Review of Integers: 7th Grade Comprehensive Guide

Master integers: step-by-step methods, definitions, and practical applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Ordering Integers
Exercise 1
Order these integers from least to greatest: -7, 3, -2, 0, 5, -4
Definition:

Integer: A whole number that can be positive, negative, or zero (..., -3, -2, -1, 0, 1, 2, 3, ...).

Ordering method:
  1. Remember: Negative numbers are always less than positive numbers
  2. Among negative numbers: the larger the absolute value, the smaller the number
  3. Zero is greater than all negative numbers but less than all positive numbers
  4. Among positive numbers: the larger the number, the greater it is
-7
-4
-2
0
3
5
-7 < -4 < -2 < 0 < 3 < 5
Given
-7, 3, -2, 0, 5, -4
Ordered
-7, -4, -2, 0, 3, 5
Least → Greatest
-7 < -4 < -2 < 0 < 3 < 5
Step 1: Identify negative numbers

-7, -2, -4 (these are all less than positive numbers and zero)

Step 2: Order negative numbers

Among -7, -4, -2: -7 is farthest from zero, so it's smallest: -7 < -4 < -2

Step 3: Place zero

Zero comes after all negatives: -7 < -4 < -2 < 0

Step 4: Order positive numbers

Among 3 and 5: 3 < 5

Step 5: Combine all

-7 < -4 < -2 < 0 < 3 < 5

-7, -4, -2, 0, 3, 5
Final answer:

From least to greatest: -7, -4, -2, 0, 3, 5

Applied rules:

• Negative numbers: Larger absolute value means smaller number

• Zero: Greater than negatives, less than positives

• Positive numbers: Larger value means greater number

2 Absolute Value
Exercise 2
Find the absolute value of: -8, 5, -3, 0, 12
Definition:

Absolute value: The distance of a number from zero on the number line, always positive.

-8
0
8
|-8| = 8
Values
-8, 5, -3, 0, 12
Absolute
8, 5, 3, 0, 12
Rule
|x| ≥ 0
Step 1: Define absolute value

Absolute value is the distance from zero, always non-negative

Step 2: Apply to each number

|−8| = 8 (distance of 8 units from 0)

|5| = 5 (distance of 5 units from 0)

|−3| = 3 (distance of 3 units from 0)

|0| = 0 (distance of 0 units from 0)

|12| = 12 (distance of 12 units from 0)

Step 3: Verify results

All absolute values are non-negative (≥ 0)

|−8| = 8, |5| = 5, |−3| = 3, |0| = 0, |12| = 12
Final answer:

Absolute values: 8, 5, 3, 0, 12

Applied rules:

• Distance concept: Absolute value measures distance from zero

• Non-negativity: |x| ≥ 0 for all real numbers x

• Positive preservation: |positive number| = same number

3 Integer Addition
Exercise 3
Calculate: (-12) + 7, (-5) + (-8), 10 + (-3)
Definition:

Integer addition: The operation of combining integers, following specific rules for positive and negative numbers.

-12
-5
(-12) + 7 = -5
(-12) + 7
-5
(-5) + (-8)
-13
10 + (-3)
7
Step 1: Different signs (negative + positive)

(-12) + 7: Subtract the smaller absolute value from the larger: 12 - 7 = 5. Use the sign of the number with larger absolute value: -5

Step 2: Same signs (both negative)

(-5) + (-8): Add absolute values: 5 + 8 = 13. Keep the common sign: -13

Step 3: Different signs (positive + negative)

10 + (-3): Subtract the smaller absolute value from the larger: 10 - 3 = 7. Use the sign of the number with larger absolute value: 7

Step 4: Verify using number line

Each result can be verified by moving along the number line

(-12) + 7 = -5, (-5) + (-8) = -13, 10 + (-3) = 7
Final answer:

(-12) + 7 = -5, (-5) + (-8) = -13, 10 + (-3) = 7

Applied rules:

• Different signs: Subtract absolute values, use sign of larger absolute value

• Same signs: Add absolute values, keep common sign

• Verification: Use number line to confirm results

Key Rules and Methods for Integers
If a > 0, then |a| = a; If a < 0, then |a| = -a
Absolute Value Definition
Addition
Same signs: add, keep sign
Different signs: subtract, keep sign of larger
Subtraction
a - b = a + (-b)
Add the opposite
Multiplication
Same signs: positive, Different: negative
Sign rules
Key definitions:

Integers: The set of whole numbers and their opposites: {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Positive integer: A whole number greater than zero (1, 2, 3, ...).

