Lesson 6.2: Generate Equivalent Fractions

Advanced lesson on generating equivalent fractions through multiplication and division methods, with detailed examples, exercises, and solutions for 4th grade students.

Lesson Examples: Problems 1 to 3
1 Multiplication Method
Example 1
Find an equivalent fraction to \(\frac{3}{5}\) by multiplying both numerator and denominator by 4.
Definition:

Equivalent fractions: Fractions that represent the same value or the same portion of a whole, even though they may look different.

Multiplication Method Steps:
  1. Identify the original fraction
  2. Choose a number to multiply both numerator and denominator by
  3. Multiply both parts by the same number
  4. Verify the result using cross multiplication
Original Fraction
\(\frac{3}{5}\)
Multiply by 4
\(\frac{3 \times 4}{5 \times 4}\)
Result
\(\frac{12}{20}\)
Step 1: Identify the original fraction

Original fraction: \(\frac{3}{5}\)

Step 2: Choose a number to multiply by

We'll multiply both parts by 4

Step 3: Multiply both numerator and denominator

Numerator: \(3 \times 4 = 12\)

Denominator: \(5 \times 4 = 20\)

Step 4: Write the equivalent fraction

\(\frac{3}{5} = \frac{12}{20}\)

Step 5: Verify using cross multiplication

\(3 \times 20 = 60\) and \(5 \times 12 = 60\)

Since both products are equal, the fractions are equivalent

\(\frac{3}{5} = \frac{12}{20}\)
Final answer:

\(\frac{3}{5} = \frac{12}{20}\)

Applied rules:

• Multiplication Property: Multiply both numerator and denominator by the same non-zero number

• Identity Property: Multiplying by \(\frac{n}{n} = 1\) doesn't change the value

• Cross Multiplication Test: \(a/b = c/d\) if \(a \times d = b \times c\)

2 Division Method
Example 2
Find an equivalent fraction to \(\frac{18}{24}\) by dividing both numerator and denominator by their greatest common factor.
Definition:

Simplifying fractions: The process of reducing a fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor (GCF).

Original Fraction
\(\frac{18}{24}\)
Find GCF
GCF(18,24) = 6
Divide by GCF
\(\frac{18 \div 6}{24 \div 6}\)
Result
\(\frac{3}{4}\)
Step 1: Find factors of numerator and denominator

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Step 2: Identify the greatest common factor

Common factors: 1, 2, 3, 6

Greatest common factor: 6

Step 3: Divide both parts by the GCF

Numerator: \(18 \div 6 = 3\)

Denominator: \(24 \div 6 = 4\)

Step 4: Write the equivalent fraction

\(\frac{18}{24} = \frac{3}{4}\)

Step 5: Verify using cross multiplication

\(18 \times 4 = 72\) and \(24 \times 3 = 72\)

Since both products are equal, the fractions are equivalent

\(\frac{18}{24} = \frac{3}{4}\)
Final answer:

\(\frac{18}{24} = \frac{3}{4}\)

Applied rules:

• Division Property: Divide both numerator and denominator by the same non-zero number

• Greatest Common Factor (GCF): Largest number that divides both parts evenly

• Simplified Form: When numerator and denominator have no common factors other than 1

3 Multiple Operations
Example 3
Generate three equivalent fractions for \(\frac{4}{6}\) using different operations.
Definition:

Multiple equivalent fractions: There are infinitely many equivalent fractions for any given fraction, generated through various multiplication or division operations.

Original
\(\frac{4}{6}\)
×2
\(\frac{8}{12}\)
×3
\(\frac{12}{18}\)
÷2
\(\frac{2}{3}\)
Step 1: Start with the original fraction

\(\frac{4}{6}\)

Step 2: Multiply by \(\frac{2}{2}\) to get first equivalent

\(\frac{4}{6} = \frac{4 \times 2}{6 \times 2} = \frac{8}{12}\)

Step 3: Multiply by \(\frac{3}{3}\) to get second equivalent

\(\frac{4}{6} = \frac{4 \times 3}{6 \times 3} = \frac{12}{18}\)

Step 4: Divide by \(\frac{2}{2}\) to get third equivalent (simplified)

\(\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}\)

Step 5: Verify one of the equivalences

Check \(\frac{4}{6} = \frac{8}{12}\): \(4 \times 12 = 48\) and \(6 \times 8 = 48\) ✓

Step 6: Verify the simplified form

Check \(\frac{4}{6} = \frac{2}{3}\): \(4 \times 3 = 12\) and \(6 \times 2 = 12\) ✓

\(\frac{4}{6} = \frac{8}{12} = \frac{12}{18} = \frac{2}{3}\)
Final answer:

Three equivalent fractions: \(\frac{8}{12}, \frac{12}{18}, \frac{2}{3}\)

Applied rules:

• Infinite Equivalents: Any fraction has infinitely many equivalent fractions

• Operation Consistency: Always apply the same operation to both numerator and denominator

• Verification: Use cross multiplication to confirm equivalences

Lesson 6.2 Concepts and Methods
\(\frac{a}{b} = \frac{a \times n}{b \times n} = \frac{a \div m}{b \div m}\)
Generating Equivalent Fractions
Multiplication Rule
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Multiply both parts by same number
Division Rule
\(\frac{a}{b} = \frac{a \div n}{b \div n}\)
Divide both parts by same number
Cross Product
\(\frac{a}{b} = \frac{c}{d}\) if \(a \times d = b \times c\)
Test for equivalence
Key definitions:

Equivalent fractions: Fractions that represent the same value or the same portion of a whole.

