Chapter 5 Understanding Integer Operations and Properties

1. Chapter 5Understanding Integer Operations and Properties Section 5.1 Addition, Subtraction, and Order Properties of Integers

2. Integers • The set of integers, I, consists of the positive integers (non-zero whole numbers), the negativeintegers, and zero. I = { …, -3, -2, -1, 0, 1, 2, 3, … } • The opposite of an integer is the mirror image of the integer around 0 on the number line. • The absolutevalue of an integer is how far away the value is from zero on the number line.

3. Interpreting the “−” Symbol • - 2 “negative 2” • - (- 2) “the opposite of negative 2” • 3 – (- 2) “3 minus negative 2”

4. Modeling Integer Addition 1. Using a Counters Model • Red counter: negative1 • Black counter: positive1 2. Using a Charged Field Model 3. Using a Number Line Model

5. Using a Counters Model (pg. 251) • Begin with an “empty” bag. • Put in 1st number. • Put in 2nd number. • Make “zero” pairs. (1 black and 1 red counter which cancel or “zero” each other out.) • Leftover counters is answer to problem.

6. Using a Counters Model • Examples a) - 1 + 4 b) -2 + (- 3) c) 4 + (- 5)

7. Using a Charged Field Model (pg. 252-3) • Start with a zero-charged field of 10 positives and 10 negatives. • Put in 1st integer. • Put in 2nd integer. • Remove zero pairs. • The sum is the charge of the resulting field. d) - 3 + 1 e) - 1 + (- 2)

8. Using a Number Line Model (pg. 253-4) For addition: • Start at 0, facing right. • Walk forward for positive, backward for negative. • Ending place is answer to problem.

9. Using a Number Line Model For addition: • Start at 0, facing right. • Walk forward for positive, backward for negative. • Ending place is answer to problem. • Examples f) 2 + (- 6) g) - 3 + (- 1)

10. Procedures for Adding Integers • Adding integers with the same sign: Add the absolute values, then use the sign of the integers. Examples: +2 + 6 + 9 = +17 -2 + -6 + -9 = -17 • Adding integers with different signs: Subtract the lesser absolute value from the greater absolute value. Give the answer the same sign as the integer with the greater absolute value. Examples: +8 + -3 = +5 -15 + 4 = -11

11. Basic Properties of Integer Addition • Additive Inverse Property For each integer a, there is a unique integer, -a, such that a + -a = 0. • Closure Property For all integers a and b, a + b is a unique integer. • Additive Identity Property Zero is the unique integer such that for each integer a, a + 0 = 0 + a = a. • Commutative Property For all integers a and b, a + b = b + a. • Associative Property For all integers a, b, and c, (a + b) + c = a + (b + c).

13. Modeling Integer Subtraction • Using a Counters Model • Using a Charged Field Model • Using the Number Line

14. Using a Counters Model (258) • Put in appropriate counters for first integer. • Add black-red pairs to represent second integer. • Then take away second integer from counters. • Remove zero pairs. • Leftover counters is answer. • Examples h) 2 – (- 5) i) - 3 – (- 4)

15. Using a Charged Field Model (259) • Start with a zero-charged field. • Put in counters for first integer. • Take out counters for second integer. • Remove zero pairs. • Leftover counters is answer. • Examples j) -3 – (- 5) k) 1 – 4

16. Using the Number Line (260) • Start at 0, facing right. • Move forward for positive, backward for negative. • Subtraction symbol means that the “walker” will change direction he/she is facing. • Ending place is answer to problem.

17. Using the Number Line • Start at 0, facing right. • Move forward for positive, backward for negative. • Subtraction symbol means that the “walker” will change direction he/she is facing. • Ending place is answer to problem. • Examples l) 2 – 5 m) - 3 – (- 4) n) 5 – (- 2)

18. A Little Theory • For all integers a, b, and c, a – b = c iff c + b = a. • Theorem: Subtracting an Integer by Adding the Opposite For all integers a and b, a – b = a + (-b). That is, to subtract an integer, add its opposite.

19. Comparing and Ordering Integers • Using the Number Line • Using Addition • Definition of Greater Than & Less Than for Integers b > a iff there is a positive integer p such that a + p = b. Also, a < b whenever b > a.