Expansion method: A technique for finding equivalent fractions by multiplying both the numerator and denominator by the same non-zero number to create larger denominators.
Equivalent fractions: Fractions that have the same value but different numerators and denominators.
Numerator: The top number in a fraction representing the number of parts taken.
Denominator: The bottom number in a fraction representing the total number of equal parts.
Scaling up: Making fractions larger while keeping the same value by multiplying both parts.
When you multiply both the numerator and denominator of a fraction by the same non-zero number, the fraction's value remains unchanged. This is because you're essentially multiplying by 1 in the form of \(\frac{n}{n}\).
Take your starting fraction, for example \(\frac{3}{4}\).
Select any number greater than 0, such as 2, 3, 4, etc.
\(\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}\)
Cross multiply: \(3 \times 8 = 24\) and \(4 \times 6 = 24\). Since both equal 24, the fractions are equivalent.
- Identify your starting fraction (e.g., \(\frac{2}{5}\))
- Determine the target denominator (e.g., convert to denominator 15)
- Calculate the multiplication factor (15 ÷ 5 = 3)
- Multiply the numerator by the factor: \(2 \times 3 = 6\)
- Multiply the denominator by the factor: \(5 \times 3 = 15\)
- Write the new fraction: \(\frac{6}{15}\)
- Verify equivalence using cross multiplication
• Only multiplying the numerator while leaving the denominator unchanged
• Using different numbers for numerator and denominator operations
• Forgetting to verify that the expanded fraction is equivalent to the original
• Confusing multiplication with addition or subtraction for creating equivalents
Expanding fractions: Creating larger equivalent fractions for comparison
Common denominators: Finding equivalent fractions with the same denominator
Standard form: Expressing fractions in required forms
Real-world applications: Scaling recipes, measurements, and ratios
Convert \(\frac{1}{3}\) to a fraction with denominator 12
\(\frac{1}{3}\)
Factor = Target denominator ÷ Original denominator = 12 ÷ 3 = 4
\(1 \times 4 = 4\)
\(3 \times 4 = 12\)
\(\frac{1}{3} = \frac{4}{12}\)
\(1 \times 12 = 12\) and \(3 \times 4 = 12\). Since both equal 12, the fractions are equivalent.
Convert \(\frac{2}{7}\) to a fraction with denominator 21
\(\frac{2}{7}\)
Factor = 21 ÷ 7 = 3
\(2 \times 3 = 6\)
\(7 \times 3 = 21\)
\(\frac{2}{7} = \frac{6}{21}\)
Cross multiply: \(2 \times 21 = 42\) and \(7 \times 6 = 42\) ✓
Convert \(\frac{5}{8}\) to a fraction with denominator 40
\(\frac{5}{8}\)
Factor = 40 ÷ 8 = 5
\(5 \times 5 = 25\)
\(8 \times 5 = 40\)
\(\frac{5}{8} = \frac{25}{40}\)
Cross multiply: \(5 \times 40 = 200\) and \(8 \times 25 = 200\) ✓
• Expansion Method: \(\frac{a}{b} = \frac{a \times n}{b \times n}\) (where n is any non-zero number)
• Factor determination: Factor = Target denominator ÷ Original denominator
• Verification: Use cross multiplication: \(\frac{a}{b} = \frac{c}{d}\) if \(a \times d = b \times c\)
\(\frac{3}{4}\)
Factor = 20 ÷ 4 = 5
\(\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}\)
\(3 \times 20 = 60\) and \(4 \times 15 = 60\) ✓
\(\frac{2}{9}\)
Factor = 36 ÷ 9 = 4
\(\frac{2}{9} = \frac{2 \times 4}{9 \times 4} = \frac{8}{36}\)
Cross multiply: \(2 \times 36 = 72\) and \(9 \times 8 = 72\) ✓
\(\frac{7}{12}\)
Factor = 48 ÷ 12 = 4
\(\frac{7}{12} = \frac{7 \times 4}{12 \times 4} = \frac{28}{48}\)
Cross multiply: \(7 \times 48 = 336\) and \(12 \times 28 = 336\) ✓
Circle models: Divide circles into equal parts and show how expansion increases parts
Rectangle models: Divide rectangles into equal sections to demonstrate expansion
Number line models: Show equivalent positions on the same number line
Set models: Use groups of objects to represent fractional amounts
Circle 1: Cut into 2 equal pieces, shade 1 piece → \(\frac{1}{2}\)
Circle 2: Cut into 4 equal pieces, shade 2 pieces → \(\frac{2}{4}\)
Both circles have the same amount shaded!
Rectangle 1: Divide into 3 equal parts, shade 2 → \(\frac{2}{3}\)
Rectangle 2: Divide into 12 equal parts, shade 8 → \(\frac{8}{12}\)
Both rectangles have the same proportion shaded!
Even though the number of parts increases, the proportion of the whole that is shaded remains the same.
Mark \(\frac{1}{2}\) and \(\frac{4}{8}\) on the same number line
They occupy the exact same position between 0 and 1!
Shows exact positioning and relative value of fractions
Helps with comparing and ordering fractions
Visual models help students understand that the expansion method doesn't change the actual amount represented by the fraction, only how it's counted. The expansion creates more, smaller parts that represent the same proportional amount.
• Builds conceptual understanding beyond procedural knowledge
• Shows that expansion creates more, smaller equivalent parts
• Connects abstract concepts to concrete representations
• Supports diverse learning styles and needs
\(\frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{8}{16}\)
\(\frac{2}{3} = \frac{4}{6} = \frac{8}{12} = \frac{16}{24}\)
\(\frac{3}{4} = \frac{6}{8} = \frac{12}{16} = \frac{24}{32}\)
Analysis: The chart shows how expansion creates larger denominators while preserving the fraction's value.
- \(\frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{8}{16}\) (All represent 0.5)
- \(\frac{2}{3} = \frac{4}{6} = \frac{8}{12} = \frac{16}{24}\) (All represent 0.667)
- \(\frac{3}{4} = \frac{6}{8} = \frac{12}{16} = \frac{24}{32}\) (All represent 0.75)