Today we'll learn how to work with numbers larger than 1 whole!
By the end of today, you'll be able to:
✓ Understand what an improper fraction is
✓ Understand what a mixed number is
✓ Convert improper fractions to mixed numbers
✓ Convert mixed numbers to improper fractions
✓ Know when to use each form
Let's start by reviewing what we already know! 🚀
A proper fraction has a numerator that is smaller than the denominator.
Examples: ½, ⅔, ¾, ⅘
These fractions are all less than 1 whole.
3 out of 4 parts = Less than 1 whole
An improper fraction has a numerator that is greater than or equal to the denominator.
Examples: 5/4, 7/3, 11/5, 8/8
These fractions represent 1 whole or more!
Don't let the name fool you! There's nothing wrong with improper fractions.
They're called "improper" because the top number is bigger than the bottom number, which is the opposite of what we usually see.
But they're perfectly correct and useful! ✅
5/4 = 1 whole circle + ¼ more!
Notice: We have more than one whole circle!
That's what makes this an improper fraction.
7/3 = 2 whole circles + ⅓ more!
Improper Fractions:
• An improper fraction has a numerator ≥ denominator
• The numerator (top) is bigger than or equal to the denominator (bottom)
• Improper fractions represent 1 whole or more
Examples:
5/4, 7/3, 11/8, 9/2
Remember: Nothing is wrong with improper fractions! They're just another way to show amounts greater than 1.
A mixed number combines a whole number and a proper fraction.
Examples: 1¼, 2⅔, 3½, 5⅗
It shows how many whole parts plus what fraction is left over.
2 = whole number (2 complete wholes)
⅓ = fraction part (one third more)
2 whole circles + ⅓ of another circle
The mixed number makes it easy to see: "I have 2 complete things plus a little bit more!"
Shows total number of parts
7 thirds total
Shows wholes + leftover
2 wholes + ⅓
7/3 and 2⅓ are exactly the same amount, just written differently!
Mixed Numbers:
• A mixed number has a whole number and a fraction together
• It shows complete wholes + a fractional part
• The whole number tells how many complete groups
• The fraction tells what's left over
Examples:
1¼, 2⅔, 3½, 5⅗
Remember: Mixed numbers and improper fractions can show the same amount!
Improper Fraction: "We have 9/4 pizzas."
Mixed Number: "We have 2¼ pizzas."
Which sounds more natural? The mixed number! It's easier to picture 2 whole pizzas plus a quarter of another one.
Improper Fraction: "The board is 17/4 feet long."
Mixed Number: "The board is 4¼ feet long."
The mixed number is easier to understand and measure!
Sometimes improper fractions are easier for calculating, but mixed numbers are easier for understanding the answer!
5 halves total
2 wholes + ½
Same circles, same amount! Just described differently! 🎨
✓ Communication: Mixed numbers are easier to understand in real life
✓ Calculations: Improper fractions are sometimes easier for math operations
✓ Comparisons: One form might make it easier to compare amounts
1. How to convert improper fractions → mixed numbers
2. How to convert mixed numbers → improper fractions
Both skills are important and useful! 🚀
"How many whole groups can I make, and what's left over?"
I have 7 fourths total.
How many groups of 4 can I make? 1 group (that's 1 whole!)
How many fourths are left over? 3 fourths
7/4 = 1¾
1 complete circle (4/4) + 3 more quarters (¾)
To convert an improper fraction to a mixed number:
Divide the numerator by the denominator!
Numerator ÷ Denominator = ?
This gives you the whole number
The remainder becomes the new numerator
The denominator stays the same
Whole number + (Remainder/Original Denominator)
11 ÷ 4 = ?
How many times does 4 go into 11?
4 goes into 11 exactly 2 times
2 × 4 = 8
11 - 8 = 3
Remainder = 3
Whole number = 2
Fraction part = 3/4 (remainder over original denominator)
11/4 = 2¾
5 × 3 = 15 ✅
5 × 4 = 20 ❌ (too big!)
Whole number = 3
17 - 15 = 2
Remainder = 2
Whole number = 3
Fraction part = 2/5
17/5 = 3⅖
3 × 7 = 21 ✅
3 × 8 = 24 ❌ (too big!)
Whole number = 7
22 - 21 = 1
Remainder = 1
Whole number = 7
Fraction part = 1/3
22/3 = 7⅓
Converting Improper Fractions to Mixed Numbers:
Method: Division!
