Journeying in Numberland
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Age
7 to 11
Challenge level
Problem
Tom and Ben are in Numberland in the district called Addition.
They have a map which looks like this:
Image
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:
Image
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.
Image
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):
Image
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?
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 are in Numberland in the district called Addition.
They have a map which looks like this:
Image
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:
Image
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.
Image
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):
Image
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?
Why do this problem?
This problem will give learners a chance to make predictions and generalisations. It also provides practice in simple addition and subtraction, and later in multiplication and division. It draws out the inverse relationship between the pairs of operations but it also encourages children to think about the order of operations.
You will need copies of this sheet, and for the second part of the problem this sheet. Squared paper might also be useful.
Possible approach
You could start by showing the first part of the problem to the whole group and by explaining the setting for the problem. A small scale version could be drawn out on the playground or on the hall floor so that the game can be played practically. The first 'journeys' of both boys could be worked out at this stage.
After this introduction, the group could work in pairs so that they are they are able to talk through their ideas with a partner, using copies of the first sheet. Encourage them to find interesting routes that use subtraction as well as addition. Routes can be drawn using different colours but pairs may well need more than one copy of the sheet. Children may need to use jottings
to keep track of their calculations and these could be done on paper or mini whiteboards, for example.
Before having a go at the second part of the problem (multiplication and division), encourage pairs to predict what they think might happen. You may feel that calculators could be used for checking results at this stage.
At the end of the lesson, bring the whole group together again to discuss their findings. They could show their most interesting and/or longest routes. Were they surprised by the results? Why do they think this happened? Although this task focuses only on numerical operations, the explanation of the results demands a very sound understanding of the number system.
Key questions
Can you find a more interesting way to go that uses subtraction as well as addition?
Do you notice anything about those answers?
Can you find a more interesting way to go that uses division as well as multiplication?
Would it be a good idea to use a calculator to check those results?
Can you explain your findings?