Project Math Access

Teaching Mathematical Concepts

Basic Number Facts and Operations

Collaborative and Inclusive Strategies

The Personal Perspective of Abraham Nemeth

The following is a summary of the recommended format to be followed in writing division problems. An outline of the steps for performing the operation will follow the summary.

On the same line as the problem identifier, the divisor is written. To represent the curved portion of the division symbol, , dots 1-3-5 are brailled. Immediately following that symbol, the dividend is brailled. No line of dots 2-5 is brailled above the dividend.

As the operation is carried out, the various digits comprising the answer (quotient) are brailled in a column to the right of the problem. This is done because it is much too difficult and inefficient to braille the quotient above the dividend, as is done in print; that would require moving the paper in and out of the braillewriter, which results in having a portion of the problem disappearing within the braillewriter. It is recommended that the entire problem should be tactually perceivable at all times while the student is performing the operation.

Following are the steps involved in the division of whole numbers without decimals. The answers may contain remainders. Following the recommended format as described above, the divisor and dividend are brailled. The paper advance key is tapped once. The carriage is positioned to the right of the problem. The divisor is divided into the proper portion of the dividend. The one-digit answer is written as the first number in the column to the right of the problem. The carriage is positioned under the units digit of the portion of the dividend which was divided. The number on the right-hand column is multiplied times the units digit of the divisor. That answer is written. If it is a two-digit number or larger, the backspace key is tapped and the tens digit is written. If a hundreds digit is required, the backspace key is tapped twice again, and that digit is written in the proper location.

Once this portion of the operation is complete, the paper is advanced two lines. The carriage is positioned in the units column of this portion of the problem. The proper procedures for subtraction are carried out. Once the result is obtained, the next number in the dividend is "brought down" and brailled immediately to the right of the previous result.

The paper is advanced one line and the carriage is positioned to the far right, under the first number in the column. The division process is repeated. The divisor is divided into the remaining number. The answer is written in the right-hand column.

The carriage is moved to the units column of the previous remainder. The digit which had just been placed in the right-hand column is multiplied times the divisor and the same process is carried out as outlined above. This procedure is repeated until no digits remain "undivided" in the quotient. If a remainder exists, that remainder is written in the right-hand column preceded by the letter "r".

To "collect" the answer in order to write it in a readable fashion, the student advances the paper two lines. The carriage is moved to the far left where the student writes ans. = followed by a blank space. The numeric indicator is brailled, followed in the same line by the digits from the right-hand column; these are brailled in sequential order, from top to bottom. If a remainder exists, the student should skip a space and write the letter, "r", followed with the digits which comprise the remainder.

#8. 32/77064 2130128 42 r2ans. = 24r2

The steps involved in the division of numbers containing decimals are similar to those outlined above. Of course, the position of the decimal point in the divisor and dividend and subsequently in the quotient must be accurate. If the divisor does not contain a decimal point, the dividend is written with the decimal point located where it is shown in the original problem.

If the divisor contains a decimal point, then it is moved to the right as many digits as necessary to not be included in the number. The decimal point in the dividend must be moved to the right as many digits as it was moved in the divisor. The dividend should be written with the decimal point located in the proper space.

The division process is carried out as outlined above. When the divisor is divided into the sequence of digits which contain the decimal point, the one-digit result must be written in the right-hand column preceded by dots 4-6 to indicate the proper location of the decimal point in the answer.

#9. 3/42.63 11212 46 .2ans. = 14.2

If the divisor and the dividend are whole numbers (without decimal points) and the answer is to be found to the nearer tenth, hundredth, thousandth, etc., the following procedures should be followed. A decimal point should be placed following the last digit of the dividend. The decimal point should be followed by a predetermined number of zeros. The number of zeros is determined by the precision of the answer. That is, if the answer is to be found to the nearer tenth, then two zeros should follow the decimal point. If the answer is to be found to the nearer hundredth, three zeros should follow the decimal point.

In this way, the division operation can be carried out to one more decimal position than the level of precision requires, in order to determine if the final decimal point should be rounded up or left as it is written. If the final digit in the column is five or greater, then it should be rounded up. If it is four or less, the decimal digit should be unchanged when the final answer is written.

#10. 17/110.00102 68068 .4120119 7ans. = 6.5