Solved Exercises on Using Models to Find Equivalent Fractions for Grade 3

Master equivalent fractions using models: visual models, area models, and fraction bars through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Area model
Exercise 1
Use area models to show that 1/2 and 2/4 are equivalent fractions. Draw two rectangles and shade them to represent each fraction.
Definition:

Area model: A visual representation using shapes to show fractions as parts of a whole area.

Area model method:
  1. Draw two identical shapes (rectangles, squares, circles)
  2. Divide the first shape according to the first fraction's denominator
  3. Shade the appropriate number of parts for the numerator
  4. Repeat for the second fraction
  5. Compare the shaded areas to verify equivalence
First fraction
1/2
Second fraction
2/4
Visual result
Equal shaded areas
Step 1: Draw two identical rectangles

Draw two rectangles of the same size

Step 2: Divide first rectangle into 2 equal parts

Draw a line down the middle of the first rectangle

Step 3: Shade 1 part of the first rectangle

Shade 1 out of 2 parts to represent 1/2

Step 4: Divide second rectangle into 4 equal parts

Draw lines to create 4 equal sections in the second rectangle

Step 5: Shade 2 parts of the second rectangle

Shade 2 out of 4 parts to represent 2/4

Step 6: Compare shaded areas

Notice that both rectangles have the same amount shaded

1/2 and 2/4 have equal shaded areas
Final answer:

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

The visual model shows that both fractions shade the same portion of their respective rectangles.

Applied rules:

• Equal Division: All parts must be equal in size for valid fraction representation

• Area Comparison: Equivalent fractions cover the same area in identical shapes

• Visual Verification: Models provide clear evidence of fraction equivalence

2 Fraction bar model
Exercise 2
Use fraction bars to find an equivalent fraction for 2/3. Show how you can divide the same length into different numbers of equal parts.
Definition:

Fraction bar: A rectangular model that represents fractions as lengths along a bar.

Original fraction
2/3
Multiplier
× 2
Equivalent fraction
4/6
Step 1: Draw a fraction bar divided into 3 equal parts

Draw a horizontal bar and divide it into 3 equal segments

Step 2: Shade 2 parts to represent 2/3

Shade 2 out of the 3 parts

Step 3: Subdivide each part into 2 equal pieces

Now each of the 3 parts is divided into 2 smaller parts

Total parts: 3 × 2 = 6 parts

Step 4: Count the shaded parts

Each of the 2 originally shaded parts is now 2 smaller parts

Shaded parts: 2 × 2 = 4 parts

Step 5: Write the equivalent fraction

4 parts shaded out of 6 total parts = 4/6

2/3 = 4/6 using fraction bar model
Final answer:

2/3 is equivalent to 4/6.

The fraction bar model shows that when we subdivide each part into 2 equal pieces, we get 4 shaded parts out of 6 total parts.

Applied rules:

• Subdivision: Dividing each part into equal smaller parts maintains the same proportion

• Proportional Scaling: When parts increase by a factor, shaded parts must increase by the same factor

• Length Representation: Fraction bars show fractions as lengths along a line

3 Circle model
Exercise 3
Draw two circles to show that 3/4 and 6/8 are equivalent fractions. Explain how the models demonstrate equivalence.
Definition:

Circle model: A circular visual representation showing fractions as sectors of a circle.

First fraction
3/4
Second fraction
6/8
Relationship
× 2/2
Step 1: Draw the first circle and divide into 4 equal parts

Draw a circle and use diameter lines to create 4 equal sectors

Step 2: Shade 3 parts of the first circle

Shade 3 out of 4 sectors to represent 3/4

Step 3: Draw the second circle and divide into 8 equal parts

Draw another circle and divide it into 8 equal sectors

Step 4: Shade 6 parts of the second circle

Shade 6 out of 8 sectors to represent 6/8

Step 5: Compare the shaded areas

Both circles have the same amount of area shaded

3/4 and 6/8 have equal shaded areas in circle models
Final answer:

3/4 and 6/8 are equivalent fractions.

The circle models demonstrate equivalence because both circles show the same proportion of the area shaded, even though they are divided into different numbers of parts.

