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

2. 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

3. 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

4. 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

5. 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

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

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

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

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

10. 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)

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

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

13. 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)

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

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

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

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

18. 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