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1.1.1

Integers and Order of Operations

Unit 1 · Integers

1.1  Integers and Order of Operations

Two minus signs in a row aren't a typo — they're an instruction. And before you calculate anything with mixed operations, you need to know which one goes first.

🎯 By the end of this you can…

  • ✓Add and subtract negative integers confidently, including double negatives
  • ✓Solve real temperature problems using signed numbers
  • ✓Find the additive and multiplicative inverse of a number
  • ✓Apply the order of operations (brackets, indices, ×÷, +−) correctly
  • ✓Insert brackets to make a calculation true

➖➖ Drag the second minus onto the first to cancel them

-6
−(
−
9)

Two negatives, dragged together, cancel into a positive.

💡 The idea

Subtracting a negative means adding

You just dragged the two minus signs together and watched them cancel. That's the whole rule: — subtracting flips direction once, and the negative sign flips it again, so two flips land you back on adding.

✏️ Worked example

The temperature in Moscow was −4°C and dropped by 7°C by midnight. What was the midnight temperature?

  1. 1
    "Dropped by 7" means subtract 7: .
  2. 2
    Both numbers push the result further negative: .

✅ Answer:

⚠️ The classic trap

"Dropped BY 7" and "dropped TO 7" are completely different questions. Dropped BY 7 means subtract 7 from whatever it was; dropped TO 7 just tells you the new value directly. Read word problems for which one it is before reaching for a calculation.

💡 The idea

Additive inverse vs. multiplicative inverse — two different ways to 'undo' a number

The additive inverse of a number is whatever you add to it to reach 0 — just flip its sign. The additive inverse of 2 is , because .

The multiplicative inverse (its reciprocal) is whatever you multiply it by to reach 1. The multiplicative inverse of 2 is , because .

🤔 What is the multiplicative inverse of −5?

🔺 Complete the addition pyramid — each box is the sum of the two below it

2
-4
-6
1

🧩 Click two numbers to bracket between them — make it equal 15

9+12÷3−1

Click the first number of the bracket.

💡 The idea

Order of operations: brackets first, then indices, then × or ÷, then + or −

You just found that bracketing forces the subtraction to happen BEFORE the division — that's the whole power of a bracket: it jumps a calculation to the front of the queue, out of its normal turn.

Yannis calculated by working strictly left to right: — wrong, because multiplication must happen before the subtraction, regardless of reading order. The correct order: first, then .

📋 Inserting brackets to make a statement true

  1. 1
    Work out what the statement currently equals, following the normal order of operations.
  2. 2
    Decide which operation needs to happen EARLIER than its normal turn to reach the target.
  3. 3
    Wrap brackets around exactly that part — brackets force their contents to go first.
  4. 4
    Re-check the whole calculation with the brackets in place.

🧩 Click two numbers to bracket between them — make it equal 40

4+3^2×5−1

Click the first number of the bracket.

🤔 Yannis says 20 − 2 × 3 + 5 = 9, working strictly left to right. Is his working correct?

🎮 Integers and order of operations — your turn

⭐ 0 XP🔥 streak 0
1Basic · 10XP
2Basic · 10XP
3Basic · 10XP
Missing integer:
4Basic · 10XP
Work out
5Basic · 10XP
Work out
6Branching · 20XP
Missing integer:
7Branching · 20XP
Calculate:
8Branching · 20XP
At midnight a town was −4°C. By 9am it rose by 7°C, then by 3pm it dropped by 11°C. Temperature at 3pm?
9Branching · 20XP
What is the additive inverse of 2?
10Branching · 20XP
What is the multiplicative inverse of 2?
11Challenge · 35XP
Work out
12Challenge · 35XP
Work out
13Challenge · 35XP
Work out
14Challenge · 35XP
Which calculation gives the LARGEST answer: , , , or ?
0 of 14 finished

🏅

You finished this lesson!

Negative numbers and order of operations are the two rules every later calculation in this book leans on — get these solid and the rest moves faster. Now go prove it against the Sentry.

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