Solved Exercises on Integer Properties in Grade 7

Master integer properties: commutative, associative, identity, inverse properties, and distributive property through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Commutative Property
Exercise 1
Verify the commutative property of addition with:
\((-7) + 12\) and \(12 + (-7)\)
Definition:

Commutative Property: The order of adding integers does not change the result: \(a + b = b + a\)

Verification method:
  1. Calculate the first expression: \(a + b\)
  2. Calculate the second expression: \(b + a\)
  3. Compare the results to confirm equality
Expression 1
\((-7) + 12\)
Expression 2
\(12 + (-7)\)
Results
\(5 = 5\)
Step 1: Calculate first expression

\((-7) + 12 = 5\) (Starting at -7, move 12 units right)

Step 2: Calculate second expression

\(12 + (-7) = 5\) (Starting at 12, move 7 units left)

Step 3: Compare results

Both expressions equal 5, confirming the commutative property

\((-7) + 12 = 12 + (-7) = 5\)
Final answer:

The commutative property holds: \((-7) + 12 = 12 + (-7) = 5\)

Applied rules:

• Commutative Property: Order doesn't matter in addition

• Integer Addition: Adding a positive moves right, negative moves left

• Verification: Both sides must yield the same result

2 Associative Property
Exercise 2
Verify the associative property of addition with:
\([(-3) + 8] + (-5)\) and \((-3) + [8 + (-5)]\)
Definition:

Associative Property: The grouping of adding integers does not change the result: \((a + b) + c = a + (b + c)\)

Grouping 1
\([(-3) + 8] + (-5)\)
Grouping 2
\((-3) + [8 + (-5)]\)
Results
\(0 = 0\)
Step 1: Calculate first grouping

\([(-3) + 8] + (-5) = [5] + (-5) = 0\)

Step 2: Calculate second grouping

\((-3) + [8 + (-5)] = (-3) + [3] = 0\)

Step 3: Compare results

Both groupings equal 0, confirming the associative property

\([(-3) + 8] + (-5) = (-3) + [8 + (-5)] = 0\)
Final answer:

The associative property holds: \([(-3) + 8] + (-5) = (-3) + [8 + (-5)] = 0\)

Applied rules:

• Associative Property: Grouping doesn't matter in addition

• Order of Operations: Solve parentheses first

• Verification: Both groupings must yield the same result

3 Identity Property
Exercise 3
Verify the identity property of addition with:
\((-15) + 0\) and \(0 + (-15)\)
Definition:

Identity Property: Adding zero to any integer gives the same integer: \(a + 0 = a\) and \(0 + a = a\)

Expression 1
\((-15) + 0\)
Expression 2
\(0 + (-15)\)
Results
\(-15 = -15\)
Step 1: Calculate first expression

\((-15) + 0 = -15\) (Adding zero doesn't change the number)

Step 2: Calculate second expression

\(0 + (-15) = -15\) (Adding zero doesn't change the number)

Step 3: Compare results

Both expressions equal -15, confirming the identity property

Step 4: Verify the result

Adding zero to any number returns the same number

\((-15) + 0 = 0 + (-15) = -15\)
Final answer:

The identity property holds: \((-15) + 0 = 0 + (-15) = -15\)

Applied rules:

• Identity Property: Zero is the additive identity

• Commutativity: Applies to both orders

• Verification: Adding zero preserves the original value

Integer Properties: Laws, Methods, and Definitions
\(a + b = b + a\)
Commutative Property
Property 1
\((a+b)+c = a+(b+c)\)
Associative Property
Property 2
\(a + 0 = a\)
Identity Property
Property 3
\(a + (-a) = 0\)
Inverse Property
Key definitions:

Integers: Whole numbers including positive, negative, and zero

Additive Inverse: The opposite of a number that sums to zero

Number Line: Visual representation of integers on a line

Complete methodology:
  1. Identify the property: Recognize which property applies to the situation
  2. Apply the rule: Use the correct mathematical expression of the property
  3. Verify the result: Check that both sides of the equation are equal
  4. Understand the concept: Grasp why the property works
Tip 1: Remember that the commutative property allows you to rearrange terms in addition.
Tip 2: The associative property lets you regroup terms for easier calculation.
Tip 3: Zero is always the identity element for addition.
Tip 4: The inverse of any number 'a' is '-a'.
Common errors: Confusing commutative with associative, forgetting that subtraction is not commutative.
Exam preparation: Memorize all integer properties, practice identifying them in expressions.
Properties to know by heart:

