Fractions: Parts of a Whole
Introduction
Fractions represent parts of a whole or parts of a group. They are essential for understanding division, ratios, proportions, and many real-world situations where we need to work with quantities that aren’t whole numbers.
From sharing a pizza among friends to measuring ingredients for cooking, fractions help us express and work with partial quantities in precise and meaningful ways.
Fraction Fundamentals
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A fraction has two parts:
3 ← Numerator (how many parts we have)
─
4 ← Denominator (how many equal parts in the whole)
This represents "3 out of 4 equal parts"Understanding Fractions Conceptually
Visual Models of Fractions
Fraction Models
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1. Area Model (Pizza/Pie):
3/4 = three-fourths
┌─────┬─────┐
│ ▓▓▓ │ ▓▓▓ │ 3 pieces shaded
├─────┼─────┤ out of 4 total
│ ▓▓▓ │ │
└─────┴─────┘
2. Linear Model (Number Line):
0───¼───½───¾───1
↑
3/4
3. Set Model (Groups):
3/4 of 8 objects = 6 objects
●●●●●●○○ (6 out of 8 are shaded)
4. Discrete Model:
3/4 means 3 out of every 4 items
●●●○ ●●●○ ●●●○ (pattern repeats)Types of Fractions
Fraction Classifications
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Proper Fractions (numerator < denominator):
3/4, 2/5, 7/8
- Less than 1 whole
- Represent part of a whole
Improper Fractions (numerator ≥ denominator):
5/4, 7/3, 9/9
- Equal to or greater than 1 whole
- Can be converted to mixed numbers
Mixed Numbers:
1¾, 2⅓, 5½
- Whole number + proper fraction
- Greater than 1
Unit Fractions:
1/2, 1/3, 1/4, 1/5...
- Numerator is 1
- Building blocks for other fractions
Equivalent Fractions:
1/2 = 2/4 = 3/6 = 4/8...
- Same value, different representation
- Multiply/divide numerator and denominator by same numberEquivalent Fractions
Finding Equivalent Fractions
Creating Equivalent Fractions
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Method 1: Multiply by 1 in fraction form
1/2 = 1/2 × 2/2 = 2/4
1/2 = 1/2 × 3/3 = 3/6
1/2 = 1/2 × 4/4 = 4/8
Visual proof:
1/2: ┌─────┬─────┐ 2/4: ┌───┬───┬───┬───┐
│ ▓▓▓ │ │ │▓▓▓│▓▓▓│ │ │
└─────┴─────┘ └───┴───┴───┴───┘
Both show the same amount shaded!
Method 2: Divide by common factors
12/16 = (12÷4)/(16÷4) = 3/4
18/24 = (18÷6)/(24÷6) = 3/4
Finding the pattern:
2/3 = 4/6 = 6/9 = 8/12 = 10/15...
Pattern: multiply both parts by 2, 3, 4, 5...Simplifying Fractions
Reducing Fractions to Lowest Terms
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Method 1: Find Greatest Common Factor (GCF)
Simplify 18/24:
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
GCF = 6
18/24 = (18÷6)/(24÷6) = 3/4
Method 2: Divide by common factors step by step
20/30 → 10/15 → 2/3
÷2 ÷5
Method 3: Prime factorization
24/36:
24 = 2³ × 3
36 = 2² × 3²
GCF = 2² × 3 = 12
24/36 = (24÷12)/(36÷12) = 2/3
Visual check:
24/36: ████████████████████████████████████
▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
2/3: ███
▓▓
Same proportion shaded!Comparing and Ordering Fractions
Comparing Fractions
Fraction Comparison Strategies
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Strategy 1: Same Denominators
Compare numerators directly
3/8 vs 5/8: Since 3 < 5, then 3/8 < 5/8
Visual:
3/8: ┌─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│ │ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┴─┘
5/8: ┌─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│▓│▓│ │ │ │
└─┴─┴─┴─┴─┴─┴─┴─┘
Strategy 2: Same Numerators
Compare denominators (smaller denominator = larger fraction)
3/4 vs 3/8: Since 4 < 8, then 3/4 > 3/8
Visual:
3/4: ┌───┬───┬───┬───┐
│▓▓▓│▓▓▓│▓▓▓│ │
└───┴───┴───┴───┘
3/8: ┌─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│ │ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┴─┘
Strategy 3: Convert to Common Denominators
Compare 2/3 and 3/4:
2/3 = 8/12
3/4 = 9/12
Since 8 < 9, then 2/3 < 3/4
Strategy 4: Convert to Decimals
2/3 = 0.667...
