Class 2 is where students stop just adding and subtracting integers and start multiplying and dividing them. The arithmetic gets faster once students memorize two sign rules — and every question in this class is an application of those rules.
The two rules you must know cold
Every product and every quotient of two signed numbers follows one of these two rules. They apply to BOTH multiplication and division.
SAME signs
positive × positive · negative × negative
result is POSITIVE (+)
DIFFERENT signs
positive × negative · negative × positive
result is NEGATIVE (−)
Shortcuts for products with 3 or more factors
Instead of applying the sign rule step by step, just count how many negative factors the product has:
EVEN number of negatives→ product is POSITIVE
ODD number of negatives→ product is NEGATIVE
Any factor is 0→ product is 0
Two rules students always forget
1. Division by 0 is undefined. Something like (−15) ÷ 0 has no answer at all. But 0 ÷ (−15) = 0 is perfectly fine. Zero on top is OK; zero on the bottom is not.
2. Parentheses matter with exponents. (−3)2 = (−3)(−3) = +9. But −32 with no parentheses means −(32) = −9. Without parentheses, you square first and apply the negative sign after.
The 25 questions below march from pure sign-rule drill through multi-digit computation and order of operations, and finish with real-life word problems where negative numbers stand for temperature drops, dives below sea level, and debt.
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A clean printable PDF with the sign-rule cheat sheet and all 25 questions — the step-by-step answers stay online for the teacher or parent to check.
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25 practice questions with hidden answers
Sign rules & quick computation (Q1–Q8)
Q1
Fill in each blank with + or −:
(a) (positive) × (negative) = ___
(b) (negative) × (negative) = ___
(c) (negative) ÷ (positive) = ___
(d) (negative) ÷ (negative) = ___
Show answer & explanation
(a) − (b) + (c) − (d) +
Why. The two rules for signs: SAME signs give a POSITIVE result, DIFFERENT signs give a NEGATIVE result. This is true for both multiplication and division. So (a) different signs → negative. (b) same signs → positive. (c) different signs → negative. (d) same signs → positive.
Q2
Compute in your head:
(a) (−4) × 6
(b) (−3) × (−9)
(c) 8 × (−7)
(d) (−5) × (−5)
Show answer & explanation
(a) −24 (b) 27 (c) −56 (d) 25
Why. Multiply the digits, then apply the sign rule. (a) 4 × 6 = 24, different signs → −24. (b) 3 × 9 = 27, same signs → +27. (c) 8 × 7 = 56, different signs → −56. (d) 5 × 5 = 25, same signs → +25.
Q3
Compute:
(a) (−48) ÷ 6
(b) (−72) ÷ (−8)
(c) 54 ÷ (−9)
(d) 0 ÷ (−15)
Show answer & explanation
(a) −8 (b) 9 (c) −6 (d) 0
Why. Divide the digits, then apply the sign rule. (a) 48 ÷ 6 = 8, different signs → −8. (b) 72 ÷ 8 = 9, same signs → +9. (c) 54 ÷ 9 = 6, different signs → −6. (d) 0 divided by any nonzero number is always 0 — no sign needed.
Q4
True or false: (−15) ÷ 0 = 0. Explain.
Show answer & explanation
False. Division by 0 is UNDEFINED.
Why. You can never divide by 0. Ask yourself "what number times 0 gives −15?" — no such number exists, because anything times 0 is 0. So −15 ÷ 0 has no answer at all; we say it is undefined. Do not confuse this with 0 ÷ (−15) = 0, which IS a valid statement.
Q5
Without computing, decide if each product is POSITIVE, NEGATIVE, or ZERO:
(a) (−2) × (−3) × (−4)
(b) (−1) × (−1) × (−1) × (−1)
(c) (−7) × 5 × 0 × (−2)
(d) (−2) × (−3) × 4 × (−5)
Show answer & explanation
(a) negative (b) positive (c) zero (d) negative
Why. Count how many NEGATIVE factors there are. An EVEN count of negatives → positive product. An ODD count → negative product. If ANY factor is 0, the whole product is 0. (a) 3 negatives (odd) → negative. (b) 4 negatives (even) → positive. (c) contains 0 → zero. (d) 3 negatives (odd) → negative.
