An Introduction of Linear and Spatial Equations Based on Nemeth Format Guidelines, Updated 2011 John Romeo, Nemeth Transcriber Full Cell Braille, Inc. April 2015

First Things First Introducing whole numbers The Nemeth whole number, unlike its literary relative, drops down into the lower part of its braille cell. Upper cell: c Lower cell: 7 Literary whole numbers: #ab #cde #fghi Nemeth whole numbers: #12 #345 #6789

Nemeth whole numbers will ordinarily be introduced to a child as early as Kindergarten. However, a few late introductions may occur in the First Grade. In the beginning, the primary student will learn to recognize, and count, literary whole numbers. Then he or she will learn—for mathematical purposes—a Nemeth whole number drops down into the lower part of its braille cell. Remember, braille text for a Kindergarten and First Grade student is written in uncontracted braille. Example: Paul has 8 pennies and Sue has 2 pennies. How many pennies do they have combined?,paul has #8 pennies and,sue has #2 Pennies4,how many pennies do they have combined8

After a student is taught to recognize and count both literary and Nemeth whole numbers, he or she will be given an introduction to the Nemeth signs of operation for addition and subtraction. During this phase of instruction, the primary student will learn to distinguish between a Nemeth minus sign and a literary hyphen—both being dots 36 ( )—by learning to recognize the braille character's purpose by how it's used in the surrounding text. Then, the Student will learn to recognize the Nemeth symbols for less than ( ), equal to ( ), and greater than ( ) so that he or she can perform comparison operations. -.k.1 "K

Read: #8 1 #9 #5+7 #12 #3+4 #5 -.k.1 "K

Remember, a contraction cannot be used in any part of a word coming into direct contact with a sign of operation or a sign of comparison (IBM §155ai, v). For the purpose of this presentation, the Introduction to Braille Manual is referenced, but for actual proofreading, cite the Nemeth Code book. Read: Eleven-eight "k twenty Sixteen.k sixteen Fourteen-five.k nine

Since some Nemeth numbers and literary punctuation marks will be represented by identical braille symbols, the Nemeth punctuation indicator — (dots 456) — is to be used for distinction. Therefore, a Nemeth punctuation indicator is to be placed before a literary punctuation mark whenever the mark comes into contact with a Nemeth number (IBM §6). _ Read:,HE A4$ #8+3 4,C Y C.T 2Y #100 8,%E SUBTRACT$ #15 F #9 4 _ _ _

A Perfect Arrangement Introducing spatial formats A Nemeth transcriber will need to learn how to properly format spatial math problems, or expressions. A spatial expression is any math expression that is displayed on two or more print lines. For example, expressions such as: 28, 30, 32, 36,?, 40 or = ? or are not to be considered spatial expressions.

Interestingly, one of these skills a transcriber needs to possess, in order to transcribe an addition and subtraction expression properly, is spatial formatting. But spatial formatting is placed— oddly enough—in Lessons 15 and 16 of AN INTRODUCTION TO BRAILLE MATHEMATICS ("IBM"). The last two lessons of the instruction manual! Why is this odd? Because, for a transcriber of mathematical material, spatial formatting is a skill that needs to be mastered as quickly as possible since a primary student's success will demand he or she be introduced to these correct formats early on.

Giving and Taking Current rules for formatting a simple spatial addition and subtraction expression 1.A blank line is to be left above and below a spatial expression (IBM §159). 2. A numeric indicator is not to be used in spatial arrangements for addition or subtraction (IBM §160). Nemeth Code: Rule XXV §185a Nemeth Code: Rule II §11b

3. A transcriber will use the separation line— 3 (dots 25). The line is to extend one cell to the left and right of the longest entry appearing above or below the arrangement (IBM §161a). 4. The sign of operation, for either addition or subtraction, is to be placed to the left one cell beyond the largest numeric entry appearing above the separation line (IBM §161c). 5. The full width of the braille page can be used for a spatial expression placed on braille lines 3 through 23 only (IBM §170). If spatial material is placed on line 24 or 25, there must be at least three blank columns between material and page numbering. Nemeth Code: Rule XXIV §178c Nemeth Code: Rule XXV §185a Nemeth Code: Rule XXIV §178b

