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Add Rational Numbers
on the Number Line

Learn to find \( p + q \) by jumping a distance \( |q| \) from \( p \) on the number line: interpret sums with integers, fractions, and decimals.

7th Grade Math

What Are Rational Numbers?

A rational number is any number that can be written as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \). This includes integers, fractions, and terminating or repeating decimals.

Integers
\( -3,\; 0,\; 7 \)
Fractions
\( \frac{2}{5},\; -\frac{3}{4} \)
Decimals
\( 0.75,\; -1.2 \)

The Number Line as a Tool

\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
\( 3 \)
 Negative Positive

The number line is a straight line where every point represents a number. Numbers increase to the right and decrease to the left. We will use it to visualize addition of rational numbers.

Why Use the Number Line?

Adding rational numbers can feel abstract, especially with negatives. The number line turns addition into physical movement: you start at one number and jump to the answer.

See the Direction
Positive \( q \) means jump right. Negative \( q \) means jump left.
See the Distance
The absolute value \( |q| \) tells you how far to jump.

The Core Rule: \( p + q \)

1
Start at \( p \). Place your finger on \( p \) on the number line.
2
Determine the distance: compute \( |q| \).
3
Choose direction: if \( q > 0 \), move right. If \( q < 0 \), move left.
4
Land on the sum: the point you reach is \( p + q \).

Visualizing \( p + q \)

\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
\( 3 \)
\( p = -1 \)
\( q = +3 \) (jump right)
\( p + q = 2 \)

Positive vs. Negative \( q \)

\( q > 0 \) (Positive) \( q < 0 \) (Negative)
Direction Right Left
Distance \( |q| \) units \( |q| \) units
Effect on sum Sum is greater than \( p \) Sum is less than \( p \)
Example \( 2 + 3 = 5 \) \( 2 + (-3) = -1 \)

Adding a Negative Number

\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
\( p = 1 \)
\( q = -3 \) (jump left)
\( p + q = -2 \)

How to Apply the Rule

1
Identify \( p \) and \( q \). In \( p + q \), \( p \) is your starting point and \( q \) is the number being added.
2
Plot \( p \) on the number line. Mark a dot at \( p \).
3
Find \( |q| \). This is your jump distance (always positive).
4
Jump in the correct direction. Right if \( q > 0 \), left if \( q < 0 \).
5
Read the landing point. That number is \( p + q \).

Example 1: \( -4 + 6 \)

\( -5 \)
\( -4 \)
\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
Start: \( -4 \)
\( +6 \) (jump 6 right)
\( = 2 \)

Example 2: \( \frac{1}{2} + \left(-\frac{3}{2}\right) \)

\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( -\frac{3}{2} \)
\( -\frac{1}{2} \)
\( \frac{1}{2} \)
\( \frac{3}{2} \)
Start: \( \frac{1}{2} \)
\( -\frac{3}{2} \) (jump left)
\( = -1 \)

Example 3: \( -2.5 + 1.8 \)

\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
Start: \( -2.5 \)
\( +1.8 \) (jump right)
\( = -0.7 \)

Real-World Interpretation

The number line model connects directly to real-life situations involving gains and losses, rises and drops, deposits and withdrawals.

Temperature
It is \( -8^\circ \)F and warms by \( 12^\circ \). New temp: \( -8 + 12 = 4^\circ \)F.
Money
You owe \$15 (balance \( -15 \)) and earn \$9. Balance: \( -15 + 9 = -6 \).
Elevation
A diver at \( -20 \) feet ascends \( 8 \) feet. Position: \( -20 + 8 = -12 \) feet.

Additive Inverse and Zero Pairs

\( -3 \)
\( -2 \)
\( -1 \)
\( 0 \)
\( 1 \)
\( 2 \)
\( 3 \)
\( p = 3 \)
\( q = -3 \)
\( = 0 \)

Common Mistakes to Avoid

✕ Wrong Direction
Jumping right when adding a negative number. Remember: negative \( q \) always means move left.
✕ Ignoring the Sign
Treating \( 5 + (-3) \) as \( 5 + 3 = 8 \) instead of \( 5 + (-3) = 2 \). The parentheses hold a negative number.
✕ Confusing Distance with Position
Writing \( |-3| = -3 \). The absolute value is always positive: \( |-3| = 3 \). It measures distance, not position.

Summary of Addition Rules

Scenario Direction Result Relative to \( p \)
\( p + q \) where \( q > 0 \) Right Greater than \( p \)
\( p + q \) where \( q < 0 \) Left Less than \( p \)
\( p + 0 \) No movement Equal to \( p \)
\( p + (-p) \) Back to origin Equal to \( 0 \)

Question 1 of 5

What is \( -3 + 7 \) on the number line?

Question 2 of 5

A submarine is at \( -120 \) feet. It ascends \( 45 \) feet. What is its new depth?

Question 3 of 5

What is \( \frac{3}{4} + \left(-\frac{5}{4}\right) \)?

Question 4 of 5

When you compute \( p + q \) on the number line, what determines the direction of your jump?

Question 5 of 5

The temperature is \( -6.5^\circ \)C. It drops another \( 3.5^\circ \)C. What is the new temperature?