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'Number Tracks' printed from http://nrich.maths.org/

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Ben's class were making number tracks and cutting them up.

number track of 0 to 12 ... ... ...

First they cut them into twos. Ben started with zero

Pairs of numbers: 0,1 2,3 4,5 etc ... ... ...

but Miles started with one:

numbers in pairs: 1,2 3,4 5,6 etc ... ... ...

Then they both added up the numbers on each piece.

Ben wrote: $0 + 1 = 1$, $2 + 3 = 5$, $4 + 5 = $... ... ...

Miles wrote: $1 + 2 = 3$, $3 + 4 = 7$, $5 + 6 =$ ... ... ...

What patterns could they see?

Alice cut her number track into $3$s and added up the numbers on each one.

numbers in threes: 0,1,2 3,4,5 etc ... ... ...

Winston made a longer number track which he cut into $5$s and he added up the numbers on each one.

numbers in fives: 0,1,2,3,4 5,6,7,8,9 etc ... ... ...

What could they discover about the sum of the numbers on their strips of number track?

What other patterns can you find?