Understanding Fractions
Fractions represent parts of a whole. As a paraprofessional, you’ll encounter fractions when helping students understand division concepts, analyzing test scores, calculating grades, and explaining mathematical concepts.
What is a Fraction?
A fraction consists of two parts:
- Numerator: The number above the line, representing the parts being considered
- Denominator: The number below the line, representing the total number of equal parts in the whole
For example, in the fraction 3/4:
- 3 is the numerator (parts being considered)
- 4 is the denominator (total equal parts)
- This represents 3 out of 4 equal parts
Types of Fractions
Proper Fractions
The numerator is less than the denominator.
Examples: 1/2, 3/5, 7/10
These fractions represent values less than 1.
Improper Fractions
The numerator is greater than or equal to the denominator.
Examples: 5/3, 7/4, 11/5
These fractions represent values greater than or equal to 1.
Mixed Numbers
A whole number and a proper fraction combined.
Examples: 2 1/3, 4 3/4, 1 5/8
These represent values greater than 1.
Converting Between Forms
Converting Improper Fractions to Mixed Numbers
- Divide the numerator by the denominator
- The quotient becomes the whole number
- The remainder becomes the new numerator
- The denominator stays the same
Example 1:
Convert 17/5 to a mixed number
Step 1: Divide 17 by 5. 17 ÷ 5 = 3 with remainder 2
Step 2: The whole number is 3
Step 3: The remainder 2 becomes the new numerator
Step 4: The denominator stays 5
Result: 17/5 = 3 2/5
Converting Mixed Numbers to Improper Fractions
- Multiply the whole number by the denominator
- Add the numerator to this product
- Put this sum over the original denominator
Example 2:
Convert 2 3/4 to an improper fraction
Step 1: Multiply 2 by 4: 2 × 4 = 8
Step 2: Add 8 + 3 = 11
Step 3: Put 11 over the denominator 4
Result: 2 3/4 = 11/4
Equivalent Fractions
Equivalent fractions represent the same value but are written with different numerators and denominators.
Creating Equivalent Fractions
Multiply or divide both the numerator and denominator by the same non-zero number.
Example 3:
Show that 2/3 and 8/12 are equivalent
Multiply 2/3 by 4/4:
2/3 × 4/4 = (2×4)/(3×4) = 8/12
Therefore, 2/3 = 8/12
Simplifying Fractions
A fraction is simplified (or in its lowest terms) when the numerator and denominator have no common factors other than 1.
Steps to Simplify a Fraction:
- Find the greatest common factor (GCF) of the numerator and denominator
- Divide both the numerator and denominator by the GCF
Example 4:
Simplify 24/36
Step 1: Find the GCF of 24 and 36, which is 12
Step 2: Divide both numerator and denominator by 12
(24 ÷ 12)/(36 ÷ 12) = 2/3
Therefore, 24/36 = 2/3 in simplest form
Operations with Fractions
Adding and Subtracting Fractions
With the Same Denominator:
Add or subtract the numerators and keep the denominator the same.
Example 5:
3/8 + 2/8 = (3+2)/8 = 5/8
7/10 – 3/10 = (7-3)/10 = 4/10 = 2/5
With Different Denominators:
- Find the least common multiple (LCM) of the denominators
- Convert each fraction to an equivalent fraction with the LCM as the denominator
- Add or subtract the numerators
- Simplify if necessary
Example 6:
2/3 + 1/4
Step 1: The LCM of 3 and 4 is 12
Step 2: Convert to equivalent fractions
2/3 = (2×4)/(3×4) = 8/12 and 1/4 = (1×3)/(4×3) = 3/12
Step 3: Add the numerators
8/12 + 3/12 = 11/12
Multiplying Fractions
Multiply the numerators together and the denominators together, then simplify if possible.
