Number: Number Sense, Operations, and Integers
Grade 7 number sense rests on a few connected ideas: divisibility, prime factorization, greatest common factor (GCF), least common multiple (LCM), powers and square roots, and the four operations with integers.
Know the vocabulary precisely. Factors divide a number evenly; multiples are products of the number and whole numbers. The GCF is the largest shared factor (useful for simplifying), while the LCM is the smallest shared multiple (useful for common denominators). Every composite number has a unique prime factorization, found with a factor tree. A perfect square has an even number of each prime factor in its factorization, so √144 = 12.
Order of operations governs every expression: brackets first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.
Integers add using zero pairs: (+5) + (−8) = −3 because five pairs cancel and three negatives remain. Subtracting a negative is adding: 4 − (−6) = 10. Multiplication and division follow sign rules: the same signs give a positive result, different signs give a negative result.
Common traps: writing −3² = 9 when exponents apply before the leading negative (−3² = −9, but (−3)² = 9); doing addition before multiplication; confusing LCM with GCF; and dropping a negative sign when rearranging.
Worked example: evaluate −5 + 3 × (−2)². Square first: (−2)² = 4. Multiply: 3 × 4 = 12. Then add: −5 + 12 = 7. Rule of thumb: simplify inside brackets, evaluate powers, then work left to right through × and ÷ before + and −.