Worksheet on Operations with Integers for Grade 3

Master operations with integers through visual strategies, step-by-step methods, and comprehensive practice worksheets.

Solution: Exercises 1 to 3
1 Integer Addition
Exercise 1
(+5) + (+3) = ?
Definition:

Integer addition: Adding positive and negative whole numbers

Positive integers: Numbers greater than zero (e.g., +5, +3)

Negative integers: Numbers less than zero (e.g., -5, -3)

Integer Addition Method:
  1. Same signs: Add absolute values and keep the common sign
  2. Different signs: Subtract smaller absolute value from larger and keep sign of larger
  3. With zero: Adding zero doesn't change the number
Problem
(+5) + (+3)
Operation
Same signs
Result
+8
+5
+
+3
=
+8
Step 1: Identify the signs

Both numbers are positive (+5 and +3)

Step 2: Apply the rule for same signs

When adding numbers with the same sign, add their absolute values

Step 3: Add the absolute values

5 + 3 = 8

Step 4: Keep the common sign

Both were positive, so the answer is +8

Step 5: Write the final answer

The answer is +8

(+5) + (+3) = +8
Final answer:

(+5) + (+3) = +8

Applied rules:

• Same signs rule: Add absolute values and keep common sign

• Positive integers: Numbers greater than zero

• Integer addition: Combine values considering direction

Tip: Think of positive numbers as "gaining" and negative numbers as "losing".
Common mistake: Forgetting to keep the sign when adding same-signed numbers.
2 Integer Subtraction
Exercise 2
(+7) - (+4) = ?
Definition:

Integer subtraction: Taking away one integer from another

Subtraction rule: Change subtraction to addition of the opposite

Problem
(+7) - (+4)
Convert
(+7) + (-4)
Result
+3
+7
-
+4
=
+3
Step 1: Convert subtraction to addition

(+7) - (+4) becomes (+7) + (-4)

Step 2: Identify the signs

One positive (+7) and one negative (-4)

Step 3: Apply the rule for different signs

Subtract smaller absolute value from larger: 7 - 4 = 3

Step 4: Keep the sign of the larger absolute value

+7 has larger absolute value, so the answer is +3

Step 5: Write the final answer

The answer is +3

(+7) - (+4) = +3
Final answer:

(+7) - (+4) = +3

Applied rules:

• Subtraction conversion: a - b = a + (-b)

• Different signs rule: Subtract and keep sign of larger absolute value

• Integer operations: Consider direction and magnitude

Tip: Always convert subtraction to addition of the opposite first.
Common mistake: Forgetting to change the sign of the number being subtracted.
3 Mixed Integer Operations
Exercise 3
(-6) + (+9) = ?
Definition:

Mixed integer operations: Adding integers with different signs

Opposite signs: One positive and one negative number

Problem
(-6) + (+9)
Different Signs
Subtract and keep sign of larger
Result
+3
-6
+
+9
=
+3
Step 1: Identify the signs

One negative (-6) and one positive (+9)

Step 2: Apply the rule for different signs

Subtract smaller absolute value from larger: 9 - 6 = 3

Step 3: Keep the sign of the larger absolute value

+9 has larger absolute value, so the answer is +3

Step 4: Write the final answer

The answer is +3

(-6) + (+9) = +3
Final answer:

(-6) + (+9) = +3

Applied rules:

• Different signs rule: Subtract and keep sign of larger absolute value

• Absolute value: Distance from zero regardless of sign

• Integer addition: Combine values considering direction

Tip: Always compare absolute values to determine which is larger.
Common mistake: Keeping the sign of the smaller absolute value instead of the larger.
Integer Operations Laws, Methods, and Strategies
a + b = c (with sign rules)
Integer Addition Formula
Rule 1
Same Signs
Add and keep sign
Rule 2
Different Signs
Subtract and keep larger sign
Rule 3
Subtraction
Change to addition of opposite
Key definitions:

Integers: Whole numbers that can be positive, negative, or zero (...-3, -2, -1, 0, 1, 2, 3...)

Positive integers: Numbers greater than zero (1, 2, 3, ...)

Negative integers: Numbers less than zero (-1, -2, -3, ...)

