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USING THE BRAILLE
MATHEMATICS CODE
+ - ÷ ≤ ≥ ∞
± ∞ ≈ < > +
÷ - ≤ ≠ √
Using the braille mathematics codeThis document is based on the braille code used in the United Kingdom for presenting mathematics in braille, and includes advice about specialist mathematical symbols and layout conventions. Written for teachers, it covers the braille needed for maths topics and conventions up to GCSE level. It is essential for young people who may use braille in public examinations to learn to recognise and understand braille maths notation and present their working and answers correctly.
“Braille Mathematics Notation” published in 2005 by the Braille Authority of the United Kingdom is a more comprehensive guide giving greater detail and covering the complexities of the maths code up to and including postgraduate study, including pure, applied maths and statistics.
1 Number notation
This section covers the braille required for positive and negative numbers, arithmetic operators, using brackets, indexes and roots.
1a) Whole numbers
Numbers are written in braille using the NUMERAL SIGN (number sign), dots 3, 4, 5 and 6 #, and the letters a to j represents the digits -9 and 0.
0 is#j 7 is #g is #a 8 is #h 2 is#b 9 is #i 3 is #c 0 is #aj 4 is #d 35 is #ce 5 is #e 00 is #ajj 6 is #f 296 is#bif
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1b) The mathematical comma
Large numbers in print are sometimes written with a comma and sometimes with a space: eg 0,000 or 0 000. In braille a mathematical comma (dot 3) is always inserted as follows:
30 000 or 30,000 #cj'jjj
25 000 or 25,000 #abe'jjj
,000 000 or ,000,000 #a'jjj'jjj
1c) Negative numbers The minus sign (dots 5,6 followed by dots 3,6) written unspaced before the numeral sign indicates a negative number:
– 4 ;-#d – 80 ;-#hj
Dots 5,6 are not used in a minus sign after an equals sign.
1d) Arithmetic operation signs
The signs for add, subtract, multiply and divide and equals always have a space BEFORE them, but are followed immediately by the number.
;6(dots 5,6 followed by dots 2,3,5) + ADD
;- (dots 5,6 followed by dots 3,6) – SUBTRACT
;8 (dots 5,6 followed by dots 2,3,6) x MULTIPLY
;4(dots 5,6 followed by dots 2,5,6) ÷ DIVIDE
;7 (dots 5,6 followed by dots 2,3,5,6) = EQUALS
For “approximately equal to” and “not equal to” see section k).
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Examples:
4 + 2 = 6 #d;6#b;7#f
4 – 2 = 2 #d;-#b;7#b
4 x 2 = 8 #d;8#b;7#h
4 ÷ 2 = 2 #d;4#b;7#b
230 – 28 + 42 – 6 = 238
#bcj;-#bh;6#db;-#f;7#bch
3 x 6 = 48 = 96 ÷ 2
#c;8#af;7#dh;7#if;4#b
1e) Using brackets with positive and negative numbers
When it is necessary to add or subtract positive and negative numbers two signs can occur together. If this occurs, the signed number should be enclosed in ROUND BRACKETS; “gh” (dots ,2,5) is used for open round brackets and “ar” (dots 3,4,5) is used for close round brackets:
2 – – 3 = 5 is #b;-<;-#c>;7#e
or
2 – –3 = 5 is #b;-<;-#c>;7#e
When multiplying or dividing positive or negative numbers, the multiplication sign can be omitted, and the division sign can be replaced with a / sign (dots 4,5,6 followed by dots 3,4) _/ – 3 ( – 6 ) = 8 ;-#c<;-#f>;7#ah
–8 / +3 = – 6 ;-#ah_/;6#c;7-#f
Often it may be necessary to use braille brackets to make the layout clear, even when they are not used in print. Additional explanation of the notation for brackets will be found later in this section.
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1f) Punctuation
When punctuation is required with mathematical expressions, the MATHEMATICAL SEPARATION SIGN “dot 6” must be used. If punctuation is used straight after a number without the separation sign, the punctuation sign could be read as the denominator of a fraction.
44. is #add,4
25, is #be,1
36? is #cf,8
55! is #ee,6
Long braille calculations need to continue from the end of one line of braille to the beginning of the next. The MATHEMATICAL HYPHEN (dot 5) is used, unspaced at the end of the line, to indicate that the rest of the calculation is on the next line.
+ 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0
#a;6#b;6#c;6#d;6#e;6#f;6#g"
;6#h;6#i;6#aj
1g) Indices or “powers”
The superscript (or power) sign is the “ING” contraction (dots 3,4,6) + followed by the number written in the lower part of the cell without the numeral sign. Thus “lower 2” (dots 2,3) denotes “squared”, “middle 3” (dots 2,5) denotes “cubed”, “lower 4” (dots 2,5,6) denotes “the power 4” and so on. The power sign follows the number, unspaced.
42 #d+2 33 #c+3 84 #h+4
5
If the superscript is negative, the minus sign (dots 5,6: dots 3,6) is placed between the superscript sign and the lower case number.
4–3 #d+;-3
If the superscript is a positive whole number with a + sign, the superscript sign must be followed by a plus sign and an ordinary number sign. 4+3 #d+;6#c
6.5 #f+#a1e
2/2 #ab+#a2
1h) Subscripts
Subscripts can be used with both numbers and letters (see algebra section) and the rules for their use are different in each case.
The subscript is the "CH" contraction “dots ,6” *followed by the appropriate number written in the lower part of the braille cell without a numeral sign.
02 (binary) #aaja*2
1i) Roots
The root sign is the “SH” contraction (dots ,4,6) % It is written BEFORE the numeral sign, unspaced. The root sign is used on its own for the square root. For the “cube root”, a “middle 3” (dots 2,5) is placed, unspaced, between the root sign and the numeral sign; for the “fourth root”, a “lower 4” (dots 2,5,6) is placed between the root sign and the numeral sign, and so on.
√64 %#fd
3√25 %3#abe
4√6 %4#af
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1j) Inequalities
These signs have a space before and after:
< means “is less than” and is dots 2,4,6 in braille: [
2 < 4 #ab[#ad
> means “is greater than” and is dots ,3,5 in braille: o
4 > 2 #ado#ab
These signs have a space before but not after them:
≤ means “is less than or equal to” and is "dots 2,4,6: dots 2,3,5,6" in braille: [7
x ≤5 ;x[7#e
-9 < n ≤20 ;-#i[n[7#bj
≥means “is greater than or equal to” and is “dots ,3,5: dots 2,3,5,6” in braille: o7
p ≥0 ;po7#aj
1k) Approximations
These two signs have a space before but not after:
≈ means “is approximately equal to” and is “dots 4,5,6: dots 2,3,5,6” in braille: _7
999 ≈000 #iii_7#ajjj
≠ means “is not equal to” and is “dot 5: dots 2,3,5,6” in braille: "7
999 ≠000 #iii"7#ajjj
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1l) Brackets
In mathematics there are three types of brackets:
( . . . . . ) ROUND BRACKETS
. . . . . CURLY BRACKETS
[ . . . . . ] SQUARE BRACKETS
In braille:
( is the “GH” contraction (dots ,2,6) and ) is the “AR” contraction (dots 3,4,5).
