Use the Visual Model to Solve in Grade 3 - Mathematics - Exercises with solutions

Master visual models: fraction circles, number lines, area models, and diagram solutions through these 10 detailed exercises with step-by-step visual approaches.

Solutions: Exercises 1 to 10
1 Fraction circles
Exercise 1
Use fraction circles to solve: Which is larger, 2/3 or 3/4?
Difficulty: Beginner Time: ~5 minutes Skills: Visual Comparison
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 comparison and equivalence.

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 (2/3)
  3. Divide the second circle into 4 equal parts and shade 3 parts (3/4)
  4. Compare the shaded areas to determine which is larger
Circle 1
2/3
Circle 2
3/4
Result
3/4 > 2/3
Step 1: Draw first circle

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

Step 2: Draw second circle

Divide into 4 equal parts, shade 3 parts: 3/4

Step 3: Compare shaded areas

3/4 covers more area than 2/3

Step 4: Conclusion

3/4 is larger than 2/3

3/4 > 2/3
Final answer:

3/4 is larger than 2/3 because the shaded area in the circle representing 3/4 is greater than the shaded area in the circle representing 2/3.

Applied rules:

• Visual comparison: Shows relative size of fractions

• 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 vs 3/4 - Visual model shows 3/4 is larger
  • 1/3 vs 1/4 - Visual model shows 1/3 is larger
  • 2/5 vs 3/5 - Same denominator, compare numerators
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 represented accurately. Equal divisions are essential for accurate comparison.

2 Number lines
Exercise 2
Use a number line to solve: Place 1/2, 1/4, and 3/4 on a number line from 0 to 1.
Difficulty: Beginner Time: ~4 minutes Skills: Linear Modeling
Definition:

Number line models: Linear visual representations that show fractions as positions on a line from 0 to 1, helping students understand fraction magnitude and order.

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

Step-by-step number line method:
  1. Draw a horizontal line from 0 to 1
  2. Divide the line into equal segments based on the denominators
  3. Place each fraction at its corresponding position
  4. Compare the positions to understand relative size
Line Segments
Divide into 4 parts
Positions
1/4, 1/2, 3/4
Order
1/4 < 1/2 < 3/4
Step 1: Draw number line

Draw a line from 0 to 1

Step 2: Divide into quarters

Divide the line into 4 equal segments

Step 3: Place fractions

1/4 at first segment, 1/2 at second, 3/4 at third

Step 4: Order fractions

From left to right: 1/4 < 1/2 < 3/4

1/4 < 1/2 < 3/4
Final answer:

On the number line: 1/4 is closest to 0, 1/2 is in the middle, and 3/4 is closest to 1. So 1/4 < 1/2 < 3/4.

Applied rules:

• Linear representation: Shows fractions as positions on a line

• Equal divisions: Number line segments must be equal

• Positioning: Fractions closer to 1 are larger

• Practice Tip: Number lines help visualize fraction magnitude

Related Examples:
  • 2/3 and 3/4 - 3/4 is farther right on number line
  • 1/5 and 2/5 - 2/5 is farther right on number line
  • 3/8 and 5/8 - 5/8 is farther right on number line
Quick Tips:
  • Draw precise divisions on number lines for accuracy
  • Fractions farther to the right are larger
  • Number lines help compare fraction sizes visually
Frequently Asked Questions:

Q: Can I use number lines for improper fractions?
A: Yes, extend the number line beyond 1 to accommodate improper fractions.

Q: How do I mark fractions with large denominators?
A: Use a ruler to make precise, equal divisions along the number line.

3 Area models
Exercise 3
Use area models to solve: Show that 2/4 and 1/2 are equivalent fractions.
Difficulty: Intermediate Time: ~4 minutes Skills: Area Modeling
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.

Step-by-step area model method:
  1. Draw two identical rectangles representing the same whole
  2. Divide the first rectangle into 4 equal parts and shade 2 parts
  3. Divide the second rectangle into 2 equal parts and shade 1 part
  4. Compare the shaded areas to verify they represent the same amount
Rectangle 1
2/4
Rectangle 2
1/2
Result
Equivalent
Step 1: Draw first rectangle

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

Step 2: Draw second rectangle

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

Step 3: Compare shaded areas

Both shaded areas cover the same portion of their respective rectangles

Step 4: Conclusion

2/4 and 1/2 are equivalent fractions

2/4 = 1/2
Final answer:

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

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 grid paper for precise area model drawing

Related Examples:
  • 3/6 = 1/2 - Same area shaded in different divisions
  • 4/8 = 1/2 - Equal shaded portions
  • 2/8 = 1/4 - Same proportional area
Quick Tips:
  • Use rectangles of the same size for fair comparison
  • Ensure divisions are equal to maintain accuracy
  • Shade the same proportion to demonstrate equivalence
Frequently Asked Questions:

Q: Can I use squares instead of rectangles?
A: Yes, squares are a special type of rectangle and work perfectly for area models.