Step 1: Divide numerator ÷ denominator
→ The answer is the whole number
Step 2: Find the remainder
→ The remainder is the new numerator
Step 3: Keep the same denominator
→ Write as: whole number + (remainder/denominator)
Example: 11/4 → 11÷4 = 2 R3 → 2¾
Problem 1: Convert 13/4 to a mixed number
Step 1: Divide 13 ÷ 4 = 3 (with a remainder)
4 × 3 = 12
Step 2: Find remainder: 13 - 12 = 1
Step 3: Write the mixed number: 3 + 1/4
Answer: 3¼
Problem 2: Convert 19/6 to a mixed number
Step 1: Divide 19 ÷ 6 = 3 (with a remainder)
6 × 3 = 18
Step 2: Find remainder: 19 - 18 = 1
Step 3: Write the mixed number: 3 + 1/6
Answer: 3⅙
Problem 3: Convert 25/8 to a mixed number
Step 1: Divide 25 ÷ 8 = 3 (with a remainder)
8 × 3 = 24
Step 2: Find remainder: 25 - 24 = 1
Step 3: Write the mixed number: 3 + 1/8
Answer: 3⅛
Problem 4: Convert 31/5 to a mixed number
Step 1: Divide 31 ÷ 5 = 6 (with a remainder)
5 × 6 = 30
Step 2: Find remainder: 31 - 30 = 1
Step 3: Write the mixed number: 6 + 1/5
Answer: 6⅕
"How many fractional parts do I have in total?"
I have 2 whole circles and ¾ of another.
Each whole has 4 quarters.
2 wholes = 2 × 4 = 8 quarters
Plus 3 more quarters = 8 + 3 = 11 quarters
2¾ = 11/4
Count all the shaded parts: 11 quarters!
(Whole × Denominator) + Numerator
Put this over the original denominator!
Whole number × Denominator
This tells you how many parts are in the wholes
Add the numerator from the fraction part
This is your new numerator
The denominator stays the same!
Whole number = 3
Denominator = 5
3 × 5 = 15
15 + 2 = 17
New numerator = 17
Denominator stays 5
3⅖ = 17/5
Whole number = 5
Denominator = 3
5 × 3 = 15
15 + 2 = 17
New numerator = 17
Denominator stays 3
5⅔ = 17/3
Whole number = 4
Denominator = 8
4 × 8 = 32
32 + 7 = 39
New numerator = 39
Denominator stays 8
4⅞ = 39/8
Converting Mixed Numbers to Improper Fractions:
Method: Multiply and Add!
Step 1: Multiply whole number × denominator
→ This shows how many parts in the wholes
Step 2: Add the numerator
→ This gives the total number of parts
Step 3: Keep the same denominator
→ Write as: (whole × denominator + numerator)/denominator
Example: 3⅖ → (3×5)+2 = 17 → 17/5
Problem 5: Convert 2⅗ to an improper fraction
Step 1: Multiply 2 × 5 = 10
Step 2: Add the numerator: 10 + 3 = 13
Step 3: Keep denominator: 5
Answer: 13/5
Problem 6: Convert 7¼ to an improper fraction
Step 1: Multiply 7 × 4 = 28
Step 2: Add the numerator: 28 + 1 = 29
Step 3: Keep denominator: 4
Answer: 29/4
Problem 7: Convert 6⅚ to an improper fraction
Step 1: Multiply 6 × 6 = 36
Step 2: Add the numerator: 36 + 5 = 41
Step 3: Keep denominator: 6
Answer: 41/6
Problem 8: Convert 9⅞ to an improper fraction
Step 1: Multiply 9 × 8 = 72
Step 2: Add the numerator: 72 + 7 = 79
Step 3: Keep denominator: 8
Answer: 79/8
Problem 9: Convert 23/7 to a mixed number
Step 1: 23 ÷ 7 = 3 (remainder)
7 × 3 = 21
Step 2: Remainder: 23 - 21 = 2
Step 3: Mixed number: 3 + 2/7
Answer: 3²⁄₇
Problem 10: Convert 12⅜ to an improper fraction
Step 1: Multiply 12 × 8 = 96
Step 2: Add: 96 + 3 = 99
Step 3: Keep denominator: 8
Answer: 99/8
Problem 11: Convert 47/6 to a mixed number
Step 1: 47 ÷ 6 = 7 (remainder)
6 × 7 = 42
Step 2: Remainder: 47 - 42 = 5
Step 3: Mixed number: 7 + 5/6
Answer: 7⅚
Problem 12: Convert 15¾ to an improper fraction
Step 1: Multiply 15 × 4 = 60
Step 2: Add: 60 + 3 = 63
Step 3: Keep denominator: 4
Answer: 63/4
Wrong: 3¼ = (3 + 4 + 1)/4 ❌
Right: 3¼ = (3 × 4 + 1)/4 = 13/4 ✅
Remember: MULTIPLY the whole by the denominator!