< . . . . . >ROUND BRACKETS
is “OW” (dots 2,4,6) and is “O” (dots ,3,5).
[ . . . . . OCURLY BRACKETS
[ is the “OF” contraction (dots ,2,3,5,6) and ] is the “WITH” contraction (dots 2,3,4,5,6).
( . . . . . )SQUARE BRACKETS
Whichever form of bracket is used in print, the same form should be used in braille.
Examples using round brackets (with numbers):
(43 – 3) x 3 <#dc;-#ac>;8#c
(2 + 4)(0 + 2) <#b;6#d><#aj;6#b>
(47 x 20)/(56 – 8) <#dg;8#bj>_/<#ef;-#h>
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Examples using curly brackets:
3 x 2 x (4 – ) = 08
#c;8[#ab;8<#d;-#a>O;7#ajh
O = , 3, 5, 7 ,o;7[#a#c#e#go
E = 2, 4, 6, 8 ,e;7[#b#d#f#ho
Square brackets are usually used to enclose a long computation.
Example using all three types in one sum:
5[2 – 8 – (3 – )] = – 20
#e(#ab;-[#ah;-<#c;-#a>o);7-#bj
2 Parts of a whole
This section covers the braille code for fractions, decimals, percentages and ratio.
2a) Fractions
Fractions are written with the numeral sign followed by an upper number for the numerator (the top number) and a lower number for the denominator (the bottom number).
/2 #a2one half /4 #a4one quarter
2/3 #b3two thirds 7/8 #g8seven eighths
In mixed numbers, the whole number is written unspaced from the fractional part.
5 3/4 #e#c4five and three quarters
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2b) Decimals
The decimal point is dot 2. It comes unspaced from the numbers before and after it. 3·4 #c1d 0·75 #j1ge ·5 #1e
The recurring sign is “dot 5” and it is placed before the digit or digits of the decimal that recur. In braille only one dot 5 is used when more than one digit recurs.
0·6. #j1"f
2·0096. #b1jji"f
0·.42857
. #j1"adbheg
2c) Percentages
The percentage sign is “middle c, p” (dots 2,5 followed by dots ,2,3,4). It is written after the number, with a space between the number and the percentage sign:
% 3p 333% #cc#a33p
50% #ej3p 7.5% #ag1e3p
2d) Ratio
The ratio sign in braille maths is dot 6 followed by dots 2,5. There is no space either before or after it.
2:3 #b,3#c
5:6:7 #e,3#f,3#g
0
3 Presentation and working of calculations One of the major difficulties with braille mathematics, even at the most basic level, is the layout of simple sums:
– the linear method (across the page) OR
– the vertical method (down the page).
Each method has advantages and disadvantages.
A set of examples for the linear and the vertical layout of Addition, Subtraction, Multiplication and Division in braille are given in detail in Appendix A.
4 Units of measurement This section describes how to express units of length, area, volume, weight and capacity. These braille units are written AFTER the number and are SPACED from the number. If the unit is represented by a single letter, such as “m” for metre(s) it must be preceded by a LETTER sign (dots 5,6), so that it will not be read as a word.
4a) Metric units of length
millimetre mm written “mm” mm
centimetre cm written “cm” cm
decimetre dm written “dm” dm
metre m written “dots 5,6 (letter sign) m” ;m
kilometre km written “km” km
Examples:
25 mm #bemm 50 cm #ejcm
8 dm #hdm 00 m #ajj;m
250 km #bejkm
When there are two units together, the general rule is to use the decimal point and the symbol of the larger unit. The basic rule is to follow the print.
.75 m #a1ge;m or
m 75 cm #a;m#gecm
4b) Metric units of weight
milligram mg written “mg” mg
decigram dg written “dg” dg
gram(me) g written “dots 5,6 (letter sign) g” ;g
kilogram kg written “kg” kg
tonne t written “dots 5,6 (letter sign) t” ;t
Examples:
675 mg #fgemg 6 dg #fdg
250 g #bej;g 50 kg #ejkg
35 t #ce;t
As with the metric units of length, it is best to use the decimal point and write the larger unit symbol. The basic rule is to follow the print.
.375 kg #a1cgekg
kg 375 g #akg#cge;g
5.695 t #e1fie;t
5 t 695 kg #e;t#fiekg
2
4c) Metric units of capacity
millilitre ml written “ml” ml 700 ml #gjjml
centilitre cl written “cl” cl 70 cl #gjcl
decilitre dl written “dl” dl 7 dl #gdl
litre l written “dots 5,6 l”;l 8 l #h;l
“Litre” can also be written as L, (capital l), this is “dots 5,6 (letter sign), dot 6 (capital sign), l” ;,l
4d) Metric units of area
For area, use the “squared” sign. In braille, this is the superscript (power) sign “ING” (dots 3,4,6) followed by lower 2 (dots 2,3) +2
The squared sign follows the unit symbol, unspaced.
These are the three most common metric units of area:
mm2 mm+2 50 mm2 #ejmm+2
cm2 cm+2 225 cm2 #bbecm+2
m2 ;m+2 400 m2 #djj;m+2
The other metric unit of area is the hectare (which is 0,000 square metres or 0,000 m2). The symbol for this is “ha”.
50 ha #ejha
3
4e) Metric units of volume
For volume, use the “cubed” sign. In braille, this is the superscript (power) sign, “ING” (dots 3,4,6) followed by a middle 3 (dots 2,5) +3
These are the three most common metric units of volume:
mm3 mm+3 000 mm3 #ajjjmm+3
cm3 cm+3 500 cm3 #ejjcm+3
m3 ;m+3 27 m3 #bg;m+3
4f) Imperial units
Imperial units of length, weight, capacity, area and volume are in Appendix B.
5 Units of time This section gives the braille code for periods of time, such as seconds, hours and days, writing times in both 2- and 24-hour clock, and writing the date.
5a) Units of time
second s written “letter sign (dots 5,6) s” ;s
45 s #de;s
minute min written “m, IN contraction” m9
5 min #aem9
4
hour h written “letter sign (dots 5,6) h” ;h
2 h #ab;h
hr written "hr" hr
3 hr #c hr
day d written “letter sign (dots 5,6) d” ;d This is used when day is written as “d” in print. If the word
is printed in full, write “dot 5, d” "dwhich is the usual contraction for day.
0 days #aj"ds 0 d #aj;d
week wk written “wk” wk
6 wk #fwk
month written in full both in print and in braille
3 months #cmon?s
year yr written “y r full stop" (dots 2,5,6) yr4
25 yr #beyr4
The full stop is necessary so that there will be no confusion with the “your” contraction.
With all these contractions, remember that the braille follows the print. If the units are written out in full in print, then they are written out in full in braille.
5
5b) Writing the time
There are two ways of writing time, either the a.m./p.m. format or the 24-hour clock format.
When times of the day in print are written with a full stop between the hours and the minutes (e.g. 5.30), they should be brailled in the same way as decimal numbers, i.e. with a dot 2.