3/4 = 0.75
Since 0.667 < 0.75, then 2/3 < 3/4
Strategy 5: Cross Multiplication
Compare a/b and c/d:
If a×d < b×c, then a/b < c/d
2/3 vs 3/4: 2×4 = 8, 3×3 = 9
Since 8 < 9, then 2/3 < 3/4Ordering Fractions
Ordering Multiple Fractions
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Order from least to greatest: 1/2, 2/3, 3/8, 5/6
Step 1: Find common denominator (LCD = 24)
1/2 = 12/24
2/3 = 16/24
3/8 = 9/24
5/6 = 20/24
Step 2: Order by numerators
9/24 < 12/24 < 16/24 < 20/24
Step 3: Convert back to original fractions
3/8 < 1/2 < 2/3 < 5/6
Number line visualization:
0───3/8───1/2───2/3───5/6───1
↑ ↑ ↑ ↑
0.375 0.5 0.667 0.833
Benchmark Strategy:
Compare to 1/2:
- 3/8 < 1/2 (since 3 < 4)
- 1/2 = 1/2
- 2/3 > 1/2 (since 4 > 3)
- 5/6 > 1/2 (since 10 > 6)
Then order within each group.Adding and Subtracting Fractions
Same Denominators
Adding/Subtracting with Same Denominators
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Rule: Add/subtract numerators, keep denominator
Addition: 2/7 + 3/7 = (2+3)/7 = 5/7
Visual:
2/7: ┌─┬─┬─┬─┬─┬─┬─┐
│▓│▓│ │ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┘
3/7: ┌─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┘
Sum: ┌─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│▓│▓│ │ │ = 5/7
└─┴─┴─┴─┴─┴─┴─┘
Subtraction: 6/8 - 2/8 = (6-2)/8 = 4/8 = 1/2
Visual:
6/8: ┌─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│▓│▓│▓│ │ │
└─┴─┴─┴─┴─┴─┴─┴─┘
Remove 2/8: ┌─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│▓│ │ │ │ │ = 4/8 = 1/2
└─┴─┴─┴─┴─┴─┴─┴─┘Different Denominators
Adding/Subtracting with Different Denominators
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Step 1: Find Least Common Denominator (LCD)
Step 2: Convert to equivalent fractions
Step 3: Add/subtract numerators
Step 4: Simplify if possible
Example: 1/3 + 1/4
Step 1: LCD of 3 and 4 = 12
Step 2: 1/3 = 4/12, 1/4 = 3/12
Step 3: 4/12 + 3/12 = 7/12
Step 4: 7/12 is already simplified
Visual proof:
1/3: ┌────┬────┬────┐
│▓▓▓▓│ │ │
└────┴────┴────┘
1/4: ┌───┬───┬───┬───┐
│▓▓▓│ │ │ │
└───┴───┴───┴───┘
Convert to twelfths:
4/12: ┌─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│▓│ │ │ │ │ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┘
3/12: ┌─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│ │ │ │ │ │ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┘
Sum: ┌─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│▓│▓│▓│▓│ │ │ │ │ │ = 7/12
└─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┘
Subtraction Example: 3/4 - 1/6
LCD = 12
3/4 = 9/12, 1/6 = 2/12
9/12 - 2/12 = 7/12Mixed Numbers
Adding/Subtracting Mixed Numbers
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Method 1: Add/subtract parts separately
2⅓ + 1¼ = (2 + 1) + (⅓ + ¼) = 3 + (4/12 + 3/12) = 3 + 7/12 = 3 7/12
