The image 
is an example taken from a recent introductory algebra textbook that is currently
being brailled by a commercial braille publishing house. The transcriber brought this example of a spatial
arrangement to the
recent NBA Meeting held in Seattle in October 2005.
The next two sections show Nemeth and UEB transcriptions, respectively. The proposed UEB transcription doesn't fit in a 40-cell width so would probably need to be addressed in some other fashion. However, this could make it difficult for the student to understand the point of "using columns."
Note that this is not advanced mathematics. It is simply an illustration of the relationship between the multiplication the student previously learned in arithmetic and the same process in algebra.
A proposed Nemeth braille transcription for the spatial math, which uses keying for the line labels, is shown below in both ASCII Braille and SimBraille©.
Cell positions:
1 2 3 4
1234567890123456789012345678901234567890
,,example #7 ,multiply3
(5x^3+x-4)(-2x^2+3x+6)_4
5x^3" + x- 4
- 2x^2"+ 3x+ 6
3333333333333333333333333333333333333
30x^3" + 6x-24 #a
15x^4" + 3x^2"-12x #b
-10x^5" - 2x^3"+ 8x^2 #c
3333333333333333333333333333333333333
-10x^5"+15x^4"+28x^3"+11x^2"- 6x-24
#a ,multiply+ by #6
#b ,multiply+ by #3x
#c ,multiply+ by -#2x^2
,,example #7 ,multiply3
(5x^3+x-4)(-2x^2+3x+6)_4
5x^3" + x- 4
- 2x^2"+ 3x+ 6
3333333333333333333333333333333333333
30x^3" + 6x-24 #a
15x^4" + 3x^2"-12x #b
-10x^5" - 2x^3"+ 8x^2 #c
3333333333333333333333333333333333333
-10x^5"+15x^4"+28x^3"+11x^2"- 6x-24
#a ,multiply+ by #6
#b ,multiply+ by #3x
#c ,multiply+ by -#2x^2
How should the spatial arrangement be done in UEB? Note that the direct solution given below is 49 cells wide even without dealing with the line labels. This is a net 12 cells or about 30% wider than the corresponding spatial arrangement in Nemeth. The 10 number signs and 6 two-cell operators add 16 cells while the special rule that a baseline indicator is not needed after a simple subscript substracts 4 cells as compared to Nemeth.
Cell positions:
1 2 3 4
1234567890123456789012345678901234567890123456789
,,example #7 ,multiply3
"<#ex9#c"6x"-#d">"<"-#bx9#b"6#cx+#f">4
#ex9#c "6 x"- #d
"- #bx9#b"6 #cx"+ #f
"333333333333333333333333333333333333333333333333
#cjx9#c "6 #fx"-#bd
#aex9#d "6 #cx9#b"-#abx
"-#ajx9#e" "- #bx9#c"6 #hx9#b
"333333333333333333333333333333333333333333333333
"-#ajx9#e"6#aex9#d"6#bhx9#c"6#aax9#b"- #fx"-#bd
,,example #7 ,multiply3
"<#ex9#c"6x"-#d">"<"-#bx9#b"6#cx+#f">4
#ex9#c "6 x"- #d
"- #bx9#b"6 #cx"+ #f
"333333333333333333333333333333333333333333333333
#cjx9#c "6 #fx"-#bd
#aex9#d "6 #cx9#b"-#abx
"-#ajx9#e" "- #bx9#c"6 #hx9#b
"333333333333333333333333333333333333333333333333
"-#ajx9#e"6#aex9#d"6#bhx9#c"6#aax9#b"- #fx"-#bd
(5x^3+x-4)(-2x^2+3x+6)_4 Nemeth "<#ex9#c"6x"-#d">"<"-#bx9#b"6#cx"6#f">4 UEB
First posted November 2, 2005.
Re-posted with minor corrections November 3, 2005.