Before you practice the work from this chapter, take some time to go over some of the multiplication and division strategies and problems we've looked at. These include:

using arrays to solve multiplication and division problems
partitioning, or splitting numbers to solve multiplication problems
using an area model to solve division problems
using a vertical algorithm to solve multiplication and division problems

With these strategies, we were able to move on to multiplication of larger numbers by two digit numbers . We were also able to solve division problems with remainder , as well as dividing by two digit numbers.

Examples

Example 1

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

Example 2

Find the product of 235 \times 14.

Worked Solution

Create a strategy

Use the standard algorithm for multiplication to find the product.

Apply the idea

Set up the vertical algorithm: \begin{array}{c} &&2&3&5 \\ &\times &&1&4 \\ \hline & &&& \\ \hline \end{array}

First we will multiply 235 by 4:

\begin{array}{c} &&{}^12&{}^23&5 \\ &\times &&1&4 \\ \hline &&9&4&0 \\ \hline \end{array}

Now we will multiply 235 by the 1 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} &&2&3&5 \\ \times&& &1&4 \\ \hline &&9&4&0 \\ & 2&3&5& 0 \\ \hline \end{array}

Add our two answers to get the final answer:

\begin{array}{c} &&2 &3 &5 \\ \times & &&1 &4 \\ \hline &&9 &4 &0 \\ +& {}^12 &3 &5 & 0 \\ \hline &3&2&9&0 \end{array}

235\times 14=3290

Example 3

Find the value of 2518\div 8.

Worked Solution

Create a strategy

Use the division algorithm.

Apply the idea

The working and steps are shown below:

A long division with 2518 is divided by 8. Ask your teacher for more information.

8 goes into 2 zero times with remainder 2.
8 goes into 25 three times with remainder 1.
8 goes into 11 one time with remainder 3.
8 goes into 38 four times with remainder 6.

2518\div 8=314 remainder 6.

Idea summary

We have different ways we can solve multiplication and division problems, and as our numbers get larger, the algorithm method can be more useful. Being able to regroup (trade), as well as find remainders with division is important.

Outcomes

ACMNA100

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

ACMNA101

Solve problems involving division by a one digit number, including those that result in a remainder

ACMNA291

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

PRACTICE: Multiplication and division

Ideas

Practice

Examples

Example 1

Example 2

Example 3

Outcomes

ACMNA100

ACMNA101

ACMNA291

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