Method 2: Convert to improper fractions
2⅓ = 7/3, 1¼ = 5/4
7/3 + 5/4 = 28/12 + 15/12 = 43/12 = 3 7/12
Subtraction with regrouping:
4⅛ - 1⅝
Can't subtract ⅝ from ⅛, so regroup:
4⅛ = 3 + 1 + ⅛ = 3 + 9/8 = 3 9/8
Now subtract:
3 9/8 - 1⅝ = (3 - 1) + (9/8 - 5/8) = 2 + 4/8 = 2½
Visual representation:
4⅛: ████ ┌─┬─┬─┬─┬─┬─┬─┬─┐
│▓│ │ │ │ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┴─┘
After regrouping:
3 9/8: ███ ┌─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│▓│▓│▓│▓│▓│▓│ + ┌─┬─┬─┬─┬─┬─┬─┬─┐
└─┴─┴─┴─┴─┴─┴─┴─┘ │▓│ │ │ │ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┴─┘Multiplying Fractions
Basic Fraction Multiplication
Multiplying Fractions Rule
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Rule: Multiply numerators, multiply denominators
a/b × c/d = (a×c)/(b×d)
Example: 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2
Visual interpretation:
"2/3 of 3/4"
Start with 3/4: ┌───┬───┬───┬───┐
│▓▓▓│▓▓▓│▓▓▓│ │
└───┴───┴───┴───┘
Take 2/3 of the shaded part:
Divide each shaded section into 3 parts, take 2:
┌─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┐
│▓│▓│ │▓│▓│ │▓│▓│ │ │ │ │
└─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┘
Result: 6 out of 12 parts = 6/12 = 1/2
Real-world example:
"2/3 of the students are girls, 3/4 of the girls play sports"
2/3 × 3/4 = 1/2 of all students are girls who play sportsMultiplying Mixed Numbers
Mixed Number Multiplication
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Method 1: Convert to improper fractions
2⅓ × 1½ = 7/3 × 3/2 = 21/6 = 3½
Method 2: Use distributive property
2⅓ × 1½ = (2 + ⅓) × (1 + ½)
= 2×1 + 2×½ + ⅓×1 + ⅓×½
= 2 + 1 + ⅓ + ⅙
= 3 + 2/6 + 1/6
= 3 + 3/6
= 3½
Area model visualization:
2⅓ × 1½ means rectangle with dimensions 2⅓ by 1½
┌─────────────┬─────┐
│ │ │
1 │ 2 × 1 │⅓×1 │
│ = 2 │=⅓ │
├─────────────┼─────┤
½ │ 2 × ½ │⅓×½ │
│ = 1 │=⅙ │
└─────────────┴─────┘
2 ⅓
Total area: 2 + ⅓ + 1 + ⅙ = 3 + 2/6 + 1/6 = 3½Simplifying Before Multiplying
Canceling Common Factors
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Instead of: 4/9 × 3/8 = 12/72 = 1/6
Cancel first: 4/9 × 3/8
↗ ↖
4÷4=1 3÷3=1
9÷3=3 8÷4=2
Result: 1/3 × 1/2 = 1/6 (much easier!)
Complex example:
15/28 × 14/45
Cancel common factors:
15/28 × 14/45
↗ ↖
15÷15=1 14÷14=1
28÷14=2 45÷15=3
Result: 1/2 × 1/3 = 1/6
This method prevents large numbers and reduces errors!Dividing Fractions
Division by Fractions
"Multiply by the Reciprocal" Rule
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Rule: a/b ÷ c/d = a/b × d/c
Example: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1½
Why does this work?
Think: "How many halves are in three-fourths?"
Visual with pizza:
3/4 pizza: ┌─┬─┬─┬─┐
│▓│▓│▓│ │
└─┴─┴─┴─┘
Each 1/2 piece: ┌──┬──┐
│▓▓│ │
└──┴──┘
How many 1/2 pieces fit in 3/4?