6. Whenever two or more spatial expressions are placed side-by-side, across the braille page, and without an alphabetic or numeric identifier ("identifier," not to be confused with "indicator") at least one blank space must be left between the end of one separation line and the beginning of the next separation line (IBM §172a). Read: ==333333== The Braille Dot Nemeth Code: Rule XXV §185c

7. Whenever a spatial expression has an identifier, the identifier must be placed on the first line of the expression (IBM §168a). One blank space is to be left between the identifier and the furthest left braille cell in the overall braille arrangement (IBM §167). Any spatial expressions with identifiers will have three blank cells between the longest element in the arrangement and the following identifier, except at the separation line, which will to have at least two blank cells between braille expressions. (IBM §172b) Read: #1_4 4 #2_4 19 #3_ =3333== =33333== = Nemeth Code: Rule XXV §185bi(1) Nemeth Code: Rule XXV §185c(2)

8. Carried whole numbers will be taught to a student early along with spatial addition in the classroom. Yet, the IBM doesn't address carried whole numbers until Lesson 15 (IBM §162a). A line for a carried whole number indicator— 7 (dots 2356) — is the same length as the separation line. The carried whole number indicator must be placed between the top line of the equation and the carried numbers. Lastly, the identifier must then be placed to the left of the top line of the equation (IBM §168b.) The Braille Dot Nemeth Code: Rule XXV §185bi(3) Nemeth Code: Rule XXV §178d

Read: #7_4 516 #8_4 429 #9_

9. Whenever cancellation is shown with a subtraction expression, a spatial arrangement is to be used (IBM §173). The whole number is to be enclosed by the opening sign of cancellation — [ ( dots 246)—and the closing sign of cancellation— ] (dots 12456). Any whole numbers will maintain their alignment for computation purposes (IBM §174). Read: #1_4 [4][ 4]9 #2_4 [8][ 3] Nemeth Code: Rule XI §60 Nemeth Code: Rule XXV §178a

As the Nemeth whole numbers fill more number places, the mathematical comma—, (dot 6)—is introduced. The mathematical comma can be a complicated little-one-cell item. At the beginning of Nemeth studies, it helps to remember that the mathematical comma replaces a literary comma— 1 (dot 2)—in all instances when a literary comma would normally come next to mathematical content, including the last number place. If not, should a literary comma be placed next to a Nemeth whole number, the literary comma could be mistaken for another digit—the whole number 1.

Read: #5_4,^! >E -P>ISON SIGNS3 "K,.K,.1_4 #6_4 #4, K #5, #7_4 8,471 #8_4 3,8[5][ 1] ,103 3,2 0 8

Let’s Multiply! Current rules for formatting a simple multiplication expression are: 1.A blank line is to be left above and below a spatial expression (IBM §159). 2. A numeric indicator is not to be used in spatial arrangements for multiplication.(IBM §160). Nemeth Code: Rule XXV §185a Nemeth Code: Rule §11b

3. A transcriber will use the separation line— 3 (dots 25). The line is to extend one cell to the left and right of the longest entry, and each separation line will need to be the same length (IBM §163). 4. The sign of operation for (dots 4, 16)—is to be placed directly next to the lower numeric entry (the multiplicand) (IBM §163). Nemeth Code: Rule XXIV §179c Nemeth Code: Rule XXIV §179b Nemeth Code: Rule XXIV §185a 5. The full width of the braille page can be used for a spatial expression placed on braille lines 3 through 23 only (IBM §170). If spatial material is placed on line 24 or 25, there must be at least three blank columns between material and page numbering.

