# Multiplication

$$\times \times \times \times \times \times \times \times \times \times$$

## Basic properties

Both commutative and distributive

• Cummutative: $a\times b = b\times a$
• Example: $2\times3 = 6 = 3\times2$
• Associative: $a\times(b\times c) = (a\times b)\times c$
• Example: $2\times(3\times5) = 30 = (2\times3)\times5$

## Application

\begin{align} 25 \times 200 &= 25 \times (2 \times 100) \\ &= (25 \times 2) \times 100 \\ &= 50 \times 100 = 5000 \end{align}

## Another

\begin{align} 8\times 25 &= (2\times2\times2)\times(5\times5) \\ &= 2\times(2\times5)\times(2\times5) \\ &= 2\times10\times10 = 2\times100 = 200. \end{align}

# Distributivity

$$a\times(b+c) = a\times b + a\times c$$

for example, $$3\times(2+4) = 3\times 2 + 3\times 4$$

## A direct model

$3\times(2+4)$ $=$  $3\times2+3\times4$

## Computation

\begin{align} 5 \times 32 &= 5 \times (30 + 2) \\ &= 5\times 30 + 5\times 2 \\ &= 150 + 10 = 160 \end{align}

Note that $5\times 30 = 150$ follows from associativity

\begin{align} 5 \times 30 &= 5 \times (3 \times 10) \\ &= (5 \times 3) \times 10 \\ &= 15 \times 10 = 150 \end{align}

## With a minus sign

\begin{align} 5 \times 28 &= 5 \times (30 - 2) \\ &= 5\times 30 - 5\times 2 \\ &= 150 - 10 = 140 \end{align}

## Multiplication tricks

All the tricks there can be explained using the algebraic properties of multiplication.

### $\times\,4$

Trick: Double, then double again.

Example: $4\times9$

• First double: $9\to18$
• Then, double again: $18\to36$

Explanation: $4\times n = (2\times2)\times n = 2\times(2\times n)$

### $\times\,5$

Trick: Multiply by 10, then cut in half.

Example: $5\times9$

• Multiply by $10$: $9\to90$
• Then, cut in half: $90\to45$

Explanation: $$5\times n = \left(\frac{1}{2}\times10\right)\times n = \frac{1}{2}\times(10\times n)$$

### $\times\,9$

Trick: Multiply by 10, then subtract the number itself.

Example: $9\times13$

• Multiply by $10$: $13\to130$
• Then, subtract $13$: $130\to117$

Explanation: $9\times n = (10-1)\times n = 10\times n - n$