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 31. Write 1 in the tens columns and carry the 3 to the hundreds column:

\begin{array}{c} &&&6&{}^31&\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 7. Write 7 in the hundreds column:

\begin{array}{c} &&&6&{}^31&\text{}^3 7&8 \\ &\times &&&&&4 \\ \hline &&&&7&1&2\\ \hline \end{array}

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

\begin{array}{c} &&&6&{}^31&\text{}^3 7&8 \\ &\times &&&&&4 \\ \hline &&2&4&7&1&2\\ \hline \end{array}

6178 \times 4 = 24\,712

Idea summary

If we use an algorithm, regrouping can be done as we solve our problem.

Arrays and area models

Let's see how we can use area models and arrays to work out division problems, numbers up to hundreds.

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Examples

Example 2

Let's use an area model to find the answer to 45 \div 3.

We set up the area model using a rectangle like this:

A rectangle with a total area of 45 and a width of 3.

We don't know what 45\div3 is straight away, we start with something we do know, like 30\div 3.

Find the area used so far if we take out 10 groups of 3.

A rectangle with a total area of 45 and a width of 3 divided into 2 rectangles. Ask your teacher for more information.

Worked Solution

Create a strategy

"Groups of" means multiply.

Apply the idea

10 groups of 3 can also be written as: 10\times 3.

\displaystyle \text{Area of left rectangle}	\displaystyle =	\displaystyle 10\times3	Multiply the sides
	\displaystyle =	\displaystyle 30

How much area is remaining?

Worked Solution

Create a strategy

The area of remaining is the area of the left rectangle. Subtract the area found in part (a) from the total area of the rectangle.

Apply the idea

\displaystyle \text{Area of left rectangle}	\displaystyle =	\displaystyle 45-30	Subtract the areas
	\displaystyle =	\displaystyle 15

The remaining area is 15.

A rectangle divided into 2 rectangles with areas of 30 and 15. Ask your teacher for more information.

What is the width of the second rectangle?

Worked Solution

Create a strategy

Use the area of the rectangle found in part (b).

Apply the idea

We know that the area of the second rectangle is 15 and the height is 3. So the width will be the area divided by the height, or 15\div3.

To solve this, we can arrange 15 circles into 3 rows:

An array of 15 circles with 3 rows and 5 columns.

There are 5 circles in each row.

The the width of the right rectangle is 5.

Using the area model above, what is 45\div3?

Worked Solution

Create a strategy

45\div3 is the total width of the rectangle in part (c).

Apply the idea

\displaystyle \text{Total width}	\displaystyle =	\displaystyle 10+5	Add the widths
	\displaystyle =	\displaystyle 15

45\div 3=15

Idea summary

We can use area models to divide one number by another number.

Strategies for division

We can use what we know about place value and partitioning numbers to solve division problems as well, as we see in this video.

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Examples

Example 3

Find the value of 396 \div 3.

Worked Solution

Create a strategy

We can partition 396 to make it easier to divide by 3.

Apply the idea

Put 396 into a place value table:

Hundreds	Tens	Ones
3	9	6

396 is made up of 3 hundreds, 9 tens and 6 ones. So: 396=300+90+6 Now we can divide each part by 3.

\displaystyle 396\div3	\displaystyle =	\displaystyle 300\div3+90\div3+6\div3	Divide each part by 3
	\displaystyle =	\displaystyle 100+30+2	Find the divisions
	\displaystyle =	\displaystyle 130+2	Add 100 and 30
	\displaystyle =	\displaystyle 132	Add 2

Idea summary

We can divide large numbers by partitioning the number, and then dividing each part of the partition.

Division algorithm

We can use a vertical algorithm to solve division, and it helps when we work with larger numbers, as this video shows.

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Examples

Example 4

Find the value of 856 \div 8.

Worked Solution

Create a strategy

Use the division algorithm.

Apply the idea

856 divided by 8 in a short division algorithm. Ask your teacher for more information.

Set up the algorithm.

8 goes into 8 once, so we put a 1 in the hundreds column.

8 goes into 5 zero times with 5 remaining, so we put a 0 in the tens column and carry the 5 to the units column.

8 goes into 56 seven times, so we put a 7 in the units column.

856 \div 8 = 107

Idea summary

Division is when we share a total into a number of groups, or find out how many items each group has. It is the reverse of multiplication.

Outcomes

MA3-5NA

selects and applies appropriate strategies for addition and subtraction with counting numbers of any size

MA3-6NA

selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation

3.05 Dividing two and three digit numbers

Ideas

Are you ready?

Examples

Example 1

Arrays and area models

Examples

Example 2

Strategies for division

Examples

Example 3

Division algorithm

Examples

Example 4

Outcomes

MA3-5NA

MA3-6NA

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