Q: What if my rectangles are different sizes?
A: The comparison won't be valid. Use rectangles of the same size for accurate comparison.

Solutions: Exercises 4 to 5
4 Bar models
Exercise 4
Use bar models to solve: Compare 3/5 and 2/3 to determine which is larger.
Definition:

Bar models: Linear visual representations that show fractions as portions of a bar, helping students understand fractional relationships through length comparison.

Note: Bar models provide a horizontal approach to understanding fraction comparison and equivalence.

Step-by-step bar model method:
  1. Draw two identical bars representing the same whole
  2. Divide the first bar into 5 equal parts and shade 3 parts (3/5)
  3. Divide the second bar into 3 equal parts and shade 2 parts (2/3)
  4. Compare the shaded lengths to determine which is larger
Bar 1
3/5
Bar 2
2/3
Result
2/3 > 3/5
Step 1: Draw first bar

Divide into 5 equal parts, shade 3 parts: 3/5

Step 2: Draw second bar

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

Step 3: Compare shaded lengths

2/3 covers more length than 3/5

Step 4: Conclusion

2/3 is larger than 3/5

2/3 > 3/5
Final answer:

2/3 is larger than 3/5 because the shaded length in the bar representing 2/3 is greater than the shaded length in the bar representing 3/5.

Applied rules:

• Linear comparison: Shows relative size of fractions through length

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

• Shading: Shaded portions represent the numerator

• Practice Tip: Use rulers for precise bar divisions

Related Examples:
  • 1/2 vs 3/4 - Bar model shows 3/4 is longer
  • 1/3 vs 1/4 - Bar model shows 1/3 is longer
  • 2/5 vs 3/5 - Same denominator, compare numerators
Quick Tips:
  • Draw bars with equal lengths for fair comparison
  • Ensure divisions are equal to maintain accuracy
  • Shade the same proportion to demonstrate equivalence
Frequently Asked Questions:

Q: What if denominators are different?
A: The visual model will show the difference in size more clearly than numerical comparison.

Q: Can I use vertical bars instead of horizontal?
A: Yes, vertical bars work just as well as horizontal bars for comparison.

5 Circle diagrams
Exercise 5
Use circle diagrams to solve: Show that 4/6 and 2/3 are equivalent fractions.
Definition:

Circle diagrams: Circular visual models divided into equal sectors to represent fractions, helping students understand fractional relationships through area comparison.

Note: Circle diagrams provide a circular approach to understanding fraction equivalence and operations.

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

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

Step 2: Draw second circle

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

Step 3: Compare shaded areas

Both shaded areas cover the same portion of the whole circle

Step 4: Conclusion

4/6 and 2/3 are equivalent fractions

4/6 = 2/3
Final answer:

4/6 and 2/3 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 compass for precise circle 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 sector 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 represented accurately. Equal divisions are essential for accurate comparison.

Solutions: Exercises 6 to 10
6 Grid models
Exercise 6
Use a 12-grid model to solve: Show that 3/4 and 9/12 are equivalent fractions.
Definition:

Grid models: Square or rectangular grids divided into equal squares to represent fractions, where equivalent fractions have the same number of shaded squares.

Final answer:

3/4 and 9/12 are equivalent fractions because they represent the same number of shaded squares in the grid model.

7 Rectangular models
Exercise 7
Use rectangular models to solve: Compare 2/3 and 5/6 to determine which is larger.
Definition:

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

Final answer:

5/6 is larger than 2/3 because it covers more area in the rectangular model.

8 Real-world problems
Exercise 8
Use visual models to solve: Sarah ate 3/4 of her pizza and Tom ate 5/6 of his pizza. Who ate more?
Definition:

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

Final answer:

Tom ate more pizza because 5/6 is larger than 3/4 when compared using visual models.

9 Mixed models
Exercise 9
Use a combination of visual models to solve: Order these fractions from least to greatest: 1/2, 3/4, 1/4, 2/3
Definition:

Mixed models: Problems involving multiple visual model types like circles, number lines, and area models to solve fraction comparison and ordering.