┌─┬─┬─┬─┐ → ┌──┬──┐ + ┌─┐
│▓│▓│▓│ │ │▓▓│ │ │▓│ (half of a 1/2 piece)
└─┴─┴─┴─┘ └──┴──┘ └─┘
Answer: 1½ pieces of size 1/2
Alternative explanation:
3/4 ÷ 1/2 = "3/4 × how many to make 1"
Since 1/2 × 2 = 1, we multiply by 2
3/4 × 2 = 6/4 = 1½Complex Division Problems
Dividing Mixed Numbers
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Problem: 2⅔ ÷ 1⅓
Step 1: Convert to improper fractions
2⅔ = 8/3
1⅓ = 4/3
Step 2: Multiply by reciprocal
8/3 ÷ 4/3 = 8/3 × 3/4 = 24/12 = 2
Answer: 2⅔ ÷ 1⅓ = 2
Real-world interpretation:
"If each serving is 1⅓ cups, how many servings in 2⅔ cups?"
Answer: 2 servings
Verification: 2 × 1⅓ = 2 × 4/3 = 8/3 = 2⅔ ✓
Word Problem Example:
"A recipe calls for 3¾ cups of flour. If you want to make ¾ of the recipe, how much flour do you need?"
3¾ × ¾ = 15/4 × 3/4 = 45/16 = 2 13/16 cups
But if the question was division:
"How many ¾-cup servings can you make from 3¾ cups?"
3¾ ÷ ¾ = 15/4 ÷ 3/4 = 15/4 × 4/3 = 60/12 = 5 servingsReal-World Applications
Cooking and Recipes
Recipe Applications
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Recipe Scaling:
Original recipe (serves 4): 2¾ cups flour
Need to serve 6 people: 6 ÷ 4 = 1½ times the recipe
Flour needed: 2¾ × 1½ = 11/4 × 3/2 = 33/8 = 4⅛ cups
Recipe Reduction:
Original recipe (serves 8): 3⅓ cups sugar
Need to serve 3 people: 3 ÷ 8 = ⅜ of the recipe
Sugar needed: 3⅓ × ⅜ = 10/3 × 3/8 = 30/24 = 5/4 = 1¼ cups
Ingredient Substitution:
Recipe calls for 2⅔ cups milk
Only have ⅓ cup measuring cup
Number of scoops: 2⅔ ÷ ⅓ = 8/3 ÷ 1/3 = 8/3 × 3/1 = 8 scoops
Cost Calculation:
Flour costs $2.40 per pound
Recipe uses ¾ pound
Cost: $2.40 × ¾ = $2.40 × 3/4 = $7.20/4 = $1.80Construction and Measurement
Construction Applications
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Board Cutting:
8-foot board, need pieces of 1⅓ feet each
Number of pieces: 8 ÷ 1⅓ = 8 ÷ 4/3 = 8 × 3/4 = 6 pieces
Remaining wood: 8 - (6 × 1⅓) = 8 - 8 = 0 feet (perfect fit!)
Paint Coverage:
1 gallon covers 400 square feet
Room area: 320 square feet
Paint needed: 320 ÷ 400 = 32/40 = 4/5 = ⅘ gallon
Tile Installation:
Room: 12½ feet × 10¼ feet
Area: 12½ × 10¼ = 25/2 × 41/4 = 1025/8 = 128⅛ square feet
Tiles are 1¼ feet × 1¼ feet each
Tile area: 1¼ × 1¼ = 5/4 × 5/4 = 25/16 square feet
Number of tiles: 128⅛ ÷ 25/16 = 1025/8 ÷ 25/16 = 1025/8 × 16/25 = 82 tilesTime and Scheduling
Time Applications
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Work Schedule:
Employee works 6¾ hours per day
Hourly wage: $18.50
Daily pay: 6¾ × $18.50 = 27/4 × $18.50 = $124.88
Project Planning:
Project takes 15½ hours total
Work 2¾ hours per day
Days needed: 15½ ÷ 2¾ = 31/2 ÷ 11/4 = 31/2 × 4/11 = 124/22 = 5 7/11 days
So need 6 full days to complete
Meeting Duration:
Meeting: 1⅓ hours
Break every ½ hour
Number of breaks: 1⅓ ÷ ½ = 4/3 ÷ 1/2 = 4/3 × 2/1 = 8/3 = 2⅔
So 2 breaks during the meeting
Travel Time:
Distance: 45¾ miles
Speed: 55 mph
Time: 45¾ ÷ 55 = 183/4 ÷ 55 = 183/4 × 1/55 = 183/220 hours
= 183/220 × 60 minutes = 49.9 minutes ≈ 50 minutesCommon Mistakes and Prevention
Typical Fraction Errors
Common Fraction Mistakes
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Mistake 1: Adding denominators
Wrong: 1/3 + 1/4 = 2/7
Correct: 1/3 + 1/4 = 4/12 + 3/12 = 7/12
Mistake 2: Cross-multiplying in addition
Wrong: 1/3 + 1/4 = (1×4 + 1×3)/(3×4) = 7/12 ← accidentally correct!