This is then followed by a.m. or p.m., which are written spaced from the number, but without a space between the two letters.
2.45 a.m. #b1dea4m4 0.5 p.m. #aj1aep4m4
Where “am” or “pm” are written without full stops they are written in braille as follows:
am written “letter sign (dots 5,6) am” ;am
pm written “letter sign (dots 5,6) pm” ;pm
2.45 am #b1de;am 0.5 pm #aj1ae;pm
In older braille texts you may come across times written with a second numeral sign separating the hours and minutes:
2.45 a.m. #b#dea4m4
If in print the time is written with a colon or space separating the hours and minutes as in 5:30 or 5 30, the hours and minutes are separated by a repeated numeral sign:
5:30 or 5 30 is written#e#cj
For 24-hour clock time, there are two formats, and the braille must follow the print.
Sometimes the time is written as one unspaced sequence of numbers. In this case, the braille follows the print
745 #agde 0030 #jjcj 0925 #jibe
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Where a 24-hour clock time uses a full stop to separate the hours and minutes a dot 2 is used as the separator in braille. Where a colon or space is used the hours and minutes are separated by a repeated numeral sign:
7.45 #ag1de 00.30 #jj1cj 09.25 #ji1be
7 45 #ag#de 00 30 #jj#cj 09 25 #ji#be
7:45 #ag#de 00:30 #jj#cj 09:25 #ji#be
When a period of time is indicated by two times joined by a hyphen the numeral sign is repeated after the hyphen:
2.30 - 4.30 #b1cj-#d1cj
4.30 - 6.30 #ad1cj-#af1cj
4.30 - 6 am #d1cj-#f;am
5c) Dates
As a general rule, the date is transcribed into braille as it is presented in print:
25 May 2005 #be,may#bjje
However, if the date is abbreviated, then the day, month and year are separated by a numeral sign, no matter what symbol is used in print, for example an oblique stroke, a hyphen or a full stop:
6.07.05 or 6/07/05 or 6-07-05 #af#jg#je If you are transcribing dates where a hyphen is used to link together two dates, for example 7-25 June, then the hyphen, “dots 3,6”, is used. The force of the numeral sign does NOT carry over the literary hyphen:
7 - 25 June #ag-#be,june
When dates are joined by a dash, the second number must also have a numeral sign:
94 – 8 #aiad--#ah
2000 – 2005 #bjjj--#bjje
7
When using an apostrophe the braille follows the print
960’s #aifj's
the ’90’s !#'ij's
or without an apostrophe “the 960s” !#aifjs
“the 990s” !#aiijs
6 Units of temperature
The degrees sign, o, (in braille, “lower j, dots 3,5,6”), is followed by capital C for Centigrade/Celsius, and capital F for Fahrenheit. These units follow the number UNSPACED.
35oC #ce0,c –5oC ;-#e0,c
70oF #gj0,f –0oF ;-#aj0,f
7 Compound units
Most compound units used in mathematics are beyond the scope of this booklet. Refer to Braille Mathematics Notation 2005, produced by BAUK, for further details.
For compound units of speed and density, the two units are separated by an oblique stroke (dots 4,5,6 followed by dots 3,4) _/
4 m/s (metres per second) in braille is#d;m_/s
NOTE: The letter sign is not necessary within compound units. For example, in metres per second the letter sign is needed before the m, but a letter sign is not needed in front of the s.
8
7a) Units of speed:
km/h (kilometres per hour) km_/h
m/s (metres per second) ;m_/s
In print miles per hour can be abbreviated as
m.p.h. or mph
In both cases the braille follows the print:
m.p.h. m4p4h4
mph ;mph
If MPH is written with print capitals and no full stops, then braille does NOT follow the print unless capitals are shown generally in the transcription.
MPH ;mph Examples:
80 km/h #hjkm_/h 33 m/s #cc;m_/s
70 m.p.h. #gjm4p4h4 70 mph #gj;mph
70 MPH #gj;mph
7b) Units of density
The most common unit of density is g/cm3. As with units of volume, use the cubed sign, which is superscript (power) sign “ING” (dots 3,4,6) followed by middle 3 (dots 2,5)
6.4 g/cm3 #f1d;g_/cm+3
9
8 Units of currency
8a) British currency
“£” in braille is the letter "l" l written before the numeral sign without a letter sign. “p” (pence) in braille is “letter sign (dots 5,6) p” ;p, and follows the number unspaced.
Examples:
50p #ej;p £50 l#ej
The mathematically correct way to write “four pounds fifty pence” is £4.50 which is l#d1ej
Sometimes you may see “£4.50p” in print, which is mathematically incorrect; however, in braille transcription the print style is followed.
A “£” sign without a number is brailled as @l
Example:
Give your answer in £
GIVEYRANSW]9@l
8b) US dollar
The dollar, $, is written as “lower d (dots 2,5,6)”, unspaced, before the numeral sign.
The cent, c, written as “letter sign (dots 5,6) c” unspaced after the number.
When using both units, use the $ sign and the decimal point to separate $ from cents.
Examples:
$04#aj 50c#ej;c $25.504#be1ej
20
8c) European Euro
The Euro (€) is “dot 4: e – dots 1,5” @e, written before the numeral sign and unspaced.
€5 or 5 Euro@e#e €7,85@e#g1he
8d) Japanese Yen
The Yen is “dot 4: y – dots ,3,4,5,6” @y, written before the numeral sign and unspaced.
00 Yen@y#ajj
8e) Other foreign currencies
For other foreign currency, please refer to British Braille 2004, published by BAUK.
9 Use of the letter sign in Algebra
This section explains why the letter sign can be needed for clarity, when single letters appear in a phrase, roman numerals are needed, or the contents of brackets could be confused with the bracket signs.
9a) Lone letters in sentences
If in doubt, use the letter sign in algebraic expressions.
Some general rules for the use of the letter sign:
If the letter is used on its own, in a sentence, then the letter sign must be used or it will be confused with the word for which that letter stands:
e.g. If x is a positive whole number
,if;xisapositive:olenumb]
The letter sign is needed before the “x” so that it will not be read as “it”.
2
When the letter begins a mathematical expression in ordinary print, such as:
p is a negative whole number
;pisanegative:olenumb]
The letter sign is needed before the letter “p” so that it will not be confused with the word for which “p” stands, “people”.
9b) Letters a - j written after a number
Always use the letter sign with letters a – j when these letters follow a number
Examples:
with letter sign: a b c d e f g h i j
3a #c;a 5b#e;b c #aa;c
without letter sign: k l m n o p q r s t u v w x y z
2x #bx 5y#ey 3z #cz
9c) Roman numerals
Roman Numerals are normally written as in literary braille. If small, they are preceded by the dots 5,6 letter sign.
i;i ii;ii iii;iii iv;iv
v;v vi;vi vii;vii viii;viii
ix;ix x;x
When labelling questions, sub-sections often use Roman Numerals, in brackets. In these cases, the standard literary brackets (lower “g” in braille – dots 2,3,5,6) are used, together with the letter sign.
