Addition and Subtraction of Positive and Negative Integers for JSS 1

Addition and Subtraction

Performance Objectives

By the end of the lesson, students should be able to:

1. add and subtract given numbers correctly;
2. state the place-value of each number in the sum or difference;
3. draw and use number line to illustrate directed numbers;
4. add and subtract positive and negative integers correctly on the number line;
5. interpret and relate positive and negative numbers to everyday activities.

Addition and Subtraction of Numbers

Addition is the process of combining two or more numbers to find their total or sum. Subtraction is the process of taking one number away from another to find the difference between them.

When performing addition and subtraction on whole numbers or decimal numbers, it is important to align the digits according to their correct positions. For decimal numbers, the decimal points must be aligned vertically before performing the operation.

Worked Examples

Example 1: Find the sum of $458,210$ and $329,845$.

To solve this, align the numbers vertically by their place values and add from right to left:

$$
\begin{array}{rcccccc}
& 4 & 5 & 8 & 2 & 1 & 0 \\
+ & 3 & 2 & 9 & 8 & 4 & 5 \\
\hline
& 7 & 8 & 8 & 0 & 5 & 5 \\
\hline
\end{array}
$$

The sum of the numbers is $788,055$.

Example 2: Subtract $145.82$ from $628.50$.

To solve this, align the decimal points vertically and subtract from right to left, borrowing where necessary:

$$
\begin{array}{rccccc}
& 6 & 2 & 8 & . & 5 & 0 \\
– & 1 & 4 & 5 & . & 8 & 2 \\
\hline
& 4 & 8 & 2 & . & 6 & 8 \\
\hline
\end{array}
$$

The difference is $482.68$.

Example 3: Find the value of $82.4 + 15.78 – 34.65$.

First, perform the addition by aligning the decimal points. Note that $82.4$ can be written as $82.40$ to match the decimal places:

$$
\begin{array}{rcccc}
& 8 & 2 & . & 4 & 0 \\
+ & 1 & 5 & . & 7 & 8 \\
\hline
& 9 & 8 & . & 1 & 8 \\
\hline
\end{array}
$$

Next, subtract $34.65$ from the result:

$$
\begin{array}{rcccc}
& 9 & 8 & . & 1 & 8 \\
– & 3 & 4 & . & 6 & 5 \\
\hline
& 6 & 3 & . & 5 & 3 \\
\hline
\end{array}
$$

The final value is $63.53$.

Place Value

Place value is the numerical value that a digit has by virtue of its position in a number.

Every digit in a number has a specific place value depending on whether it is to the left or right of the decimal point. The table below shows the place value names for whole numbers and decimals:

Worked Examples

Example 1: Find the sum of $342.5$ and $58.19$, and state the place value of each digit in the final answer.

First, perform the addition:

$$
\begin{array}{rcccc}
& 3 & 4 & 2 & . & 5 & 0 \\
+ & & 5 & 8 & . & 1 & 9 \\
\hline
& 4 & 0 & 0 & . & 6 & 9 \\
\hline
\end{array}
$$

The sum is $400.69$. The place value of each digit in $400.69$ is stated below:

1. The digit $4$ is in the Hundreds ($H$) position, representing $400$.
2. The first digit $0$ is in the Tens ($T$) position, representing $0$.
3. The second digit $0$ is in the Units ($U$) position, representing $0$.
4. The digit $6$ is in the tenths ($t$) position, representing $\frac{6}{10}$ or $0.6$.
5. The digit $9$ is in the hundredths ($h$) position, representing $\frac{9}{100}$ or $0.09$.

Example 2: Subtract $74.8$ from $125.35$, and state the place value of the digit $5$ in the difference.

First, perform the subtraction:

$$
\begin{array}{rcccc}
& 1 & 2 & 5 & . & 3 & 5 \\
– & & 7 & 4 & . & 8 & 0 \\
\hline
& & 5 & 0 & . & 5 & 5 \\
\hline
\end{array}
$$

The difference is $50.55$. The place values of the digit $5$ in the result are:

1. The first $5$ (in $50$) is in the Tens ($T$) position, representing $50$.
2. The second $5$ (in $0.5$) is in the tenths ($t$) position, representing $\frac{5}{10}$ or $0.5$.
3. The third $5$ (in $0.05$) is in the hundredths ($h$) position, representing $\frac{5}{100}$ or $0.05$.

Directed Numbers and the Number Line

Directed numbers are numbers that have both a size and a specific direction, represented as either positive or negative.

Positive numbers are numbers greater than zero, written with a plus sign ($+$) or with no sign at all. Negative numbers are numbers less than zero, always written with a minus sign ($-$). Zero is neither positive nor negative; it acts as the starting point or origin.

A number line is a straight line with numbers placed at equal intervals. It is used to visualize directed numbers and perform basic operations. On a standard horizontal number line, positive numbers lie to the right of zero, and negative numbers lie to the left of zero.

Observe the diagram of a standard number line below.

Addition and Subtraction of Positive and Negative Integers

Integers are whole numbers that can be positive, negative or zero.