Q6
Compute:
(a) (−1)7
(b) (−1)100
(c) (−2)3
(d) (−2)4
Show answer & explanation
(a) −1 (b) 1 (c) −8 (d) 16
Why. For (−1) raised to a power: ODD exponent → −1, EVEN exponent → +1. So (−1)7 = −1 (7 is odd) and (−1)100 = 1 (100 is even). For (−2): (−2)3 = (−2)(−2)(−2) = 4·(−2) = −8. (−2)4 = (−2)(−2)(−2)(−2) = 4·4 = 16 (four negatives make positive).
Q7
Which is bigger: (−3)2 or −32? Show why.
Show answer & explanation
(−3)2 = 9 is bigger. −32 = −9.
Why. The parentheses change EVERYTHING. (−3)2 means (−3) × (−3) = 9 (positive, because same signs). But −32 with no parentheses means −(32) — you square 3 first, THEN apply the negative sign. So −32 = −(9) = −9. This is one of the most common mistakes on tests — always look for the parentheses.
Q8
Fill in the missing number:
(a) (−6) × ___ = −42
(b) ___ ÷ (−4) = 9
(c) (−8) × ___ = 56
Show answer & explanation
(a) 7 (b) −36 (c) −7
Why. Work backwards. (a) 42 ÷ 6 = 7 (digits). Signs: negative × ? = negative, so ? must be POSITIVE. Answer 7. (b) 9 × (−4) = −36 (check: −36 ÷ −4 = 9 ✓). (c) 56 ÷ 8 = 7. Signs: negative × ? = positive, so ? must be NEGATIVE. Answer −7.
Products & quotients with multi-digit integers (Q9–Q16)
Q9
(−13) × 12 = ?
Show answer & explanation
−156
Why. 13 × 12 = 156 (compute normally: 13 × 10 = 130, 13 × 2 = 26, 130 + 26 = 156). Different signs → negative. Final: −156.
Q10
(−25) × (−16) = ?
Show answer & explanation
400
Why. Same signs → positive. 25 × 16 = 400 (25 × 16 = 25 × 4 × 4 = 100 × 4 = 400). Final: +400.
Q11
(−144) ÷ (−12) = ?
Show answer & explanation
12
Why. Same signs → positive. 144 ÷ 12 = 12 (since 12 × 12 = 144). Final: +12.
Q12
225 ÷ (−15) = ?
Show answer & explanation
−15
Why. Different signs → negative. 225 ÷ 15: notice 15 × 15 = 225, so 225 ÷ 15 = 15. Final: −15.
Q13
(−8) × 25 × (−4) = ?
Show answer & explanation
800
Why. Two negatives (even count) → positive result. Regroup for easy multiplication: 25 × (−4) = −100, then (−8) × (−100) = 800. Or: (−8) × (−4) = 32, then 32 × 25 = 800.
Q14
Evaluate: (−2)5
Show answer & explanation
−32
Why. (−2)5 = (−2)(−2)(−2)(−2)(−2). Group in pairs: (−2)(−2) = 4, (−2)(−2) = 4. So we have 4 × 4 × (−2) = 16 × (−2) = −32. Or use the shortcut: 25 = 32, and an ODD number of negatives → negative. So (−2)5 = −32.
Q15
(−360) ÷ (−9) ÷ (−5) = ? (compute left to right)
Show answer & explanation
−8
Why. Division is LEFT TO RIGHT, one step at a time. Step 1: (−360) ÷ (−9) = 40 (same signs → positive; 360 ÷ 9 = 40). Step 2: 40 ÷ (−5) = −8 (different signs → negative; 40 ÷ 5 = 8). Final: −8.
Q16
Fill in the missing number: (−234) ÷ ___ = −18
Show answer & explanation
13
Why. Signs first: negative ÷ ? = negative, so ? must be POSITIVE. Digits: 234 ÷ ? = 18, so ? = 234 ÷ 18 = 13. Check: (−234) ÷ 13 = −18 ✓ (since 13 × 18 = 234).