(iii)7;iii7 (ix)7;ix7 (xiv)7;xiv7
22
The use of capital letters in mathematical braille follows the new rules of British Braille 2004. The double capital indicator “dot 6 dot 6” is now used for a sequence of two or more capital letters. Single capital letters standing alone now require a letter sign before the capital sign .
III,,iii IX,,ix XIV,,xiv
I ;,I V;,v X ;,x
Note: Everything which follows the numeral sign is a number, everything which follows the capital sign is a capital letter, and everything that follows the letter sign is a lower case letter until there is a space or another sign.
9d) Brackets
When using curly brackets, and the letter “o” is listed within the bracket, then a letter sign has to be written before the letter “o” so as not to confuse it with the closing curly bracket, which is also a letter “o”. Thus:
a e i o u[aei;ouo
When using square brackets and the first term is the letter “a”, then a letter sign has to be inserted or the first terms will be read “of a” as the open square bracket is the “OF” contraction in braille. Thus:
[a e i o u](;aeiou)
10 Algebraic fractions and the oblique stroke
The oblique stroke “/ ” , in braille, is dots 4,5,6 followed by the “ST” contraction, “dots 3,4” _/ and it is frequently used in algebra, especially when writing fractions. It becomes necessary when either the numerator or the denominator or both are letters. Fractions in algebra cannot be written as they would be with numerical numerators and denominators.
23
Examples:
x or /x #a_/x
a3 a_/#cor;a_/#c
ab
is written in braille asa_/bor;a_/b
x2 is written in braille as#a_/x+2
If the numerator or the denominator of the algebra fraction is made up of two or more terms, these terms have to be joined together using round brackets: open bracket, <is “dots ,2,6” in braille and the close bracket, >, is “dots 3,4,5” These brackets must be used in braille even if they are not used in print.
(x + y)/z may be written in print as x + yz but in braille the brackets must
be used as follows:
(x + y)/z<x;6y>_/z
x/(y + z) may be written in print as xy + z but in braille the brackets must
be used as follows:
x/(y + z)x_/<y;6z>
(w + x)/(y + z) may be written in print as w + xy + z but in braille the brackets
must be used as follows:
(w + x)/(y + z)<w;6x>_/<y;6z>
Similarly ym x n requires brackets in braille:
y/(m x n)y./<m;8n>
24
11 Examples of algebra layout
11a) Simple operations3a + 4a = 7a #c;a;6#d;a;7#g;a
5x + 3x = 8x #ex;6#cx;7#hx
0z – 6z = 4z #ajz;-#fz;7#dz
20p – 5p = 5p #bjp;-#ep;7#aep
11b) Simple equations2x = 8 #bx;7#h
4a = 2 #d;a;7#ab
x2 = 25 x+2;7#be
a + 3 = a;6#c;7#aa
p – 4 = 9 p;-#d;7#i
2c2 = 32 #b;c+2;7#cb
3a + 5 = 2a + #c;a;6#e;7#b;a;6#aa
11c) Examples using brackets(6x)(3y) <#fx><#cy>
(3a + 5b)(6a – 3b) <#c;a;6#e;b><#f;a;-#c;b>
x(x2) x<x+2>
a(a3) with letter sign ;a<a+3>
a(a3) with no letter sign a<a+3>
(4x + 4y)/2 <#dx;6#dy>_/#b
2(x – 3) #b<x;-#c>
x – (2/x) = 5 x;-<#b_/x>;7#e
(x + )/2 +(x + 2)/3<x;6#a>_/#b;6<x;6#b>_/#c
25
√x + y %<x;6y>
a2x + ax2 = ax(a + x) a+2x;6ax+2;7ax<a;6x>
(ax + b)(cx + d) <ax;6b><cx;6d>
11d) Examples with superscripts
xn x+n xn+2 x+n;6#b x½ x+#a2
11e) Examples with subscripts
If the numerical subscript refers to a letter, then the subscript is written as a lower case number, unspaced from the letter.
x2x2
This also applies if the subscript is a negative number.
y- y;-1
If the subscript is a letter, then the subscript sign must ALWAYS be used and the letters written as upper case.
xn x*n x2n x*#bn
11f) Arrows used in simple mappings
The arrow used in braille, is “dots 2,5 followed by dots ,3,5” 3owith a space before but not after.
x →x + x3ox;6#a
26
11g) Function
The letter “f” “dots ,2,4” is used to denote function, followed by the argument.
y = f(x – a)y;7f<x;-a>
11h) Proportionality
The sign, “is proportional to”, is ∝in print. In braille itis “middle c (dots 2,5) followed by lower g (dots 2,3,5,6) written unspaced”. 37
In braille the “proportion to” sign is positioned with a space before but not after the sign.
y ∝x2y37x+2
12 Geometry
12a) Degree sign
The degree sign in geometry, used as an angle measurement, is the same as the degree sign used for temperature. It is “lower j (dots 3,5,6)” 0 written, unspaced, after the number.
Examples:
90o #ij0 360o #cfj0 45o #de0
12b) Angle
In print, the angle sign is written in one of two ways. Angle CAB can be written ∠CAB or CÂB. In braille, the angle sign is “dots 4,5,6 followed by dots 2,4,6”.
This is followed by the capital letters labelling the angle:
∠ CAB_[,,cabCÂB_[,,cab
27
12c) Triangle sign
In print, the sign for “triangle” is a small triangle before the three letters, “triangle ABC” is shown as ∆ABC. In braille, this is written as “dots ,2,4,6 followed by dots ,4,5” (which is the “ED” contraction followed by “d”) $d.This is followed by the capital letters labelling the triangle:
∆ABC$d,,abc
12d) “Parallel to” sign
In print, the “is parallel to” sign is two parallel lines. This sign is placed between the two pairs of letters.
AB is parallel to CD is written as AB || CD.
In braille, the “is parallel to” sign is “dots 4,5,6: dots 4,5,6” __with a space before, but not afterwards.
AB || CD,,ab__,,cd
Note that AB is normally brailled as ,,ab in maths. In exceptional cases itcan be brailled as ,a,b if it is thought liable in the context to be misreadas the capitalised word “ABOUT”. The same applies to other such cases, e.g. CD (COULD).
12e) Therefore sign
In geometry theorems and proofs the “therefore” sign, which in print is ∴ is often used. In braille it is “dot 6: dots ,6” and looks just like the print symbol ,*
Leave a space between this sign and what follows.
Example:
∴∠ABC = 90o,*_[,,abc;7#ij0
28
12f) Greek letters
The sign to indicate a capital Greek letter is “dots 4,5,6”.
The sign to indicate a small Greek letter is “dots 4,6”.
The level of geometry covered in this booklet uses only π and θ, but it is useful to recognise the commonly used letters of the Greek Alphabet:
Letter English Capital Small Capital Small Version Greek Greek Braille Braille
Alpha a Α α _a .a
Beta b Β β _b .b
Gamma g Γ γ _g .g
Delta d ∆ δ _d .d
Theta th Θ θ _? .?
Mu m Μ µ _m .M
Pi p Π π _p .p
Sigma s Σ σ _s .s
Omega o (long) Ω ω _w .w
A full list can be found in Braille Mathematics Notation 2005.