Absolute value: Distance from zero regardless of sign (|−5| = 5, |+5| = 5)

Opposite: The number with the opposite sign (opposite of +5 is -5)

Integer operations methodology:
  1. Addition with same signs: Add absolute values, keep common sign
  2. Addition with different signs: Subtract smaller absolute value from larger, keep sign of larger
  3. Subtraction: Convert to addition of the opposite number
  4. Always consider direction: Positive moves right, negative moves left
Tip 1: Think of a number line: positive numbers go right, negative numbers go left.
Tip 2: Always convert subtraction to addition of the opposite first.
Tip 3: Compare absolute values to determine which number is "larger" in magnitude.
Tip 4: Remember that adding a negative is the same as subtracting.
Common errors: Forgetting to change signs when converting subtraction, keeping the wrong sign, misidentifying which absolute value is larger.
Key properties: Commutative for addition (a+b=b+a), Associative for addition ((a+b)+c=a+(b+c)), and Identity (a+0=a).
Essential rules for integer operations:

• Same signs: Add absolute values and keep common sign

• Different signs: Subtract smaller from larger and keep sign of larger

• Subtraction: Convert to addition of the opposite

• Absolute value: Distance from zero regardless of sign

• Direction: Positive moves right, negative moves left

Comprehensive Summary

Definitions: Integers are whole numbers that can be positive, negative, or zero. Absolute value is the distance from zero regardless of sign. Opposites are numbers with the same absolute value but different signs.

Core Rules: For addition with same signs, add absolute values and keep common sign. For addition with different signs, subtract smaller absolute value from larger and keep sign of larger absolute value. For subtraction, convert to addition of the opposite number.

Methods: 1) Same signs: Add and keep sign. 2) Different signs: Subtract and keep larger sign. 3) Subtraction: Convert to addition of opposite. 4) Always consider direction and magnitude.

Examples: (+5) + (+3) = +8, (+7) - (+4) = (+7) + (-4) = +3, (-6) + (+9) = +3.

Tips: Use number lines for visualization. Always convert subtraction to addition first. Compare absolute values to determine which is larger.

Common Pitfalls: Forgetting to change signs when converting subtraction, keeping the wrong sign after operations, misidentifying which absolute value is larger.

Memorization Aids: "Same signs, add and keep; Different signs, subtract and keep the sign of the larger."

Integer Operations Strategies
📊
Same Signs Addition
(+5) + (+3) = +8 → Add absolute values, keep sign
When signs match, add and keep
Different Signs Addition
(-6) + (+9) = +3 → 9-6=3, keep sign of larger
Subtract, keep larger's sign
Subtraction Conversion
(+7) - (+4) = (+7) + (-4) = +3
Change to addition of opposite
Verification Method
Check direction and magnitude make sense
Does answer fit the problem context?
Integer Operations Steps
1
Identify Signs
2
Convert if Subtraction
3
Apply Rule
4
Keep Correct Sign
5
Verify
Visualization of Integer Operations
Breaking down integer operations into step-by-step processes makes calculations manageable and understandable.
Practice with different strategies to strengthen integer operations skills!
Solution: Exercises 4 to 5
4 Integer Subtraction with Negatives
Exercise 4
(-8) - (-3) = ?
Definition:

Subtracting negatives: When subtracting a negative number, it becomes addition

Double negative: Subtracting a negative is equivalent to adding a positive

Problem
(-8) - (-3)
Convert
(-8) + (+3)
Result
-5
-8
-
-3
=
-5
Step 1: Convert subtraction to addition

(-8) - (-3) becomes (-8) + (+3) (subtracting a negative becomes adding)

Step 2: Identify the signs

One negative (-8) and one positive (+3)

Step 3: Apply the rule for different signs

Subtract smaller absolute value from larger: 8 - 3 = 5

Step 4: Keep the sign of the larger absolute value

-8 has larger absolute value, so the answer is -5

Step 5: Write the final answer

The answer is -5

(-8) - (-3) = -5
Final answer:

(-8) - (-3) = -5

Applied rules:

• Subtraction conversion: a - (-b) = a + b

• Different signs rule: Subtract and keep sign of larger absolute value

• Double negative: Subtracting a negative is adding

Tip: Remember "subtracting a negative is adding a positive".
Common mistake: Not changing the double negative correctly.
5 Complex Integer Operation
Exercise 5
(+4) + (-7) + (+2) = ?
Definition:

Complex integer operations: Multiple operations with integers in one expression

Order of operations: Work left to right for addition and subtraction

Problem
(+4) + (-7) + (+2)
Step 1
(+4) + (-7) = -3
Step 2
(-3) + (+2) = -1
Step 1: Work left to right - first operation

(+4) + (-7): Different signs, subtract 7-4=3, keep sign of larger (-7), so -3

Step 2: Continue with next operation

(-3) + (+2): Different signs, subtract 3-2=1, keep sign of larger (-3), so -1

Step 3: Write the final answer

The answer is -1

(+4) + (-7) + (+2) = -1
Final answer:

(+4) + (-7) + (+2) = -1

Applied rules:

• Order of operations: Work left to right for same precedence

• Multiple operations: Solve one operation at a time

• Integer addition: Apply rules for each pair of numbers

Tip: Solve one operation at a time, working left to right.
Common mistake: Trying to solve all operations at once instead of sequentially.
Integer Operations Laws, Methods, Definitions, and Expressions
a + b = c (with sign rules applied)
Integer Addition Formula
Key definitions:

Integers: Whole numbers that can be positive, negative, or zero (...-3, -2, -1, 0, 1, 2, 3...)

