Solved Exercises on Comparing Fractions Visually in Grade 3

Master comparing fractions visually: pie charts, bar models, number lines, and comparison strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Pie Chart Comparison
Exercise 1
Compare 1/4 and 1/3 using pie charts. Which fraction is larger?
Definition:

Fraction: A part of a whole, represented as numerator/denominator

Numerator: The number of parts taken

Denominator: The total number of equal parts

Visual comparison method:
  1. Draw two identical circles (pies)
  2. Divide the first circle into 4 equal parts, shade 1
  3. Divide the second circle into 3 equal parts, shade 1
  4. Compare the shaded areas visually
vs
Step 1: Draw pie for 1/4

Pie divided into 4 equal slices, 1 slice is shaded

Shaded area = 25% of the pie

Step 2: Draw pie for 1/3

Pie divided into 3 equal slices, 1 slice is shaded

Shaded area ≈ 33.3% of the pie

Step 3: Compare visually

The 1/3 pie has a larger shaded portion than the 1/4 pie

1/3 > 1/4
Final answer:

1/3 is larger than 1/4 because when taking 1 part from a smaller number of total parts (3 vs 4), each part is larger

Applied rules:

• Visual comparison: Larger shaded area means larger fraction

• Same numerator rule: With same numerator, larger denominator means smaller fraction

• Pie model: Useful for comparing fractions with same numerator

2 Bar Model Comparison
Exercise 2
Compare 2/5 and 3/5 using bar models. Which fraction is larger?
Definition:

Bar model: Rectangular representation showing parts of a whole

Same denominator: Fractions with equal total parts

2/5
vs
3/5
Step 1: Draw bars of equal length

Both bars represent the same whole (same denominator)

Step 2: Shade 2 out of 5 parts for first fraction

2/5 = 40% of the bar

Step 3: Shade 3 out of 5 parts for second fraction

3/5 = 60% of the bar

Step 4: Compare the shaded lengths

The 3/5 bar has more shaded area than the 2/5 bar

3/5 > 2/5
Final answer:

3/5 is larger than 2/5 because when the denominators are the same, the fraction with the larger numerator is greater

Applied rules:

• Same denominator rule: Compare numerators directly

• Bar model: Shows proportional relationships clearly

• Visual comparison: Longer shaded bar represents larger fraction

3 Number Line Comparison
Exercise 3
Compare 1/2 and 2/4 using a number line. Are they equal?
Definition:

Number line: Visual representation showing fractions as points

Equivalent fractions: Different fractions representing the same value

1/2 2/4 0 1
Step 1: Draw a number line from 0 to 1

Mark the positions of 1/2 and 2/4

Step 2: Locate 1/2 on the number line

1/2 is exactly halfway between 0 and 1

Step 3: Locate 2/4 on the number line

2/4 = 1/2, so it's at the same position

Step 4: Compare positions

Both fractions are at the same point on the number line

1/2 = 2/4
Final answer:

1/2 and 2/4 are equivalent fractions because they represent the same point on the number line

Applied rules:

• Equivalent fractions: Same position on number line means same value

• Simplification: 2/4 reduces to 1/2 by dividing both by 2

• Number line: Shows relationship between fractions spatially

Rules and methods, laws,...
When numerator is same: Larger denominator = Smaller fraction
Same Numerator Rule
Rule 1
1/2 > 1/3 > 1/4 > 1/5
Same numerator, larger denominator is smaller
Rule 2
2/5 < 3/5 < 4/5
Same denominator, larger numerator is bigger
Rule 3
2/4 = 1/2
Equivalent fractions
Key definitions:

Comparison: Determining if one fraction is greater than, less than, or equal to another

Visual models: Pie charts, bar models, number lines to represent fractions

Equivalent fractions: Different fractions with the same value

Complete methodology:
  1. Identify the fractions: Look at numerators and denominators
  2. Choose visual model: Pie for same numerators, bar for same denominators, number line for equivalence
  3. Draw the model: Accurately represent each fraction
  4. Compare visually: Determine which is larger, smaller, or equal
  5. Write conclusion: Use >, <, or = symbols
Tip 1: Use pie charts when fractions have the same numerator.
Tip 2: Use bar models when fractions have the same denominator.
Tip 3: Use number lines to check if fractions are equivalent.
Tip 4: Remember: More parts mean smaller pieces when the whole is the same.
Common errors: Forgetting that larger denominators mean smaller pieces, confusing numerators and denominators.
Key insight: The denominator tells how many equal parts the whole is divided into.
Formulas to know by heart:

• Same numerator: 1/a > 1/b when a < b

• Same denominator: a/c > b/c when a > b

• Equivalent: a/b = (a×n)/(b×n) for any n ≠ 0

Solution: Exercises 4 to 5
4 Mixed Comparison
Exercise 4
Compare 3/4 and 2/3 using visual models. Which fraction is larger?
Definition:

Mixed comparison: Fractions with different numerators and denominators

3/4
2/3
Step 1: Convert to common denominator

3/4 = 9/12 and 2/3 = 8/12

Step 2: Compare numerators

Since 9 > 8, then 9/12 > 8/12

Step 3: Conclusion

Therefore 3/4 > 2/3

3/4 > 2/3
Final answer:

3/4 is larger than 2/3. Visually, the 3/4 pie has more shaded area than the 2/3 pie.

