| Percent change | Divide change by the original amount. | A price from $40 to $46 rises 6/40 = 15%. |
| Unit rate | Express both quantities on a common one-unit basis. | 180 miles in 3 hours gives 60 miles per hour. |
| Distributive law | Multiply each term inside parentheses. | 3(2x − 5) = 6x − 15. |
| Complement | Count everything except the unwanted event. | If 3 of 10 equally likely outcomes fail, success probability is 7/10. |
| Area versus perimeter | Area covers a surface; perimeter measures its boundary. | A 3 by 8 rectangle has area 24 and perimeter 22. |
| Mean total | Recover a total by multiplying mean by count. | Five values with mean 12 sum to 60. |
| Opposite signs | Subtract magnitudes when adding numbers with opposite signs; keep the sign of the larger magnitude. | −7 + 12 = 5. Checking 12 − 7 confirms a positive result. |
| Inequality reversal | Multiplication or division by a negative number reverses the order relation. | −2x > 8 becomes x < −4. Test x = −5: 10 > 8. |
| Scale and area | A length scale factor r produces an area factor r² for similar figures. | A copy with lengths tripled has area nine times as large, not three times. |
| Triangle area | Use one half times a base times its perpendicular height, not an unrelated sloping side. | Base 10 cm and perpendicular height 6 cm give 30 cm². |
| Probability without replacement | After a draw, update both the favorable count and the total count. | With 3 red and 2 blue tokens, two reds in two draws have probability (3/5)(2/4) = 3/10. |
| Slope and intercept — grade 9 | In y = mx + b, m is change in y per change in x; b is y when x = 0. | Points (2,9) and (5,21) give m = 12/3 = 4 and b = 1. |
| Pythagorean theorem — grade 9 | For a right triangle with legs a and b and hypotenuse c, a² + b² = c². | Legs 6 and 8 give c² = 100, so the positive length c is 10. |
| Scientific notation — grade 9 | Write a nonzero magnitude as a coefficient at least 1 and less than 10 times a power of 10. | 0.00072 = 7.2 × 10^−4. Moving four decimal places back checks the value. |