Journeys in Numberland
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Problem
Tom and Ben were in Numberland in the county of Addition.
They had a map which looked like this:
They were at point B and they began 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 added two for every square, and if they go South (down on the map), they take away two for each square.
First they made these journeys:
The blue line shows Tom's journey and the green line shows Ben's.
How many points did they have each when they reached 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.
After they had explored in the county of Addition in Numberland, Tom and Ben went on to the county of Multiply.
Here they had a new map which looked like this (here are two copies of the map):
They explored here too. Each time they started at B with $10$ points and made their way to E. Try lots of journeys yourself.
What do you notice about the journeys this time?
Can you explain why this happens?
Getting Started
Using a pencil and paper to keep a running total of your score as you move along the path might help.
How will you make sure you remember the paths you've tried?
How will you remember the score each time?
Student Solutions
Although this problem 'just' involved some calculating, explaining the results was a little more tricky. Charlie, Tom, James and Jonathan from Avenue Junior School looked at the journeys that Tom and Ben made:
Tom starts at B then goes right and gets $5$ more which takes him to $15$.
He then goes up four adding $8$ to his score taking him up to $23$.
He then goes right and gets $5$ more taking him to $28$.
So Tom's score ends up at $28$.
Ben starts at B and goes to the left two squares taking his score to $0$.
He then goes up two squares taking his score to $4$.
He then goes four to the right adding $20$ to his score so he now has $24$.
He then goes up two squares taking his score to $28$.
So Ben's score is $28$, which, oddly enough, is the exact same score as Tom.
Jessica from Egerton Primary School described how her class found lots of different ways of going from B to E on the grid:
We found that in all four of our journeys we got $28$. The whole of the class got $28$ at least two times. Aidan and I used red and blue crayon to mark which way we went. Aidan and I think that we got $28$ because when you go one way you always have to go back the same way.
Harry, who goes to St Anne's Primary School, had a go at the routes on all three grids. He wrote:
grid 1) We have found out that they both ended up on $28$. It doesn't matter how long or short their trial is, it will end up on the same answer.
grid 2) On this grid we have found out that the answer is the same as on the first grid. However, the grid can be as big or as small as you like but you will still have an answer of $28$.
grid 3) On the final grid the answer is always $800,000$.
Harry also played around with the grids a bit, changing addition to multiplication and subtraction to division or vice versa. This changed the answer but the answer was still always the same, no matter what route you took.
Jack and Skye from Swavesey Primary had a go at the third grid. They said:
No matter what way you go, you will always end up with the same anwser. We tried a few ways using our computer screen following the journey with our fingers, this was on the multiplication and divide square.
Every time we did it we ended up with the anwser of $800000$.
So we tried even more journeys and always got the same answer of $800000$.
Above, Jessica began to explain why she thought she always got $28$ on the first two grids. The Maths Galaxy Explorers from North Walsham Junior School clearly thought very hard about this:
We spent our week trying to solve the Journeys in Numberland mathematical challenge. Here is our solution:
Whichever route or sequence you take, you will always get the same answer, as long as the grid is the same. This is because every time there is a times and a divide, or an add and a takeaway, and every time they cancel each other.
For example if you go right for a times by $2$ and left for a divide by $2$, they cancel each other out.
For example: $3 \times 2 = 6$ but if you divide it back ... $6 \div2 = 3 $ (which is just halving by the way! and $\times2$ is doubling!) It just cancels out your calculation.
Or $10 + 9 = 19$ OPPOSITE $19 - 9 =10$ tada!!! (It's like dividing and times, it's just undoing again, just like the undo button on your computer!) .
Well done too to Barbara, Cong and Nazra from Arnhem Wharf Primary School who also realised that they would always get the same score for a particular grid.
Teachers' Resources
Journeys in Numberland
Tom and Ben were in Numberland in the county of Addition.
They had a map which looked like this:
They were at point B and they began 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 added two for every square, and if they go South (down on the map), they take away two for each square.
First they made these journeys:
The blue line shows Tom's journey and the green line shows Ben's.
How many points did they have each when they reached 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.
After they had explored in the county of Addition in Numberland, Tom and Ben went on to the county of Multiply.
Here they had a new map which looked like this (here are two copies of the map):
They explored here too. Each time they started at B with $10$ points and made their way to E. Try lots of journeys yourself.
What do you notice about the journeys this time?
Can you explain why this happens?
Why do this problem?
Possible approach
Key questions
Possible extension
You could ask learners to find routes which give the smallest or largest possible answer, staying within the grid. Or they could find routes which involve exactly two subtractions, for example. Alternatively, you could challenge some learners to create their own grid on which different routes between the same starting and end points produce different answers. Why does their grid work? (A combined add-subtract and multiply-divide grid leads to the need for brackets.)
Possible support
Some children might benefit from starting on the Stage 1 version of this problem: The Add and Take-away Path.