Solved Exercises on Visual Models for Equivalent Fractions in Grade 4

Master equivalent fractions through visual models: circle models, bar models, number lines, and area models with these 5 detailed exercises.

Rules, Methods, and Definitions
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Equivalent Fractions Rule
Key Definitions:

Equivalent Fractions: Fractions that represent the same part of a whole

Visual Model: A picture or diagram that shows what a fraction looks like

Numerator: The top number showing how many parts we have

Denominator: The bottom number showing how many equal parts make the whole

Visual Modeling Methods:
  1. Circle Model: Divide a circle into equal parts
  2. Bar Model: Divide a rectangle into equal sections
  3. Number Line: Mark fractions on a line
  4. Area Model: Use grids to show parts of a whole
Rule 1
\(\frac{1}{2} = \frac{2}{4}\)
Multiply both by 2
Rule 2
\(\frac{2}{3} = \frac{4}{6}\)
Multiply both by 2
Rule 3
\(\frac{3}{4} = \frac{6}{8}\)
Multiply both by 2
Tip 1: Draw the same size model for both fractions.
Tip 2: Count the total parts and shaded parts.
Tip 3: Shaded areas should be the same size.
Solution: Exercises 1 to 3
1 Circle Model
Exercise 1
Show that 1/2 = 2/4 using circle models
Definition:

Circle Model: A circle divided into equal parts where some parts are shaded

Method:
  1. Draw two identical circles
  2. Divide first circle into 2 equal parts, shade 1
  3. Divide second circle into 4 equal parts, shade 2
  4. Compare shaded areas
1/2
2/4
Step 1: Draw first circle

Circle divided into 2 equal parts, 1 shaded = 1/2

Step 2: Draw second circle

Circle divided into 4 equal parts, 2 shaded = 2/4

Step 3: Compare

Both shaded areas cover the same amount of space

1/2 = 2/4 ✓
Final answer:

1/2 = 2/4 because both fractions represent the same portion of the whole

Applied rules:

• Multiplication rule: Multiply numerator and denominator by same number

• Visual verification: Equal shaded areas mean equivalent fractions

• Same whole: Both models must be the same size

2 Bar Model
Exercise 2
Show that 2/3 = 4/6 using bar models
Definition:

Bar Model: A rectangular bar divided into equal segments

2/3
4/6
Step 1: Draw first bar

Bar divided into 3 equal parts, 2 shaded = 2/3

Step 2: Draw second bar

Bar divided into 6 equal parts, 4 shaded = 4/6

Step 3: Compare lengths

Shaded portions are the same length

2/3 = 4/6 ✓
Final answer:

2/3 = 4/6 because both fractions represent the same portion of their respective wholes

Applied rules:

• Equivalent fraction rule: 2/3 × 2/2 = 4/6

• Equal representation: Same proportion of the whole

• Visual comparison: Bars must be same length for accurate comparison

3 Number Line Model
Exercise 3
Show that 1/4 = 2/8 using a number line
Definition:

Number Line Model: A line divided into equal segments representing fractions

1/4
2/8
0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 1
Step 1: Draw number line

Create a line from 0 to 1 with 8 equal divisions

Step 2: Mark 1/4

Since 1/4 = 2/8, mark at the same point as 2/8

Step 3: Verify positions

Both fractions occupy the same position on the line

1/4 = 2/8 ✓
Final answer:

1/4 = 2/8 because both fractions represent the same point on the number line

Applied rules:

• Position rule: Equivalent fractions occupy the same position

• Conversion rule: 1/4 = (1×2)/(4×2) = 2/8

• Linear representation: Distance from zero determines value

Solution: Exercises 4 to 5
4 Area Model
Exercise 4
Show that 3/5 = 6/10 using area models
Definition:

Area Model: A grid or rectangle divided into equal parts

3/5
6/10
Step 1: Create first grid

Grid of 5 squares, shade 3 = 3/5

Step 2: Create second grid

Grid of 10 squares, shade 6 = 6/10

Step 3: Compare shaded areas

Both grids show the same proportion shaded

3/5 = 6/10 ✓
Final answer:

3/5 = 6/10 because both fractions represent the same portion of their respective areas

Applied rules:

• Area equivalence: Same proportion of total area

• Multiplication rule: 3/5 × 2/2 = 6/10

• Grid modeling: More divisions provide more precise representations

5 Comparison Model
Exercise 5
Show that 2/6 = 1/3 using visual models and simplification
Definition:

Simplification: Reducing a fraction to its lowest terms

2/6
1/3
Step 1: Draw first model

Rectangle divided into 6 parts, 2 shaded = 2/6

Step 2: Draw second model

Rectangle divided into 3 parts, 1 shaded = 1/3

Step 3: Simplify 2/6

2/6 = (2÷2)/(6÷2) = 1/3

Step 4: Verify visually

Both models show the same proportion shaded

2/6 = 1/3 ✓
Final answer:

2/6 = 1/3 because when simplified, both fractions represent the same value

Applied rules:

• Simplification rule: Divide both numerator and denominator by GCD

• Visual confirmation: Equivalent fractions show same proportion

• GCD concept: Greatest Common Divisor reduces to simplest form

Questions & Answers

Question: I don't understand why 1/2 and 2/4 are the same amount. Don't 2/4 have bigger numbers so it should be more?

Answer: Great question! The numbers being bigger doesn't mean the fraction is larger. Think of it this way:

  • 1/2 means you take 1 piece out of 2 total pieces
  • 2/4 means you take 2 pieces out of 4 total pieces

If you cut a pizza in half (2 pieces) and eat 1 piece, you ate 1/2 of the pizza.

If you cut the same pizza into 4 pieces and eat 2 pieces, you still ate the same amount!

The key insight is that when you multiply both the numerator and denominator by the same number, you're just cutting the same whole into smaller, equal pieces. The amount you get doesn't change.

Question: How do I know which visual model to use when comparing fractions? Should I always draw circles?

Answer: Different visual models work better for different situations:

  • Circle models: Great for parts of a whole like pizza, pie, or cake
  • Bar models: Good for comparing lengths or amounts
  • Number lines: Best for seeing which fraction is larger or smaller
  • Area models: Helpful for grid-based problems

The most important thing is that whatever model you choose, both fractions must refer to the same size whole. The model should make the comparison clear and obvious.

Start with the model that makes the most sense for the problem context!

Question: My child is having trouble seeing that 3/4 and 6/8 are equivalent. How can I help them visualize this better?

Answer: Try this hands-on approach:

  1. Take a piece of paper and fold it in half twice to make 4 equal parts
  2. Shade 3 of those parts to show 3/4
  3. Now fold the same paper in half three times to make 8 equal parts
  4. You'll see that the same amount is covered by 6 of the 8 parts

You can also use food items like chocolate bars, crackers, or sandwiches. Cut one in 4 pieces and eat 3, then cut another identical item in 8 pieces and eat 6. They're the same amount!

The key is showing that even though there are more pieces, the total amount eaten is the same.