An Introduction of Linear and Spatial Equations Based on Ne - PowerPoint Presentation

Slide1

An Introduction of Linear and Spatial Equations Based on Nemeth Format Guidelines, Updated 2011

John Romeo, Nemeth TranscriberFull Cell Braille, Inc.April 2015Slide2

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 #fghiNemeth whole numbers: #12 #345 #6789 Slide3

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 combined8Slide4

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

"KSlide5

Read:

#8 1 #9

#5+7

#12

#3+4 #5

-

.k

.1

"KSlide6

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 sixteenFourteen-five .k nineSlide7

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

_

_

_Slide8

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 3 + 4 =

? or

869 - 342 are not to be considered

spatial expressions.

But an expression such as

Line 1 will be.

Line 2

Line 3

 Slide9

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. Slide10

Giving and Taking

Current rules for formatting a simple spatial addition and subtraction expression

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 Slide11

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 Slide12

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

:

683

768 98 +212 - 31 - 4

333333

==

333333

==

33333

895

737

94

The Braille Dot

Nemeth Code:

Rule XXV §185c Slide13

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_4

23

+

8 - 7 - 5

=

3333

== =

33333

== =

33333

12 12

18

Nemeth Code:

Rule XXV

§185bi(1)

Nemeth Code:

Rule XXV

§185c(2) Slide14

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 Slide15

Read:

1 1 1 777777 777777 777777#7_4 516 #8_4 429 #9_4 335

+247 +354 +528

333333 333333 333333

763 783 683Slide16

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

:

3 14

7

13

#1_4

[

4

]

[

4

]

9

#2_4

[

8

]

[

3

]

3

- 8 6

- 1 7

2

33333333333 33333333333

3 6 3

6 6

1

Nemeth Code:

Rule XI §60

Nemeth Code:

Rule XXV §178a Slide17

As the Nemeth whole numbers fill more number places, themathematical

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. Slide18

Read:

#5_4 ,^! >E -P>ISON SIGNS3 "K, .K,

.1_4

#6_4 #4

,

395+617 .K #5

,012 1 1 77777777 4 11#7_4 8,471 #8_4 3

,

8[5][ 1] + 632 - 6 4 3 33333333

3333333333333 9,103

3,2 0 8Slide19

Let’s

Multiply! Current rules for formatting a simple multiplication expression are:

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 Slide20

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 multiplication

—

@*

(

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.Slide21

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 Slide22

Read: (Gray box lines are for illustrative purposes only to show a blank line.)

======================================== 5 77777

#1_4 49 #2_4 138 #3_4 7,625

@*

6 @*25 @*131 33333 333333 333333333

294 690 7,625

276 228,750 333333 762,500 3450 333333333

998,875========================================Slide23

Divide and ConquerCurrent 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.

Slide24

Read:

#7_4 #5o

25

#8_4

#30

o300#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.Slide25

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 Slide26

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.Slide27

Read:

8 20 r"

25

3333 3333333333

#1_4 5o40 #2_4 31o645

40 62

3333 3333333333 0 25Slide28

Half 'N' HalfCurrent 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. Slide29

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). Slide30

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 Slide31

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.Slide32

The Braille Dot

Read

:

#5_4 ?2

/

3# #6_4 6_?5

/12_# -?1/2#

+4_?1

/4 _# 33333333 333333333333