7.1 Fractions
Key Takeaways
- A fraction represents a part of a whole, with the numerator counting parts and the denominator telling how many equal parts make one whole; unit fractions (1/b) are the building blocks of all fractions.
- Equivalent fractions are produced by multiplying or dividing both numerator and denominator by the same non-zero number; the cross-products test (a/b = c/d iff a·d = b·c) is the definitive equality check.
- Comparing fractions is efficient with benchmarks 0, 1/2, and 1, or with common denominators; adding and subtracting require common denominators, while multiplication uses 'part of a part' area models and division answers 'how many groups of the divisor fit in the dividend.'
- Mixed numbers combine whole numbers and proper fractions; improper fractions (numerator ≥ denominator) convert to mixed numbers by division with remainder and back by multiplication plus addition.
- Real-world fraction problems in Georgia P-5 classrooms include sharing pizzas, dividing recipe quantities, measuring lengths, and partitioning rectangular area models for multiplication.
Fractions as Numbers
A fraction names a part of a whole. In elementary grades, students first meet fractions by partitioning a rectangle or a circle into equal parts. The denominator (bottom number) tells how many equal parts make one whole; the numerator (top number) tells how many of those parts are being counted. So 3/4 means "3 of 4 equal parts."
A unit fraction has a numerator of 1 (1/2, 1/3, 1/4, ...). Every fraction is a sum of unit fractions: 3/4 = 1/4 + 1/4 + 1/4. This idea is the bridge from whole-number addition to fraction addition.
Fractions on a Number Line
Fractions are numbers, so they live on the number line. To plot 3/4, partition the segment from 0 to 1 into 4 equal parts and count 3 parts from 0. The endpoint of that count is the location of 3/4. A fraction greater than 1 (like 5/4) lands past 1; a fraction equal to 1 (4/4) lands exactly on 1.
Equivalent Fractions
Two fractions are equivalent if they name the same quantity. The key principle: multiplying or dividing both numerator and denominator by the same non-zero number does not change the fraction's value.
Worked Example: Generate fractions equivalent to 2/3.
Multiply top and bottom by 2, 3, and 4:
| Multiplier | Numerator | Denominator | Equivalent Fraction |
|---|---|---|---|
| ×2 | 4 | 6 | 4/6 |
| ×3 | 6 | 9 | 6/9 |
| ×4 | 8 | 12 | 8/12 |
All of 2/3, 4/6, 6/9, 8/12 name the same point on the number line. To simplify, divide top and bottom by a common factor: 8/12 ÷ 4/4 = 2/3.
The cross-products test is the definitive equality check: a/b = c/d if and only if a·d = b·c. For 2/3 vs 4/6: 2·6 = 12 and 3·4 = 12, so the fractions are equal.
Comparing Fractions
Students compare fractions using three strategies: same numerator, same denominator, or benchmarks.
| Pair | Strategy | Result |
|---|---|---|
| 2/5 vs 2/7 | Same numerator: smaller denominator means bigger pieces | 2/5 > 2/7 |
| 3/8 vs 5/8 | Same denominator: bigger numerator wins | 3/8 < 5/8 |
| 3/8 vs 5/9 | Compare to 1/2: 3/8 < 1/2, 5/9 > 1/2 | 3/8 < 5/9 |
| 3/4 vs 5/6 | Common denominator 12: 9/12 vs 10/12 | 3/4 < 5/6 |
Benchmark fractions 0, 1/2, and 1 give quick estimates. A fraction with numerator close to denominator is close to 1; a fraction with numerator about half the denominator is near 1/2.
Adding and Subtracting Fractions
Like denominators: add or subtract numerators; keep the denominator.
3/8 + 2/8 = 5/8. The denominator names the size of the pieces; it does not change when pieces combine.
Unlike denominators: build a common denominator using equivalent fractions. For 1/4 + 2/3, use 12 as the common denominator: 1/4 = 3/12 and 2/3 = 8/12, so the sum is 11/12.
Multiplying Fractions
Multiplication answers "part of a part." The area model makes this visible: shade 2/3 of a rectangle, then shade 1/2 of that shaded region. The doubly-shaded region is 1/3 of the whole, so 2/3 × 1/2 = 1/3.
Algorithm: multiply numerators, multiply denominators. 2/3 × 1/2 = (2·1)/(3·2) = 2/6 = 1/3 after simplifying.
A whole number times a fraction is repeated addition: 3 × 1/4 = 1/4 + 1/4 + 1/4 = 3/4. A fraction times a whole number shrinks the whole: 1/3 × 6 = 2.
Dividing Fractions
Division answers "how many groups of the divisor fit in the dividend?" For 3/4 ÷ 1/8, ask: how many 1/8-pieces fit in 3/4? Since 3/4 = 6/8, the answer is 6.
The algorithm: keep the first fraction, change division to multiplication, flip the divisor (reciprocal). 3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6.
Common-error alert: young students often invert the wrong fraction. Reinforce that only the divisor (the second fraction) is flipped.
Mixed Numbers and Improper Fractions
A mixed number combines a whole number and a proper fraction, like 2 1/3. An improper fraction has a numerator greater than or equal to its denominator, like 7/3. They name the same quantity.
- To convert 7/3 to a mixed number: divide 7 by 3 → quotient 2 remainder 1, so 2 1/3.
- To convert 2 1/3 to an improper fraction: 2·3 + 1 = 7, over 3, so 7/3.
Real-World Fraction Problems
Georgia elementary classrooms use fractions in authentic contexts:
- Sharing pizzas: 3 pizzas, 4 friends → each gets 3/4 pizza.
- Adjusting recipes: 1/2 cup of sugar doubled is 1 cup; halved is 1/4 cup.
- Measuring lengths: a board 5 1/2 ft long cut into 4 equal pieces gives 11/8 ft per piece (or 1 3/8 ft).
- Partitioning gardens: a 6 m × 3 m garden with 1/3 planted in carrots gives 6 m² of carrots.
When students draw an area model or number line, the operation becomes visible; when they only memorize the algorithm, the meaning is lost.
Worked Example: Adding with Unlike Denominators
Suppose a class plants 1/4 of a garden in tomatoes and 2/3 in peppers. What fraction of the garden is planted in vegetables?
Step 1 — Find a common denominator. The least common multiple of 4 and 3 is 12. Step 2 — Build equivalent fractions. 1/4 = 3/12 and 2/3 = 8/12. Step 3 — Add numerators, keep the denominator. 3/12 + 8/12 = 11/12.
So 11/12 of the garden is planted in vegetables, leaving 1/12 for paths or other crops. This three-step routine (common denominator, equivalent fractions, add numerators) works for both addition and subtraction of fractions with unlike denominators.
Which expression is equivalent to 4/6?
A student compares 3/5 and 5/9 using benchmarks and common denominators. Which statement is correct?
A recipe calls for 3/4 cup of flour. The cook wants to make 1/2 of the recipe. How much flour is needed?