Applied rules:

• Circular Representation: Fractions can be shown as parts of a circle

• Equal Proportion: Equivalent fractions represent the same proportion of the whole

• Visual Proof: Models provide concrete evidence of fraction equivalence

Using Models to Find Equivalent Fractions: Complete Guide
\(\frac{a}{b} = \frac{a \times n}{b \times n}\)
Equivalent Fraction Formula
Area Model
Shaded regions
Rectangle/circle
Bar Model
Segment lengths
Linear representation
Set Model
Counted items
Groups of objects
Key definitions:

Equivalent Fractions: Fractions that represent the same value or amount, even though they have different numerators and denominators.

Visual Model: A physical or drawn representation that helps understand mathematical concepts.

Area Model: A rectangular or circular model that shows fractions as parts of a whole area.

Fraction Bar: A linear model that represents fractions as lengths along a bar.

Proportional Reasoning: Understanding that equivalent fractions represent the same proportion of a whole.

Model-Based Equivalent Fraction Process:
  1. Select Appropriate Model: Choose the best visual model for the problem
  2. Draw Initial Fraction: Create the visual representation of the original fraction
  3. Modify the Model: Subdivide or regroup parts to create new fractions
  4. Count New Parts: Determine the new numerator and denominator
  5. Verify Equivalence: Confirm that the shaded/selected areas are equal
  6. Record Results: Write the equivalent fraction equation
Tip 1: Always ensure all parts in your model are equal in size.
Tip 2: Use graph paper to help create equal divisions in area models.
Tip 3: When subdividing, multiply both numerator and denominator by the same number.
Tip 4: Compare the shaded areas directly to verify equivalence.
Tip 5: Practice with physical manipulatives like fraction tiles or circles.
Common errors: Unequal divisions, not maintaining the same proportion, miscounting parts.
Success strategies: Precise drawings, systematic subdivision, visual comparison.
Model Rules:

• Equal Parts Rule: All divisions in the model must be equal for valid fraction representation

• Proportion Preservation: When subdividing, the proportion of shaded area must remain the same

• Multiplication Consistency: When multiplying parts, apply the same factor to both numerator and denominator

• Visual Verification: The shaded areas in equivalent fractions should appear equal

Solution: Exercises 4 to 5
4 Grid model
Exercise 4
Use a grid model to find an equivalent fraction for 1/3. Draw a grid and show how you can create more equal parts while maintaining the same proportion.
Definition:

Grid model: A rectangular array of squares used to represent fractions as parts of a whole grid.

Original fraction
1/3
Multiplier
× 3
Equivalent fraction
3/9
Step 1: Draw a 1×3 grid (1 row, 3 columns)

Create a rectangle divided into 3 equal squares in a row

Step 2: Shade 1 square to represent 1/3

Shade 1 out of the 3 squares

Step 3: Subdivide each square into 3 smaller squares

Divide each of the 3 squares into 3 smaller squares

This creates a 3×3 grid with 9 total squares

Step 4: Shade the corresponding squares

The originally shaded square is now 3 smaller squares

Shade these 3 smaller squares

Step 5: Write the equivalent fraction

3 squares shaded out of 9 total squares = 3/9

1/3 = 3/9 using grid model
Final answer:

1/3 is equivalent to 3/9.

The grid model shows that when we subdivide each of the 3 parts into 3 smaller parts, we get 3 shaded squares out of 9 total squares.

Applied rules:

• Grid Subdivision: Each original square is divided into the same number of smaller squares

• Proportional Counting: The number of shaded squares increases proportionally

• Array Representation: Grids provide a systematic way to visualize fraction equivalence

5 Real-world model
Exercise 5
Use a real-world context to show that 2/5 and 4/10 are equivalent. Draw a picture representing a garden with 5 sections where 2 are planted with flowers. Then show how to divide each section into 2 parts and find the equivalent fraction.
Definition:

Real-world model: A contextual representation that connects fraction concepts to practical situations.

