Long Multiplication

You are already skilled at using your times tables for single digits. But what happens when you need to multiply bigger numbers, like $2 \times 34$?
Long multiplication is a powerful strategy that breaks big problems down into small, easy steps, using the times tables you already know!

A Pattern with Tens

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To get ready for long multiplication, let's explore a pattern. What happens when we multiply a number by 10, 20, 30, or other multiples of 10?

3 groups of 10 ($3 \times 10$):

$\textcolor{colordef}{3}\times \textcolor{colorprop}{1}0$ $=$ $\textcolor{colordef}{3}\times$

$=$ $+$ $+$

$=$ $\textcolor{olive}{3}0$
3 groups of 20 ($3 \times 20$):

$\textcolor{colordef}{3}\times \textcolor{colorprop}{2}0$ $=$ $\textcolor{colordef}{3}\times$

$=$ $+$ $+$

$=$ $\textcolor{olive}{6}0$

Do you see a pattern emerging?

$3 \times 4 = 12 \quad \rightarrow \quad 3 \times 40 = 120$
$3 \times 5 = 15 \quad \rightarrow \quad 3 \times 50 = 150$
$3 \times 6 = 18 \quad \rightarrow \quad 3 \times 60 = 180$

The pattern is that multiplying by a multiple of 10 is like using the basic times table, but then adding a zero to the end of the answer!

Proposition The "Add a Zero" Rule

To multiply by a multiple of 10, you can multiply the non-zero digits first, then place a zero at the end of your answer.$\begin{aligned}\textcolor{colordef}{3}\times \textcolor{colorprop}{1}&=\textcolor{olive}{3}\\\textcolor{colordef}{3}\times \textcolor{colorprop}{2}&=\textcolor{olive}{6 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{3}&=\textcolor{olive}{9 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{4}&=\textcolor{olive}{12 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{5}&=\textcolor{olive}{15 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{6}&=\textcolor{olive}{18 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{7}&=\textcolor{olive}{21 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{8}&=\textcolor{olive}{24 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{9}&=\textcolor{olive}{27 }\\\end{aligned}$ $\quad \rightarrow \quad$ $\begin{aligned}\textcolor{colordef}{3}\times \textcolor{colorprop}{1}0&=\textcolor{olive}{3}0\\\textcolor{colordef}{3}\times \textcolor{colorprop}{2}0&=\textcolor{olive}{6}0\\\textcolor{colordef}{3}\times \textcolor{colorprop}{3}0&=\textcolor{olive}{9}0\\\textcolor{colordef}{3}\times \textcolor{colorprop}{4}0&=\textcolor{olive}{12}0\\\textcolor{colordef}{3}\times \textcolor{colorprop}{5}0&=\textcolor{olive}{15}0\\\textcolor{colordef}{3}\times \textcolor{colorprop}{6}0&=\textcolor{olive}{18}0\\\textcolor{colordef}{3}\times \textcolor{colorprop}{7}0&=\textcolor{olive}{21}0\\\textcolor{colordef}{3}\times \textcolor{colorprop}{8}0&=\textcolor{olive}{24}0\\\textcolor{colordef}{3}\times \textcolor{colorprop}{9}0&=\textcolor{olive}{27}0\\\end{aligned}$

Multiplying Step-by-Step

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How can we solve $2 \times 34$? The key is to break 34 into its place values: 30 (tens) and 4 (ones).

$2 \times 34$	$=$	$2 \times$
	$=$	$2 \times$ $+ 2 \times$
	$=$	$+$
	$=$

$\quad \rightarrow \quad$ $\begin{aligned}[t]2 \times 34&=2 \times (30 + 4) \\&=(2 \times 30) + (2 \times 4) \\&=60 + 8\\&=68\\\end{aligned}$The column method helps us organize this process neatly.

Method Long Multiplication by One-Digit Numbers

Calculate:$$34\times 2 $$

Step 1: Set up the problem. Write the first number on top and the second number below, lining up the place values.
Step 2: Multiply the ones. Multiply the ones digit of the top number by the bottom number. ($4 \times 2 = 8$)
Step 3: Multiply the tens. Multiply the tens digit of the top number by the bottom number. ($3 \text{ tens} \times 2 = 6 \text{ tens}$, which is 60).
Step 4: Add the results together. Sum the partial products to find the final answer. ($8 + 60 = 68$)

\(\textcolor{colordef}{3}\times \textcolor{colorprop}{1}0\)	\(=\)	\(\textcolor{colordef}{3}\times\)
	\(=\)	\(+\) \(+\)
	\(=\)	\(\textcolor{olive}{3}0\)

\(\textcolor{colordef}{3}\times \textcolor{colorprop}{2}0\)	\(=\)	\(\textcolor{colordef}{3}\times\)
	\(=\)	\(+\) \(+\)
	\(=\)	\(\textcolor{olive}{6}0\)

\(2 \times 34\)	\(=\)	\(2 \times\)
	\(=\)	\(2 \times\) \(+ 2 \times\)
	\(=\)	\(+\)
	\(=\)