Fractions · Lesson 3 of 9
Fractions Beyond One and Mixed Numbers
“Group fractional units into wholes and move between fractions and mixed numbers.”
• Locate fractional lengths greater than one on a number line. • Compare the numerator and denominator to identify amounts below, equal to, or above one. • Group fractional units into complete wholes and a leftover part. • Convert fractions to mixed numbers and mixed numbers to fractions. • Recognise fractions that represent whole numbers exactly.
Keep counting after one whole
A fractional unit remains the same size even when we count enough copies to pass one whole. Three half-unit lengths make one whole and another half. Six fifth-unit lengths make one whole and another fifth. On a number line, these lengths continue past 1; we must keep the spacing between consecutive fractional units unchanged.
A denominator of 5 means that five fifths make one whole. If the numerator is smaller than 5, we have less than a whole; if it is 5, we have exactly one whole; if it is greater, we have more than one. The same reasoning works for any positive denominator. It is counting the pieces needed to make a whole, not a comparison of unrelated numbers.
| Relationship | Meaning | Example |
|---|---|---|
| Numerator < denominator | Less than one whole | 3/5 < 1 |
| Numerator = denominator | Exactly one whole | 5/5 = 1 |
| Numerator > denominator | More than one whole | 7/5 > 1 |
Problem
Classify 2/3, 5/5, 3/2, 4/3, and 7/5 relative to 1.
- 1.For each fraction, compare the count of pieces with the number needed for a whole.
- 2.Two thirds is less than one because two is less than three. Five fifths equals one.
- 3.Three halves, four thirds, and seven fifths exceed one because their numerators exceed their denominators.
Separate complete wholes from leftover pieces
To understand 7/2, picture seven half-pieces. Every two half-pieces make a whole, so three pairs form three wholes and one half-piece remains. We can write the amount as 3 + 1/2 or more compactly as 3 1/2. This second form makes the whole part and the leftover fractional part easy to see.
A number written with a whole-number part and a fractional part less than one, such as 2 2/3. The parts are added: 2 2/3 means 2 + 2/3.
Problem
Write 8/3 as a mixed number.
- 1.Each whole needs three thirds.
- 2.Group eight thirds as three thirds + three thirds + two thirds.
- 3.The first two groups each make one whole; two thirds remain.
- 4.Therefore 8/3 = 2 + 2/3 = 2 2/3.
Problem
Write 47/9 as a mixed number.
- 1.Each whole requires nine ninths. Five wholes use 5 × 9 = 45 ninths.
- 2.Subtract the used pieces from the total count: 47 − 45 = 2 ninths remain.
- 3.Thus 47/9 = 5 2/9. The leftover numerator 2 is smaller than the denominator 9.
This grouping is exactly whole-number division with a remainder. Divide the numerator by the denominator: the quotient counts the complete wholes and the remainder counts leftover pieces. The denominator does not change because the leftover pieces still have their original fractional size. If the remainder is zero, the fraction is a whole number and no fractional part is needed.
Problem
Does 8/4 need to be written as a mixed number?
- 1.Every four quarter-pieces make one whole.
- 2.Eight quarters form two complete groups of four, leaving no quarter-pieces.
- 3.So 8/4 = 2. Write the whole number 2; a nonzero fractional part would change the amount.
3 1/4 means 3 + 1/4. It does not mean 3 × 1/4. Its fractional part must be less than one, so 2 5/3 has not yet collected all of its complete wholes.
Turn the wholes back into fractional units
The conversion also works in reverse. If you have three wholes and three quarters, each whole can be replaced by four quarters. The three wholes supply twelve quarters, and the fractional part supplies another three. The resulting single fraction counts all fifteen quarters. We change the way we describe the amount, not the amount itself.
Problem
Write 3 3/4 as one fraction.
- 1.Read 3 3/4 as 3 + 3/4.
- 2.Replace each of the three wholes by four quarters: 3 × 4 = 12 quarter-pieces.
- 3.Add the three quarter-pieces already in the fractional part: 12 + 3 = 15.
- 4.Therefore 3 3/4 = 15/4. The denominator stays 4 because every piece is a quarter.
In general, multiply the whole part by the denominator to count the pieces inside the wholes. Then add the fractional numerator and keep the denominator. The letters in the relationship below simply stand for the numbers in a mixed fraction. The numerator of the fractional part is smaller than its denominator.
Problem
Write 7 2/3 as a fraction, then check by grouping.
- 1.Seven wholes contain 7 × 3 = 21 thirds.
- 2.Include the extra two thirds: 21 + 2 = 23 thirds. Thus 7 2/3 = 23/3.
- 3.Grouping 23 thirds into groups of three gives seven whole groups with two thirds left, returning to 7 2/3.
Quiz
Which fraction is greater than 1?
How many complete whole units are in 7/2?
Which mixed number equals 11/5?
Which fraction equals 3 1/4?
What is the simplest way to write 12/3?
Why does 9/4 become 2 1/4?
Practice Problems
- Use a number line to show 3/2 and the lengths 6/5, 7/5, 8/5, and 9/5.
- Find the number of complete wholes in 7/2, 4/3, 7/3, 8/3, 11/5, and 9/4. Explain one answer with grouped pieces.
- Write 9/2, 9/5, 21/19, 47/9, 12/11, and 19/6 as mixed numbers.
- Write 3 1/4, 7 2/3, 9 4/9, 3 1/6, 2 3/11, and 3 9/10 as fractions.
- Find a fraction greater than one that equals a whole number. Explain why it has no leftover fractional part.
- A student writes 11/4 as 1 7/4. Explain why the fractional part still contains a whole and rewrite the quantity correctly.
- Draw nine quarter-rotis and label the same quantity as a fraction and a mixed number.
Key Takeaways
• Counting fractional units can produce lengths and amounts greater than one. • A numerator below, equal to, or above the denominator gives a fraction below, equal to, or above one. • Group as many pieces as the denominator into each whole. • Whole-number division gives the whole part and the leftover numerator. • To convert a mixed number, count the pieces in its wholes and add the remaining pieces.