Numerator: Top number in a fraction, representing the number of parts taken.

Denominator: Bottom number in a fraction, representing the total number of equal parts.

Greatest Common Factor (GCF): The largest number that divides two or more numbers evenly.

Simplest form: A fraction is in simplest form when the numerator and denominator have no common factors other than 1.

Two Methods for Generating Equivalent Fractions:
  1. Multiplication Method: Multiply both numerator and denominator by the same non-zero number
  2. Division Method: Divide both numerator and denominator by the same non-zero number (to simplify)
Tip 1: Always multiply or divide both numerator AND denominator by the same number.
Tip 2: Use visual models (circles, rectangles) to verify equivalence.
Tip 3: Cross multiplication is the fastest way to check if two fractions are equivalent.
Tip 4: The simplest form of a fraction has no common factors between numerator and denominator except 1.
Common errors: Forgetting to multiply/divide both parts, not checking for simplification, confusing equivalent with equal.
Key insight: Equivalent fractions represent the same point on a number line.
Essential formulas:

• Generating equivalent fractions: \(\frac{a}{b} = \frac{a \times n}{b \times n}\)

• Testing equivalence: \(\frac{a}{b} = \frac{c}{d}\) if \(a \times d = b \times c\)

• Simplifying: \(\frac{a}{b} = \frac{a \div \text{GCF}}{b \div \text{GCF}}\)

Lesson Practice: Problems 4 to 5
4 Finding Missing Parts
Example 4
Find the missing number: \(\frac{5}{6} = \frac{?}{24}\)
Definition:

Finding missing parts: Use the relationship between equivalent fractions to determine unknown numerators or denominators.

Given
\(\frac{5}{6} = \frac{?}{24}\)
Find multiplier
\(6 \times ? = 24\)
Apply to numerator
\(5 \times 4 = 20\)
Result
\(\frac{20}{24}\)
Step 1: Determine how the denominator changed

From 6 to 24: \(24 \div 6 = 4\)

So we multiplied the denominator by 4

Step 2: Apply the same multiplier to the numerator

Original numerator: 5

New numerator: \(5 \times 4 = 20\)

Step 3: Write the complete equivalent fraction

\(\frac{5}{6} = \frac{20}{24}\)

Step 4: Verify using cross multiplication

\(5 \times 24 = 120\) and \(6 \times 20 = 120\)

Since both products are equal, the fractions are equivalent

\(\frac{5}{6} = \frac{20}{24}\)
Final answer:

The missing number is 20, so \(\frac{5}{6} = \frac{20}{24}\)

Applied rules:

• Consistency Rule: Whatever operation is done to the denominator must also be done to the numerator

• Multiplier Identification: Find the factor by which one denominator was multiplied to get the other

• Cross Multiplication Check: Verify the result by cross multiplying

5 Complex Simplification
Example 5
Simplify \(\frac{36}{48}\) to its simplest form using the division method.
Definition:

Simplest form: A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.

Start
\(\frac{36}{48}\)
÷12
\(\frac{3}{4}\)
Check
No common factors
Step 1: Find factors of numerator and denominator

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Step 2: Identify the greatest common factor

Common factors: 1, 2, 3, 4, 6, 12

Greatest common factor: 12

Step 3: Divide both parts by the GCF

Numerator: \(36 \div 12 = 3\)

Denominator: \(48 \div 12 = 4\)

Step 4: Check if in simplest form

Factors of 3: 1, 3

Factors of 4: 1, 2, 4

Only common factor is 1, so \(\frac{3}{4}\) is in simplest form

Step 5: Verify the original equivalence

Cross multiplication: \(36 \times 4 = 144\) and \(48 \times 3 = 144\) ✓

\(\frac{36}{48} = \frac{3}{4}\)
Final answer:

\(\frac{36}{48} = \frac{3}{4}\) in simplest form

Applied rules:

• Greatest Common Factor: Divide by the largest common factor for fastest simplification

• Simplest Form Check: When GCD of numerator and denominator is 1

• Verification: Always check using cross multiplication

Detailed Lesson Summary: Lesson 6.2 - Equivalent Fractions
\(\frac{a}{b} = \frac{a \times n}{b \times n} = \frac{a \div m}{b \div m}\)
Core Principles
Key Definitions:

Equivalent fractions: Different fractions that represent the same value or the same portion of a whole. They occupy the same position on a number line.

Multiplication method: A technique for generating equivalent fractions by multiplying both the numerator and denominator by the same non-zero number.