Order of operations with signed integers (Q17–Q20)
Q17
Evaluate: (−3) × 4 + (−2) × (−5)
Show answer & explanation
−2
Why. PEMDAS: do the multiplications FIRST, then add. (−3) × 4 = −12. (−2) × (−5) = +10. Now add: −12 + 10 = −2. Final: −2.
Q18
Evaluate: 20 ÷ (−4) − (−6) × 3
Show answer & explanation
13
Why. PEMDAS: division and multiplication first, left to right. 20 ÷ (−4) = −5 (different signs). (−6) × 3 = −18 (different signs). Now the expression becomes −5 − (−18) = −5 + 18 = 13. Final: +13.
Q19
Evaluate: (−2)3 × 5 − (−4)2
Show answer & explanation
−56
Why. PEMDAS: exponents first, then multiplication, then subtraction. (−2)3 = −8 (odd exponent → negative). (−4)2 = 16 (even exponent → positive). Now: −8 × 5 − 16 = −40 − 16 = −56. Final: −56.
Q20
Evaluate: ( (−12) + 4 ) × (−3) ÷ 6
Show answer & explanation
4
Why. Parentheses first: (−12) + 4 = −8. Then left to right: (−8) × (−3) = 24 (same signs). Finally 24 ÷ 6 = 4. Final: +4.
Word problems — real-life signed products & quotients (Q21–Q25)
Q21
The temperature drops 3 °F every hour. If it is 5 °F right now, what will the temperature be 6 hours from now?
Show answer & explanation
−13 °F
Why. "Drops 3 °F per hour" means the change is −3 °F/hour. Over 6 hours, total change = −3 × 6 = −18 °F. New temperature = 5 + (−18) = −13 °F.
Q22
A submarine is diving at 12 meters per minute. It started at sea level (0 m). Where is it 8 minutes later? (positive = above sea level, negative = below)
Show answer & explanation
−96 meters (96 m below sea level)
Why. Diving at 12 m per minute means the rate is −12 m/min. Over 8 minutes, total change = −12 × 8 = −96 m. Starting at 0, new depth = 0 + (−96) = −96 m.
Q23
A stock lost $84 in value over 7 days at a steady rate. What was the average change per day?
Show answer & explanation
−$12 per day
Why. "Lost $84" means total change = −$84. Steady rate over 7 days: average = −84 ÷ 7 = −$12 per day. The negative sign tells us the stock LOST value on average.
Q24
Amara owes her brother $6. She owes him for 4 more toys at the same price. What integer represents her total debt after buying the extra toys?
Show answer & explanation
−$30 (she owes $30 total)
Why. Debt is represented by NEGATIVE numbers. She already owes −$6 (that's 1 toy). Four more toys at −$6 each = 4 × (−6) = −$24. Total debt = −6 + (−24) = −$30. She owes her brother $30 total.
Q25
A hot-air balloon is descending at a rate of 8 feet per second. If it needs to descend 480 feet to land, how many seconds until it lands?
Show answer & explanation
60 seconds (1 minute)
Why. "Descending 8 feet per second" is a rate of −8 ft/sec. Total descent needed = −480 ft. Time = distance ÷ rate = (−480) ÷ (−8) = +60 seconds. Same signs → positive time, which makes physical sense (time can't be negative). The balloon lands after 60 seconds, or 1 minute.
Printable version
The same 25 questions plus a one-page sign-rule cheat sheet, ready to print for homework or classroom use.
Download PDF ↓
About the Young Fermats Pre-Algebra course
Class 2 is one of 48 rolling classes in our Young Fermats Pre-Algebra program for grade 5–6 students (ages 10–12). The full arc runs from integers on the number line through systems of linear equations and probability — the direct on-ramp to Algebra 1 in 7th or 8th grade. Classes are small-group (max 6 students), 120 minutes per week, and students can start any Monday because the syllabus is rolling.
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Ready to see the whole 48-class arc?
Class 2 is a taste — Young Fermats runs 48 rolling classes, students can join any Monday, and the syllabus takes them from integers through systems of equations. Small-group (max 6), 120 minutes a week, at 226 W 79th Street. First class is free. Cancel any time with 15 days' notice.
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