Using Sass Variables with CSS3 Media Queries

Similar to Philipp Zedler's answer, you can do it with a mixin. That lets you have everything in a single file if you want.

    @mixin styling($base-width) {
        // your SCSS here, e.g.
        #Contents {
            width: $base-width;
        }
    }

    @media screen and (max-width: 1170px) {
        @include styling($base-width: 960px);
    }
    @media screen and (min-width: 1171px) {
        @include styling($base-width: 1160px);
    }

This is simply not possible. Since the trigger @media screen and (max-width: 1170px) happens on the client-side.

Achieving your expected result would only be possible if SASS grabbed all rules and properties in your stylesheet containing your $base_width variable and copied/changed them accordingly.

Since it won't work automatically you could do it by hand like this:

@media screen and (max-width: 1170px)
      $base_width: 960px // you need to indent it to (re)set it just within this media-query
      // now you copy all the css rules/properties that contain or are relative to $base_width e.g.
      #wrapper
          width: $base_width
          ...

@media screen and (min-width: 1171px)
    $base_width: 1160px
      #wrapper
          width: $base_width
          ...

This is not really DRY but the best you can do.

If the changes are the same every time you could also prepare a mixin containing all the changing values, so you wouldn't need to repeat it. Additionally you can try to combine the mixin with specific changes. Like:

@media screen and (min-width: 1171px)
    +base_width_changes(1160px)
    #width-1171-specific-element // additional specific changes, that aren't in the mixin
        display: block

And the Mixin would look like this

=base_width_changes($base_width)
    #wrapper
        width: $base_width