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Published 2008 Revised 2013
Model  Calculation  Outcome 
Adding puffs of hot air  Adding a positive number  Increase (in height) 
Adding sandbags  Adding a negative number  Decrease (in height) 
Subtracting puffs of hot air  Subtracting a positive number  Decrease (in height) 
Subtracting sandbags  Subtracting a negative number  Increase (in height) 
Model  Calculation  Outcome 
Buy good players  Adding a positive number  Increase (in league position) 
Buy bad players  Adding a negative number  Decrease (in league position) 
Sell good players  Subtracting a positive number  Decrease (in league position) 
Sell bad players  Subtracting a negative number  Increase (in league position) 
$2 + 1 = 3$
$3  2 = 1$
$3  1 = 2$

Top + Bottom = Whole
Whole  Top = Bottom
Whole  Bottom = Top

$4 + (^1) = 3$
$3  4 =$ $^1$
$3  (^1) = 4$

Top + Bottom = Whole
Whole  Top = Bottom
Whole  Bottom = Top

$5 + (^2) = 3$
$3  5 =$ $^2$
$3  (^2) = 5$

Top + Bottom = Whole
Whole  Top = Bottom
Whole  Bottom = Top

This is $^4$
$1 + (5) =$ $^4$
$^4  1 =$ $^5$
$^4  (^5) = 1$

Top + Bottom = Whole
Whole  Top = Bottom
Whole  Bottom = Top

$^1 + (^3) =$ $^4$
$^4  (^1) =$ $^3$
$^4  (^3) =$ $^1$

Top + Bottom = Whole
Whole  Top = Bottom
Whole  Bottom = Top

$6 + (^2)$  =  ? 
=  
$6 + (^2)$  =  4 
$4 + (^6)$  =  ? 
=  
$4 + (^6)$  =  $^2$ 
$^2 + (^3)$  =  ? 
=  
$^2 + (^3)$  =  $^5$ 
We can subtract a negative number from a negative number and end up with a negative solution:
This represents $^8$ 
And this represents $^8$
subtract $^3$

Leaving us with $^5$ 
represents $5$ 
but so does this 
represents $^4$ 
but so does this 