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Journeying in Numberland

Stage: 2 Challenge Level: Challenge Level:1

Tom and Ben are in Numberland in the district called Addition.
They have a map which looks like this:
map

They are at point B and they begin their journey with ten points.
For every square they walk to the right on the map, they add five.
For every square they walk to the left on the map, they take away five.
If they go North (up on the map), they add two for every square, and if they go South (down on the map), they take away two for every square.


First they make these journeys:
 map

The blue line shows Tom's journey and the green line shows Ben's.
How many points do they have each when they reach E?
Do you notice anything?

Here is a different grid for you to make up some journeys of your own, beginning at B and ending at E.
map

You can download and print off this sheet which has two copies of the grid map.
What do you notice about your different journeys?
Can you explain your observations?


After they had explored in the district called Addition in Numberland, Tom and Ben go on to the district called Multiply.
Here they have a new map which looks like this (here are two copies of the map):
map


They explore here too. Each time they start at B with $10$ points and make their way to E. Try lots of journeys yourself.

What do you notice about the journeys this time?
Can you explain why this happens?