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Spatial Arrangements for Multiplication Spatial Arrangements for Division

Nemeth: Spatial Arrangements for Division

There are a number of division signs in print. This lesson focuses only on the right curved division symbol. More information about the other forms of the division sign can be found in the Nemeth Code Book. (The Nemeth Braille Code for Mathematics and Scientific Notation, 1972 Revision).

Key Vocabulary:

Table 1: Right-curved division sign examples
right-curved division sign 333
o
12 divided by 4 = 3    3
 3333
4o12

The right curved division sign in print is represented with dots 1-3-5 (O) in braille. The separation line (dots 2-5 in series, 333) must be placed on the line above the division symbol. The separation line must begin in the column containing the division symbol and extend one cell to the right of the entire arrangement. The NI is not used.

Additional separation lines within the division problem (throughout the partial product and differences) must begin in the column containing the division symbol and extend one cell beyond the width of the entire arrangement. All separation lines within the problem must be the same length.

Several methods exist for writing long division problems in print and several exist for Nemeth as well. The Nemeth Code book provides examples of each. The teacher should determine which method will be best to use with the student.

Traditional Method

For purposes of this lesson, we will use the term "traditional method" to describe the method that looks similar to print format. Separation lines are used within the partial product and Nemeth Code rules are followed. Students should be familiarized with this method because it will be found in textbook examples and other formally transcribed materials. This method is illustrated on page 9 of the Craig book and below.

Table 2: Traditional method
long division 270 divided by 5 = 54    54
 33333
5o270
  25
 33333
   20
   20
 33333
 



The division problem is worked as it is in print. The traditional sequence of dividing is followed by placing the quotient above the separation line. The paper is moved in and out of the braille writer as the problem is worked. When this format is used, the partial product will disappear into the brailler when the student moves the paper to braille the quotient. This method may feel tedious to some.

Alternate Method

The alternate method described in Strategies for Developing Mathematics Skills in Students Who Use Braille on pages 46-48 does not make use of the separation lines. The right-curved division symbol is used. With this method, the quotient is placed to the right of the problem as it is worked. This method does not follow code, but it ensures that the numerals in the problem are visible to the student at all times.

Table 3: Alternate method
long division 270 divided by 5 = 54  5o270
   25   #5

    20
    20   4

ans4 .k #54

This example can be viewed on graph paper through the link: figure 3 on graph paper

The steps are as follows:

  1. Braille the divisor, curved division symbol and dividend on the first line.
  2. Advance the paper one line. Press the spacer key to move the embosser head to the right of the problem.
  3. Divide the divisor into the dividend and braille the Numeric Indicator with the one digit (partial) quotient.
  4. Backspace to position the embosser head under the dividend.
  5. Multiply (the divisor by the partial quotient) and braille the partial product.
  6. Advance the paper two lines. This will leave a blank line in place of the separation line.
  7. Subtract the partial product from the dividend and braille the difference, aligning the numerals in the appropriate columns.
  8. Advance the paper one line. Move the embosser head to the right and align it under partial quotient.
  9. Repeat the division process and braille the single digit partial quotient. The numeric indicator is not used and the numeral is brailled directly below the first partial quotient.
  10. Repeat from step 4 until the division process is complete.
  11. Advance the paper two lines and braille the complete quotient. Collect the digits in the answer column by starting at the top. The Numeric Indicator serves as a marker for the starting point. Braille the digits in sequence as they appear from top to bottom. Do not add them. Use the format ans. = ### to write the final quotient. If a remainder exists, braille the letter r, followed by the numerals which comprise the remainder.

Additional alternate methods exist for writing out long division problems. A method similar to the one outlined above, called the T-method uses separation lines as well as vertical bars to separate the problem from the answers. An example of this method can be seen on page 12 in the Craig book.

Division of Numerals with Decimals

The formats discussed thus far apply to division of whole numbers. When the divisor or dividend are numerals containing decimals, the position of the decimal point must be considered in the arrangement. When a decimal point or comma occurs in the dividend of a division arrangement, a blank column of cells should be left in the corresponding location throughout the arrangement, except in the separation lines.

