Solution: Page 264-266 (Explorations)

Page 264: Border Sums

Q2. Complete the grids.

Grid 1 (Border Sum +4)

• Corner -10, -5, 9.
• Need to fill empty cells so rows/cols sum to +4.
• Top Row: $-10 + x + y = 4$.
• This requires trial and error based on the full grid structure provided in the book.
  • Note: Without the exact visual of empty cells, we solve logic generally.
  • If Left Column is $-10 + \dots + 9 = 4$, missing middle is $+5$.

Page 265: Figure it Out

1. Integers between pairs (increasing order):

• a. 0 and -7: $-6, -5, -4, -3, -2, -1$.
• b. -4 and 4: $-3, -2, -1, 0, 1, 2, 3$.
• c. -8 and -15: $-14, -13, -12, -11, -10, -9$.
• d. -30 and -23: $-29, -28, -27, -26, -25, -24$.

2. Three numbers sum to -8:

• Example: $-2, -3, -3$.
• Example: $-10, +1, +1$.

3. Dice Sums (Faces: -1, 2, -3, 4, -5, 6)

• Smallest Sum: $(-5) + (-5) = -10$.
• Largest Sum: $6 + 6 = 12$.
• Impossible Numbers:
  • Can we get 11? Need $5+6$ (No 5). Need $6+5$ (No 5). Need $4+7$ (No 7). So 11 is impossible.
  • (Requires checking all combinations).

Page 266: Sequences

6. Complete the sequences:

• a. -40, -34, -28, -22… (Adding +6 each time)
  • Next: -16, -10, -4.
• b. 3, 4, 2, 5, 1, 6, 0, 7… (Pattern: Alternate +1, -2, +3, -4…)
  • Actually, look at alternating terms:
  • Odd pos: 3, 2, 1, 0, -1, -2. (Subtract 1)
  • Even pos: 4, 5, 6, 7, 8, 9. (Add 1)
  • Next 3 terms: -1, 8, -2.
• c. …, 12, 6, 1, -3, -6…
  • 12 to 6 (-6)
  • 6 to 1 (-5)
  • 1 to -3 (-4)
  • -3 to -6 (-3)
  • Next: -6 to ? (-2) $\rightarrow$ -8.
  • Next: -8 to ? (-1) $\rightarrow$ -9.
  • Previous terms: Need to subtract -7 ($19 \to 12$) and -8 ($27 \to 19$). So starts with 27, 19.