3.02 Groups and arrays

Lesson

We can rewrite  repeated addition  as multiplication. Let's try this problem to practice.

Examples

Example 1

a

Write this addition as a multiplication: 12+12+12+12+12+12+12+12

Worked Solution
Create a strategy

We can rewrite the addition as a multiplication using: \text{Number of groups}\times\text{Amount in each group}

Apply the idea

We have 8 groups of 12. We can rewrite this as multiplication: 12+12+12+12+12+12+12+12=8\times 12

b

Find the value of 8 \times 12.

Worked Solution
Create a strategy

Use skip counting or a multiplication table.

Apply the idea

By skip counting by 12 eight times, we have:12, \,24, \,36, \,48, \,60, \,72, \,84, \,96

8 \times 12=96

Idea summary

If we add something over and over, it's like having many groups of that number. When we have groups of something, we call that multiplication, and we use \times for multiplication.

6+6+6+6+6 is the same as 5 groups of 6, or 5 \times 6.

Arrays as products

In this video we see how rows and columns (called an array), can be used to solve multiplication.

Examples

Example 2

Which of these number sentences describe the array?

There may be more than one correct answer.

A
16 \times 8 =2
B
8 \times 2 = 16
C
16 \times 2 =8
D
2 \times 8 =16
Worked Solution
Create a strategy

If we multiply the number of rows by the number of columns, we find the total number of squares.

Apply the idea

The number sentences which describe the array are options B and D.

Idea summary

We get the same answer whichever way we look at our array.

Groups and arrays for division

We can also see how arrays can be used to solve division.

Examples

Example 3

We can divide 84 by 6 by drawing 84 dots in 6 rows.

a

To work this out we can count groups of 6 until we reach 84.

But that could take a long time if we go 1 group of 6 at a time, so let's count up in larger groups.

If we first count 10 groups of 6, how many dots will we have used?

Worked Solution
Create a strategy

We need to multiply 6 rows by 10 columns using dots.

Apply the idea

Draw 10 columns of the 6 dots to get 60 dots:

So we have used 60 dots.

b

How many dots are remaining when we take away the first 60?

Worked Solution
Create a strategy

Subtract the number of dots in part (a) from the total dots.

Apply the idea

We need to subtract 60 dots from part (a) from the 84 total dots. We can use a vertical algorithm.

\begin{array}{c} &&8&4 \\ &-&6&0 \\ \hline &&2&4 \\ \hline \end{array}

c

How many columns of 6 dots will we have in the group of 24?

Worked Solution
Create a strategy

Divide the number of groups by the number of dots in each column.

Apply the idea

We can also see that there are 4 columns in the array below:

d

Here is the complete array.

Using this, what is 84 \div 6?

Worked Solution
Create a strategy

Count the number of columns of dots in the array.

Apply the idea

There are 10 columns in the left part of the array, and 4 columns in the right part.

So altogether there are 10+4=14 columns of dots in the array.

This means that: 84 \div 6=14

Idea summary

We can use an array to show multiplication (groups of) and division (sharing into groups of), which helps us to write out the numerical problems.