Table 4: Decimals and commas in long division
long division 91,441.62 divided by 18 = 5,080.09     5,080.09
  33333333333
18o91,441.62
   90
  33333333333
    1 44
    1 44
  33333333333
        1 62
        1 62
  33333333333
 


The spatial arrangement for long division appears as it would in formally transcribed material (table 5). The dividend contains both a comma and a decimal point and these also appear in the quotient. The comma and decimal point are not repeated in the partial product and difference and a blank cell is left instead.

Table 5: Decimals in long division
long division 93.24 divided by 18 = 5.18     5.18
  3333333
18o93.24
   90
  3333333
    3.2
    1.8
  3333333
    1.44
    1.44
  3333333
 

When students are being introduced to the concept of division of decimals, it may be preferable to have the student place the decimal point in the partial product as the problem is worked (table 6). Although this would not be done with formal transcription of mathematical materials, it is permissible in the classroom to help the student learn to compute these problems. There will be variation depending on how the teacher requires the student to perform the computation. Additional explanation is found in the Craig book.

Caret

Carets are used to indicate repositioning of the decimal point to facilitate computation. The caret is represented by dots 4-5-6, 1-2-6 ( _<). When the caret is used, one blank cell is left under the original decimal point and two blank cells are left under the caret in the partial product.

Table 6: Long division with caret
long division with caret 2.60 divided by .4 = 6.5        6 .5
    33333333
.4_<o2.6_<0
     2 4
    33333333
       2  0
       2  0
    33333333
 

 

 

 

 

 

 

 

 


The caret in long division is used to reposition the decimal in both the divisor and the dividend (table 6). Use of the caret in actual computation may not be required by the teacher. Like other Nemeth indicators, it is essential that the blind child be given the opportunity to learn the meaning and use of this indicator since it might appear in his or her mathematics textbook and other formally transcribed mathematical materials. Not teaching these symbols may leave the child at a disadvantage when the symbols appear in formal assessment materials.


Remainders

When a dividend can no longer be divided and a numeral smaller than the dividend is left, the resulting numeral is known as the remainder. In long division problems with remainders, the remainder is written on the line with the quotient, and preceded by the letter r. In Nemeth, the remainder is written by brailling the letter r followed by the Multipurpose Indicator (dot 5, " ). The remainder follows the Multipurpose Indicator. The Multipurpose Indicator will be studied in more depth in a later lesson. One blank cell is left after the quotient and before the letter r. The separation line must extend one cell to the right of the width of the entire problem. A quotient with a remainder can extend the width of the spatial arrangement by several cells. The letter r can appear as upper or lower case and this must be represented in braille.

Table 7: Long division with remainders
long division 286 divided by 4 = 71 remainder 2 Traditional Method Alternate Method
   71 r"2
 333333333
4o286
  28
 333333333
    6
    4
 333333333
    2
4o286
  28   #7

    6
    4   1

    2
       r2

ans4 .k #71r2


Problem Identifiers

Problem identifiers frequently appear on worksheets and within textbook exercises. The problem identifier can be in the form of sequential numbers or letters or both.

When a problem identifier is used with a spatial arrangement for long division, the numeral or alphabetic identifier must be positioned on the same braille line as the divisor, regardless of its position in print. One column of blank cells should be left between the last symbol appearing in the identifier and the left-most symbol in the overall arrangement of the braille problem. The last symbol of the identifier is often a Literary period and the Punctuation Indicator must be used before the Literary period.

Table 8: Problem identifiers with long division
long division number twelve, 198 divided by 22 = 9            9
        33333
#12_4 22o198
         198
        33333
           0
 


Practice 03.b

Try these long division problems. Use both the traditional and alternate methods for each. Check your work against the answer key.

 
176 divided by 16 126 divided by 9 156 divided by 13
108 divided by 5 529 divided  by 2 74.8 divided by 6.4