How to Explain Rational Numbers to Seventh Graders

"Wait, how can 3 − 7 have an answer? You can't take 7 from 3!"

Actually, you can. The answer is −4. And understanding how to work with negative numbers opens up a whole new world of mathematics.

Seventh grade is when students fully embrace the rational number system—all numbers that can be written as fractions, including negative fractions and decimals. Let's explore how to make this expanded number system make sense.

Why Rational Numbers Matter in Seventh Grade

Rational numbers describe real situations that whole numbers can't:

• Temperature: It's −5°F outside
• Elevation: Death Valley is 282 feet below sea level (−282 ft)
• Finance: A debt of $500 (−$500)
• Football: Lost 8 yards (−8 yards)
• Stock market: Down 2.5 points

Beyond real-world contexts, rational numbers are essential for:

• Solving equations
• Graphing on the coordinate plane
• Understanding slope
• All of algebra

What Are Rational Numbers?

The Definition

A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

Examples of Rational Numbers

Integers:        -5, -2, 0, 3, 17
                 (can be written as -5/1, 3/1, etc.)

Fractions:       1/2, -3/4, 7/8, -11/5

Terminating      0.25 = 1/4
Decimals:        -0.6 = -3/5
                 2.75 = 11/4

Repeating        0.333... = 1/3
Decimals:        0.1666... = 1/6
                 -0.272727... = -3/11

What's NOT Rational?

Non-terminating, non-repeating decimals:

π = 3.14159265358979...  (never ends, never repeats)
√2 = 1.41421356237...    (never ends, never repeats)

These are irrational numbers—they cannot be written as fractions.

The Number Line Extended

Visualizing Negative Numbers

    ←─────────────────────────────────────────────→
    -5   -4   -3   -2   -1    0    1    2    3    4    5
                              ↑
                           zero

    ←── Negative numbers     Positive numbers ──→
        (less than zero)     (greater than zero)

Key Observations

• Zero is neither positive nor negative
• Numbers increase as you move right
• Numbers decrease as you move left
• Opposites are the same distance from zero: 4 and −4

Absolute Value

Absolute value is the distance from zero (always positive or zero).

|5| = 5    (5 is 5 units from zero)
|-5| = 5   (-5 is also 5 units from zero)
|0| = 0    (0 is 0 units from zero)

|-3.7| = 3.7
|2/3| = 2/3

Comparing Rational Numbers

On the Number Line

The number farther RIGHT is greater:

    -4   -3   -2   -1    0    1    2    3
     ●                   ●
    -4         <         1

-4 < 1 because -4 is left of 1

Comparing Negatives

More negative = smaller value:

-8 < -3  (−8 is farther left)
-2 > -7  (−2 is farther right)

Think: −2 is closer to zero, so it's greater.

Comparing Rational Numbers in Different Forms

Convert to the same form:

Compare -3/4 and -0.8

-3/4 = -0.75
-0.8 = -0.8

-0.75 > -0.8  (closer to zero)

So -3/4 > -0.8

Adding Rational Numbers

Same Signs: Add Absolute Values, Keep the Sign

Positive + Positive = Positive
5 + 3 = 8

Negative + Negative = Negative
(-5) + (-3) = -8

Different Signs: Subtract Absolute Values, Take Sign of Larger

(-7) + 4 = ?

Step 1: Find absolute values: |-7| = 7, |4| = 4
Step 2: Subtract: 7 - 4 = 3
Step 3: Sign of larger absolute value: negative
Answer: -3

Another example:

8 + (-3) = ?

Step 1: |8| = 8, |-3| = 3
Step 2: Subtract: 8 - 3 = 5
Step 3: Sign of larger: positive
Answer: 5

Number Line Model

Think of moving on the number line:

• Positive numbers: move RIGHT
• Negative numbers: move LEFT

-4 + 6 = ?

Start at -4, move 6 units RIGHT:
    -4   -3   -2   -1    0    1    2
     ●───────────────────────────→●

Land on 2. So -4 + 6 = 2

Adding with Fractions and Decimals

Same rules apply:

-2.5 + 1.8 = ?
|2.5| > |1.8|, so result is negative
2.5 - 1.8 = 0.7
Answer: -0.7

-3/4 + 1/2 = ?
= -3/4 + 2/4
= (-3 + 2)/4
= -1/4

Subtracting Rational Numbers

The Key Rule

Subtracting is adding the opposite.

a - b = a + (-b)

Examples

5 - 8 = 5 + (-8) = -3

-3 - 5 = -3 + (-5) = -8

-4 - (-7) = -4 + 7 = 3

2 - (-6) = 2 + 6 = 8

Why Subtracting a Negative Gives Addition

Think of it as "taking away a debt."

