Let's see how we use the partition, array, and area method to multiply 2 digit numbers by 1 digit numbers, then see how we could use an algorithm to solve our problem.

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Examples

Example 2

Let's use an area model to find 77 \times 3.

Fill in the areas of each rectangle.

A rectangle with a length of 77 and a height of 3 divided into 4 rectangles. Ask your teacher for more information.

Worked Solution

Create a strategy

For each rectangle, use the formula \text{Area} = \text{length} \times \text{width}.

Apply the idea

Area of top left rectangle: 70 \times 2= 140

Area of top right rectangle: 7 \times 2= 14

Area of bottom left rectangle: 70 \times 1= 70

Area of bottom right rectangle: 7 \times 1= 7

What is the total area of all four rectangles?

Worked Solution

Create a strategy

Add the areas from part (a) together using a place value table.

Apply the idea

Put the numbers in a place value table:

Hundreds	Tens	Ones
1	4	0
	7	0
	1	4
		7

Now we can add the numbers down each column and regroup where needed:

	Hundreds	Tens	Ones
	1	4	0
		7	0
		1	4
+	1	1	7
=	2	3	1

4+7 = 11 so we put a 1 in the ones column and carry the 1 to the tens column.
Then 4+7+1+1=13 so we put a 3 in the tens column and carry the 1 to the hundreds column.

So the total area is 231.

Find 77 \times 3.

Worked Solution

Create a strategy

Use the answer from part (b).

Apply the idea

The rectangle from part (a) has a length of 77 and a width of 3. So we can find its area using 77\times 3. But since we already found its area in part (b) to be 231, we get: 77\times 3 = 231

Idea summary

We have different ways we can solve multiplication problems, such as the area method and arrays. But as our numbers get larger, the algorithm method can be more useful.

Multiply 3 digit numbers

We can also use the strategies we've seen above to multiply larger numbers, so let's see how the area method can be used. Then we solve the same problem, using a vertical algorithm. Which do you prefer?

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Examples

Example 3

Find 6178 \times 4.

Worked Solution

Create a strategy

Use standard algorithm method for multiplication.

Apply the idea

Write the product in a vertical algorithm:

\begin{array}{c} &&6&1&7&8 \\ &\times &&&&4 \\ \hline \\ \hline \end{array}

Start from the far right. Multiplying 4 by 8, we have 32 which can be written as 3 tens and 2 units. Write 2 underneath 4 and carry the 3 to the tens column:

\begin{array}{c} &&&6&1&\text{}^3 7&8 \\ &\times &&&&&4 \\ \hline &&&&&&2\\ \hline \end{array}

Then move to the left. Multiply 4 by 7 and add the 3 to get 28+3=31. Write 1 in the tens columns and carry the 3 to the hundreds column:

\begin{array}{c} &&&6&\text{}^3 1&\text{}^3 7&8 \\ &\times &&&&&4 \\ \hline &&&&&1&2\\ \hline \end{array}

Move to the left again. Multiply 4 by 1 and add 3 to get 4+3=7. Write 7 in the hundreds column:

\begin{array}{c} &&& 6&\text{}^3 1&\text{}^3 7&8 \\ &\times &&&&&4 \\ \hline &&&&7&1&2\\ \hline \end{array}

Multiply 4 by 6 to get 24. Write 2 in the ten thousands column and 4 in the thousands column:

\begin{array}{c} &&& 6&\text{}^3 1&\text{}^3 7&8 \\ &\times &&&&&4 \\ \hline &&2&4&7&1&2\\ \hline \end{array}

6178 \times 4=24\,712

Idea summary

We can use several different methods to solve multiplication problems, but we don't always know if we will need to do any regrouping or trading. If we use an algorithm, regrouping can be done as we solve our problem.

Outcomes

ACMNA100

Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

ACMNA291

Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

3.03 Multiplication by single digits

Ideas

Are you ready?

Examples

Example 1

Multiply 2 digit numbers

Examples

Example 2

Multiply 3 digit numbers

Examples

Example 3

Outcomes

ACMNA100

ACMNA291

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