This method doesn't always work: 1/2 + 1/3 ≠ (1×3 + 1×2)/(2×3) = 5/6
Correct: 1/2 + 1/3 = 3/6 + 2/6 = 5/6 ← happens to match, but wrong method
Mistake 3: Multiplying denominators in multiplication
Wrong: 2/3 × 1/4 = 2/(3×4) = 2/12 = 1/6
Correct: 2/3 × 1/4 = (2×1)/(3×4) = 2/12 = 1/6 ← same answer, wrong process
Mistake 4: Not simplifying answers
Problem: 3/4 + 1/4 = 4/4
Wrong: Leave as 4/4
Correct: 4/4 = 1
Mistake 5: Improper conversion of mixed numbers
Wrong: 2⅓ = (2×3)/3 = 6/3 = 2
Correct: 2⅓ = (2×3 + 1)/3 = 7/3
Prevention Strategies:
- Always find common denominators for addition/subtraction
- Remember: multiply straight across for multiplication
- Always simplify final answers
- Use visual models to check reasonableness
- Practice conversion between mixed and improper fractionsBuilding Fraction Fluency
Conceptual Understanding First
Fraction Learning Progression
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Stage 1: Concrete Understanding
- Use manipulatives (fraction bars, circles)
- Real-world contexts (pizza, chocolate bars)
- Visual models and drawings
- Part-whole relationships
Stage 2: Equivalent Fractions
- Pattern recognition (1/2 = 2/4 = 3/6...)
- Simplifying fractions
- Finding common denominators
- Comparing fractions
Stage 3: Operations
- Addition/subtraction with like denominators
- Addition/subtraction with unlike denominators
- Multiplication concepts and algorithms
- Division concepts and algorithms
Stage 4: Applications
- Word problems
- Mixed numbers
- Real-world contexts
- Connections to decimals and percents
Daily Practice Routine:
1. Visual warm-up (5 minutes): Identify fractions from pictures
2. Equivalent practice (10 minutes): Create equivalent fractions
3. Operation focus (15 minutes): Practice one operation type
4. Problem solving (10 minutes): Real-world applications
5. Reflection (5 minutes): What did we learn?Conclusion
Fractions are fundamental to mathematical understanding, bridging the gap between whole numbers and the continuous nature of real-world quantities. They provide the foundation for understanding ratios, proportions, algebra, and advanced mathematical concepts.
Fractions: Complete Understanding
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Conceptual Understanding:
✓ Multiple representations (visual, numerical, contextual)
✓ Part-whole relationships
✓ Equivalence concepts
Procedural Fluency:
✓ Finding equivalent fractions
✓ Comparing and ordering
✓ All four operations with fractions
Strategic Competence:
✓ Choosing appropriate methods
✓ Estimation with fractions
✓ Problem-solving approaches
Adaptive Reasoning:
✓ Why algorithms work
✓ When to use different representations
✓ Connections to other mathematical concepts
Productive Disposition:
✓ Confidence with fractions
✓ Appreciation for precision
✓ Persistence through complex problemsMaster fractions well, and you’ll have powerful tools for mathematical reasoning that extend far beyond arithmetic. Whether working with ratios in science, proportions in art, or rates in business, fractions provide essential mathematical language for describing and working with the continuous quantities that surround us in daily life.