Final answer:

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

10 Complex models
Exercise 10
Use visual models to solve: Complete the pattern: 1/2, 2/4, 3/6, ___, ___
Definition:

Complex models: Advanced problems that require understanding of fraction relationships, patterns, and equivalences using visual representations.

Final answer:

The pattern continues: 1/2, 2/4, 3/6, 4/8, 5/10 (all equivalent to 1/2)

Detailed Summary of Using Visual Models to Solve
Key Definitions:

Visual models: Concrete representations of mathematical concepts using shapes, diagrams, or pictures that help students understand abstract ideas.

Fraction circles: Circular visual models divided into equal parts to represent fractions.

Number lines: Linear visual representations showing fractions as positions on a line.

Area models: Rectangular visual representations divided into equal parts to show fractions.

Bar models: Linear visual representations showing fractions as portions of a bar.

Core Rules and Principles:
  1. Equal Division Rule: All parts in a visual model must be equal in size to accurately represent fractions
  2. Whole Representation: Visual models must represent the same whole for accurate comparison
  3. Shading Principle: Shaded portions represent the numerator of the fraction
  4. Comparison Method: Use visual models to compare fraction sizes by examining shaded areas or lengths
Basic Example: 1/2 = 2/4 - Visual models show same shaded area
Advanced Example: 3/4 > 2/3 - Visual models show 3/4 covers more area
Real-World Application: Visual models help understand fraction relationships in daily life
Memory Tip: "What you see is what you get" - Visual models make fractions tangible
Common Mistakes: Unequal divisions, comparing different wholes, inaccurate shading, misinterpreting visual comparisons.
Memorization Tips: Practice with physical models, remember that equal parts are essential, and use multiple visual model types for verification.
Key Procedures:

• Creating Visual Models: Draw equal divisions and shade appropriate parts

• Comparing Fractions: Use visual models to compare shaded areas or lengths

• Verifying Equivalence: Check that visual models show same portion of whole

• Ordering Fractions: Use visual models to arrange by size

Rules and Methods for Using Visual Models to Solve
\(\text{Visual Model} = \frac{\text{Shaded Parts}}{\text{Total Equal Parts}}\)
Visual Fraction Representation
Method 1
Fraction Circles
Circular models
Method 2
Number Lines
Linear models
Method 3
Area Models
Rectangular models

Key Takeaways

  • Visual models make abstract fraction concepts concrete
  • All parts in a visual model must be equal in size
  • Shaded portions represent the numerator
  • Visual models help compare fraction sizes effectively
  • Multiple model types can verify the same concept
  • Visual models support understanding of fraction equivalence

Questions & Answers

Question: Why do I need to use visual models when I can just compare numbers?

Answer: Visual models help you understand what fractions really mean:

  • They show you the actual size of fractions as portions of a whole
  • They make it easier to see which fraction is larger or smaller
  • They help you understand why fractions are equivalent
  • They build a foundation for more complex fraction work

Visual models make abstract concepts concrete and easier to understand.

Example: It's easier to see that 3/4 is larger than 1/2 when you can see the actual shaded areas.

Question: How can I help my child master visual models?

Answer: Here are effective strategies to master visual models:

  • 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 fraction concepts
  • Pattern recognition: Help them notice fraction relationships and equivalences

Start with simple fractions like 1/2, then gradually introduce more complex concepts.

Consistent practice with visual models builds strong conceptual understanding.

Question: What if I make mistakes when drawing the visual models?

Answer: Making mistakes is part of learning! Here's how to handle them:

  • Check your work: Make sure all parts are equal in size
  • Count carefully: Verify you're shading the correct number of parts
  • Use tools: Rulers, compasses, or templates for precision
  • Redraw if needed: Don't be afraid to start over with a cleaner model

The key is to ensure equal divisions and accurate shading to represent fractions correctly.

Practice will help you improve your visual model accuracy over time.

Model Glossary

Visual Model
A concrete representation of mathematical concepts using shapes, diagrams, or pictures.
Fraction Circle
A circular visual model divided into equal parts to represent fractions.
Number Line
A linear visual representation showing fractions as positions on a line from 0 to 1.
Area Model
A rectangular visual representation divided into equal parts to show fractions.
Bar Model
A linear visual representation showing fractions as portions of a bar.

Visual Models Educational Team

Elementary Mathematics Specialists & Curriculum Developers

Our team of experienced elementary math teachers and curriculum specialists creates research-based, student-friendly visual model 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 using visual models to solve fraction problems.