Example 7:
2/5 × 3/4
Multiply the numerators: 2 × 3 = 6
Multiply the denominators: 5 × 4 = 20
Result: 2/5 × 3/4 = 6/20 = 3/10
Dividing Fractions
To divide fractions, multiply by the reciprocal of the second fraction.
Example 8:
3/4 ÷ 2/5
Step 1: Find the reciprocal of 2/5, which is 5/2
Step 2: Multiply 3/4 by 5/2
3/4 × 5/2 = (3×5)/(4×2) = 15/8 = 1 7/8
Comparing Fractions
Method 1: Convert to a Common Denominator
- Find the LCM of the denominators
- Convert each fraction to an equivalent fraction with this common denominator
- Compare the numerators
Example 9:
Compare 2/3 and 3/5
Step 1: The LCM of 3 and 5 is 15
Step 2: Convert to equivalent fractions
2/3 = (2×5)/(3×5) = 10/15 and 3/5 = (3×3)/(5×3) = 9/15
Step 3: Compare numerators: 10 > 9
Therefore, 2/3 > 3/5
Method 2: Cross Multiplication
- Cross multiply the fractions
- Compare the products
Example 10:
Compare 4/7 and 5/9
Cross multiply: 4 × 9 = 36 and 7 × 5 = 35
Compare: 36 > 35
Therefore, 4/7 > 5/9
Classroom Applications of Fractions
Scenario 1: Grading Papers
A student gets 18 questions correct out of 24 questions on a quiz. What fraction of questions did they answer correctly? What is this as a simplified fraction?
Fraction: 18/24
Simplify by dividing both numbers by their GCF, which is 6:
(18 ÷ 6)/(24 ÷ 6) = 3/4
The student answered 3/4 of the questions correctly.
Scenario 2: Dividing Class Time
A teacher needs to divide a 60-minute class period. If the teacher wants to spend 1/4 of the time on review, 1/2 on new material, and the rest on practice, how many minutes will be spent on practice?
Review: 60 × 1/4 = 15 minutes
New material: 60 × 1/2 = 30 minutes
Practice: 60 – 15 – 30 = 15 minutes
As a fraction of the total time: 15/60 = 1/4
Tips for Teaching Fractions
- Use visual models: Pie charts, rectangular models, and number lines help students visualize fractions.
- Relate to real life: Use examples like pizza slices, sharing candies, or measuring ingredients.
- Begin with unit fractions: Start with fractions that have 1 as the numerator (like 1/2, 1/3, 1/4).
- Emphasize equivalence: Help students understand that fractions can look different but represent the same value.
- Use manipulatives: Fraction tiles, paper folding, and other hands-on materials help build conceptual understanding.
Common Fraction Misconceptions and Errors
- Adding denominators: Students often incorrectly add both numerators and denominators (e.g., 1/2 + 1/3 = 2/5).
- Comparing fraction sizes: Students may think 1/8 is larger than 1/4 because 8 is larger than 4.
- Improper simplification: Dividing numerator and denominator by different numbers.
- Misunderstanding division: Failing to use the reciprocal when dividing fractions.
Key Points to Remember
- A fraction represents parts of a whole where the denominator shows the total number of equal parts.
- To add or subtract fractions, you need a common denominator.
- To multiply fractions, multiply numerators and multiply denominators.
- To divide fractions, multiply by the reciprocal of the divisor.
- Simplify fractions by dividing the numerator and denominator by their greatest common factor.
- Equivalent fractions represent the same value despite having different numerators and denominators.
Interactive Quiz: Fractions
1. What is 3/8 + 2/8 in simplest form?
2. Convert the mixed number 2 1/3 to an improper fraction.
3. Simplify the fraction 18/24.
4. Calculate 2/3 × 3/5.
5. Which of these fractions is largest?
6. A teacher spends 2/5 of class time on lecture and 1/4 on group work. What fraction of class time remains for other activities?
7. Calculate 3/4 ÷ 1/2.
8. If a student gets 14 problems correct out of 20 problems, what fraction of problems did they get correct in simplest form?