Understanding Equivalent Fractions for Kids in Grade 3 - Mathematics - Exercises with solutions

Master equivalent fractions: visual representations, fraction models, cross multiplication, and kid-friendly methods through these 10 detailed exercises.

Solutions: Exercises 1 to 10
1 Visual fraction models
Exercise 1
Draw fraction circles to show that 1/2 and 2/4 are equivalent fractions.
Difficulty: Beginner Time: ~5 minutes Skills: Visual Representation
Definition:

Equivalent fractions: Fractions that represent the same value or occupy the same position on a number line, even though they have different numerators and denominators.

Note: Two fractions are equivalent if they represent the same portion of a whole. Visual models help students understand this concept concretely.

Step-by-step visualization method:
  1. Draw two identical circles representing the same whole
  2. Divide the first circle into 2 equal parts and shade 1 part
  3. Divide the second circle into 4 equal parts and shade 2 parts
  4. Compare the shaded areas to see they are equal
Circle 1
1/2
Circle 2
2/4
Step 1: Draw first circle

Divide circle into 2 equal parts, shade 1 part: 1/2

Step 2: Draw second circle

Divide circle into 4 equal parts, shade 2 parts: 2/4

Step 3: Compare shaded areas

Both shaded areas cover the same amount of space

1/2 = 2/4
Final answer:

1/2 and 2/4 are equivalent fractions because they represent the same portion of the whole.

Applied rules:

• Visual representation: Shows fractions as portions of a whole

• Equal division: Parts must be equal in size to compare accurately

• Shading: Shaded portions represent the numerator

• Practice Tip: Use manipulatives or drawings to visualize fraction equivalence

Related Examples:
  • 1/3 = 2/6 - Same portion of whole divided differently
  • 1/4 = 2/8 - Visual representation shows equal areas
  • 2/3 = 4/6 - Different divisions, same shaded area
Quick Tips:
  • Draw circles or rectangles to represent the same whole
  • Ensure divisions are equal to maintain accuracy
  • Shade the same proportion to demonstrate equivalence
Frequently Asked Questions:

Q: Can I use rectangles instead of circles?
A: Yes, rectangles, squares, or any shape can be used as long as the whole is divided equally.

Q: What if my divisions aren't equal?
A: The fractions won't be equivalent. Equal divisions are essential for accurate comparison.

2 Finding equivalent fractions
Exercise 2
Find two equivalent fractions for 3/5 by multiplying the numerator and denominator by the same number.
Difficulty: Intermediate Time: ~4 minutes Skills: Multiplication, Fraction Manipulation
Definition:

Finding equivalent fractions: Multiply both numerator and denominator by the same non-zero number to get an equivalent fraction.

Note: This method maintains the value of the fraction while changing its representation. The relationship between numerator and denominator stays the same.

Step-by-step multiplication method:
  1. Choose a non-zero number to multiply both numerator and denominator
  2. Multiply the numerator by this number
  3. Multiply the denominator by the same number
  4. Verify the fractions are equivalent by cross multiplication
Original Fraction
3/5
Multiplied by 2
6/10
Multiplied by 3
9/15
Step 1: Multiply by 2

3/5 × 2/2 = (3×2)/(5×2) = 6/10

Step 2: Multiply by 3

3/5 × 3/3 = (3×3)/(5×3) = 9/15

Step 3: Verify equivalence

For 3/5 and 6/10: 3×10 = 30 and 5×6 = 30, so 30 = 30 ✓

3/5 = 6/10 = 9/15
Final answer:

Two equivalent fractions for 3/5 are 6/10 and 9/15.

Applied rules:

• Multiplication property: Multiply both parts by the same number

• Equivalence verification: Cross multiply to confirm equality

• Non-zero multiplier: Cannot multiply by zero

• Practice Tip: Always verify equivalence using cross multiplication

Related Examples:
  • 2/3 = 4/6 = 6/9 - Multiplied by 2 and 3
  • 1/4 = 2/8 = 3/12 - Multiplied by 2 and 3
  • 4/7 = 8/14 = 12/21 - Multiplied by 2 and 3
Quick Tips:
  • Always multiply both numerator and denominator by the same number
  • Any non-zero number can be used for multiplication
  • Cross multiplication confirms the equivalence
Frequently Asked Questions:

Q: Can I divide instead of multiply?
A: Yes, dividing both numerator and denominator by the same number also gives equivalent fractions.

Q: What if I multiply numerator and denominator by different numbers?
A: The resulting fraction won't be equivalent to the original.

