Integers, Rational Numbers, and Integer Operations

Integers and Rational Numbers

Integers (Date: 9/8/26)

• Definition: Whole numbers, their opposites, and zero.

Classification Examples (Integer? Yes or No)

• $612$: Yes
• $\frac{8}{9}$: No
• $0$: Yes
• $-43$: Yes

Rational Numbers

• Definition: Numbers that can be expressed as a fraction in the form $\frac{a}{b}$, where $b \neq 0$.

Forms of Rational Numbers

• Fractions: Examples include $\frac{2}{3}$ and $-5\frac{1}{8}$.
• Decimals: Examples include $4.5$ and $-1.01$.
• Integers: Examples include $0$ and $-2$.

Classification Examples (Rational Number? Yes or No)

• $\frac{4}{5}$: Yes
• $104$: Yes (can be written as $\frac{104}{1}$)
• $-\frac{8}{17}$: Yes
• $3.97$: Yes

Graphing on a Number Line

• Values to Plot:
  • Point $A = 2\frac{1}{2}$
  • Point $B = 0.5$
• Number Line Representation:
  • On a number line spanning from $-5$ to $5$:
  • Point $B$ is plotted at $0.5$ (halfway between $0$ and $1$).
  • Point $A$ is plotted at $2\frac{1}{2}$ ($2.5$, halfway between $2$ and $3$).

Addition of Integers

Addition Rules (Date: 9/9/26)

• Same Signs: If the signs are the same, add the numbers and keep the sign.
• Different Signs: If the signs are different, subtract the smaller absolute value from the larger absolute value and use the sign of the larger absolute value.

Step-by-Step Examples

• Example 1: $5 + 14 = 19$
  • Rule: Same signs, both are positive. Add numbers, keep the positive sign.
• Example 2: $-8 + (-23) = -31$
  • Rule: Same signs, both are negative. Add $8 + 23 = 31$ and keep the negative sign.
• Example 3: $22 + (-6) = 16$
  • Rule: Different signs.
  • Calculation: $|22| - |-6| = 22 - 6 = 16$.
  • Sign: The sign of $22$ is positive, so the answer is positive ($16$).
• Example 4: $9 + (-27) = -18$
  • Rule: Different signs.
  • Calculation: $|27| - |9| = 27 - 9 = 18$.
  • Sign: The sign of $27$ is negative, so the answer is negative ($-18$).

Addition Practice Table ("Try these")

Subtraction of Integers

Rules for Subtracting Integers (Date: 9/10)

1. Add the opposite! (KCC Method)
   • K (Keep): Keep the $1\text{st}$ number.
   • C (Change): Change subtraction ($-$) to addition ($+$).
   • C (Change): Change the sign of the $2\text{nd}$ number to its opposite.
   • Important Rule: KCC is for subtraction only!
2. Then follow the addition rules.

Demonstrative Examples of KCC

• Example 1: $17 - 18$
  • Apply KCC: Keep $17$, Change -$ to +$, Change$18$to$-18.\n - Rewritten form: 17 + (-18)\n- **Example 2:** 15 - (-2)\n - Apply KCC: Keep 15$, Change$-$ to $+$, Change $-2$ to $2$.
  • Rewritten form: $15 + 2$

KCC Rewriting Practice

• Problem: $13 - 17$ $\rightarrow$ Re-write using KCC: $13 + (-17)$
• Problem: $-11 - 12$ $\rightarrow$ Re-write using KCC: $-11 + (-12)$
• Problem: $12 - (-8)$ $\rightarrow$ Re-write using KCC: $12 + 8$
• Problem: $-8 - (-38)$ $\rightarrow$ Re-write using KCC: $-8 + 38$

Subtraction Practice Table ("Try these", Date: 9/16)