Wrong: 2⅗ = 13/10 ❌
Right: 2⅗ = 13/5 ✅
The denominator NEVER changes!
Wrong: 17/5 = 3 ❌
Right: 17/5 = 3⅖ ✅
Don't forget to include the leftover fraction!
Question 1: Is 5/3 an improper fraction?
YES! The numerator (5) is greater than the denominator (3).
Question 2: Does 4⅗ have a whole number part?
YES! The whole number is 4. The fraction part is ⅗.
Question 3: Do 9/4 and 2¼ represent the same amount?
YES! They are the same amount, just written in different forms.
9/4 = 2¼ (9 ÷ 4 = 2 remainder 1)
✓ Multiplying fractions
✓ Dividing fractions
✓ Making calculations easier
✓ Comparing fractions with different denominators
✓ Measuring in real life
✓ Describing amounts (like pizzas)
✓ Final answers to word problems
✓ Making the amount easier to understand
Be able to use BOTH forms! Sometimes you'll need to convert back and forth in the same problem!
A recipe calls for 2¾ cups of flour. You want to write this as an improper fraction to make it easier to triple the recipe later. What improper fraction equals 2¾?
Step 1: 2 × 4 = 8
Step 2: 8 + 3 = 11
Step 3: Keep denominator = 4
Answer: 11/4 cups of flour
Now it's easy to triple: 11/4 × 3 = 33/4 cups!
Sarah ran 13/4 miles. Her coach wants to know how many full miles she ran and what fraction of a mile was left over. Convert to a mixed number.
Step 1: 13 ÷ 4 = 3 (remainder)
4 × 3 = 12
Step 2: 13 - 12 = 1
Step 3: 3 + 1/4
Answer: Sarah ran 3¼ miles
She ran 3 complete miles plus ¼ of another mile!
You need boards that are each 3⅝ feet long. You want to calculate the total length if you need 4 boards. First, convert 3⅝ to an improper fraction.
Step 1: 3 × 8 = 24
Step 2: 24 + 5 = 29
Step 3: Keep denominator = 8
Answer: 29/8 feet per board
Now you can easily multiply: 29/8 × 4 = 116/8 = 14½ feet total!
Improper Fractions:
• Numerator ≥ denominator
• Represent 1 or more wholes
• Examples: 5/4, 7/3, 11/5
Mixed Numbers:
• Whole number + proper fraction
• Easy to understand in real life
• Examples: 1¼, 2⅓, 3⅝
Same Amount, Different Forms!
7/3 = 2⅓ (they're equal!)
Use Division!
1. Divide numerator ÷ denominator
2. Quotient = whole number
3. Remainder = new numerator
4. Same denominator
Example:
13/4 = 3¼
Multiply & Add!
1. Multiply whole × denominator
2. Add the numerator
3. Put over same denominator
Example:
3¼ = 13/4
Problem 13: Convert 35/6 to a mixed number
35 ÷ 6 = 5 R5
Answer: 5⅚
Problem 14: Convert 8⅗ to an improper fraction
(8 × 5) + 3 = 40 + 3 = 43
Answer: 43/5
Problem 15: Convert 41/7 to a mixed number
41 ÷ 7 = 5 R6
Answer: 5⁶⁄₇
Problem 16: Convert 10⅞ to an improper fraction
(10 × 8) + 7 = 80 + 7 = 87
Answer: 87/8
Challenge: If you have 3⅔ pizzas and eat ⅔ of a pizza, how many pizzas do you have left? (Hint: Convert to improper fractions first!)
Step 1: Convert 3⅔ to improper: (3×3)+2 = 11/3
Step 2: Subtract: 11/3 - 2/3 = 9/3
Step 3: Simplify: 9/3 = 3/1 = 3
Answer: 3 whole pizzas left!
Today You Mastered:
✓ What improper fractions are
✓ What mixed numbers are
✓ How to convert improper fractions to mixed numbers
✓ How to convert mixed numbers to improper fractions
✓ When to use each form
✓ Real-world applications
Coming Up Next:
Adding & subtracting mixed numbers
Multiplying & dividing mixed numbers
Solving real-world problems with mixed numbers
You're doing incredible work! Keep it up! 🌟🎯🚀