• Commutative: \(a + b = b + a\)

• Associative: \((a + b) + c = a + (b + c)\)

• Identity: \(a + 0 = a\)

• Inverse: \(a + (-a) = 0\)

• Closure: Sum of integers is always an integer

Solution: Exercises 4 to 5
4 Inverse Property
Exercise 4
Verify the inverse property of addition with:
\((-9) + 9\) and \(17 + (-17)\)
Definition:

Inverse Property: Adding an integer to its opposite (additive inverse) gives zero: \(a + (-a) = 0\)

Expression 1
\((-9) + 9\)
Expression 2
\(17 + (-17)\)
Results
\(0 = 0\)
Step 1: Calculate first expression

\((-9) + 9 = 0\) (Starting at -9, moving 9 units right reaches 0)

Step 2: Calculate second expression

\(17 + (-17) = 0\) (Starting at 17, moving 17 units left reaches 0)

Step 3: Verify the result

Both expressions equal zero, confirming the inverse property

\((-9) + 9 = 17 + (-17) = 0\)
Final answer:

The inverse property holds: \((-9) + 9 = 0\) and \(17 + (-17) = 0\)

Applied rules:

• Inverse Property: Every integer has an additive inverse

• Definition of Opposite: The additive inverse of 'a' is '-a'

• Verification: Always equals zero

5 Distributive Property
Exercise 5
Verify the distributive property with:
\(3 \times [(-4) + 7]\) and \(3 \times (-4) + 3 \times 7\)
Definition:

Distributive Property: Multiplication distributes over addition: \(a \times (b + c) = a \times b + a \times c\)

Expression 1
\(3 \times [(-4) + 7]\)
Expression 2
\(3 \times (-4) + 3 \times 7\)
Results
\(9 = 9\)
Step 1: Calculate first expression

\(3 \times [(-4) + 7] = 3 \times [3] = 9\)

Step 2: Calculate second expression

\(3 \times (-4) + 3 \times 7 = (-12) + 21 = 9\)

Step 3: Verify the result

Both expressions equal 9, confirming the distributive property

Step 4: Understand the concept

Multiplying by a sum is the same as multiplying each term separately

\(3 \times [(-4) + 7] = 3 \times (-4) + 3 \times 7 = 9\)
Final answer:

The distributive property holds: \(3 \times [(-4) + 7] = 3 \times (-4) + 3 \times 7 = 9\)

Applied rules:

• Distributive Property: Multiplication distributes over addition

• Order of Operations: Parentheses first in the first expression

• Verification: Both sides must equal the same value

Integer Properties: Comprehensive Guide
\(a + b = b + a\)
Commutative Property
Key definitions:

Commutative Property: The order of adding integers does not affect the sum

Associative Property: The grouping of adding integers does not affect the sum

Identity Property: Adding zero to any integer leaves it unchanged

Inverse Property: Adding an integer to its opposite results in zero

Distributive Property: Multiplication distributes over addition

Complete methodology:
  1. Analyze the expression: Recognize the structure (sum, grouping, etc.)
  2. Choose the property: Determine which property applies to the situation
  3. Apply the property: Use the correct mathematical expression of the property
  4. Verify the result: Confirm that both sides of the equation are equal
Tip 1: Remember that subtraction is NOT commutative or associative.
Tip 2: The associative property is helpful for mental math by regrouping terms.
Tip 3: Always verify your results by calculating both sides of the equation.
Tip 4: Practice with various integer combinations to strengthen understanding.
Common errors: Confusing properties, applying them to subtraction/multiplication, forgetting negative signs.
Exam preparation: Memorize all properties, practice identifying them in complex expressions.
Properties to know by heart:

• Commutative: \(a + b = b + a\)

• Associative: \((a + b) + c = a + (b + c)\)

• Identity: \(a + 0 = a\)

• Inverse: \(a + (-a) = 0\)

• Distributive: \(a \times (b + c) = a \times b + a \times c\)

Exercise with Visualization: Integer Operations
Exercise 6: Integer Number Line Operations
Consider the following integer operations on a number line:
\(f_1(x) = x + 5\)
\(f_2(x) = x + (-3)\)
\(f_3(x) = x + 0\)

Analysis: The chart shows how adding integers affects positions on a number line.