Division method: A technique for generating equivalent fractions by dividing both the numerator and denominator by the same non-zero number.

Greatest Common Factor (GCF): The largest number that divides two or more numbers evenly without remainder.

Simplest form: A fraction is in simplest form when the numerator and denominator have no common factors other than 1.

Cross multiplication: A method to verify equivalence by multiplying diagonally: if \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\).

Core Methods:

Multiplication Method:

  1. Identify the original fraction you want to create an equivalent for
  2. Choose a number to multiply both numerator and denominator by
  3. Multiply both parts of the fraction by the chosen number
  4. Verify equivalence using cross multiplication

Division Method:

  1. Identify the original fraction you want to simplify
  2. Find the factors of both the numerator and denominator
  3. Determine the GCF of the numerator and denominator
  4. Divide both parts of the fraction by the GCF
  5. Verify equivalence using cross multiplication
Tip 1: Always perform the same operation (multiply or divide) on both numerator and denominator to maintain equivalence.
Tip 2: Use visual models (rectangles, circles) to understand why equivalent fractions represent the same amount.
Tip 3: Cross multiplication is the most reliable way to check if two fractions are equivalent: if \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\).
Tip 4: To check if a fraction is in simplest form, ensure the GCD of numerator and denominator is 1.
Tip 5: Use divisibility rules to quickly identify common factors (numbers ending in 0, 2, 4, 6, 8 are divisible by 2; sums of digits divisible by 3 indicate divisibility by 3).
Core Rule: The fundamental principle is that \(\frac{a}{b} = \frac{a \times n}{b \times n} = \frac{a \div m}{b \div m}\) where n, m ≠ 0. This preserves the fraction's value while changing its appearance.
Practical Application: Equivalent fractions are essential for adding/subtracting fractions with different denominators, comparing fractions, and solving real-world problems.
Essential Rules and Formulas:

• Multiplication Property: \(\frac{a}{b} = \frac{a \times n}{b \times n}\) where n ≠ 0

• Division Property: \(\frac{a}{b} = \frac{a \div n}{b \div n}\) where n ≠ 0

• Simplification Formula: \(\frac{a}{b} = \frac{a \div \text{GCF}(a,b)}{b \div \text{GCF}(a,b)}\)

• Equivalence Test: \(\frac{a}{b} = \frac{c}{d}\) if and only if \(a \times d = b \times c\)

• Simplest Form Test: A fraction is in simplest form when \(\text{GCD}(\text{numerator}, \text{denominator}) = 1\)

Simple Example

Create an equivalent fraction to \(\frac{1}{4}\) by multiplying by 3:

\(\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)

Verification: \(1 \times 12 = 12\) and \(4 \times 3 = 12\) ✓

Intermediate Example

Simplify \(\frac{15}{25}\) to simplest form:

Factors of 15: 1, 3, 5, 15

Factors of 25: 1, 5, 25

GCF = 5

\(\frac{15}{25} = \frac{15 \div 5}{25 \div 5} = \frac{3}{5}\)

Verification: \(15 \times 5 = 75\) and \(25 \times 3 = 75\) ✓

Advanced Example

Find the missing number: \(\frac{7}{8} = \frac{?}{40}\)

From 8 to 40: \(40 \div 8 = 5\)

Apply to numerator: \(7 \times 5 = 35\)

\(\frac{7}{8} = \frac{35}{40}\)

Verification: \(7 \times 40 = 280\) and \(8 \times 35 = 280\) ✓

Key Takeaways for Lesson 6.2:
  • Equivalent fractions have the same value but different appearances
  • Multiplication method creates larger equivalent fractions
  • Division method simplifies fractions to smaller forms
  • Always perform the same operation on both numerator and denominator
  • Use cross multiplication to verify equivalences
  • Simplification makes fractions easier to work with
  • There are infinitely many equivalent fractions for any given fraction
  • The simplest form is reached when GCD of numerator and denominator is 1
Lesson 6.2 Assessment
Practice Problems

Solve the following problems using the methods learned in Lesson 6.2:

  1. Find an equivalent fraction to \(\frac{5}{7}\) by multiplying both parts by 6.
  2. Simplify \(\frac{20}{30}\) to its simplest form.
  3. Find the missing number: \(\frac{3}{8} = \frac{?}{32}\)
  4. List three equivalent fractions for \(\frac{2}{9}\).
  5. Are \(\frac{10}{15}\) and \(\frac{14}{21}\) equivalent? Show your work.
Answer Key:

1. \(\frac{5}{7} = \frac{30}{42}\) (multiply both by 6)

2. \(\frac{20}{30} = \frac{2}{3}\) (divide both by GCF=10)

3. The missing number is 12, so \(\frac{3}{8} = \frac{12}{32}\)

4. Three equivalent fractions: \(\frac{4}{18}, \frac{6}{27}, \frac{8}{36}\) (or others like \(\frac{10}{45}, \frac{12}{54}\))

5. Yes, they are equivalent: \(10 \times 21 = 210\) and \(15 \times 14 = 210\), so cross products are equal.