Why this matters for FAST: Operations with rational numbers are fundamental to Grade 7 math. Students must confidently work with negative numbers in all four operations to succeed on FAST and prepare for algebra.
Students think 5 - (-3) should be less than 5. This is WRONG! Subtracting a negative is the same as adding a positive.
"Subtracting a negative is like removing a debt - it makes you richer! 5 - (-3) = 5 + 3 = 8. Think: If I take away owing $3, I'm better off by $3!"
Students think (-3) + (-5) = 8 because "two negatives make a positive."
"The rule about two negatives making a positive ONLY applies to multiplication and division! For addition, think of a number line: (-3) + (-5) moves left 3, then left 5 more = -8. Adding two debts makes MORE debt!"
Students add denominators when adding fractions with negative numbers.
"The rules for fractions don't change with negatives! Find common denominators for adding/subtracting, multiply straight across for multiplying. The negative sign follows the same rules as with integers."
Ask: "What's the temperature if it's 5 degrees and drops 8 degrees?" Connect to real-world context. Answer: -3 degrees.
"For adding: Same signs - add and keep the sign. Different signs - subtract and keep the sign of the larger absolute value. For subtracting: Change to adding the opposite!"
Adding Integers
| Problem | Rule | Answer |
|---|---|---|
| (-4) + (-6) | Same signs: add, keep negative | -10 |
| (-7) + 3 | Different signs: subtract, larger is negative | -4 |
| 8 + (-5) | Different signs: subtract, larger is positive | 3 |
Subtracting: Change to adding the opposite!
5 - (-3) = 5 + 3 = 8 | -4 - 7 = -4 + (-7) = -11
Multiplication and Division Rules
| Signs | Result | Examples |
|---|---|---|
| Positive x Positive | Positive | 4 x 3 = 12 |
| Negative x Negative | Positive | (-4) x (-3) = 12 |
| Positive x Negative | Negative | 4 x (-3) = -12 |
| Negative x Positive | Negative | (-4) x 3 = -12 |
Same signs = Positive | Different signs = Negative
This applies to division too: (-12) / (-4) = 3 | (-12) / 4 = -3
"The sign rules are the same for fractions and decimals! First, determine the sign using our rules, then perform the operation as usual."
Examples with Fractions and Decimals
(-1/2) + (-3/4) = (-2/4) + (-3/4) = -5/4
(-2/3) x (3/4) = -6/12 = -1/2
(-2.5) + 1.8 = -0.7
(-3.2) x (-1.5) = 4.8
Work through these together:
"What is (-5) x (-4) + (-10)?"
A) -30 B) 10 C) 30 D) -10
Correct answer: B) 10. First, (-5) x (-4) = 20 (negative times negative = positive). Then 20 + (-10) = 10.
For struggling students: Use number lines and integer chips. Focus on addition and subtraction before moving to multiplication/division. Use real-world contexts like temperature, debt, and elevation.
For advanced students: Challenge with multi-step order of operations problems. Introduce expressions with variables and negative coefficients.
For home: Send Parent Activity sheet. Families can explore temperature changes, bank account balances, and sports statistics with negative numbers.