Properties of Multiplication

Increments in Products

Consider the multiplication $23 \times 27$.

• If we increase the first number (23) by 1, the new product is $24 \times 27$.
• This effectively adds one more “set” of 27 to the total.
• Increase = 27.

However, what if both numbers increase? Let the numbers be $a$ and $b$. If we increase both by 1, we get $(a+1)(b+1)$.

The product increases by the sum of the original numbers plus 1.

Visualizing Distributivity

The Distributive Property allows us to break numbers down. $a(b+c) = ab + ac$

We can visualize this as a grid of dots. If we have $a$ rows and $(b+c)$ columns, we can split them into a block of $b$ columns and a block of $c$ columns.

Generalizing to Two Binomials

What happens if we multiply two sums? $(a + m)(b + n)$

We distribute each term in the first bracket to every term in the second bracket:

1. Multiply $a$ by $b \rightarrow ab$
2. Multiply $a$ by $n \rightarrow an$
3. Multiply $m$ by $b \rightarrow mb$
4. Multiply $m$ by $n \rightarrow mn$

$(a + m)(b + n) = ab + an + mb + mn$