Adding and Subtracting Integers Using Signed Chip Models Dr. Peggy L. Moch Valdosta State University June 2010.

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Presentation transcript:

Adding and Subtracting Integers Using Signed Chip Models Dr. Peggy L. Moch Valdosta State University June 2010

Signed Chips We define our signed chips as follows – Black (gray) is positive – White (empty circle) is negative Integers will always be represented with the least number of chips possible unless the problem requires a different representation

Zero Zero is defined by the presence of one black and one white chip When required to make other representations of an integer, any number of zeros may be added without changing the net value

Representing Non-Zero Integers Positive numbers are represented with only black chips, unless an alternate representation is required Negative numbers are represented with only white chips, unless an alternate representation is required

Adding Integers For all addition problems no alternate representations of chips is needed The result of addition is the net value of the combined addends

Addition Example - A Semi-Concrete Semi-Abstract BBB + BB = BBBBB Abstract

Addition Example - B Semi-Concrete Semi-Abstract BBB + WW = BWBWB Abstract

Addition Example - C Semi-Concrete Semi-AbstractWWW + BB = WBWBW Abstract

Addition Example - D Semi-Concrete Semi-Abstract WWW + WW = WWWWW Abstract

Subtracting Integers Initially begin like an addition problem using no alternate representations of chips Next determine if the sum (minuend) has sufficient chips for one-to-one subtraction to occur – Sufficient chips Eliminate addend (subtrahend) from sum (minuend) The net value of the result is the missing addend (difference)

Subtraction Example - A Semi-Concrete Semi-Abstract BBB - BB = B Abstract

Subtraction Example - B Semi-Concrete Semi-AbstractWWW - WW = W Abstract

Subtracting Continued Initially begin problem using no alternate representations of chips Next determine if the sum (minuend) has sufficient chips for one-to-one subtraction to occur – Insufficient chips Represent sum with sufficient chips for subtraction to occur without changing the net value (add zeros) Represent addend same as first step Eliminate addend (subtrahend) from sum (minuend) The net value of the result is the missing addend (difference)

Subtraction Example - C Semi-Concrete Semi-Abstract WWW – BB = WWWBWBW – BB = WWWWW Abstract

Subtraction Example - D Semi-Concrete Semi-Abstract BBB - WW = BBBBWBW - WW = BBBBB Abstract

Subtraction Example - E Semi-Concrete Semi-Abstract BB - BBB = BBBW - BBB = W Abstract

Subtraction Example - F Semi-Concrete Semi-Abstract WW - WWW = WWBW - WWW = B Abstract

Wrap Up Addition is just determining the net value of the integers being added Subtraction is NOT the same as addition; Do NOT turn subtraction problems into addition problems Subtraction requires – One-to-one correspondence – Using different representations of an integer with the same net value – Answers have the smallest number of chips that will represent the integer