29
12g) Circle formulae
In formulae a letter sign following π is not needed, as the letters will revert to ordinary small letters.
Examples of formulae:
C = πd ,c;7.pd
C = 2πr ,c;7#b.pr
A = πr2 ,a;7.pr+2
V = 4/3 πr3 ,v;7#d3.pr+3
12h) Sine, cosine and tangent signs
The three signs for sine, cosine and tangent are followed by the sign for the angle in question, or the angle itself (in degrees) written unspaced after the trigonometry sign.
Sine (sin). The braille is “the ED contraction (dots ,2,4,6) followed by the letter “s” (dots 2,3,4)” $s
sinθ $s.? sin 45o $s#de0
Cosine (cos). The braille is “the ED contraction (dots ,2,4,6) followed by the letter “c” (dots ,4) $c
cos θ $c.? cos 60o $c#fj0
Tangent (tan). The braille is “the ED contraction (dots ,2,4,6) followed by the letter “t” (dots 2,3,4,5)” $t
tan θ $t.? tan 35o $t#ce0
Examples:
Sine Rule: a /sin A = b/sin B = c/sin C a_/$s,a;7b_/$s,b;7c_/$s,c
Area of a triangle: 2 absinC
#a2ab$s,c
30
13 Co-ordinates, vectors and matrices
13a) Co-ordinates
Co-ordinates are written in round brackets. Open bracket ( is the “GH” contraction in braille (dots ,2,6) and close bracket ) is the “AR” contraction in braille (dots 3,4,5).
The two numbers are separated by a space and NOT, as in print, by a comma.
(3, 4)<#c#d> When using the four quadrants, the appropriate + or - sign is placed, unspaced, before the number. A comma, however, should be used when the second co-ordinate is signed either + or -, following the mathematical separator sign (dot 6).
(-3, 5) <;-#c#e>
BUT (-3, +5) <;-#c,1;6#e>
Examples
P(, 2) ,p<#a#b>
Q(-2, -) ,q<;-#b,1;-#a>
R(2, -) ,r<#b,1;-#a>
These rules apply for all co-ordinate work.
13b) Vectors →
The vector between points A and B, is usually written in print as AB
The superscript arrow is coded in braille as “dots 4,6: dots 2,5: dots ,3,5” .3Oand it follows the capital letters, unspaced.
In the example above, A→
B is coded as,,ab.3O
3
The notation for writing a column vector is straightforward and there are two major rules about layout:
The vector brackets (which in braille are the “FOR” sign, all six dots, placed at the beginning and the end of each number in the vector), MUST be underneath each other.
The numeral signs for the two numbers in the vector also must be underneath each other. If the numbers are signed, then, on occasions, this can present problems with the layout.
56
=#e= +4–3 =;6#d=
=#f= =;-#c=
0–2
=#j= 0 4 =#aj=
=;-#b= =#d=
+2 0
=;6#ab= –8+5
=;-#ah=
=#j= =;6#e=
When shape ABCD is translated by a certain vector to A′B′C′D′, the “dash” symbol is used. In braille this is “dot 4, followed by the “IN” contraction (dots 3,5)” @9 unspaced.
A′B′C′D′,,a@9b@9c@9d@9
If ABCD is translated to A"B"C"D" the “double dash” is written as “dot 4, followed by “IN” “IN”.
32
13c) Matrices
The rules for column vectors also apply to matrices.
Examples:
4 2 5
=#d#b=
=#a#e=
–3 0 0 +4
=;-#c#j=
=#j;6#d=
The last example is rather complex. There are spaces throughout the matrix and these need to be counted carefully so that the brackets and the numeral signs are under each other.
+ 0 –9 0 –3 +8 +0 –5 0
=;6#a#j;-#i=
=#j;-#c;6#ah=
=;6#aj;-#e#j=
33
14 Handling data
14a) Data collection and tally charts
In braille, use the following notation for tally charts: “dots 4,5,6 up to four times and then complete the five with a middle c – dots 2,5”.
is written in braille as____3
An example of the correct lay out is illustrated below, and shows an imaginary tally chart of scores when a die is rolled 36 times
Score Tally Totals
7
2 6
3 4
4 0
5 6
6 3
In braille this is set out using “dot 3” for tracking dots
Scoret,ytotals
#a'''''''____3__''''''''#g
#b'''''''____3_'''''''''#f
#c'''''''____''''''''''''#d
#d'''''''____3____3'''''#aj
#e'''''''____3_'''''''''#f
#f'''''''___'''''''''''''#c
34
14b) Stem and leaf diagrams
In print:
| 4 8 3 22 | 6 5 03 | 5
The vertical bar line in braille is “dots 4,5,6” SPACED from the numbers
#a_#d#h#c#b
#b_#f#e#j
#c_#a#e#a
14c) Standard deviation and mean - ∑
In braille, the capital sigma, Σ, is written as “dots 4,5,6 (capital Greek letter sign), s” _s
As the capital Greek letter sign only relates to the following letter, in the exampleΣfx, the Σ can be followed by the f and the x with no letter sign(s) needed.
Σfx in braille is written as _sfx
x (mean) is written as x:
14d) Probability
The probability of an event A is P(A), where A denotes the event
In braille, this follows the print “dot 6: P: open bracket “GH” contraction: dot 6: A: close bracket “AR” contraction”.
P(A) is ,p<,a>
35
Appendix AThis appendix gives examples of how to set out mathematical questions, calculations and answers in braille.
1 Number arrays
If numbers are displayed in columns and rows, the golden rule is that units must always line up under each other. Some children working at lower reading levels will prefer double line spacing.
2 3 4 5 #a #b #c #d #e
2 3 4 5 #aa #ab #ac #ad #ae
2 22 23 24 25 #ba #bb #bc #bd #be
In arrays that involve hundreds, tens and units, up to two spaces may be needed to ensure that the units line up.
00 0 02 #ajj #aja #ajb
200 20 202 #bjj #bja #bjb
If the array includes numbers that require the mathematical comma eg 0 000, an extra space to take account of this is needed, so that the units still all line up.
Similarly in arrays that involve decimals, the decimal points and units must all line up.
4.3 5.4 5.6 #d1c #e1d #ae1f
4.03 5.04 5.06 #d1jc #e1jd #ae1jf
.5 .25 .75 #1e #1be #1ge
The units of negative numbers in an array should also line up, so space for the minus sign “dots 3,6” need to be factored in.
36
If numbers in print are listed separated by commas eg , 2, 3, 4, DO NOT include commas in the braille version, as these could be confused with the decimal point.