3 Cross multiplication
Exercise 3
Use cross multiplication to determine if 2/3 and 4/6 are equivalent fractions.
Difficulty: Intermediate Time: ~3 minutes Skills: Cross Multiplication, Comparison
Definition:

Cross multiplication: For fractions a/b and c/d, if a×d = b×c, then the fractions are equivalent.

Note: This method provides a mathematical way to verify fraction equivalence without needing visual models or converting to decimals.

Step-by-step cross multiplication method:
  1. Identify the numerators and denominators of both fractions
  2. Multiply the numerator of the first fraction by the denominator of the second
  3. Multiply the denominator of the first fraction by the numerator of the second
  4. Compare the two products to determine equivalence
Fractions to Compare
2/3 and 4/6
Cross Products
2×6 = 12 and 3×4 = 12
Result
Equivalent
Step 1: Set up the fractions

We have 2/3 and 4/6

Step 2: Cross multiply

2 × 6 = 12 and 3 × 4 = 12

Step 3: Compare products

Since 12 = 12, the fractions are equivalent

Step 4: Conclusion

2/3 = 4/6 because their cross products are equal

2/3 = 4/6
Final answer:

Yes, 2/3 and 4/6 are equivalent fractions because 2×6 = 3×4 (both equal 12).

Applied rules:

• Cross multiplication: Multiply diagonally across the fractions

• Equality test: If cross products are equal, fractions are equivalent

• Verification: Provides mathematical proof of equivalence

• Practice Tip: Cross multiplication is the fastest way to check equivalence

Related Examples:
  • 3/4 and 6/8: 3×8 = 24 and 4×6 = 24 → Equivalent
  • 1/2 and 3/7: 1×7 = 7 and 2×3 = 6 → Not equivalent
  • 5/10 and 1/2: 5×2 = 10 and 10×1 = 10 → Equivalent
Quick Tips:
  • Cross multiplication is the most reliable method for checking equivalence
  • Always multiply numerator of first by denominator of second
  • Then multiply denominator of first by numerator of second
Frequently Asked Questions:

Q: Does cross multiplication work for all fractions?
A: Yes, it works for any two fractions regardless of their values.

Q: Can I use cross multiplication to compare more than two fractions?
A: Compare pairs of fractions one at a time using cross multiplication.

Solutions: Exercises 4 to 5
4 Comparing fractions
Exercise 4
Which fraction is larger: 3/4 or 5/8? Show your work.
Definition:

Comparing fractions: Determining which fraction represents a larger portion of a whole by using various methods like common denominators or cross multiplication.

Note: To compare fractions with different denominators, you need to find a common basis for comparison.

Step-by-step comparison method:
  1. Find a common denominator or use cross multiplication
  2. If using common denominators, convert both fractions
  3. Compare the numerators of the converted fractions
  4. State which fraction is larger based on the comparison
Fractions to Compare
3/4 and 5/8
Using LCD
6/8 and 5/8
Result
3/4 > 5/8
Step 1: Find common denominator

The LCD of 4 and 8 is 8

Step 2: Convert 3/4 to eighths

3/4 = (3×2)/(4×2) = 6/8

Step 3: Compare numerators

6/8 vs 5/8 → Since 6 > 5, then 6/8 > 5/8

3/4 > 5/8
Final answer:

3/4 is larger than 5/8 because when converted to eighths, 3/4 equals 6/8, and 6/8 > 5/8.

Applied rules:

• Common denominators: Convert fractions to have the same denominator

• Numerator comparison: Larger numerator means larger fraction

• Least common denominator: Smallest common multiple of denominators

• Practice Tip: Cross multiplication is another method: 3×8 = 24, 4×5 = 20, so 24 > 20

Related Examples:
  • 2/3 vs 3/5: Convert to 10/15 vs 9/15 → 2/3 > 3/5
  • 1/2 vs 3/4: Convert to 2/4 vs 3/4 → 3/4 > 1/2
  • 4/6 vs 2/3: Both equal 2/3 → They are equivalent
Quick Tips:
  • Convert to common denominators when comparing
  • Or use cross multiplication: compare a×d and b×c for a/b vs c/d
  • For fractions with same numerators, smaller denominator means larger fraction
Frequently Asked Questions:

Q: What if denominators are the same?
A: Simply compare numerators - larger numerator means larger fraction.

Q: What if numerators are the same?
A: The fraction with the smaller denominator is larger.

