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Fractions Cancellations

Nemeth: Fractions

Download the embossable version of all examples on this page: examples file in BRF format

Simple Fractions

Fractions in mathematics can be written in several ways. The horizontal bar or line (—) is commonly used in elementary mathematics and is represented by dots 3-4 (/). The diagonal line or slash (/) is represented by dots 4-5-6, 3-4 (_/). Examples in this lesson will utilize the horizontal line only. The Nemeth Braille Code for Mathematics and Scientific Notation, 1972 Revision should be consulted for information on the use of the diagonal line.

A simple fraction is defined as one whose numerator and denominator contains no fractions. The simple-fraction indicators (opening: dots 1-4-5-6, ? and closing: dots 3-4-5-6, #) are used to enclose a simple fraction in which the print fraction line is horizontal.

Table 1 - Simple Fractions
simple fraction 2 over 3 simple fraction 2+1 over 3+4 simple fraction 2+1 over 3 simple fraction a+b over c
?2/3# ?2+1/3+4# ?2+1/3# ?a+b/c#

Mixed Numbers

A mixed number is an expression which begins with a numeral and is followed by a simple fraction whose numerator and denominator are both numerals. If the expression contains any letters, it is not considered a mixed number in Nemeth. The mixed fraction indicator (opening: dots 4-5-6, 1-4-5-6, _? and closing: dots 4-5-6, 3-4-5-6, _#) is used to enclose the fractional part of the mixed number.

Table 2 - Mixed Numbers
mixed number 2 and 2 over 3 mixed number 8 and 3 over 4 mixed number 1 and 1 over 2
#2_?2/3_# #8_?3/4_# #1_?1/2_#

Complex Fractions

A complex fraction is one whose numerator or denominator or both, contains a simple fraction. The horizontal line for complex fractions is made with dots 6, 3-4 (,/). The complex fraction indicators opening: dots 6, 1-4-5-6 (,?) and closing: dots 6, 3-4-5-6 (,#) are used to enclose the complex fraction.

Table 3 - Complex Fractions
complex fraction numerator 4 over 5, denominator 10 complex fraction numerator 8, denominator 6 over 7
,??4/5#,/10,# ,?8,/?6/7#,#

Spatial Arrangements

When introducing fractions, it may be preferable to use spatial arrangements. This will help the child to develop the concept of the numeral on the “top” or “bottom” of the fraction and will be useful as the principles of invert and multiply are taught. Once the child is familiar with fractions, the linear arrangement should be used as it is much less complex and can be read quickly by the braille reader.

In a spatial arrangement of fractions, the horizontal line is represented as dots 2-5 in series (33). This line should be the same length as the longest expression above or below it. The appropriate fraction indicators are used with the fraction line and the Numeric Indicator is used with the numerator and denominator since each is preceded by a space.

Table 4 - Spatial Arrangements
Simple Fraction
Complex Fraction
simple fraction 2 over 3 complex fraction numerator 1 over 2, denominator 5

Addition and Subtraction of Fractions

Fractions are written in linear format for addition and subtraction problems. The total problem is arranged spatially and aligned vertically. Transcribing rules dictate that the numerator should be right justified in the numerator column and the denominator should be left justified in the denominator column. This means that a space may need to be left after the opening fraction indicator or before the closing fraction indicator in order to align the numerals by place value. In the classroom, these transcribing rules may not be strictly adhered to as it may be more efficient for the student to braille the fractions in a spatial arrangement without concern for precise vertical alignment.

Table 5 - Addition and Subtraction
fraction 3 over 8 + fraction 4 over 8 = fraction 7 over 8 fraction 2 over 16 + fraction 11 over 16 = fraction 13 over 16
  This example illustrates vertical alignment of the numerators. A space left after the opening fraction indicator provides for alignment of the numerators to facilitate their addition.
 ? 2/16#

The fractions are aligned vertically and the sign of operation is placed immediately to the left of the opening fraction indicator on the line just above the separation line. The separation line extends one cell on either side of the problem.

In arrangements containing mixed numbers, the whole numbers are aligned vertically with the tens, ones, etc. falling in the appropriate column. The sign of operation is placed one cell to the left of the widest number.

Table 6 - Addtion and Subtraction of Mixed Numbers
11 and 5 over 12 minus mixed number 3 and 3 over 12 = mixed number 8 and 5 over 12
- 3_?3/12_#

The procedures described above apply to fractions with common denominators. The lowest common denominator must be found before the fractions can be added or subtracted.

Table 7 - Finding the LCD
addition of mixed number and simple fraction  
  1_?3/4_# .k 1_?3/4_#
 +  ?1/2 # .k   ?2/4 #
              1_?5/4_# .k #2_?1/4_#
text description: mixed number one and three-fourths plus simple fraction one-half equals mixed number one and three-fourths plus two-fourths equals sum of one and five-fourths reduced to two and one-fourth

In the above problem, the lowest common denominator is found for the fraction. The fractions are added and the answer is reduced to lowest terms. Although the Numeric Indicator is not used in the spatial arrangement, it is required before the final fraction after the equals sign.

On pages 48-53 in Strategies for Developing Mathematics Skills in Students Who Use Braille, additional methods for adding and subtracting fractions are discussed. These methods are similar to the methods studied earlier for spatial arrangements of the four basic operations. Lines are skipped in place of using the separation line and the final answer is brailled on a separate line using the format “ans. = ###”. These methods and the methods discussed for multiplication and division of fractions should be studied carefully and considered as options when teaching the youngster to perform calculations on the brailler.

It is important to note that these methods differ greatly from the strict transcribing rules described in the code book. These alterations in format allow the student to perform calculations on the brailler more easily and quickly. Transcription of classroom materials for student use should adhere to rules outlined in the code book.

Practice 04.a

Braille the following. Check your work against the answer key.

Practice 04.a
simple fraction 3 over 7 simple fraction 6 over 10
mixed number 1 and one-eighth complex fraction one-half over 3