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Australia
Year 7

1.01 The four operations

Lesson

Introduction

There are four main operations in arithmetic - addition, subtraction, multiplication, and division. They are all kinds of binary operation, where binary means "two" - they each combine two numbers together in different ways to produce a result. We use them to form expressions and equations with numbers:

An image showing parts of an equation with  labels. Ask your teacher for more information.

Addition

The most fundamental of these is addition, which uses the + symbol. The expression 5+3 loosely means "5 with 3 more", which is the same as the number 8. Think about it as moving from left to right along a number line:

A number line from 0 to 10. Numbers 5 and 8 are marked. Ask your teacher for more information.

Starting at 5 and taking 3 steps right lands on 8.

Exploration

If you know how all the single digit numbers add together, you can add numbers of any size by using one of these strategies. Practice your single digit addition here:

Loading interactive...

The symbol + indicates that two numbers are added together.

For larger numbers, we can use a place value table like this:

...HundredsTensUnits
...
+...
\text{Result}...

Write the two numbers you want to add together, one above the other, in the first two rows. Make sure to match their units, tens, hundreds, and so on in the same column. Add the numbers from right to left, adding 1 to the result in the next column along if your sum is 10 or more - we say that we carry the 1.

We can add more than two numbers together in a single place value table. If any column adds up to more than 10, we need to carry the number of tens into the next column - for example, if a column adds to a number between 40 and 49, we need to carry the 4 into the next column.

Another strategy is to use regrouping, where we split each number up into convenient pieces. We can then add the numbers together in a different order, since reversing the order that numbers appear in an addition expression doesn't change the answer.

Examples

Example 1

Find 720+250.

Worked Solution
Create a strategy

Use a place value table.

Apply the idea

Set up the vertical algorithm and add the numbers down each column:

HundredsTensUnits
720
+250
\text{Result}970

720+250=970

Reflect and check

We also could have solved this question by regrouping.

720 is equal to 7 hundreds and 2 tens. 250 is equal to 2 hundreds and 5 tens.

\displaystyle 720+250\displaystyle =\displaystyle 7\text{ hundreds}+2\text{ tens}+2\text{ hundreds}+5\text{ tens}Group the hundreds and tens
\displaystyle =\displaystyle 9\text{ hundreds}+7\text{ tens}Add the hundreds and tens
\displaystyle =\displaystyle 970Write as a single number
Idea summary

We can add two or more numbers together in a single place value table. If any column adds up to more than 10, we need to carry the number of tens into the next column.

We can use regrouping to add large numbers, where we split each number up into convenient pieces such as thousands, hundreds, tens, and ones.

Subtraction

The next operation is subtraction, which uses the - symbol, and is essentially the opposite of addition. The expression 5-3 loosely means "5 with 3 less", which is the same as the number 2. Think about it as moving from right to left along a number line:

A number line from 0 to 10. Numbers 2 and 5 are marked. Ask your teacher for more information.

Starting at 5 and taking 3 steps left lands on 2.

A subtraction equation is always related to an addition equation, like this:

13-7=6 is related to 6+7=13 is related to 13-6=7.

We can perform subtraction using a place value table as well. Write the first number above the second, and work from right to left, subtracting the bottom number from the top number. If the bottom number is larger than the top number in any column, add 10 to the top number and take 1 away from the top number in the next column.

Examples

Example 2

Evaluate 68\,248-194.

Worked Solution
Create a strategy

We can use a place value table.

Apply the idea

Write the numbers in the first two rows of the place value table and subtract the units:

Ten thousandsThousandsHundredsTensUnits
68248
-194
=4

In the tens column, since 9 is larger than 4, we borrow 10 from the hundreds column and perform the subtraction:

Ten thousandsThousandsHundredsTensUnits
681{}^148
-194
=54

Finish by subtracting the numbers in the remaining columns:

Ten thousandsThousandsHundredsTensUnits
681{}^148
-194
=68054

68\,248-194=68\,054

Idea summary

We can perform subtraction using a place value table. Write the first number above the second, and work from right to left, subtracting the bottom number from the top number.

Multiplication

Multiplication uses the symbol \times, and is related to addition. The expression 5\times3 loosely means "3 groups of 5", which is the same as the number 5+5+5=15. Notice that the number 3 indicates how many groups of 5 there are.

There is a connection between multiplication and area - you need 15 small squares to cover a rectangle that is 5 units long and 3 units high:

A rectangle 5 units long and 3 units high made up of 15 small squares. Ask your teacher for more information.

Just like addition, reversing the order that numbers appear in a multiplication expression doesn't change the answer - a rectangle that is 3 units long and 5 units high requires the same number of small squares to cover it:

A rectangle 3 units long and 5 units high made up of 15 small squares. Ask your teacher for more information.

Exploration

Explore this applet to see how two single digit numbers multiply together:

Loading interactive...

