Grade 7 · Grade 7: Proportional Reasoning and Rational Numbers · Operations with Rational Numbers
Operations with Rational Numbers
Add, subtract, multiply and divide signed rationals with confidence, and convert them to decimal form.
Learning objectives
- Add and subtract rational numbers on the number line, using p − q = p + (−q).
- Multiply and divide signed rationals and explain the sign rules.
- Convert rational numbers to terminating or repeating decimals.
- Solve real-world problems using all four operations.
AERO Mathematics alignment
AERO.M7.NS.5
Add and subtract rational numbers, representing addition and subtraction on a horizontal or vertical number line, and understand that p − q = p + (−q).
AERO.M7.NS.6
Multiply and divide rational numbers, applying sign rules and understanding that quotients are rational when the divisor is not zero.
AERO.M7.NS.7
Convert a rational number to a decimal using long division, and know that the decimal form terminates or eventually repeats.
AERO.M7.NS.8
Solve real-world and mathematical problems involving the four operations with rational numbers.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
At 06:00 the temperature is −3 °C. By noon it has risen 11 degrees; by midnight it has fallen 15 degrees.
Think about it
What is the midnight temperature, and how can the number line show every step?
Hint: Subtracting is the same as adding the opposite.
EXPLAIN
Working with signed rationals
Addition and subtraction
On the number line, adding a positive moves right and adding a negative moves left. Subtraction is addition of the opposite: p − q = p + (−q). So 4 − (−3) = 4 + 3 = 7.
Multiplication and division
| Signs | Result | Example |
|---|---|---|
| positive × positive | positive | 3 × 4 = 12 |
| positive × negative | negative | 3 × (−4) = −12 |
| negative × negative | positive | (−3) × (−4) = 12 |
| Division | Same rules as multiplication | −12 ÷ 4 = −3 |
The rule for two negatives follows from the distributive property: since 3 × 0 = 0 and 0 = 4 + (−4), consistency forces (−3) × (−4) to be positive.
Decimal form
Long division converts any rational number into a decimal that terminates, such as 3/8 = 0.375, or repeats, such as 1/3 = 0.333…. Both are rational because they came from a ratio of integers.
Where this is used
Bank statements, altitude changes, temperature swings and scores in games all rely on signed arithmetic.
INVESTIGATION
Try it yourself
Write a one-week account log with deposits and withdrawals including at least three negatives. Compute the running balance, then verify the final balance in a single expression using addition of opposites.
Watch
Adding and Subtracting Integers
Math Antics · 11:23
Makes the sign rules feel obvious using the number line.
Before you watch: What is the opposite of −6?
- Compute −8 − (−3).
- Explain why a negative times a negative is positive.
Interactive simulation · PhET
Number Line: Distance
Model subtraction as distance and direction between two points.
While you explore
- How does the sim show that 4 − (−3) equals 7?
- When is the distance between two numbers equal to their sum?
Key vocabulary
- Repeating decimal
- A decimal with digits that go on in a pattern forever.
- Additive inverse
- The opposite of a number.
Practice questions
0/1 correct
Level 1 · Criterion A
What is −8 − (−5)?
MYP criterion tasks
Level 2 · Criterion C
Explain, using the number line, why 6 − (−4) equals 10.
Reflect & track
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