2 Alternative layout of arithmetic
2a) Using round brackets in multiplications
Sometimes a braille sum can be usefully adapted and abbreviated by using round brackets:
Print .....................Braille adaptation changes
- 2 x 3 = - 6 - 2 ( 3) = - 6 ;-#b<#c>;7-#f
- 3 x - 4 = 2 - 3 ( - 4) = 2 ;-#c<;-#d>;7#ab
6 - -3 = 9 6 - ( - 3) = 9 #f;-<;-#c>;7#i
- 2 - + 4 = - 6 - 2 - (+ 4) = - 6 ;-#b;-<;6#d>;7-#f
- 2 x - 4 = 8 (- 2) (- 4) = 8 <;-#b><;-#d>;7#h
2b) Using the slash sign in divisions
Sometimes using the slash sign “dots 4,5,6” followed by dots 3,4 as an alternative to the divide sign
6 ÷ 2 = 8 6 / 2 = 8 #af_/#b;7#h
6 ÷ - 4 = - 4 6 / - 4 = - 4 #af_/;-#d;7-#d
- 20 ÷ - 5 = 4 - 20 / - 5 = 4 ;-#bj_/;-#e;7#d
3 Indexes and powers
The power sign is “dots 3,4,6” followed by the appropriate number in the lower part of the braille cell:
72 02 #g+2#aj+2
73 03 #g+3#aj+3
74 04 #g+4#aj+4
37
3a) Negative indices4-2 6-2 #d+;-2#f+;-2
4-3 6-3 #d+;-3#f+;-3
4-4 6-4 #d+;-4#f+;-4
3b) Positive whole number indices6+3 0+3 #f+;6#c #aj+;6#c
6+ 0 + #f+;6#aa #aj+;6#aa
3c) Decimal indices6.5 0.5 #f+#a1e #aj+#a1e
62.5 02.5 #f+#b1e #aj+#b1e
3d) Fractional indices6½ 0½ #f+#a2 #aj+#a2
6¾ 0¾ #f+#c4 #aj+#c4
4 Square and cube roots
The root sign is “dots ,4,6” before the numeral sign, unspaced. For cubed roots, the root sign is followed immediately by middle 3.
√ 9 = 3 %#I;7#c
√ 16 = 4 %#af;7#d
3√ 8 = 2 %3#h;7#b
3√ 27 = 3 %3#bg;7#c
38
4a) Worked examples with powers and roots
24 - 23 = 6 - 8 = 8
#b+4;-#b+3;7#af;-#h;7#h
22 x 23 = 22+3 = 25
#b+2;8#b+3;7#b+#b;6#c;7#b+5
22 ÷ 23 = 22-3 = 2-
#b+2;4#b+3;7#b+#b;-#c;7#b+;-1
22 x 23 = 2(2+3) = 25
#b+2;8#b+3;7#b+<#b;6#c>;7#b+5
22 ÷ 23 = 2(2-3) = 2-
#b+2;4#b+3;7#b+<#b;-#c>;7#b+;-1
√ 64 / √ 4 = 4
%#fd_/%#d;7#d
5 Chart layout with simple inequalities
Time to travel to school Number of pupils (t minutes) 0 < t ≤ 5…………………………… 5 5 < t ≤ 0………………………….29 0 < t ≤ 20………………………….57
,"t6travel6s*ool,numb](pupils
7;tm9utes7
#j[t[7#e'''''''#e
#e[t[7#aj''''''#bi
#aj[t[7#bj'''''#eg
39
6 Linear layout of number calculations
Each new sum should be numbered, and indented 2 spaces from the left margin (start in cell 3). Continuation lines should start at the margin (cell )
Use double line spacing between sums for students, if they are not fluent braille readers.
. 62 + 3 = 65 #a4#fb;6#c;7#fe
2. 62 - 2 = 60 #b4#fb;-#b;7#fj
3. 70 x 2 = 40 #c4#gj;8#b;7#adj
4. 70 ÷ 2 = 35 #d4#gj;4#b;7#ce
7 Simple calculations using linear braille
7a) Addition
2 + 32 = 5 + 2 = 53 #ba;6#cb;7#ea;6#b;7#ec
(i) add the tens (2 + 30)(ii) add the units (5 + 2)(iii) = 53
28 + 35 = 58 + 5 = 63#abh;6#ce;7#aeh;6#e;7#afc
(i) add the hundreds and the tens (28 + 30)(ii) add the units (58 + 5)(iii) = 63
40
7b) Subtraction
64 - 2 = 44 - = 43
#fd;-#ba;7#dd;-#a;7#dc
(i) subtract the tens (64 - 20)(ii) subtract the units (44 - )(iii) = 43
654 - 228 = 454 - 28 = 434 - 8 = 426
#fed;-#bbh;7#ded;-#bh;7#dcd"
;-#h;7#dbf
(i) subtract the hundreds (654 - 200)(ii) subtract the tens (454 - 20)(iii) subtract the units (434 - 8)(iv) = 426
6,24 - ,53 = 5,24 - 53= 5,4 - 53 = 5,09 - 3 = 5,088
#f'bda;-#a'aec;7#e'bda;-#aec"
;7#e'ada;-#ec;7#e'jia;-#c;7#e'jhh
7c) Multiplication
22 x 3 = 60 + 6 = 66#bb;8#c;7#fj;6#f;7#ff
(i) multiply the tens (20 x 3)(ii) multiply the units (2 x 3)(iii) add tens to units (60 + 6)(iv) = 66
4
633 x 5 = 3,000 + 50 + 5= 3,00 + 50 + 5 = 3,60 + 5= 3,65
#fcc;8#e;7#c'jjj;6#aej;6#ae"
;7#c'ajj;6#ej;6#ae;7#c'afj;6#e"
;7#c'afe
(i) multiply the hundreds (600 x 5) = 3,000(ii) multiply the tens (30 x 5) = 50(iii) multiply the units (3 x 5) = 5(iv) add together using linear method
Another example:
249 x 3 = 600 + 20 + 27= 700 + 20 + 27 = 740 + 7 = 747
#bdi;8#c;7#fjj;6#abj;6#bg"
;7#gjj;6#bj;6#bg;7#gdj;6#g;7#gdg
7d) Division
44 ÷ 2 in print would be solved as 2 ) 4 4 2 2
In braille, change to linear layout: 2 44 = 22
#b#dd;7#bb(i) 2 “into” 4 tens = 2 (tens)(ii) 2 “into” 4 units = 2 (units)(iii) = 22
42
842 ÷ 6 in print would be solved as 6 )842 40 r. 2
In braille, change to linear layout: 6 842 = 40 r. 2
#f#hdb;7#adjr4#b (i) 6 “into” 8 (hundreds) = (hundreds)(ii) carry over 2 hundreds in your mind(iii) 6 “into” 24 (tens) = 4 (tens)(iv) 6 “into” 2 will not go = 0 (units)(v) remainder of 2 units = r. 2
8 Long multiplication
6 x 3 = (6 x 0) + (6 x 3)6 x 0 = 606 x 3 = 30 + 8 = 4860 + 48 = 200 + 8 = 208= 208
#af;8#ac;7<#af;8#aj>;6<#af;8#c>
#af;8#aj;7#afj
#af;8#c;7#cj;6#ah;7#dh