Operations on integers can be performed easily by moving along the number line. The direction of movement depends on the operation and the sign of the number being added or subtracted.

The rules for moving on the number line are:

1. To add a positive integer, move to the right.
2. To subtract a positive integer, move to the left.
3. To add a negative integer, move to the left (this is equivalent to subtracting a positive number: $a + (-b) = a – b$).
4. To subtract a negative integer, move to the right (this is equivalent to adding a positive number: $a – (-b) = a + b$).

Worked Examples

Example 1: Find the sum of $-3$ and $5$ using a number line.

To solve $-3 + 5$:

1. Start at $-3$ on the number line.
2. Since $5$ is a positive integer being added, move $5$ units to the right.
3. The movement lands on $2$.

Therefore, $-3 + 5 = 2$.

Observe the illustration of this movement on the number line below.

Example 2: Find the value of $2 – 6$ using a number line.

To solve $2 – 6$:

1. Start at $2$ on the number line.
2. Since $6$ is being subtracted, move $6$ units to the left.
3. The movement lands on $-4$.

Therefore, $2 – 6 = -4$.

Example 3: Find the value of $-1 + (-4)$ using a number line.

Adding a negative number is equivalent to subtraction, so $-1 + (-4)$ is the same as $-1 – 4$:

1. Start at $-1$ on the number line.
2. Move $4$ units to the left.
3. The movement lands on $-5$.

Therefore, $-1 + (-4) = -5$.

Example 4: Find the value of $-2 – (-5)$ using a number line.

Subtracting a negative number is equivalent to adding a positive number, so $-2 – (-5)$ is the same as $-2 + 5$:

1. Start at $-2$ on the number line.
2. Move $5$ units to the right.
3. The movement lands on $3$.

Therefore, $-2 – (-5) = 3$.

Everyday Application of Positive and Negative Integers

Directed numbers are used in many real-life situations to represent opposite states or directions. Common everyday applications include:

1. Temperature: Temperatures above freezing point ($0^\circ\text{C}$) are represented as positive integers, while temperatures below freezing point are represented as negative integers.
2. Business and Finance: Profits, deposits, and earnings are represented as positive integers. Losses, withdrawals, debts, and overdrafts are represented as negative integers.
3. Elevation: Heights above sea level are positive integers, while depths below sea level are negative integers.

Worked Examples

Example 1: The temperature of a cold room in a fish market in Lagos is $-5^\circ\text{C}$. If the temperature rises by $8^\circ\text{C}$ during the afternoon, what is the new temperature?

The initial temperature is $-5^\circ\text{C}$. A rise in temperature means addition:

$$
-5 + 8 = 3
$$

The new temperature is $3^\circ\text{C}$.

Example 2: Aminu has an account balance of $-N1,200$ at a local bank, which means he owes the bank money. If he deposits $N5,000$ into his account, what will his new balance be?

The initial balance is $-1200$. A deposit means addition:

$$
-1200 + 5000 = 3800
$$

His new account balance is $N3,800$.

Example 3: A submarine is operating at a depth of $150\text{ metres}$ below sea level. It then climbs up by $60\text{ metres}$. What is its new position relative to sea level?

Depth below sea level is represented as a negative number, so the initial position is $-150\text{ metres}$. Climbing up means addition:

$$
-150 + 60 = -90
$$

The new position is $-90\text{ metres}$, which means the submarine is $90\text{ metres}$ below sea level.

Exercises

Section A: Multiple Choice Questions

Choose the correct option for each of the following questions:

1. What is the sum of $12,450$ and $8,790$?
   1. $20,240$
   2. $21,240$
   3. $21,140$
   4. $20,140$
2. In the number $45.89$, what is the place value of the digit $9$?
   1. Tens
   2. tenths
   3. hundredths
   4. Units
3. Using a number line, what is the value of $-4 + 7$?
   1. $-11$
   2. $3$
   3. $-3$
   4. $11$
4. What is the value of $-2 – (-5)$?
   1. $-7$
   2. $-3$
   3. $3$
   4. $7$
5. A thermometer reads $-3^\circ\text{C}$. If the temperature drops by $4^\circ\text{C}$, what is the new temperature?
   1. $1^\circ\text{C}$
   2. $-7^\circ\text{C}$
   3. $-1^\circ\text{C}$
   4. $7^\circ\text{C}$

Section B: Theory Questions

Solve the following problems step-by-step:

1. Find the difference between $845.2$ and $129.65$, and state the place value of each digit in the result.
2. Draw a horizontal number line from $-8$ to $8$ and use it to illustrate the following operations:
   1. $-2 + 6$
   2. $3 – 7$
   3. $-1 + (-5)$
3. A trader in Onitsha market made a loss of $N4,500$ on Monday, and a profit of $N12,000$ on Tuesday.
   1. Write down each of these transactions as a directed number.
   2. Calculate the trader’s net financial position at the end of Tuesday.
4. An elevator in a building starts at the ground floor (represented as $0$). It goes down $2$ floors to the basement parking lot, and then goes up $7$ floors. What floor does the elevator end up on? Show your calculations using directed numbers.