If you have $10 and someone removes a $5 debt you owed:

10 - (-5) = 10 + 5 = $15

Taking away something negative is a positive change!

With Fractions and Decimals

-1.5 - 2.3 = -1.5 + (-2.3) = -3.8

3/4 - (-1/2) = 3/4 + 1/2 = 3/4 + 2/4 = 5/4

-2/3 - 1/6 = -4/6 - 1/6 = -5/6

Multiplying Rational Numbers

The Sign Rules

Positive × Positive = Positive     (+)(+) = +
Negative × Negative = Positive     (-)(−) = +
Positive × Negative = Negative     (+)(−) = −
Negative × Positive = Negative     (−)(+) = −

Memory trick: Same signs = positive, different signs = negative.

Why Negative × Negative = Positive

Pattern approach:

3 × -2 = -6
2 × -2 = -4
1 × -2 = -2
0 × -2 = 0
-1 × -2 = ?

The pattern increases by 2 each time:
-6, -4, -2, 0, 2

So -1 × -2 = 2

Opposite of opposite:
−1 means "take the opposite." So (−1)(−1) means "take the opposite of the opposite" = original = positive.

Multiplying Decimals and Fractions

(-0.3)(0.5) = -0.15
(different signs → negative)

(-2/3)(-3/4) = 6/12 = 1/2
(same signs → positive)

(-1.2)(-0.5) = 0.6
(same signs → positive)

Multiple Factors

Count the negatives:

• Even number of negatives → positive result
• Odd number of negatives → negative result

(-2)(3)(-4)(-1) = ?

Three negatives (odd) → negative
2 × 3 × 4 × 1 = 24
Answer: -24

(-2)(-3)(-4)(-5) = ?

Four negatives (even) → positive
2 × 3 × 4 × 5 = 120
Answer: 120

Dividing Rational Numbers

Same Sign Rules as Multiplication

Positive ÷ Positive = Positive     12 ÷ 3 = 4
Negative ÷ Negative = Positive     (-12) ÷ (-3) = 4
Positive ÷ Negative = Negative     12 ÷ (-3) = -4
Negative ÷ Positive = Negative     (-12) ÷ 3 = -4

Dividing Fractions

Remember: Multiply by the reciprocal.

(-2/3) ÷ (4/5) = (-2/3) × (5/4)
               = -10/12
               = -5/6

(-3/4) ÷ (-1/2) = (-3/4) × (-2/1)
                = 6/4
                = 3/2

Mixed Operations

Follow order of operations (PEMDAS/BODMAS):

-8 + 4 × (-2) = ?

Step 1: Multiply first: 4 × (-2) = -8
Step 2: Add: -8 + (-8) = -16

(-6) ÷ 2 - 3 = ?

Step 1: Divide first: (-6) ÷ 2 = -3
Step 2: Subtract: -3 - 3 = -6

Converting Between Forms

Fraction to Decimal

Divide numerator by denominator:

3/8 = 3 ÷ 8 = 0.375

-5/6 = -5 ÷ 6 = -0.8333... = -0.83̄

Decimal to Fraction

Terminating decimals:

0.75 = 75/100 = 3/4

-0.4 = -4/10 = -2/5

Repeating decimals:

0.333... = 1/3

0.272727... = 27/99 = 3/11

Recognizing Equivalent Forms

-0.25 = -1/4 = -25%

0.6̄ = 2/3

-1.5 = -3/2 = -1 1/2

Hands-On Activities

Temperature Math

Use weather data:

• "It was −5°F in the morning and rose 12 degrees. What's the temperature now?"
• "The high was 8°F and the low was −7°F. What's the difference?"