  • \(f_1(x) = x + 5\) (Moves 5 units right)
  • \(f_2(x) = x + (-3)\) (Moves 3 units left)
  • \(f_3(x) = x + 0\) (No movement, identity property)

Questions & Answers

Question: I don't understand how the commutative property works with negative integers. Why does \((-7) + 12\) equal \(12 + (-7)\)?

Answer: Great question! The commutative property works the same way regardless of whether the integers are positive or negative. Let me break it down:

  • For \((-7) + 12\): Start at -7 on the number line and move 12 units right → lands at 5
  • For \(12 + (-7)\): Start at 12 on the number line and move 7 units left → lands at 5

The order of the numbers doesn't matter in addition - you end up at the same position on the number line. Whether you start with the negative number and add a positive, or start with the positive and add a negative, the final result is the same.

Think of it like walking: if you walk 7 steps backward then 12 steps forward, you'll end up in the same place as if you walked 12 steps forward then 7 steps backward.

Question: When I need to calculate \([(-3) + 8] + (-5)\), how do I know if I can change the grouping?

Answer: Yes, you can change the grouping because of the associative property of addition! This property states that \((a + b) + c = a + (b + c)\).

  • Your original expression: \([(-3) + 8] + (-5) = 5 + (-5) = 0\)
  • With changed grouping: \((-3) + [8 + (-5)] = (-3) + 3 = 0\)

Both approaches give the same result. The associative property is especially useful for mental math because you can regroup numbers to make calculations easier. For example, if you had \(17 + (-8) + 8\), you could group the last two terms first since \((-8) + 8 = 0\), making the calculation simpler: \(17 + 0 = 17\).

Important note: This only works for addition (and multiplication), not for subtraction!

Question: What is the difference between the identity property and the inverse property? They both involve zero somehow.

Answer: You're right that both properties involve zero, but they work differently:

  • Identity Property: Adding zero to any number gives back the original number: \(a + 0 = a\). Zero is called the "identity element" because it preserves the identity of the number.
  • Inverse Property: Adding a number to its opposite gives zero: \(a + (-a) = 0\). The opposite \((-a)\) is called the "additive inverse" of \(a\).

Examples:

  • Identity: \((-15) + 0 = -15\) (zero preserves the number)
  • Inverse: \((-15) + 15 = 0\) (the number and its opposite cancel out)

Think of the identity property as "nothing changes," while the inverse property is "everything cancels out."

Question: Can I use the distributive property with subtraction? Like in \(3 \times [(−4) + 7]\), could I also write it as \(3 \times (7 - 4)\)?

Answer: Yes, you can rewrite \((-4) + 7\) as \(7 - 4\) since they're equivalent expressions. And yes, the distributive property works with subtraction too!

Here's how both approaches work:

  • Original: \(3 \times [(-4) + 7] = 3 \times 3 = 9\)
  • Rewritten: \(3 \times (7 - 4) = 3 \times 7 - 3 \times 4 = 21 - 12 = 9\)

The distributive property works for subtraction as well: \(a \times (b - c) = a \times b - a \times c\).

However, remember that subtraction is NOT commutative or associative, so you need to be careful with the order. The distributive property is one of the few properties that works with subtraction.

Question: Are there any properties that work for subtraction like they work for addition?

Answer: Unfortunately, subtraction does NOT have the nice properties that addition has:

  • Not Commutative: \(a - b ≠ b - a\) (except when \(a = b\))
  • Not Associative: \((a - b) - c ≠ a - (b - c)\)
  • No Identity Element: While \(a - 0 = a\), it's not true that \(0 - a = a\)

However, you can sometimes use addition properties by rewriting subtraction as addition of negatives:

Instead of \(a - b\), think of it as \(a + (-b)\), then you can apply addition properties.

For example: \((a - b) + c = (a + (-b)) + c = a + ((-b) + c)\) using the associative property.

This is why we often convert subtraction problems to addition problems when working with properties.