Integers are whole numbers — positive, negative, and zero. In this unit you'll master multiplying and dividing with negative signs, square and cube roots, index notation, and the correct order of operations.
Watch: signed integers orbit on a number line from −10 to +10. Positives glow pink, negatives glow blue.
Always ask: what's inside brackets? What are the indices? THEN multiply/divide, THEN add/subtract.
🔥 Temperature Number Line Visualizer
Click on the number line to place Temperature A (🌠️ red) and Temperature B (❄️ blue). See the signed difference between them!
Click once for City A, click again for City B, then alternates. Drag to reposition.
How to calculate signed difference:
Difference = Temp A − Temp B (with sign!)
e.g. A = −4°C, B = 9°C → Difference = −4 − 9 = −13°C (A is 13° colder than B)
Absolute (unsigned) gap = |−4 − 9| = 13°C
📋 Ex 1: Sign Rules Sorter
Click the correct sign (+ or −) for each multiplication or division. You get immediate feedback!
🔎 Ex 2: Roots Challenge
Type the answer for each root question, then press Check Answers.
📈 Ex 3: Powers Builder
Choose a base and an exponent to see the expanded form and result!
Base (a):
Exponent (n):
Select a base and exponent above...
🔢 Ex 4: BIDMAS with Negatives
Work out each expression using BIDMAS. Type your answer and press Check.
🌎 Ex 5: Real-World Negatives
Solve each word problem. Type your answer (just the number, e.g. −12 or 15) then press Check.
📊 Ex 6: Ordering Challenge
Click the values below in order from smallest to largest to place them in the slots.
Click a placed value to remove it. Fill all 8 slots then check!
Smallest→→Largest
📝 Practice Questions
Show all your working! Use BIDMAS and sign rules carefully.
Calculate: (−6) × (−3)
Calculate: (−5) × 4
Calculate: 24 ÷ (−8)
Calculate: (−36) ÷ (−9)
Find the value of 7²
Find the value of 4³
Find √81
Find ∛8
Find √121
Find ∛216
Evaluate: 3 + (−2) × 5
Evaluate: (4 + (−6))²
Evaluate: √(3² + 4²)
Evaluate: −2³ + 10
Evaluate: (−1) × (−2) × (−3)
The temperature drops 5°C each hour. After 4 hours, write the total change as a multiplication. What is the answer?
A diver is at 0 m and descends 9 m every minute for 3 minutes. Write as a multiplication. What is their depth?
False. (−3)² = (−3)×(−3) = +9. But −3² = −(3×3) = −9. The brackets make the difference!
🔥 Challenge: Integers Word Problems
Show full working for every question!
The temperature in a city is −8°C on Monday. It falls by the same amount each day and reaches −20°C on Thursday (3 days later). Write a multiplication to find the daily change. What is it?
A submarine is at −60 m. It divides its ascent equally over 5 minutes to reach the surface (0 m). How far does it rise each minute? Write as a division with directed numbers.
A bank account has a balance of −£15. Money is deposited and withdrawn several times: +£30, then −£8, then −£4. What is the final balance? Show each step.
The temperature outside is −3°C. Inside a greenhouse it is 4 times warmer (the inside temperature is 4 times greater than the outside temperature in absolute value, but positive). Write an expression and find the inside temperature. Hint: inside = 4 × |outside|.
A number squared gives the same result whether you square the number or its negative: (n)² = (−n)². Show this is true for n = 5 and n = 7. Then explain why this always works using sign rules.
Find all possible integer values of n where n² < 50. List them all, including negatives.
Jamal owes his parents £3 a week for pocket money he borrowed. After 6 weeks he starts earning £5 a week and repays £3 of it each week. How many weeks after he starts earning does it take to clear the debt? Show your working.
The formula for the area of a square is A = s². If s = −7 cm (a directed distance), explain why the area is still positive. Calculate the area.
Evaluate: 5² − (−3)² × 2 + ∛27 using correct BIDMAS. Show each operation step by step.
Change = −20 − (−8) = −12°C over 3 days. Daily change = −12 ÷ 3 = −4°C per day.
Total ascent = 0 − (−60) = +60 m over 5 minutes. Distance per minute = +60 ÷ 5 = +12 m per minute. In directed terms: −60 ÷ (−5) = +12 m/min.
Start: −15. After +30: −15 + 30 = +15. After −8: 15 − 8 = +7. After −4: 7 − 4 = +£3.
5² = 25 and (−5)² = (−5)×(−5) = +25 ✓. 7² = 49 and (−7)² = +49 ✓. Why: squaring always means two negatives, and negative × negative = positive, so the result is always positive.