Original situation
2/5 planted
Subdivision
× 2
New situation
4/10 planted
Step 1: Draw the original garden with 5 equal sections

Draw a rectangular garden divided into 5 equal parts

Step 2: Plant flowers in 2 sections

Shade or mark 2 out of the 5 sections to represent 2/5 planted

Step 3: Subdivide each section into 2 equal parts

Divide each of the 5 sections into 2 smaller sections

Now there are 5 × 2 = 10 total sections

Step 4: Determine how many new sections are planted

Each of the 2 originally planted sections is now 2 smaller sections

Planted sections: 2 × 2 = 4 sections

Step 5: Write the equivalent fraction

4 sections planted out of 10 total sections = 4/10

2/5 of the original garden = 4/10 of the subdivided garden
Final answer:

2/5 and 4/10 are equivalent fractions.

The garden model demonstrates that when each section is subdivided, the proportion of planted area remains the same.

Applied rules:

• Contextual Modeling: Real-world situations make abstract concepts concrete

• Proportional Scaling: The same factor applies to both the numerator and denominator

• Practical Application: Equivalent fractions appear in many real-life scenarios

Advanced Model-Based Fraction Concepts
\(\frac{a}{b} = \frac{c}{d} \iff ad = bc\)
Cross Multiplication Rule
Advanced definitions:

Proportional Relationship: A relationship where the ratio between corresponding parts remains constant.

Scaling Factor: The number by which both numerator and denominator are multiplied to create an equivalent fraction.

Visual Verification: Using models to confirm mathematical calculations of fraction equivalence.

Model Transformation: Changing the representation while preserving the fractional value.

Equivalent Sets: Collections of objects that represent the same fractional relationship.

Advanced Model-Based Methodology:
  1. Model Selection: Choose the most appropriate visual model for the problem
  2. Initial Setup: Create the visual representation of the given fraction
  3. Transformation Planning: Determine how to modify the model to create equivalent fractions
  4. Systematic Modification: Apply the same transformation to all parts of the model
  5. Verification: Check that the new model represents the same proportion
  6. Generalization: Extend the concept to other similar problems
Tip 1: When using models, always verify that the proportions remain the same.
Tip 2: Practice switching between different types of models to strengthen understanding.
Tip 3: Use models to predict equivalent fractions before calculating.
Tip 4: Connect visual models to symbolic representations.
Tip 5: Use models to verify your calculations with cross multiplication.
Advanced errors: Not maintaining equal subdivisions, not applying transformations uniformly, misinterpreting visual representations.
Advanced strategies: Multiple model approaches, verification methods, systematic transformations.
Advanced Fraction Properties:

• Transitive Property: If a/b = c/d and c/d = e/f, then a/b = e/f

• Reflexive Property: Every fraction is equivalent to itself

• Symmetric Property: If a/b = c/d, then c/d = a/b

• Proportional Scaling: Multiplication factor applies to both numerator and denominator

• Model Consistency: All representations of equivalent fractions must show the same proportion

Questions & Answers

Question: Why do I need to use models? Can't I just multiply the top and bottom numbers?

Answer: Great question! Models help you understand WHY equivalent fractions work:

  • Models show you visually that the fractions represent the same amount
  • They help you understand the concept before memorizing the procedure
  • Models provide a way to check if your multiplication is correct
  • They make abstract ideas more concrete and understandable

Once you understand the concept with models, you can use multiplication more confidently!

Question: My child is struggling to draw equal parts in their models. How can I help them be more accurate?

Answer: Here are some helpful strategies:

  • Use graph paper to help create equal divisions
  • Start with simple divisions (halves, thirds) before moving to more complex ones
  • Use a ruler to ensure straight lines
  • Practice folding paper to visualize equal parts
  • Use fraction manipulatives or printables for hands-on practice

The key is to emphasize that all parts must be the same size for the fraction to be valid!

Question: Can I use models to find equivalent fractions with really big numbers like 5/7 = ?/21?

Answer: For large numbers, models become difficult to draw precisely, but you can still understand the concept:

  • Imagine dividing each of 7 parts into 3 equal pieces (since 7×3=21)
  • Each of the 5 shaded parts would become 3 smaller parts
  • So 5×3 = 15 parts would be shaded
  • Therefore: 5/7 = 15/21

Models work best for smaller numbers, but they help you understand the pattern that applies to all equivalent fractions!