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Consecutive Negative Numbers

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Take, for example, four consecutive negative numbers, say $$^-7, ^-6, ^-5, ^-4$$ Now place $+$ and/or $-$ signs between them.

e.g.
           

There are many more possibilities. Try to list all of them.

Now work out the solutions to the various calculations.

e.g. 
           

Choose a different set of four consecutive negative numbers and repeat the process.

Take a look at both sets of solutions. Notice anything?
Can you explain any similarities?

Can you predict some of the solutions you will get when you start with a different set of four consecutive negative numbers?
Test out any conjectures you may have.

Can you explain and justify your findings?