🎯⭐ INTERACTIVE LESSON

Operations with Integers

Learn step-by-step with interactive practice!

Operations with Integers - Complete Interactive Lesson

Part 1: Numbers Go Both Ways 🔢

Numbers Go Both Ways 🔢

An integer is any whole number, including zero and the negatives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ (no fractions or decimals!).

You already meet negative numbers all the time:

Temperature: it can be $-8$ degrees on a cold morning
Money: owing your friend $5 is like having $-5$ dollars
Elevation: a diver $20$ meters below sea level is at $-20$ m

The key idea is direction. On a number line, positive means moving right and negative means moving left. In this lesson you'll learn to add, subtract, multiply, and divide integers with confidence.

Absolute Value: How Far From Zero 📏

The absolute value of a number is its distance from zero on the number line. Distance is never negative, so absolute value is always positive or zero. We write it with bars: $|\ \ |$ .

$|5| = 5$ (the number $5$ is $5$ steps from zero)
(the number is also steps from zero)

⭐ The Two Rules for Adding Integers

When you add integers, first look at the signs.

Rule 1 — Same Signs: Add the absolute values and keep the shared sign.

$5 + 3 = 8$ (both positive → answer positive)
$-5 + (-3) = -8$ (both negative → answer negative)

Quick Concept Check ✅

Let's make sure the adding rules stuck before we move on.

Part 2: Worked Examples: Adding ➕

Worked Examples: Adding ➕

Example 1 — Same signs: $-6 + (-4)$

Both signs are negative, so add the absolute values: $6 + 4 = 10$
Keep the shared sign (negative): answer $= -10$ ✅

Part 3: Guided Practice: Pick the Answer

Guided Practice: Pick the Answer 🎯

Work each one carefully. Decide whether to use the same-sign or different-sign rule!

Finish Each Statement 🧩

Choose the option that correctly completes each sentence about integer operations.

Part 4: Integers in the Real World 🌎

Integers in the Real World 🌎

Negative numbers describe anything that can go below a starting point — money owed, temperatures below zero, or going underground.

Temperature change: At dawn it was $-5$ °C. By noon it rose $8$ degrees.

New temperature: $-5 + 8 = 3$ °C 🌡️

Bank account: Jordan has $20, then spends $32 (overdraft!).

Balance: , so Jordan owes $12 (a balance of -$12).

Part 5: Putting It All Together

🏆 Putting It All Together

You've now worked with all four operations on integers. The trickiest part is keeping track of signs — here's a summary to lock it in:

Operation	Key Rule	Example
Add ➕	Same signs: add & keep sign. Different signs: subtract & use sign of larger	$-5 + 3 = -2$
Subtract ➖	Add the opposite: $a - b = a + (-b)$

This is super useful for adding integers with different signs, because we'll compare which number is "bigger" in distance from zero.

Rule 2 — Different Signs: Subtract the smaller absolute value from the larger, and take the sign of the number with the larger absolute value.

$7 + (-3)$ : subtract $7 - 3 = 4$ , and $7$ wins (it's bigger), so the answer is $+4$
$-7 + 3$ : subtract $7 - 3 = 4$ , and $-7$ wins, so the answer is $-4$

Situation	What to do	Sign of answer
Same signs	Add the absolute values	Keep the shared sign
Different signs	Subtract absolute values	Sign of the larger one

Think of it like a tug-of-war: positives pull right, negatives pull left, and the stronger side wins!

Example 2 — Different signs: $9 + (-12)$

The signs differ, so subtract absolute values: $12 - 9 = 3$
$-12$ has the larger absolute value, so the answer is negative: answer $= -3$ ✅

Worked Examples: Subtracting ➖

The golden rule for subtraction: Add the opposite!

$a - b = a + (-b)$

Change the subtraction to addition, then flip the sign of the second number. Now use the adding rules you already know.

Example 1: $5 - 8$

Add the opposite: $5 + (-8)$
Different signs → subtract: $8 - 5 = 3$ , and $-8$ wins → answer $= -3$ ✅

Example 2: $-3 - 7$

Add the opposite: $-3 + (-7)$
Same signs → add: $3 + 7 = 10$ , keep negative → answer $= -10$ ✅

Example 3: $4 - (-6)$ (watch the double negative!)

Add the opposite of $-6$ , which is $+6$ : $4 + 6$
answer $= 10$ ✅

Tip: Two minus signs next to each other turn into a plus: $4 - (-6) = 4 + 6$ .

Your Turn ✍️

Solve each one. Type just the number, including a minus sign if the answer is negative (like $-7$ ).

Box 1: $-6 + (-9) = ?$
Box 2: $8 + (-15) = ?$
Box 3: $-2 - (-10) = ?$

20 - 32 = 20 + (-32) = -12

20 - 32 = 20 + (- 32) = - 12

Multiplying for repeated change: A submarine descends $15$ meters each minute for $4$ minutes. Descending is negative:

$4 \times (-15) = -60$ , so it ends at $-60$ meters (different signs → negative).

The math is the same as before — just decide which operation the story describes, then apply the integer rules.

Solve the Story Problems 📖

Type just the number, with a minus sign if the answer is negative.

Box 1: The temperature is $-3$ °C and drops another $6$ degrees. What is the new temperature?
Box 2: A diver is at $-12$ m and rises $5$ m. What is the diver's new depth?
Box 3: A game costs $7$ points each time you lose. If you lose $3$ times, what is your total score change? (Use multiplication.)

One More Real-World Problem 💳

The big shortcut for $\times$ and $\div$ : count the negative signs. An even number of negatives gives a positive answer; an odd number of negatives gives a negative answer. For example, $(-2) \times (-3) \times (-1)$ has three negatives (odd), so the result is negative: $-6$ .

Final Challenge: Mixed Operations 🎓

These mix every operation together. Decide on the right rule for each — take your time!