5 Fraction circles
Exercise 5
Draw fraction circles to show that 2/3 and 4/6 are equivalent fractions.
Definition:

Fraction circles: Circular visual models divided into equal parts to represent fractions, helping students understand fractional relationships.

Note: Fraction circles provide a concrete way to visualize fraction equivalence and comparison.

Step-by-step fraction circle method:
  1. Draw two identical circles representing the same whole
  2. Divide the first circle into 3 equal parts and shade 2 parts
  3. Divide the second circle into 6 equal parts and shade 4 parts
  4. Compare the shaded areas to verify they represent the same amount
Circle 1
2/3
Circle 2
4/6
Result
Equivalent
Step 1: Draw first circle

Divide into 3 equal parts, shade 2 parts: 2/3

Step 2: Draw second circle

Divide into 6 equal parts, shade 4 parts: 4/6

Step 3: Compare shaded areas

Both shaded areas cover the same portion of the whole circle

2/3 = 4/6
Final answer:

2/3 and 4/6 are equivalent fractions because they represent the same portion of the whole circle.

Applied rules:

• Visual representation: Shows fractions as portions of a whole

• Equal division: Parts must be equal in size to compare accurately

• Shading: Shaded portions represent the numerator

• Practice Tip: Use physical fraction circles or draw precise divisions

Related Examples:
  • 1/2 = 3/6 - Same portion of whole divided differently
  • 1/3 = 2/6 - Visual representation shows equal areas
  • 3/4 = 6/8 - Different divisions, same shaded area
Quick Tips:
  • Draw circles with precise divisions using a compass or template
  • Ensure each part is exactly equal in size
  • Shade the same proportion to demonstrate equivalence
Frequently Asked Questions:

Q: Can I use other shapes besides circles?
A: Yes, rectangles, squares, or any shape can be used as long as the whole is divided equally.

Q: What if my divisions aren't equal?
A: The fractions won't be equivalent. Equal divisions are essential for accurate comparison.

Solutions: Exercises 6 to 10
6 Number line representation
Exercise 6
Show on a number line that 1/2 and 3/6 are equivalent fractions.
Definition:

Number line representation: Placing fractions on a number line to show their positions and equivalence, where equivalent fractions occupy the same position.

Note: Number lines provide a linear model for understanding fraction equivalence and comparison.

Final answer:

1/2 and 3/6 are equivalent fractions because they occupy the same position on the number line.

7 Area models
Exercise 7
Draw area models to show that 2/4 and 1/2 are equivalent fractions.
Definition:

Area models: Rectangular visual representations divided into equal parts to show fractions, where equivalent fractions have the same shaded area.

Note: Area models provide a rectangular approach to understanding fraction equivalence and operations.

Final answer:

2/4 and 1/2 are equivalent fractions because they represent the same shaded area of equal wholes.

8 Real-world problem
Exercise 8
Sarah ate 2/3 of her pizza and Tom ate 4/6 of his pizza. Who ate more pizza?
Definition:

Real-world fraction problems: Word problems that involve fractions in practical situations, requiring students to apply fraction concepts to everyday scenarios.

Note: These problems help students understand the practical applications of fraction equivalence and comparison.

Final answer:

Sarah and Tom ate the same amount of pizza because 2/3 = 4/6.

9 Missing numerator
Exercise 9
Find the missing numerator: 3/4 = ?/8
Definition:

Missing numerator problems: Finding unknown values in equivalent fraction equations by using the principle that equivalent fractions have proportional numerators and denominators.

Note: To find a missing numerator, determine the relationship between the denominators and apply the same operation to the numerator.

Final answer:

The missing numerator is 6, so 3/4 = 6/8.

10 Mixed operations
Exercise 10
Order these fractions from least to greatest: 1/2, 2/3, 3/4, 1/4
Definition:

Mixed operations: Problems involving multiple fraction concepts like ordering, comparing, and finding equivalences in a single task.

Note: To order fractions, convert them to have common denominators or use cross multiplication to compare pairs.

Final answer:

From least to greatest: 1/4, 1/2, 2/3, 3/4

Detailed Summary of Equivalent Fractions for Kids
Key Definitions:

Equivalent fractions: Fractions that represent the same value or occupy the same position on a number line, even though they have different numerators and denominators.

Numerator: The top number in a fraction that tells us how many parts we have.

Denominator: The bottom number in a fraction that tells us how many equal parts the whole is divided into.

Visual models: Pictures, shapes, or objects that help us see what fractions look like.

Cross multiplication: A method to check if two fractions are equivalent by multiplying diagonally across the fractions.

