Distributivity says the same thing without any multiplying:
(23 + 1) × 27 = 23 × 27 + 1 × 27 = 23 × 27 + 27
Why it happens: 24 × 27 means twenty-four 27s. Twenty-three of them make 23 × 27; the twenty-fourth is one extra 27. Adding 1 to the first factor adds one more copy of the second factor.
Q2.
What if the second number (27) is increased by 1?
Answer
Then the product increases by 23, the first number.
Why it happens: this is the identity a (b + c) = ab + ac with a = 23, b = 27, c = 1. Twenty-three rows of 28 dots is twenty-three rows of 27 dots with one extra dot in every row — that is 23 extra dots.
Why it happens: raising the first factor adds a whole extra row (27 dots), raising the second adds a whole extra column (23 dots), and the single corner dot where the new row meets the new column gets counted once more — that is the “+ 1”.
Q4.
Do you see a pattern that could help generalise our observations to the product of any two numbers?
Answer
Yes. Write the two numbers as a and b. Then
a (b + 1) = ab + a (a + 1) b = ab + b (a + 1)(b + 1) = ab + a + b + 1
Putting 23 for a and 27 for b gives exactly the three increases above: 23, 27 and 51.
Tip: the general statement is worth more than the three answers, because it works for every pair of numbers at once — including negative ones. Later in the chapter it grows into Identity 1: (a + m)(b + n) = ab + mb + an + mn.