The67Sprint — LearningMap: Operations with Integers

🔢 The Number Line Framework

Zero is the origin — positives sit to the right, negatives to the left
Any number further right on the number line is always the greater number
Absolute value is the distance of a number from zero — always positive: |−5| = 5

➕➖ Addition & Subtraction Rules

Same signs: add the numbers and keep the common sign
Different signs: subtract smaller absolute value from larger, keep the bigger one's sign
Subtracting an integer = adding its additive inverse: 7 − (−7) = 7 + 7 = 14

SAME SIGN DIFF SIGN INVERSE

✖️➗ Multiplication & Division Signs

Same sign × same sign (or ÷) = positive
Different sign × different sign (or ÷) = negative
Token Model: green = +1, red = −1 — a green+red pair forms a "zero pair" that cancels to 0

Memory Cheat Code

Adding two negatives still gives a negative (−3 + −2 = −5); multiplying two negatives gives a positive (−3 × −2 = +6). Don't mix the two rules up!

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C7_Mat_Pt2_Ch2_LearningMap_OperationsWithIntegers Hero

⚡ Sprint Facts — Operations with Integers

Rule of Zero Adding zero never changes a number's value

Commutativity Swapping multiplier and multiplicand doesn't change the product

Even/Odd Negatives Even number of negative signs → positive answer; odd → negative

Brahmagupta's Rule Subtracting a negative from a positive equals adding two positives

Coin Movement Model Final position P = a + b — right is positive, left is negative

Zero Pair (+1) + (−1) = 0 — adding zero pairs never changes a number's value

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🔑 Key Terms

Integers

The complete set of whole numbers, their negative counterparts, and zero

Additive Inverse

The number that, added to a given number, results in zero — e.g. inverse of 18 is −18

Zero Pair

One positive token and one negative token that cancel each other to equal zero

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🏆 Olympiad & Challenger Edge

Collatz Conjecture (Integer Version)

If even, divide by 2; if odd, multiply by −3 and add 1. Track sequences (e.g. starting at −7) to find repeating loops like {−2, −1, 4, 2, 1}.

Diophantine Equations — "Pibs"

Find combinations of alien currency coins solving 13x − 9y = 85, using only +13 and −9 value coins.

Magic Grids & Border Sums

Find missing integers in a 3×3 grid so the border sum or diagonal rows equal a specific positive or negative target.

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❌ Missing Units & Statements

Word problems must end with a final statement and unit — e.g. "−18°C" or "Loss of ₹1000" — or marks are cut.

❌ Poor Step-Marking

In long-answer questions, explicitly write what's given, show the equation, and box the final answer.

🏦 Bank Accounts

Credits (deposits) act as positive integers; debits (withdrawals, penalties) act as negative integers.

🛗 Bela's Building

Ground floor is 0. Going up is positive (+2); heading to the basement is negative (−3).

🏔️ Geography

Mount Everest: +8848 m above sea level. Mariana Trench: −11034 m below sea level.

🌡️ Temperature — Ladakh

Winter temperatures in Ladakh routinely fall below 0°C to negative figures.

🚨 Sign Confusion

−3 + (−2) is NOT +5. Adding two negatives still gives a negative: −5. Don't confuse addition rules with multiplication rules.

🚨 Forgetting the Sign

Getting the number right but dropping the final + or − sign costs marks in every exam.

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🔗 Logic Chain — Integer Subtraction via Token Model

Step 1 — Identify Value

Start with the initial value (e.g. 4) and the required subtraction (e.g. subtract −12)

🔢

Step 2 — Add Zero Pairs

Not enough negative tokens exist, so add 12 positive and 12 negative tokens (a zero pair)

🟢🔴

Step 3 — Remove Tokens

Remove the required 12 negative tokens as instructed by the subtraction

➖

Step 4 — Count Remainder

Count the remaining tokens to find the final result: 4 + 12 = 16

✅

🦉

Mentor Byte

"Never try to memorize integer rules like a robot! Integers are simply a language for direction. Use the Token Model or visualize Bela's Building elevator in your mind. If you take away a 'debt' (subtracting a negative), you are actually becoming richer (adding a positive). Once you can see the math happening in the real world, the + and − signs will never confuse you again."

💛 Parent Tip

Watch out for the Wind Trap of integers — the sign rule mix-up. Ask your child: "Am I adding or multiplying?" before they solve any integer problem. This one question catches most exam errors.