#afj;6#dh;7#bjj;6#h;7#bjh
;7#bjh
43
27 x 3 = (27 x 0) + (27 x 3)= 27 x 0 = 270 = 27 x 3 = 60 + 2= 80 + = 8= 270 + 8 = 350 + = 35= 35
#bg;8#ac;7<#bg;8#aj>;6<#bg;8#c>
;7#bg;8#aj;7#bgj
;7#bg;8#c;7#fj;6#ba
;7#hj;6#a;7#ha
;7#bgj;6#ha;7#cej;6#a;7#cea
;7#cea
9 Long division
Example: 294 ÷ 3 in print would be solved as 2 2 r. 8 3 ) 2 9 4 2 6 3 4 2 6 8
In braille change to a linear layout: 3 294 = 22 r. 8
(i) 3 “into” 29 (tens) = 2 (tens) r. 3(ii) carry over 3 (tens) in your mind(iii) 3 “into” 34 = 2 (units) r. 8(iv) remainder of 8 units = r. 8
(v) = 22 r. 8
294 ÷ 3 change to:
3 294 #ac#bid
3 29 = 2 r. 3 #ac#bi;7#br4#c
3 34 = 2 r. 8 #ac#cd;7#br4#h
= 22 r. 8 ;7#bbr4#h
44
10 Fractions
10a) Adding fractions
Example:
2 2
+ 3 4
+ 5 8
= 0 2
+ 4
+ 8
= 0 4 8
+ 2 8
+ 8
= 0 7 8
#b#a2;6#c#a4;6#e#a8
;7#aj#a2;6#a4;6#a8
;7#aj#d8;6#b8;6#a8;7#aj#g8
10b) Subtracting fractions
24 2 3
- 0 5 6
- 3 3
#bd#b3;-#aj#e6;-#c#a3
= 24 4 6
- 0 5 6
- 3 2 6 ;7#bd#d6;-#aj#e6;-#c#b6
= 4 6
- 5 6
- 2 6
;7#aa#d6;-#e6;-#b6
= 4 6
- 7 6
;7#aa#d6;-#g6
= 4 6
- 6
;7#aa#d6;-#a#a6
= 0 4 6
- 6
;7#aj#d6;-#a6
= 0 3 6
= 0 2
;7#aj#c6;7#aj#a2
45
10c) Multiplying fractions
2
x 4
= 8 #a2;8#a4;7#a8
4
x 2 3
x 4 5
#a4;8#b3;8#d5
= 2 2
x 4 5
;7#b12;8#d5
= 8 60
= 4 30
= 2 5
;7#h60;7#d30;7#b15
2 2
x 3 4
#b#a2;8#c#a4
= 5 2
x 3 4
;7#e2;8#ac4
= 65 8
= 8 8
;7#fe8;7#h#a8
10d) Dividing fractions
2 2
÷ 4
#b#a2;4#a4
= 5 2
÷ 4
;7#e2;4#a4
= 5 2
x 4
;7#e2;8#d1
= 20 2
= 0 ;7#bj2;7#aj
6 2 3
÷ 2
#f#b3;4#a2
= 20 3
÷ 2
;7#bj3;4#a2
= 20 3
x 2
= 40 3
;7#bj3;8#b1;7#dj3
= 3 3
;7#ac#a3
46
11 Decimals
11a) Adding decimals
20.3 + 26.4 #bj1c;6#bf1d
= 40.3 + 6.4 ;7#dj1c;6#f1d
= 46.3 + 0.4 ;7#df1c;6#j1d
= 46.7 ;7#df1g
6.35 + 27.2 #af1ce;6#bg1ba
= 36.35 + 7.2 ;7#cf1ce;6#g1ba
= 43.35 + 0.2 ;7#dc1ce;6#j1ba
= 43.55 + 0.0 ;7#dc1ee;6#j1ja
= 43.56 ;7#dc1ef
11b) Subtracting decimals
86.3 - 29.2 #hf1c;-#bi1b
= 66.3 - 9.2 ;7#ff1c;-#i1b
= 57.3 - 0.2 ;7#eg1c;-#j1b
= 57. ;7#eg1a
49.6 - 23.7 #di1f;-#bc1g
= 29.6 - 3.7 ;7#bi1f;-#c1g
= 26.6 - 0.7 ;7#bf1f;-#j1g
= 25.9 ;7#be1i
47
11c) Multiplying decimalsExample: 8.3 x 2. #h1c;8#b1a
Remove the decimal point before working out, change to
83 x 2 #hc;8#ba
= 80 x 20 = 600 ;7#hj;8#bj;7#afjj
and 80 x = 80 &#hj;8#a;7#hj
and 3 x 2 = 63 &#c;8#ba;7#fc
= 600 + 80 + 63 ;7#afjj;6#hj;6#fc
= 680 + 63 ;7#afhj;6#fc
= 743 ;7#agdc
(now re-introduce the decimal point for final answer)
= 7.43 ;7#ag1dc
11d) Dividing decimals
One approach is to change the sum to an equivalent sum where you are dividing by a whole number:
20. 4 ÷ 0.2 #bj1d;4#j1b
But 20.4 ÷ 0.2 = 204 ÷ 22 204 #b#bjd
2 204 = 02 #b#bjd;7#ajb
20.4 ÷ 0.03 #bj1d;4#j1jc But 20.4 ÷ 0.03 = 2040 ÷ 33 2040 = 0680 #c#bjdj;7#jfhj
So 20.4 ÷ 0.03 = 680 #bj1d;4#j1jc;7#fhj
20.4 ÷ 3 #bj1d;4#c
3 20.4 = 06.8 #c#bj1d;7#jf1h
20.4 ÷ 3 = 6.8 #bj1d;4#c;7#f1h
48
12 Vertical column layout for calculations
To produce vertical layout, the braillist needs to back space, which is a skill in itself. In choosing between linear and vertical layouts teachers need to be aware of the braille skills required and to develop these without compromising the development of the pupil’s mathematical understanding.
. Digits, tens, hundreds etc are vertically aligned in columns.2. Horizontal lines are brailled in middle ‘c’s (dots 2,5) 3. No closure line required.4. Operational signs should be placed as in print BUT often it is preferable to have the operation signs before the numeral signs.5. Competent braillists may be able to work without the numeral signs. However, the omission of numeral signs should be explained beforehand. 6. Tens, hundreds are ‘carried over’ mentally.
12a) Vertical layout for whole numbers
63 + 5 63 #fc
+ 5 ;6#ae
78 333
#gh
236 + 58 236 carry over #bcf
+ 58 tens and hundreds ;6#eh
294 mentally 3333
#bid
63 - 5 63 ‘borrow’ #fc
- 5 mentally ;-#ae
48 333
#dh
236 - 58 236 #bcf
- 58 ;-#eh
78 3333
#agh
49
22 x 3 22 #bb
x 3 ;8#c
66 333
#ff
24 x 2 24 #bd
x 2 Multiply by the ;8#ab
48 units first, then 333
+ 240 by the tens. #dh
288 ;6#bdj
3333
#bhh
294 ÷ 3 22 r. 8 #bbr4#h
3 )294 3333
26 #ac#bid
34 bf
26 33
8 cd
bf
33
h
Long division is very difficult in braille.
The numeral signs can be omitted when the student is a competent braillist.
Some students prefer to braille the answer down the side, underneath the divisor, as this is technically easier than returning to the top line. The final answer needs to be identified clearly eg 294 / 13 = 22 r 8.
