

Start facing North. Turn right: East. Turn right: South. Turn left: East.

So after right, right, left you face East.

Why this works: track the facing as a single word and change it after every turn.

What does a left turn do?

It turns you a quarter circle the other way.

Facing North, a left turn makes you face West. Facing West, it makes you face South.

Does the direction of the turn depend on the map?

No. Left and right always depend on the way you face at that moment.

So always update your facing after every turn.

How do you plot the path?

Start at a dot. Draw a line for each walk in the direction you face.

Label every corner with a letter, and write the distance beside each line.

How do you find the final direction?

Draw a line from the start to the end. Then look at where the end lies from the start.

If the end is above and to the right, it is North-East.

Can you see one solved?

Rani starts facing East and walks 4 km. She turns right and walks 4 km.

Right of East is South. So she ends up below and to the right of the start. The end point is South-East of the start.

Why this works: East then South always places the end below and right of the start.

How do you find the distance?

Split the path into total East-West movement and total North-South movement.

Subtract walks that go opposite ways. Then you get a right-angled triangle.

Which rule gives the distance?

The distance is the longest side of that triangle.

Use the rule: distance squared equals side one squared plus side two squared.

Which triangles come up most?

3-4-5, 6-8-10, 5-12-13 and 8-15-17. Their sides are whole numbers.

Learn these and you can skip the squaring.

Can you see one solved?

Mohan walks 9 km North, then turns right and walks 12 km.

The two sides are 9 and 12. This is 3 times the 3-4-5 triangle, so the distance is 15 km.

Why this works: 9 and 12 are 3 times 3 and 4, so the long side is 3 times 5, which is 15.

Can you try a path with an opposite walk?

Sita walks 8 km East, then 3 km West, then 4 km North.

The East-West total is 8 – 3 = 5 km East. The North-South total is 4 km North.

The distance squared is 5 x 5 + 4 x 4 = 41, so it is a little more than 6 km. She is North-East of the start.

Why this works: the West walk cancels part of the East walk before you use the triangle rule.

Can you try a 5-12-13 case?

Kabir walks 5 km South and then 12 km West.

The sides are 5 and 12, so the distance is 13 km. He ends South-West of the start.

Why this works: 5-12-13 is a whole-number triangle, so no squaring is needed.

How do you handle two walkers?

Start both from the same dot. Draw each path in a different colour.

Then compare the two end points.

What do you compare?

Look at where one end point lies from the other.

For example, if one person is above and to the right, that person is North-East of the other.

Can you see one solved?

Leena walks 10 km North. Divya walks 10 km West from the same point.

Divya is below and to the left of Leena. So Divya is South-West of Leena.

Why this works: from Leena, Divya is 10 km down and 10 km left.

What mistakes cost marks?

Forgetting that a turn depends on the way you now face.

Answering from the start when the question asks from the end point.

How do you stay quick?

Keep the compass in the corner of your sketch. Mark every corner with a letter.

Use the common right triangles for distance.

How should you practise?

Solve three walking puzzles a day and sketch every one.

Then try the quiz below and read every explanation.