6. The identifier is to be placed on the first line of the expression (IBM §168c). 7. Carried numbers are used with spatial multiplication. However, the IBM doesn't address this directly for multiplication. Remember, a line of carried number indicators is the same length as the separation line and is to be placed between the carried numbers and the first line of multiplication. Lastly, the identifier is to be placed on the first line of the multiplication. Nemeth Code: Rule XXV §185bi

Read: (Gray box lines are for illustrative purposes only to show a blank line.) ======================================== #1_4 49 #2_ @* , , , ,875 ========================================

Divide and Conquer Current rules for formatting a simple spatial division expression are: A primary student will learn division as soon as he or she has an understanding of simple multiplication expressions. Then he or she will learn the opposite of multiplication is division. A simple division expression can be easily transcribed into a linear format. The division symbol— O (dots 135)—must be placed between the divisor and the dividend (IBM §164a). (There are other division symbols, but more often than not, at this level, it is this symbol). If the division expression shows only a divisor and a dividend consisting only of numerals, comma, or decimal point, the expression is not to be regarded as spatial expression. Therefore, the separation line shown in print can be omitted (IBM §164g). The numeric indicator is to be used at the beginning of an expression whenever a spatial format is not used.

Read: #7_4 #5o25 #8_4 #30o300 #9_4 #105o420 For the example above, the divisor is on the left and the dividend is on the right. However, if the expression had been a division sentence, such as 15 ÷ 3 = 5, the division language would be reversed. The 15 would now be called the dividend, and the 3 would now be called the divisor. The quotient is the answer.

1.A blank line is to be left above and below a spatial expression (IBM §159). 3. A transcriber will use the separation line— 3 (dots 25). A separation line is to begin at the division sign. It is to extend one cell beyond the overall arrangement, and all separation lines are to be the same length. (IBM §164b). 2. A numeric indicator is not to be used in a spatial arrangement for division (IBM §160). Nemeth Code: Rule XXV §185a Nemeth Code: Rule II §11b(3) Nemeth Code: Rule XXV §180c

6. If a letter r is used to show a remainder in print, the remainder is to be preceded by a space and followed by a multipurpose indicator— " (IBM §172a). 5. The identifier is to be placed on the first line of the dividend. (IBM §168d). Nemeth Code: Rule XXIV §185a Nemeth Code: Rule XXV §185bii Nemeth Code: Rule XXIV §180e 4. The full width of the braille page can be used for a spatial expression placed on braille lines 3 through 23 only (IBM §170). If spatial material is placed on line 24 or 25, there must be at least three blank columns between material and page numbering.

Read: 8 20 r" #1_4 5o40 #2_4 31o

Half 'N' Half Current rules for formatting a simple spatial addition or subtraction fractional expression are: For the purpose of this presentation, of primary grade linear and spatial material formatting, I will address only addition and subtraction of simple and mixed number fractions. The opening fraction indicator is— ? (dots 1456) and the closing fraction indicator is— # (dots 3456) (IBM §71). The horizontal fraction line is shown by a: / (dots 34) (IBM §72). Notice the numeric indicators are not used. This is because a fraction indicator will tell the primary student the material is numeric.

Read: ?7/8# ?9/15# ?25/761# Read: #1_?1/2_# #14_?7/19_# #25_?23/105_# When transcribing a mixed number fraction, the opening mixed number fraction indicator— _? (dots 456, 1456) and the closing mixed number fraction indicator— _# (dots 456, 3456)—will be used (IBM §74).

1.A blank line is to be left above and below a spatial expression (IBM §159). 3. The sign of operation for an addition or subtraction expression is to be placed to the left and one cell beyond the longest fraction entry that appears above the separation line. (IBM §161c). 2. The numeric or alphabetic identifier is to be placed on the first line of the addition or subtraction expression (IBM §168a). Although the whole numbers will now be fractions, this will not change the placement of the identifier. Nemeth Code: Rule XXV §185a Nemeth Code: Rule XXV §185bi Nemeth Code: Rule XXIV §178c

5. The numerator and denominator are to be written directly next to the fraction line— / (dots 34)—even though this will cause a minor re-alignment of the numbers for computation (IBM §161f). Nemeth Code: Rule XXV §185a Nemeth Code: Rule XXIV §178e 4. The full width of the braille page can be used for a spatial expression placed on braille lines 3 through 23 only (IBM §170). If spatial material is placed on line 24 or 25, there must be at least three blank columns between material and page numbering.

The Braille Dot Read: #5_4 ?2/3# #6_4 6_?5/12_# -?1/2# +4_?1/4 _#