Core Rules and Principles:
  1. Multiplication Property: Multiply both numerator and denominator by the same non-zero number to get an equivalent fraction
  2. Division Property: Divide both numerator and denominator by the same non-zero number to get an equivalent fraction
  3. Cross Multiplication Test: For fractions a/b and c/d, if a×d = b×c, then the fractions are equivalent
  4. Common Denominator Rule: Fractions can be compared by converting them to have the same denominator
Basic Example: 1/2 = 2/4 = 3/6 - Multiply both parts by 2, 3, etc.
Visual Memory: Draw circles or rectangles to see equivalent fractions
Real-World Application: 2/3 of a pizza = 4/6 of a pizza - Same amount eaten
Memory Tip: "Whatever you do to the bottom, do to the top!"
Common Mistakes: Forgetting to multiply/divide both numerator and denominator, comparing fractions without common denominators, confusing numerator and denominator.
Memorization Tips: Practice with visual models, remember the cross multiplication rule, and use the "do the same to top and bottom" principle.
Key Procedures:

• Creating Equivalent Fractions: Multiply or divide both parts by the same number

• Checking Equivalence: Use cross multiplication or convert to common denominators

• Ordering Fractions: Find common denominators and compare numerators

• Simplifying Fractions: Divide both parts by their common factors

Rules and Methods for Equivalent Fractions
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Finding Equivalent Fractions
Method 1
Visual Models
Circle/rectangle models
Method 2
Cross Multiplication
Check equivalence
Method 3
Common Denominators
For comparison

Key Takeaways

  • Equivalent fractions represent the same portion of a whole
  • Multiply both numerator and denominator by the same number to find equivalents
  • Cross multiplication confirms fraction equivalence
  • Visual models help understand fraction relationships concretely
  • Common denominators enable direct comparison of fractions
  • Always do the same operation to both numerator and denominator
  • Practice with fun examples makes learning easier

Questions & Answers

Question: I don't understand why 1/2 and 2/4 are the same. Don't they have different numbers?

Answer: Great question! Even though 1/2 and 2/4 have different numbers, they represent the same amount:

  • Think of a pizza cut in half - 1 piece of 2 is 1/2 of the pizza
  • Now imagine the same pizza cut into 4 pieces - 2 pieces of 4 is 2/4 of the pizza
  • Both 1/2 and 2/4 give you the same amount of pizza!

The key is that when you multiply both the top number (numerator) and bottom number (denominator) by the same number, the fraction value stays the same.

Example: 1/2 × 2/2 = (1×2)/(2×2) = 2/4

Question: How can I help my child understand equivalent fractions better?

Answer: Here are effective strategies to teach equivalent fractions:

  • Use manipulatives: Fraction circles, bars, or pizza cutouts make abstract concepts concrete
  • Draw pictures: Have your child draw rectangles or circles divided into different parts
  • Real-world examples: Use food, money, or measuring cups to show equivalent amounts
  • Pattern recognition: Help them notice that 1/2 = 2/4 = 3/6 = 4/8

Start with simple fractions like 1/2 = 2/4, then gradually introduce more complex equivalences.

Consistent practice with visual models builds strong conceptual understanding.

Question: How do I know if two fractions are equivalent without drawing pictures?

Answer: You can use the cross multiplication method:

  • Take two fractions like 1/2 and 2/4
  • Multiply diagonally: 1×4 = 4 and 2×2 = 4
  • If the products are equal (4 = 4), the fractions are equivalent!

This is a quick mathematical way to check equivalence without visual models.

Another method is to see if you can multiply or divide both parts of one fraction by the same number to get the other fraction.

Fraction Glossary

Equivalent Fractions
Fractions that represent the same value or portion of a whole, such as 1/2 and 2/4.
Numerator
The top number in a fraction that tells us how many parts we have.
Denominator
The bottom number in a fraction that tells us how many equal parts the whole is divided into.
Cross Multiplication
A method to check if two fractions are equivalent by multiplying diagonally: a/b = c/d if a×d = b×c.
Least Common Denominator (LCD)
The smallest common multiple of the denominators of two or more fractions, used for comparison and operations.

Math Foundations Educational Team

Elementary Mathematics Specialists & Curriculum Developers

Our team of experienced elementary math teachers and curriculum specialists creates research-based, student-friendly fraction resources. All content is aligned with Common Core State Standards and reviewed by mathematics education experts to ensure accuracy and pedagogical effectiveness for 3rd grade learners.