Negative integer: A whole number less than zero (-1, -2, -3, ...).

Absolute value: The distance of a number from zero on the number line, denoted |x|.

Opposite numbers: Two numbers that are the same distance from zero but on opposite sides of zero.

Number line: A visual representation of numbers arranged in order from left to right.

Zero pair: A positive and negative number that add to zero (e.g., +3 and -3).

Integer operations methodology:
  1. Addition: If signs are the same, add and keep the sign; if different, subtract and keep the sign of the larger absolute value
  2. Subtraction: Change subtraction to addition of the opposite number
  3. Multiplication: Multiply absolute values; if signs are the same, result is positive; if different, result is negative
  4. Division: Similar to multiplication, divide absolute values and apply sign rules
Tip 1: Remember: negative numbers decrease as they move away from zero.
Tip 2: Absolute value is always positive or zero.
Tip 3: When adding integers with different signs, subtract their absolute values.
Tip 4: Use a number line to visualize integer operations.
Common errors: Forgetting sign rules, confusing absolute value with opposites, not properly handling double negatives.
Success strategies: Understanding sign rules, using number lines, checking answers with opposite operations.
Essential integer principles:

• Ordering: On a number line, numbers increase from left to right

• Sign rules: Operations follow specific rules based on signs of operands

• Distance concept: Absolute value measures distance from zero

• Closure: Integers are closed under addition, subtraction, and multiplication

a + (-a) = 0
Additive Inverse
|a| = a if a ≥ 0, -a if a < 0
Absolute Value Function
Solution: Exercises 4 to 5
4 Integer Subtraction
Exercise 4
Calculate: (-15) - (-8), 12 - (-5), (-9) - 4
Definition:

Integer subtraction: The operation of finding the difference between two integers, accomplished by adding the opposite of the subtrahend.

-15
-7
(-15) - (-8) = (-15) + 8 = -7
(-15) - (-8)
(-15) + 8 = -7
12 - (-5)
12 + 5 = 17
(-9) - 4
(-9) + (-4) = -13
Step 1: Apply subtraction rule

Change subtraction to addition of the opposite: a - b = a + (-b)

Step 2: Calculate (-15) - (-8)

(-15) - (-8) = (-15) + 8 = -7 (different signs, subtract, keep sign of larger absolute value)

Step 3: Calculate 12 - (-5)

12 - (-5) = 12 + 5 = 17 (same signs, add and keep sign)

Step 4: Calculate (-9) - 4

(-9) - 4 = (-9) + (-4) = -13 (same signs, add and keep sign)

(-15) - (-8) = -7, 12 - (-5) = 17, (-9) - 4 = -13
Final answer:

(-15) - (-8) = -7, 12 - (-5) = 17, (-9) - 4 = -13

Applied rules:

• Subtraction to addition: Change sign of subtrahend and add

• Double negative: Subtracting a negative is the same as adding a positive

• Sign rules: Apply addition rules after converting to addition

5 Integer Multiplication
Exercise 5
Calculate: (-6) × 4, (-3) × (-7), 8 × (-2)
Definition:

Integer multiplication: The operation of repeated addition with specific rules for determining the sign of the product.

0
Same signs: positive product, Different: negative product
(-6) × 4
-24
(-3) × (-7)
21
8 × (-2)
-16
Step 1: Apply multiplication sign rules

Same signs → positive product, Different signs → negative product

Step 2: Calculate (-6) × 4

Different signs: (-6) × 4 = -(6 × 4) = -24

Step 3: Calculate (-3) × (-7)

Same signs: (-3) × (-7) = 3 × 7 = 21

Step 4: Calculate 8 × (-2)

Different signs: 8 × (-2) = -(8 × 2) = -16

(-6) × 4 = -24, (-3) × (-7) = 21, 8 × (-2) = -16
Final answer:

(-6) × 4 = -24, (-3) × (-7) = 21, 8 × (-2) = -16

Applied rules:

• Sign rules: Same signs produce positive, different signs produce negative

• Multiplication: Multiply absolute values first, then apply sign rule

• Verification: Check that sign rules were applied correctly

Comprehensive Guide: Review of Integers
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Set of Integers
Key definitions:

Integers: The set of whole numbers and their opposites, including zero.

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

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

Absolute value: The distance of a number from zero on the number line, always non-negative.

Opposite numbers: Two numbers that have the same absolute value but different signs.

Number line: A visual representation showing the order of numbers from least to greatest.

