Using Arrays to Multiply Fractions in Grade 3

Learn how to use arrays as a visual tool to understand and solve fraction multiplication problems effectively.

Fraction Array Fundamentals
1 Core Concepts
Key Definitions:

Fraction: A number representing a part of a whole (e.g., 1/2, 3/4).

Array: A rectangular arrangement of objects in equal rows and equal columns.

Unit Fraction: A fraction with numerator 1 (e.g., 1/2, 1/3, 1/4).

Proper Fraction: A fraction where the numerator is smaller than the denominator.

Equivalent Fractions: Fractions that represent the same value (e.g., 1/2 = 2/4).

Numerator: The top number in a fraction (how many parts we take).

Denominator: The bottom number in a fraction (how many equal parts the whole is divided into).

Fraction Multiplication with Arrays:

When multiplying fractions, we're finding a part of a part:

\(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\)

This means we take one-half of one-third.

Part of Part: Fraction multiplication finds a part of a fractional part
Visual Connection: Arrays help show how fractions interact
Multiplication Rule: Multiply numerators and denominators separately
2 Array Components for Fractions
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
Fraction Multiplication Formula
Numerators
Multiply Top
Parts taken
Denominators
Multiply Bottom
Total parts
Result
New Fraction
Final answer
Simplify
Reduce
Lowest terms
Tip 1: Draw arrays to visualize how fractions overlap.
Tip 2: Multiply numerators together and denominators together.
Tip 3: Simplify the result if possible.
Tip 4: Use visual models to verify your answer.
Building Fraction Arrays
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
Fraction Multiplication Process
Fraction Array Construction Steps:

Step 1: Identify the two fractions to multiply

Step 2: Draw an array representing the first fraction

Step 3: Subdivide the array based on the second fraction

Step 4: Identify the overlapping section

Step 5: Count the overlapping parts

Step 6: Write the multiplication equation

Fraction Array Process:
  1. Set Up: Draw an array representing the first fraction
  2. Subdivide: Further divide the array based on the second fraction
  3. Identify: Locate the overlapping region
  4. Count: Count the overlapping sections
  5. Calculate: Use the formula (numerator×numerator)/(denominator×denominator)
  6. Verify: Check that your answer matches the expected result
1/2 × 1/3
(1×1)/(2×3)
1/6
2/3 × 1/4
(2×1)/(3×4)
2/12 = 1/6
1/2 × 3/4
(1×3)/(2×4)
3/8
2/5 × 3/4
(2×3)/(5×4)
6/20 = 3/10
Fraction Array Rules:

• Multiply numerators together to get the new numerator

• Multiply denominators together to get the new denominator

• Count systematically to avoid missing or double-counting objects

• Simplify the result by finding common factors

• The final fraction should be in lowest terms

Fraction Array Examples
3 Basic Fraction Arrays
Example 1: 1/2 × 1/3 Array

Explanation: Take 1/2 of 1/3 = 1/6

Calculation: (1×1)/(2×3) = 1/6

Equation: 1/2 × 1/3 = 1/6

Construction Steps:
  1. Draw a 2×3 grid (based on denominators)
  2. Shade 1/2 of the grid (1 row out of 2)
  3. Within that shaded area, shade 1/3 (1 column out of 3)
  4. Count: 1 out of 6 total sections is fully shaded
4 Advanced Fraction Arrays
Example 2: 2/3 × 3/4 Array

Explanation: Take 2/3 of 3/4 = 6/12 = 1/2

Calculation: (2×3)/(3×4) = 6/12 = 1/2

Equation: 2/3 × 3/4 = 1/2

Process:

Draw a 3×4 grid (based on denominators)

Shade 2/3 of the grid (2 rows out of 3)

Within that shaded area, shade 3/4 (3 columns out of 4)

Count: 6 out of 12 total sections are fully shaded

Example 3: 1/4 × 2/5 Array

Explanation: Take 1/4 of 2/5 = 2/20 = 1/10

Calculation: (1×2)/(4×5) = 2/20 = 1/10

Equation: 1/4 × 2/5 = 1/10

Process:

Draw a 4×5 grid (based on denominators)

Shade 1/4 of the grid (1 row out of 4)

Within that shaded area, shade 2/5 (2 columns out of 5)

Count: 2 out of 20 total sections are fully shaded

Questions & Answers

Question: How can I help 3rd graders understand the concept of multiplying fractions?

Answer: Use concrete visual representations and real-world analogies:

  • Concrete Manipulatives: Use physical objects like fraction tiles or circles
  • Visual Models: Draw arrays, circles, or rectangles divided into equal parts
  • Real-World Context: Use examples like "half of a quarter of a pizza"
  • Hands-On Activities: Have students physically divide objects into equal groups
  • Step-by-Step Process: Focus on understanding before memorizing rules

Emphasize that multiplying fractions is finding a part of a fractional part, which is a natural extension of whole number multiplication.

Question: Why do we multiply both the top and bottom numbers when multiplying fractions?

Answer: When multiplying fractions, we multiply both numerators and denominators because:

  • Numerators: Tell us how many parts we're taking (so we multiply these)
  • Denominators: Tell us how many equal parts the whole is divided into (so we multiply these too)
  • Visual Thinking: The array shows us how the parts combine
  • Consistency: This rule works for all fraction multiplication

Think of it as finding a part of a part - both the number of parts we take and the size of each part change.

Question: How can I practice fraction multiplication with my child at home?

Answer: Here are engaging ways to practice fraction multiplication at home:

  • Cooking Together: Use recipes to practice fractional amounts
  • Visual Representations: Draw arrays or use graph paper to create visual models
  • Art Projects: Create visual representations of fraction multiplication
  • Game Time: Play fraction-based board games or card games
  • Household Items: Use toys, blocks, or coins to create arrays
  • Story Problems: Create fun word problems based on your child's interests

Keep sessions short (10-15 minutes) and make it fun. Encourage your child to explain their thinking!