Applied rules:

• Common denominator method: Convert fractions to compare numerators

• Visual confirmation: Pie charts show relative sizes

• Cross multiplication: 3×3 > 4×2 (9 > 8)

5 Equivalent Fractions
Exercise 5
Show that 2/6 and 1/3 are equivalent using visual models.
Definition:

Equivalent fractions: Fractions that represent the same value

2/6
1/3
Step 1: Simplify 2/6

2/6 = (2÷2)/(6÷2) = 1/3

Step 2: Compare shaded areas

Both pies show approximately 33.3% shaded

Step 3: Verify with number line

Both fractions occupy the same position on the number line

Step 4: Conclusion

2/6 and 1/3 represent the same value

2/6 = 1/3
Final answer:

2/6 and 1/3 are equivalent fractions. Both represent one-third of the whole.

Applied rules:

• Simplification: Divide numerator and denominator by their GCD

• Visual confirmation: Same shaded area means same value

• Proportional reasoning: 2 parts of 6 equals 1 part of 3

Fractions Comparison Methods and Strategies
Same Denominator: a/c vs b/c → Compare a vs b
Direct Comparison
Key definitions:

Comparison: The process of determining if one fraction is greater than, less than, or equal to another

Visual models: Pictorial representations that help understand fractional relationships

Equivalent fractions: Fractions that represent the same value despite different numerators and denominators

Complete methodology:
  1. Analyze the fractions: Identify numerators and denominators
  2. Choose the best method: Same denominator, same numerator, or convert to common denominator
  3. Select visual model: Pie chart, bar model, or number line based on fraction types
  4. Draw and compare: Create accurate visual representations
  5. Verify results: Use mathematical operations to confirm
Tip 1: When denominators are the same, compare numerators directly.
Tip 2: When numerators are the same, the fraction with the smaller denominator is larger.
Tip 3: Use cross multiplication: a/b vs c/d → Compare a×d vs b×c.
Tip 4: Always draw models to scale for accurate visual comparisons.
Common errors: Confusing larger denominators with larger fractions, miscounting parts in visual models.
Exam preparation: Practice converting fractions to common denominators, master visual model creation.
Formulas to know by heart:

• Same denominator: a/c > b/c if a > b

• Same numerator: a/b > a/c if b < c

• Cross multiplication: a/b vs c/d → Compare a×d vs b×c

• Conversion: a/b = (a×n)/(b×n) for any n ≠ 0

Fraction Comparison Techniques

Comparison Techniques: Various methods to compare fractions visually and mathematically.

  • Pie charts: Best for comparing fractions with same numerator
  • Bar models: Ideal for fractions with same denominator
  • Number lines: Perfect for checking equivalence
  • Cross multiplication: Reliable mathematical method

Questions & Answers

Question: I don't understand why 1/3 is bigger than 1/4 when 3 is smaller than 4. Can you explain this?

Answer: Great question! This is a common source of confusion. Think of it this way:

  • If you have a pizza cut into 3 pieces, each piece is larger than if the same pizza was cut into 4 pieces
  • The denominator tells you how many equal pieces the whole is divided into
  • More pieces (larger denominator) means each piece is smaller
  • So 1/3 means 1 large piece, while 1/4 means 1 small piece

Visual representation: A pie chart showing 1/3 vs 1/4 clearly demonstrates that 1/3 covers more area.

Memory trick: "More pieces, smaller pieces" - so when comparing fractions with the same numerator, the larger denominator creates smaller pieces.

Question: How do I know which visual model to use when comparing fractions?

Answer: Here's a systematic approach to choosing the right visual model:

  • Same numerator: Use pie charts to compare different-sized pieces (e.g., 1/3 vs 1/5)
  • Same denominator: Use bar models to compare number of pieces (e.g., 2/6 vs 4/6)
  • Different numerators and denominators: Use number lines or convert to common denominators
  • Checking equivalence: Use number lines to see if fractions occupy the same position

The key is to match the visual model to the type of comparison you need to make. Pie charts work well for seeing piece size, bar models for counting pieces, and number lines for precise positioning.

Tip: Start with the visual model that matches what's similar between your fractions!

Question: My child struggles with comparing fractions that have different numerators and denominators. What strategies can I teach them?

Answer: Here are effective strategies for comparing fractions with different numerators and denominators:

  1. Convert to common denominator: Change both fractions to have the same denominator, then compare numerators
  2. Cross multiplication: For a/b vs c/d, compare a×d vs b×c
  3. Use benchmarks: Compare to familiar fractions like 1/2, 1/4, or 1
  4. Visual estimation: Use mental pictures of pie charts or bar models

Example: To compare 2/3 and 3/5, convert to 10/15 and 9/15. Since 10 > 9, then 2/3 > 3/5.

Start with visual models to build understanding, then transition to mathematical methods as confidence grows. Practice with real-world examples like pizza slices or chocolate bars!