Observing Patterns in the Tables

1st Table:

• Left Column:

  • The multiplier $2$ is multiplied by integers (positive, zero, and negative).
  • Pattern:
    • As the number decreases by 1 (from $4, 3, 2, 1, 0, -1, -2$), the result decreases by 2 for each step.
    • Zero multiplication results in $0$, showing $2 \times 0 = 0$.
    • Negative multiplication inverts the sign of the result: $2 \times (-n) = -2n$.

• Right Column:

  • The multiplier is $-2$ instead of $2$.
  • Pattern:
    • The result remains consistent in magnitude but alternates in sign compared to the left column.
    • $-2 \times 4 = -8$, $-2 \times (-4) = 8$.

2nd Table:

• Powers of 10:
  • The multiplier $2$ is multiplied by powers of 10 ($10^4, 10^3, \ldots, 10^{-2}$).
  • Pattern:
    • Each power of 10 increases or decreases the result by a factor of 10.
    • For positive exponents ($10^4, 10^3, 10^2, \ldots$), the result gets multiplied by 10 each time.
    • For negative exponents ($10^{-1}, 10^{-2}$), the result divides by 10, leading to decimal values $0.2, 0.02$.

Summary of Patterns:

1. Left Column (1st Table): Multiplying by $2$ results in an arithmetic sequence decreasing by 2 as numbers move from positive to negative.
2. Right Column (1st Table): Multiplying by $-2$ inverts the sign of the result while maintaining magnitude.
3. 2nd Table (Powers of 10): Multiplying by powers of 10 results in scaling up or down by a factor of 10 for each power.

Would you like a deeper analysis of one table or an explanation of these concepts in mathematical terms?

Questions to Explore:

1. Why does multiplying by zero always result in zero?
2. What happens if we multiply $2$ by higher negative powers of $10$ (e.g., $10^{-3}$)?
3. Can the pattern in the 1st table apply to other multipliers (e.g., $3$ or $4$)?
4. How does the pattern in powers of 10 relate to scientific notation?
5. What changes in the patterns if the multiplier is a fraction (e.g., $\frac{1}{2}$)?

Tip:

Arithmetic sequences and powers of 10 are foundational in algebra and help simplify larger calculations using patterns!