50
12b) Vertical calculations with decimals
. Decimal points should be vertically aligned.2. Operational signs should be placed as in print BUT often it is preferable
to have the operation signs before the numeral signs.
22.24 #bb1bd
+ 2.35 ;6#ba1ce
43.59 333333
#dc1ei
203.5 #bjc1ea
+ 3.30 ;6#c1cj
206.8 3333333
#bjf1ha
364.80 #cfd1hj
- 363.56 ;-#cfc1ef
.24 3333333
#a1bd
22.3 #bb1c
x 3 ;8#c
66.9 33333
#ff1i
Remember that braille is written from left to right. The sighted computations, in vertical layout, which are no problem in print, require the brailler to be manipulated from right to left requiring much back spacing and checking.
Cubarithms and Tactiles can be used to teach vertical layout.
5
13 Confusion between decimal points, fractions and literary and mathematical punctuationDecimal point is dot 2.Literary comma is dot 2.Mathematical comma is dot 3.Literary apostrophe is dot 3.Mathematical continuation sign is dot 5.Mathematical separation sign is dot 6.Literary and mathematical capital sign is dot 6.
Examples:
,402.502 and #a'djb1ejb&
,402,502 #a'djb'ejb
Use dot 6 before a full stop after ordinary mathematical numbers (but not question numbers, etc). Without the dot 6, . can look the same as
4 and
2. can look the same as 2 4
.
. and 4
#a4 #a4
2. and 2 4
#b4 #b4
So . should be written as #a,4 and 2. should be written as #b,4 , although the dot 6 is not required for question numbers.
Likewise .0 could easily be confused with 0
.0 and 0
#a1j #a10
52
It is easy to see how the following pairs could be misread:
6.0 6.0 #f1aj #fa1j
6 0
6 00
#f110 #f100
Understanding these potential confusions can help a maths teacher judge when a pupil has made a braille mistake but understood the mathematical concept.
Remember that in braille lists the comma is not required. This avoids the confusion of “6,” being written as
#f,1
numeral sign 6 dot 6 followed by comma dot 2
and looking similar to 6 9
#f9
Never split large numbers over two lines.
Never split digits from operator signs.
Always use the continuation sign (dot 5) when calculations continue onto the following line:
2 + 32 + + 2 + 4 + 6.0 + 3
#ba;6#cb;6#a;6#b;6#d;6#f1j"
;6#c
53
Appendix B Imperial units of measure
1 Units of length
Print Braille Print example Braille example
inch 9* 9 inch #i9*
inches 9*es 9 inches #i9*es
in 9 9 in #i9
ins 9s 9 ins #i9s
" _ 9" #i_
foot foot 6 foot #ffoot
ft ft 2 ft #bft
' . 2' #b.
2 ft 6 in #bft#f9
2' 6" #b.#f_
yard y>d
yd yd 4 yd #dyd
yds yds 4 yds #dyds
mile mile
miles miles 0 miles #ajmiles
2 Units of area
Print Braille Print example Braille example
in2 9+2 25 in2 #be9+2
ins2 9s+2 25 ins2 #be9s+2
ft2 ft+2 8 ft2 #hft+2
yd2 yd+2 4 yd2 #dyd+2
54
3 Units of volume
Print Braille Print example Braille example
in3 9+3 27 in3 #bg9+3
ins3 9s+3 27 ins3 #bg9s+3
ft3 ft+3 8 ft3 #hft+3
yd3 yd+3 25 yd3 #abeyd+3
4 Units of weight
Print Braille Print example Braille example
ounce
oz oz 8 oz #hoz
pound
lb lb 3lb 2 oz #clb#aboz
stone
st st 0 st 7 lb #ajst#glb
hundredweight
cwt cwt 5 cwt #aecwt
ton ton 00 ton #ajjton
5 Units of capacity
Print Braille Print example Braille example
pint p9t pint #AP9T
pt PT 4 pt #dpt
gallon gallon 2 gallon #bgallon
gal gal 5 gal #egal
3 gal 5 pt #cgal#ept
55
Indexaddition 2, 39, 48algebraic fractions 23algebraic layout 24 -26, angle sign 26approximately equal to 6area 2arrays 35brackets 3, 7, 22capacity 2centigrade, Celsius 7centimetres 0,2, 3circle formulae 29compound units 7, 8common confusions 5, 52coordinates 30cosine 29currency 9data charts 33, 34dates 6decimals 9, 46degree sign 7, 26density 8division 2, 36, 4, 43, 49equals sign 2feet and inches 53fractions 8, 23, 44, 45function sign 26geometry 26 - 29grammes Greek letters 28hyphen, mathematical 4imperial measures 53indices 4, 36, 37inequalities 6, 38length 0letter sign 20 -22linear layout 39 - 47litres 2mass mathematical comma 2, 5maths hyphen 4, 5maths separation sign 4, 5
matrices 32 mean 34measurement 0minus sign 2minutes 3money 9multiplication 2, 36, 40, 42, 49negative numbers 2not equal to 6oblique stroke 22, 23operation signs 2parallel to sign 27percentages 9pounds, pence 9pounds and ounces 53powers 4, 36, 37probability 34proportional to sign 26punctuation 4, 5, 52ratios 9roman numerals 2roots 5, 37, 38seconds 3separation sign 4set notation 8sine, cosine and tangent 29speed 8square roots 5, 37, 38standard deviation 34stem and leaf diagrams 34subtraction 2, 40, 48subscripts 5superscripts 5tally charts 33temperature 7therefore sign 27time 3 - 6triangle sign 27vectors 30 - 3vertical layout 48 - 50volume 3weight
56
Resources
Braille mathematics notation 2005 This 68 page book is the official statement of the rules of braille mathematics code, authorised by the Braille Authority of the UK. NEW 2005 Edition.£5.00 (Inc. VAT)
British Braille (2004)Compiled and authorised by the Braille Authority of the United Kingdom, this book is the standard reference giving the rules of the Standard English Braille as used in the UK. This 2004 edition replaces the previous edition published in 992, and incorporates additions, amendments and corrections to the rules. It also supersedes the leaflet "Capitals in British Braille", TC20533-36, published in 998. Available in print - TC20242, capitalised braille - TC2024 and non-capitalised braille - TC29.£7.25 (Inc. VAT)
Both books are available from RNIB Customer Services. Telephone 0845 702 353 or email [email protected].
RNIB sells a range of mathematical equipment that is either designed to be easy to see or to be used by touch. This includes the cubarithm for teaching the layout of braille arithmetic and a Maths concept kit for teaching mathematical concepts to children and young people with sight loss.
For a free catalogue of our learning products please contact RNIB Customer Services on 0845 702 353 or email [email protected]
Other titles in this series:
TC20909 Using the Braille French Code
TC20910 Using the Braille German Code
TC20911 Using the Braille Spanish Code
TC21085P Using the Braille Science Code
ISBN
1 85878 664 9
978 1 85878 664 3
TC21086P
2007
Produced by RNIB on behalf of
RNIB/VIEW Mathematics Curriculum Group
