Equivalent fractions: Fractions that represent the same part of a whole, e.g., \(\frac{a}{b} = \frac{na}{nb}\) where n is any non-zero number
- Draw visual representations of both fractions
- Check if they represent the same portion
- Use simplification to verify equivalence
Draw a circle divided in half and shade one half. Draw another circle divided into fourths and shade two fourths.
Both shaded areas represent the same portion of the circle.
\(\frac{2}{4} = \frac{2÷2}{4÷2} = \frac{1}{2}\)
\(\frac{1}{2} = \frac{2}{4}\)
• Visual modeling: Draw pictures to understand fraction relationships
• Equivalence: Multiplying or dividing both numerator and denominator by the same number creates an equivalent fraction
• Verification: Always check with simplification
Common denominator: When fractions have the same denominator, the one with the larger numerator is greater
Denominators are 3 and 4. LCM(3,4) = 12
\(\frac{2}{3} = \frac{2×4}{3×4} = \frac{8}{12}\)
\(\frac{3}{4} = \frac{3×3}{4×3} = \frac{9}{12}\)
Since 8 < 9, we have \(\frac{8}{12} < \frac{9}{12}\)
\(\frac{2}{3} < \frac{3}{4}\)
• Common denominator rule: When denominators are equal, compare numerators
• LCM method: Find smallest common multiple of denominators
• Conversion: Multiply both numerator and denominator by same factor
Cross multiplication: For \(\frac{a}{b}\) and \(\frac{c}{d}\), compare \(a×d\) with \(b×c\)
For \(\frac{a}{b}\) vs \(\frac{c}{d}\): multiply \(a×d\) and \(b×c\)
\(\frac{3}{5}\) vs \(\frac{2}{3}\): \(3×3\) vs \(5×2\)
\(3×3 = 9\) and \(5×2 = 10\)
Since \(9 < 10\), we have \(\frac{3}{5} < \frac{2}{3}\)
LCD of 5 and 3 is 15: \(\frac{9}{15}\) vs \(\frac{10}{15}\) → \(\frac{9}{15} < \frac{10}{15}\) ✓
\(\frac{3}{5} < \frac{2}{3}\)
• Cross multiplication: If \(ad < bc\), then \(\frac{a}{b} < \frac{c}{d}\)
• Efficiency: Cross multiplication is faster than finding LCD for large denominators
• Verification: Always verify results using another method
Fraction: A number representing part of a whole, expressed as \(\frac{\text{numerator}}{\text{denominator}}\)
Numerator: The top number indicating how many parts are taken
Denominator: The bottom number indicating total equal parts in the whole
Equivalent fractions: Different fractions that represent the same value
Proper fraction: A fraction where the numerator is smaller than the denominator
Improper fraction: A fraction where the numerator is greater than or equal to the denominator
Like fractions: Fractions with the same denominator
Unlike fractions: Fractions with different denominators
- Introduction: Review basic fraction concepts and vocabulary
- Visual demonstration: Use visual models to show fraction relationships
- Introduce methods: Present three comparison methods with examples
- Guided practice: Work through examples together
- Independent practice: Students solve problems independently
- Assessment: Evaluate understanding through exercises
• If denominators are equal: \(\frac{a}{c} > \frac{b}{c}\) when \(a > b\)
• Cross multiplication: \(\frac{a}{b} > \frac{c}{d}\) when \(ad > bc\)
• If numerators are equal: \(\frac{a}{b} > \frac{a}{c}\) when \(b < c\)
• Equivalent fractions: \(\frac{a}{b} = \frac{ka}{kb}\) for any non-zero k
Mixed number: A number combining a whole number and a proper fraction, e.g., \(a\frac{b}{c} = \frac{ac+b}{c}\)
Both mixed numbers have the same whole number part (1), so we compare the fractional parts.
\(\frac{1}{3}\) vs \(\frac{1}{4}\)
When numerators are equal, the fraction with the smaller denominator is larger.
Since 3 < 4, we have \(\frac{1}{3} > \frac{1}{4}\)
Since the whole numbers are equal and \(\frac{1}{3} > \frac{1}{4}\), we have \(1\frac{1}{3} > 1\frac{1}{4}\)
\(1\frac{1}{3} > 1\frac{1}{4}\)
• Same whole numbers: Compare fractional parts when whole numbers are equal
• Equal numerators: The fraction with the smaller denominator is larger
• Verification: Convert to improper fractions to confirm: \(\frac{4}{3} = 1.33...\) vs \(\frac{5}{4} = 1.25\)
Complex comparison: Comparing fractions where neither numerator nor denominator is the same
Since denominators and numerators are different, use cross multiplication.
\(5 × 5 = 25\) and \(8 × 3 = 24\)
Since \(25 > 24\), we have \(\frac{5}{8} > \frac{3}{5}\)
LCD of 8 and 5 is 40: \(\frac{25}{40}\) vs \(\frac{24}{40}\) → \(\frac{25}{40} > \frac{24}{40}\) ✓
\(\frac{5}{8} > \frac{3}{5}\)
• Complex comparisons: Use cross multiplication when neither numerator nor denominator matches
• Verification: Always check with an alternative method
• Multiple methods: Cross multiplication and common denominator should yield the same result
Proper fraction: Numerator is smaller than denominator (\(\frac{a}{b}\) where \(a < b\))
Improper fraction: Numerator is greater than or equal to denominator (\(\frac{a}{b}\) where \(a ≥ b\))
Mixed number: Combination of whole number and proper fraction (\(a\frac{b}{c}\))
Like fractions: Fractions with the same denominator
Unlike fractions: Fractions with different denominators
Equivalent fractions: Fractions that represent the same value despite having different numerators and denominators
- Identify fraction types: Determine if proper, improper, or mixed numbers
- Choose appropriate method: Common denominator, cross multiplication, or decimal conversion
- Apply the method: Follow specific steps for the chosen technique
- Verify the result: Use an alternative method to confirm accuracy
- Express final answer: Write comparison using appropriate inequality symbol
• Common denominator: \(\frac{a}{b} = \frac{an}{bn}\) for any non-zero n
• Cross multiplication: \(\frac{a}{b} > \frac{c}{d}\) if and only if \(ad > bc\)
• Mixed to improper: \(a\frac{b}{c} = \frac{ac+b}{c}\)
• Same numerator: \(\frac{a}{b} > \frac{a}{c}\) if \(b < c\)
• Same denominator: \(\frac{a}{c} > \frac{b}{c}\) if \(a > b\)
\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}\)
Analysis: As the numerator and denominator increase proportionally while maintaining the relationship where numerator is one less than denominator, the value approaches 1.
- \(\frac{1}{2} = 0.5\) (50% of whole)
- \(\frac{2}{3} ≈ 0.667\) (66.7% of whole)
- \(\frac{3}{4} = 0.75\) (75% of whole)
- \(\frac{4}{5} = 0.8\) (80% of whole)
- \(\frac{5}{6} ≈ 0.833\) (83.3% of whole)