Elevation Adventures

Mountain peak: 14,000 ft
Sea level: 0 ft
Ocean trench: -35,000 ft

"What's the difference between the peak and the trench?"
14,000 - (-35,000) = 14,000 + 35,000 = 49,000 ft

Football Gains and Losses

Play 1: +8 yards
Play 2: -3 yards
Play 3: -2 yards
Play 4: +15 yards

Total: 8 + (-3) + (-2) + 15 = 18 yards

Bank Account Simulation

Start with $100. Track deposits (+) and withdrawals (−):

Starting balance: $100
Deposit: +$50     → $150
Withdrawal: -$75  → $75
Bill payment: -$30 → $45
Interest: +$2     → $47

Integer Chips

Use two-color chips (or drawings):

• Red = negative
• Yellow = positive
• One red + one yellow = zero pair (cancel out)

Show -3 + 5:

Start: ● ● ●  (three red = -3)
Add:   ○ ○ ○ ○ ○  (five yellow = +5)

Cancel zero pairs:
● ● ● ○ ○ ○ ○ ○
↓ ↓ ↓
(three pairs cancel)

Left over: ○ ○ (two yellow = +2)

So -3 + 5 = 2

Common Mistakes and How to Fix Them

Mistake 1: Wrong Sign When Subtracting

Error: 5 − 8 = 3 (forgot to make it negative)

Fix: Use number line. Start at 5, move 8 left. You pass zero and land at −3.

Mistake 2: Double Negative Confusion

Error: 4 − (−3) = 1 (subtracted instead of added)

Fix: "Minus a negative" = "plus a positive." Draw it: subtracting −3 is like taking away a leftward arrow, which moves you right. 4 + 3 = 7.

Mistake 3: Multiplying Signs Wrong

Error: (−3)(−4) = −12 (forgot two negatives make positive)

Fix: Use the rule chart. Same signs = positive. Or count: 2 negatives (even) = positive.

Mistake 4: Absolute Value Sign Errors

Error: |−7| = −7

Fix: Absolute value is DISTANCE. Distance is never negative. |−7| asks "How far is −7 from zero?" Answer: 7 units.

Mistake 5: Comparing Negatives Backwards

Error: −8 > −3 (thinking "8 is bigger than 3")

Fix: Use the number line. −8 is farther left (farther from zero in the negative direction), so it's less. Or think: "Which would you rather have, −$8 or −$3?" −$3 is better (less debt), so −3 > −8.

Visual Models

Number Line Jumps

Calculate: -2 + 5

    -3   -2   -1    0    1    2    3    4
          ●─────────────────────→●
          Start                  End

Answer: 3

Zero Pairs

-4 + 4 = 0

● ● ● ●     +     ○ ○ ○ ○     =     0
(-4)              (+4)              (zero pairs)

Multiplication Pattern

   3 × (-2) = -6
   2 × (-2) = -4    ↑
   1 × (-2) = -2    ↑  +2 each time
   0 × (-2) = 0     ↑
  -1 × (-2) = 2     ↑
  -2 × (-2) = 4     ↑

Connecting to Other Concepts

To the Coordinate Plane

The coordinate plane uses rational numbers:

• Point (−3, 4) is in Quadrant II
• Point (2, −5) is in Quadrant IV
• Understanding negative coordinates requires rational number fluency

To Algebraic Equations

Solving equations involves rational operations:

x + 5 = 3
x = 3 - 5
x = -2

To Slope

Slope calculations involve rational numbers:

Slope = (y₂ - y₁)/(x₂ - x₁)

Points (1, -3) and (4, 3):
Slope = (3 - (-3))/(4 - 1) = 6/3 = 2

To Scientific Notation

Scientific notation uses negative exponents:

0.00045 = 4.5 × 10⁻⁴

Practice Ideas for Home

Temperature Tracking

• Record daily highs and lows
• Calculate temperature changes
• Compare to other cities

Sports Statistics

• Football: track total yards (gains and losses)
• Golf: calculate score relative to par (+3, −2)
• Hockey: plus/minus ratings

Financial Literacy

• Track allowance (income vs. spending)
• Understand that debt is negative
• Calculate net worth (assets − debts)

Elevator Math

• Ground floor = 0
• Basement = negative floors
• "Go down 3 floors from floor 2" = 2 + (−3) = −1

Calculator Exploration

• Find the (−) or +/− key
• Verify calculations
• Explore patterns

The Bottom Line

Rational numbers complete the number system that students use for the rest of mathematics. Once they're comfortable with positives, negatives, fractions, and decimals all mixed together, they're ready for algebra.

Key takeaways:

1. Rational numbers include all integers, fractions, and terminating/repeating decimals
2. Adding same signs: add and keep the sign
3. Adding different signs: subtract and use the sign of the larger absolute value
4. Subtracting: add the opposite
5. Multiplying/dividing: same signs = positive, different signs = negative

When seventh graders master rational number operations, they can solve equations, graph points, and tackle real-world problems involving debt, temperature, elevation, and more. It's math that truly expands their world.