Positive integers: Numbers greater than zero (1, 2, 3, ...)

Negative integers: Numbers less than zero (-1, -2, -3, ...)

Absolute value: Distance from zero regardless of sign (|−5| = 5, |+5| = 5)

Opposite: The number with the opposite sign (opposite of +5 is -5)

Integer operations methods:
  1. Same signs addition: Add absolute values, keep common sign
  2. Different signs addition: Subtract smaller from larger, keep larger's sign
  3. Subtraction: Convert to addition of the opposite
  4. Multiple operations: Work left to right, solving one at a time
  5. Number line method: Visualize movement left/right for direction
Tip 1: Always convert subtraction to addition of the opposite first.
Tip 2: Use number lines to visualize the direction of movement.
Tip 3: Compare absolute values to determine which number is "larger" in magnitude.
Tip 4: For multiple operations, solve one at a time from left to right.
Common errors: Forgetting to change signs when converting subtraction, keeping the wrong sign after operations, not comparing absolute values correctly.
Key properties: Commutative for addition (a+b=b+a), Associative for addition ((a+b)+c=a+(b+c)), and Identity (a+0=a).
Essential integer operations laws:

• Same signs law: Add absolute values and keep common sign

• Different signs law: Subtract smaller from larger and keep sign of larger

• Subtraction law: Convert to addition of the opposite

• Absolute value law: Distance from zero regardless of sign

• Order law: Work left to right for multiple operations

Detailed Integer Operations Summary

Conceptual Understanding: Integers include positive numbers, negative numbers, and zero. The absolute value represents the distance from zero regardless of direction. Operations with integers require considering both magnitude and direction.

Strategic Approaches: For addition with same signs, add absolute values and keep the common sign. For addition with different signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.

Algorithm Development: Subtraction should always be converted to addition of the opposite number. This simplifies the process and reduces confusion about sign changes.

Order of Operations: When multiple operations appear, work from left to right, solving one operation at a time.

Verification Techniques: Use number line visualization to verify that the direction and magnitude of the result make sense in the context of the problem.

Common Challenges: Students often struggle with converting subtraction to addition, especially with double negatives. Systematic approaches and visualization techniques help address these challenges.

Practical Applications: Real-world contexts like temperature changes, financial transactions, or elevation changes make integer operations meaningful and relevant.

Integer Operations Visual Guide
⚙️
Same Signs Addition
(+5) + (+3) = +8 → Add absolute values, keep sign
When signs match, add and keep
Different Signs Addition
(-6) + (+9) = +3 → 9-6=3, keep sign of larger
Subtract, keep larger's sign
Subtraction Conversion
(+7) - (+4) = (+7) + (-4) = +3
Change to addition of opposite
Double Negative
(-8) - (-3) = (-8) + (+3) = -5
Subtracting negative = adding positive
Integer Operations Process
1
Identify Signs
2
Convert if Subtraction
3
Apply Rule
4
Keep Sign
5
Verify
Integer Operations Conceptual Understanding
Breaking down integer operations into step-by-step processes helps students understand the underlying mathematical concepts.
Master these foundational concepts for future math success!

Questions & Answers

Question: I get confused when subtracting negative numbers. Why does (-8) - (-3) become (-8) + (+3)?

Answer: Great question! When you subtract a negative number, it's the same as adding the positive version. Think of it like this:

If you have $8 debt (which is -8) and someone takes away $3 of your debt (subtract -3), you now owe $5 less debt, which is the same as gaining $3.

So: (-8) - (-3) = (-8) + (+3) = -5

Remember: "Subtracting a negative is adding a positive!"

Question: My child struggles with integer operations. How can I help them visualize the concept?

Answer: Here are some visualization strategies:

  • Number line: Draw a horizontal line with zero in the middle, positive numbers to the right, negative to the left
  • Temperature analogy: Going up increases (positive), going down decreases (negative)
  • Money analogy: Positive = money earned, negative = debt or money owed
  • Arrow movements: Point arrows right for positive, left for negative

Start with concrete examples before moving to abstract operations. Practice with real-world scenarios like temperature changes or bank account balances.

Question: How do I know which number has the larger absolute value?

Answer: Absolute value is just the distance from zero, ignoring the sign! To compare absolute values:

For (-6) and (+9):

  • Absolute value of -6 is 6
  • Absolute value of +9 is 9
  • Since 9 > 6, +9 has the larger absolute value

Remember: |-6| = 6 and |+9| = 9, so you just compare the numbers without their signs!