Multiplication can be visualised using a rectangular grid of squares where the height is the first number and the width is the second number. Then the number of squares in the grid is the answer.

Once you know how all the single digit numbers multiply together, you can use a place value table to multiply large numbers together. Write one number above the other, and multiply the top numbers by the bottom unit, working right to left.

Examples

Example 3

Evaluate 30\times5.

Worked Solution
Create a strategy

We can use a place value table.

Apply the idea

Write the numbers in the first two rows of the place value table:

HundredsTensUnits
30
\times5
=0

Start from the far right. 5\times 0=0 so we put a 0 in the units column. 5\times 3=15 so we put a 5 in the tens column and a 1 in the hundreds column:

HundredsTensUnits
30
\times5
=150

30\times 5=150

Idea summary

We can use a place value table to multiply large numbers together. Write one number above the other, and multiply the top numbers by the bottom unit, working right to left.

Larger multiplications

To multiply even larger numbers, we use the bottom units digit in exactly the same way. Once that is finished, we write a 0 below our result in the units column and multiply with the bottom tens digit, then add our results at the end.

Another way to multiply large numbers is to use the area model. We imagine a large rectangle split up into smaller ones by place value.

Examples

Example 4

Evaluate 708\times63.

Worked Solution
Create a strategy

Using the place value table, where we need two rows for multiplication, and a final row for adding the results.

Apply the idea

Write the numbers in the first two rows of the table:

Ten thousandsThousandsHundredsTensUnits
708
\times 63
\text{Working}
\text{ }
\text{Result}

Start from the far right. Multiply 3 by each digit in the top number, regrouping where necessary:

Ten thousandsThousandsHundredsTensUnits
7{}^208
\times 63
\text{Working} 2 1 2 4
\text{ }
\text{Result}

Write 0 in the units column below 4, and multiply 6 by each digit in the top number, regrouping where necessary:

Ten thousandsThousandsHundredsTensUnits
7{}^408
\times 63
\text{Working}2 1 2 4
\text{ }4248 0
\text{Result}

Finish the multiplication by adding the results in the third and fourth row. If the result in a column is larger than 9 write the units digit below and carry the tens digit to the next column:

Ten thousandsThousandsHundredsTensUnits
7{}^408
\times 63
\text{Working}{}^12 1 2 4
\text{ }4248 0
\text{Result} 4 46 0 4

708\times63=44\,604

Reflect and check

We can also use an area model to solve this multiplication as you will see in the video.

Idea summary

To multiply even larger numbers, we multiply by the units digit and then we write a 0 below our result in the units column and multiply with the tens digit. Then we add our results at the end.

Another way to multiply large numbers is to use the area model. We draw a large rectangle split up into smaller ones by place value.

Division

The fourth operation is division, which uses the \div symbol, and is essentially the opposite of multiplication. The expression 15\div3 loosely means "15 split into 3 groups", which is the same as the number 5.

Exploration

Explore this applet to see how to divide numbers up to 144:

Loading interactive...

We are dividing the number of squares in the grid by the number of columns to get the number of rows in the grid.

A division equation is (almost) always related to a multiplication equation: 20\div5=4 is related to \\ 4\times5=20 is related to 20\div4=5.

When dividing larger numbers we will use a procedure called short division. We write the number we are dividing by (the divisor) on the left and the number we are dividing (the dividend) on the right, draw a curved line between them, and a bar across to the right. We will be writing our result (called the quotient) above the bar.

An image showing parts of a short division. Ask your teacher for more information.

Together this forms a division tableau. We then work from left to right, performing smaller divisions and carrying the remainder along.

Examples

Example 5

Evaluate 1616\div 4.

Worked Solution
Create a strategy

Use short division.

Apply the idea
A short division with 1616 being divided by 4. Ask your teacher for more information.

Form the short division tableau with 1616 as the dividend and 4 as the divisor.

A short division with 1616 being divided by 4. Ask your teacher for more information.

We cannot divide 1 by 4, so we carry the 1 across to the next number along to form 16.

A short division with 1616 being divided by 4. Ask your teacher for more information.

4 can fit into 16\,\,4 times since 4\times 4=16. So we put the 4 in the hundreds place.

A short division with 1616 being divided by 4. Ask your teacher for more information.

Again, we cannot divide 1 by 4, so we carry the 1 across to the next number along to form 16. We put a 0 in the tens place.

A short division with 3961 being divided by 17. Ask your teacher for more information.

4 can fit into 16\,\,4 times so we put the 4 in the units place.

Since there is no remainder at this step, we conclude that 1616\div 4=404

Idea summary

When dividing larger numbers we will use short division.

An image showing parts of a short division. Ask your teacher for more information.

We write the number we are dividing by (the divisor) on the left and the number we are dividing (the dividend) on the right. We will be writing our result (called the quotient) above the bar.

We then work from left to right, performing smaller divisions and carrying the remainder along.

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