When we start to work with larger numbers, it helps to have some different strategies to use, including the area method, partitioning and long multiplication. Let's review a strategy with smaller numbers, so then we can use it with larger numbers.

Examples

Example 1

Calculate 10\times 18 by completing an area model.

Firstly, complete the missing values in the area model.

\times	10
10
8

Worked Solution

Create a strategy

Multiply the number at the top of the column by the number on the left of the row.

Apply the idea

\times	10
10	100
8	80

Now add the values you found in part (a) to calculate 10\times 18.

Worked Solution

Create a strategy

Add the two results from part (a).

Apply the idea

\displaystyle 10\times 18	\displaystyle =	\displaystyle 100+80	Add the values
	\displaystyle =	\displaystyle 180

Idea summary

We can use an area model to multiply 2 two digit numbers.

Multiplication

If we are multiplying a number in the thousands, look out for some ways to make it easier, like we do in this video.

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Examples

Example 2

Find 2\times 869.

Worked Solution

Create a strategy

Use the standard algorithm method for multiplication.

Apply the idea

Write the product in a vertical algorithm:

\begin{array}{c} &&8&6&9 \\ &\times &&&2 \\ \hline \\ \hline \end{array}

Start from the far right. Multiply to get 2\times 9 = 18. So we write the 8 in the ones place and carry the 1 to the tens place:

\begin{array}{c} &&8&{}^16&9 \\ &\times &&&2 \\ \hline &&&&8 \\ \hline \end{array}

Move left and multiply to get 2\times 6=12. Then we add the 1 to get 13. So we write the 3 in the tens place and carry the 1 to the hundreds place:

\begin{array}{c} &&{}^18&{}^16&9 \\ &\times &&&2 \\ \hline &&&3&8 \\ \hline \end{array}

Move left and multiply to get 2\times 8 = 16. Then we add the 1 to get 17. So we write the 1 in the thousands place and the 7 in the hundreds place:

\begin{array}{c} &&&{}^18&{}^16&9 \\ &\times & &&&2 \\ \hline &&1&7&3&8 \\ \hline \end{array}

869\times2=1738

Idea summary

The standard algorithm method is useful for multiplying large numbers.

Long multiplication

The area model is great, but we can also perform long multiplication. Let's see how they compare.

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Examples

Example 3

Find 739 \times 63.

Worked Solution

Create a strategy

Use long multiplication.

Apply the idea

First we will multiply 739 by 3.

3 \times 9 = 27 so we put the 7 in the ones place and carry the 2 to the tens place.

\begin{array}{c} &&7&{}^23&9 \\ &\times &&6&3 \\ \hline &&&& 7 \\ \hline \end{array}

3 \times 3=9 then add the carried over 2 to get 11. Put the 1 in the tens place and carry the 1 to the hundreds place.

\begin{array}{c} &&{}^17&{}^23&9 \\ &\times &&6&3 \\ \hline &&&1& 7 \\ \hline \end{array}

3 \times 7=21 then add the carried over 1 to get 22. Put the 2 in the hundreds place and 2 in the thousands place.

\begin{array}{c} &&&{}^17&{}^23&9 \\ &\times &&&6&3 \\ \hline &&2&2&1& 7 \\ \hline \end{array}

Now we will multiply 739 by the 6 in the tens place. We will write our answer underneath our previous answer.

Since we are multiplying by a number in the tens place we will place a 0 in the units place.

\begin{array}{c}&&&7&3&9 \\ &\times &&&6&3 \\ \hline &&2&2&1& 7 \\ &&&&& 0 \\ \hline \end{array}

6\times 9=54 so we put a 4 in the tens place and carry the 5 to the tens place.

\begin{array}{c}&&&7&{}^53&9 \\ &\times &&&6&3 \\ \hline &&2&2&1& 7 \\ &&&&4& 0 \\ \hline \end{array}

6\times 3=18 then add the carried over 5 to get 23. Put a 3 in the hundreds place and carry the 2 to the hundreds place.

\begin{array}{c}&&&{}^27&{}^53&9 \\ &\times &&&6&3 \\ \hline &&2&2&1& 7 \\ &&&3&4& 0 \\ \hline \end{array}

6\times7=42 then add the carried over 2 to get 44. Put a 4 in the thousands place and put 4 to the ten thousands place.

\begin{array}{c}&&&&{}^27&{}^53&9 \\ &\times &&&&6&3 \\ \hline &&&2&2&1& 7 \\ &&4&4&3&4& 0 \\ \hline \end{array}

Add our two answers to get the final answer:

\begin{array}{c}&&&&7&3&9 \\ &\times &&&&6&3 \\ \hline &&&2&2&1& 7 \\ &+&4&4&3&4& 0 \\ \hline &&4&6&5&5&7 \end{array}

739\times 63=46\,557

Idea summary

As our numbers get larger, it might be easier to use long multiplication. We can still use the area method and partitioning, but long multiplication allows us to set out our problem more easily.

Multiplication in written problems

The words and the context they're used in can give us clues to how we need to tackle a problem.

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Examples

Example 4

A rock climber descends from the top of a cliff face to the ground in 5 stages, dropping 106 metres each time.

What is the height of the cliff face?

Worked Solution

Create a strategy

Multiply the number of times the rock climber drops down by the distance the rock climber drops each time.

Apply the idea

The climber drops 106 metres 5 times.

We can break 106 up into 100 and 6, and multiply them by 5.

\displaystyle \text{Height}	\displaystyle =	\displaystyle 100\times 5 + 6 \times 5	Multiply both 100 and 6 by 5
	\displaystyle =	\displaystyle 500+30	Find the multiplications
	\displaystyle =	\displaystyle 530 \text{ metres}	Add the numbers

Idea summary

When we have a written problem, all we need to do is find the keywords to help us understand what we need to work out.

Outcomes

AC9M6N09

use mathematical modelling to solve practical problems, involving rational numbers and percentages, including in financial contexts; formulate the problems, choosing operations and efficient calculation strategies, and using digital tools where appropriate; interpret and communicate solutions in terms of the situation, justifying the choices made

4.02 Larger multiplications

Ideas

Are you ready?

Examples

Example 1

Multiplication

Examples

Example 2

Long multiplication

Examples

Example 3

Multiplication in written problems

Examples

Example 4

Outcomes

AC9M6N09

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