Zero property: Zero is neither positive nor negative, serving as the boundary between positive and negative numbers.

Closure property: The result of an operation on integers is always an integer.

Complete integer operations methodology:
  1. Addition: Same signs → add and keep sign; Different signs → subtract and keep sign of larger absolute value
  2. Subtraction: Convert to addition by adding the opposite of the subtrahend
  3. Multiplication: Multiply absolute values; same signs → positive, different signs → negative
  4. Division: Divide absolute values; same signs → positive, different signs → negative
  5. Ordering: Use number line or compare absolute values for negative numbers
  6. Absolute value: Distance from zero, always non-negative
Tip 1: For addition with different signs, subtract absolute values and take the sign of the number with larger absolute value.
Tip 2: Remember: subtracting a negative number is the same as adding a positive number.
Tip 3: Use the phrase "same signs, positive result; different signs, negative result" for multiplication and division.
Tip 4: When ordering negative numbers, the one with the larger absolute value is actually smaller.
Tip 5: Always check your work by using the inverse operation.
Common errors: Forgetting sign rules, confusing addition and subtraction rules, not properly handling absolute values, misapplying sign rules for multiplication and division.
Success strategies: Understanding sign rules, using number lines, checking answers, practicing regularly.
Key concepts: Sign rules, absolute value, number line ordering, closure properties.
Essential integer principles:

• Sign rules: Operations follow specific rules based on signs of operands

• Ordering: On number line, numbers increase from left to right

• Absolute value: Always non-negative, measures distance from zero

• Closure: Results of operations on integers are always integers

• Inverses: Every integer has an additive inverse that sums to zero

• Identity elements: 0 for addition, 1 for multiplication

• Verification: Use inverse operations to check results

a + b = c ⟺ a = c - b
Addition/ Subtraction Relationship
a × b = c ⟺ a = c ÷ b (b ≠ 0)
Multiplication/ Division Relationship
|a| = |-a|
Absolute Value Property

Questions & Answers

Question: I get confused about when to make my answer positive or negative. How do I remember the sign rules?

Answer: This is a very common confusion! Here's a helpful way to remember:

  • Addition: If signs are the same, add and keep the sign. If signs are different, subtract and keep the sign of the number with the larger absolute value.
  • Multiplication/Division: Count the number of negative signs. If there's an even number of negatives, the result is positive. If there's an odd number of negatives, the result is negative.

For multiplication and division, think of it this way:

  • Positive × Positive = Positive
  • Negative × Negative = Positive (two negatives make a positive)
  • Positive × Negative = Negative
  • Negative × Positive = Negative

A helpful phrase to remember: "Same signs, positive result; different signs, negative result" for multiplication and division.

For addition, think of it as combining debts and credits. If you have debt (-) and more debt (-), you have more debt. If you have debt (-) and credit (+), you need to see which is larger.

Question: How can I help my child understand negative numbers better?

Answer: Negative numbers can be challenging! Here are effective strategies:

  1. Use real-world examples: Temperature below zero, bank account debts, elevations below sea level
  2. Number lines: Draw horizontal or vertical number lines to visualize movement
  3. Concrete manipulatives: Use colored chips where one color represents positive and another represents negative
  4. Storytelling: Create stories about gaining and losing money, or moving forward and backward
  5. Practice regularly: Consistent practice builds familiarity and comfort
  6. Connect to prior knowledge: Relate to subtraction concepts they already understand

Emphasize that negative numbers are just as real as positive numbers, just representing quantities in the opposite direction or below a reference point.

Start with simple comparisons and ordering before moving to operations. Patience is key as this concept takes time to fully develop.

Encourage your child to verbalize their thinking process when solving problems to help identify misconceptions.

Question: What's the difference between subtracting a negative number and adding a positive number?

Answer: Actually, subtracting a negative number is mathematically equivalent to adding a positive number! This is because of the definition of subtraction:

Subtraction definition: a - b = a + (-b)

So when you have a - (-b), this becomes a + (-(-b)) = a + b.

For example: 5 - (-3) = 5 + 3 = 8

This happens because subtracting a negative number means removing a debt or reversing a loss, which effectively increases the total. Think of it as "taking away a loss" which is the same as "gaining".

The key insight is that subtracting a negative number moves you to the right on the number line, just like adding a positive number does.

This is why we often say "subtracting a negative is the same as adding a positive" - they are mathematically equivalent operations that produce the same result.

This concept is fundamental to understanding how integer operations work and forms the basis for more advanced algebraic manipulations.