Lesson

## Ideas

It's time to look back over how we can use a vertical algorithm to solve  addition  and  subtraction  problems. Remember to:

• line digits up by place value columns

• start adding, or subtracting, from the units place, then move left

• estimate your answer first to get an idea of what to expect

We also looked at how to use a vertical algorithm with trading, or regrouping. For  addition, we regroup  to the higher place value, but for  subtraction, we regroup  down to the next lower place value.

### Examples

#### Example 1

Find the value of 34\,246+3213.

Worked Solution
Create a strategy

Use the standard algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & &3 &4 &2 &4 &6 \\ &+ & &3 &2 &1 &3 \\ \hline & \\ \hline \end{array}

Add the units column first: 6 + 3 = 9.

\begin{array}{c} & &3 &4 &2 &4 &6 \\ &+ & &3 &2 &1 &3 \\ \hline & & & & & &9 \\ \hline \end{array}

Add the tens column: 4 + 1 = 5

\begin{array}{c} & &3 &4 &2 &4 &6 \\ &+ & &3 &2 &1 &3 \\ \hline & & & & &5 &9 \\ \hline \end{array}

Add the hundreds column: 2 + 2 = 4

\begin{array}{c} & &3 &4 &2 &4 &6 \\ &+ & &3 &2 &1 &3 \\ \hline & & & &4 &5 &9 \\ \hline \end{array}

Add the thousands column: 4 + 3 = 7

\begin{array}{c} & &3 &4 &2 &4 &6 \\ &+ & &3 &2 &1 &3 \\ \hline & & &7 &4 &5 &9 \\ \hline \end{array}

Add the ten thousands column: 3 + 0 = 3

\begin{array}{c} & &3 &4 &2 &4 &6 \\ &+ & &3 &2 &1 &3 \\ \hline & & 3 &7 &4 &5 &9 \\ \hline \end{array}

34\,246 + 3213 = 37\,459

#### Example 2

Find the value of 49\,481 + 49\,100.

Worked Solution
Create a strategy

Use the standard algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & & &4 &9 &4 &8 &1 \\ &+ & &4 &9 &1 &0 &0 \\ \hline & \\ \hline \end{array}

Add the units column first.\begin{array}{c} & & &4 &9 &4 &8 &1 \\ &+ & &4 &9 &1 &0 &0 \\ \hline & & & & & & &1\\ \hline \end{array}

In the tens column, we get 8+0=8.\begin{array}{c} & & &4 &9 &4 &8 &1 \\ &+ & &4 &9 &1 &0 &0 \\ \hline & & & & & &8 &1\\ \hline \end{array}

In the hundreds column, we get 4+1=5.\begin{array}{c} & & &4 &9 &4 &8 &1 \\ &+ & &4 &9 &1 &0 &0 \\ \hline & & & & & 5&8 &1\\ \hline \end{array}

In the thousands column we get 9 + 9 = 18. So we bring down 8 and carry the 1 to the ten thousands place.\begin{array}{c} & & &\text{}^1 4 &9 &4 &8 &1 \\ &+ & &4 &9 &1 &0 &0 \\ \hline & & & & 8& 5&8 &1\\ \hline \end{array}

For the ten thousands place we get 1+4+4=9.\begin{array}{c} & & &\text{}^1 4 &9 &4 &8 &1 \\ &+ & &4 &9 &1 &0 &0 \\ \hline & & & 9& 8& 5&8 &1\\ \hline \end{array}

So the answer is:49\,481 + 49\,100 = 98\,581

#### Example 3

Find the value of 55\,872-2619.

Worked Solution
Create a strategy

Use the standard algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & & &5 &5 &8 &7 &2 \\ &- & & &2 &6 &1 &9 \\ \hline & \\ \hline \end{array}

In the ones column we can see that 2 is less than 9, so we need to trade 1 ten from the tens place.

So we get 12-9=3 in the ones column and 7 tens becomes 6 tens in the first row.\begin{array}{c} & && 5&5 &8 &6 &\text{}^1 2 \\ &- && &2 &6 &1 &9 \\ \hline & && & & & &3\\ \hline \end{array}

In the tens column, we have 6-1=5:\begin{array}{c} & && 5&5 &8 &6 &\text{}^1 2 \\ &- && &2 &6 &1 &9 \\ \hline & && & & & 5 &3\\ \hline \end{array}

In the hundreds column, we have 8-6=2:\begin{array}{c} & && 5&5 &8 &6 &\text{}^1 2 \\ &- && &2 &6 &1 &9 \\ \hline & && & & 2& 5 &3\\ \hline \end{array}

In the thousands column, we have 5-2=3:\begin{array}{c} & && 5& 5 &8 &6 &\text{}^1 2 \\ &- && &2 &6 &1 &9 \\ \hline & && & 3& 2& 5 &3\\ \hline \end{array}

And for the ten thousands place 5-0=5:\begin{array}{c} & && 5& 5 &8 &6 &\text{}^1 2 \\ &- && &2 &6 &1 &9 \\ \hline & && 5& 3& 2& 5 &3\\ \hline \end{array}

So the answer is:55\,872 - 2619 = 53\,253

Idea summary

A vertical algorithm is very useful when we starting working with more digits. Even though our numbers may be bigger, the problems are solved the same way, just with more steps. For addition, we need to regroup, or trade to the higher place value, but for subtraction we regroup down to the next lower place.

### Outcomes

#### VCMNA185

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