6.2 Fractions, Decimals & Percentages

Fractions, Decimals, and Percentages: Conceptual Models and Tripartite Equivalence

Fractions, decimals and percents run through both Stanford 10 mathematics subtests. In Mathematics Problem Solving they are part of the Number Sense & Operations cluster; in Mathematics Procedures, a real Intermediate 2 report shows Computation with Decimals (12 items) and Computation with Fractions (10 items) as two of the three content clusters. Students need to move easily among representations: visual models, fractions, decimals and percents in real-world problems.

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Conceptual Models of Fractions

Understanding fractions begins with three distinct conceptual models that appear frequently in pictorial item stimuli:

1. Area / Region Model: A continuous geometric shape (circle, rectangle, grid) is partitioned into equal-sized regions. The denominator defines the total number of equal sub-regions, while the numerator indicates the number of shaded or selected sub-regions.
2. Set Model: A collection of discrete objects (e.g., counters, stars, geometric tokens) represents the whole. The denominator represents the total count of items in the set, and the numerator represents the count of a specific subset possessing a designated characteristic.
3. Number Line / Length Model: A linear distance between whole numbers is subdivided into equal segments. A fraction represents a specific coordinate point measured from the origin (0). This model becomes increasingly important in upper elementary and middle school because it shows that fractions are numbers with exact positions on the number line, not just two stacked whole numbers.

$\text{Number Line Partition (Fourths):} \quad 0 \longleftrightarrow \frac{1}{4} \longleftrightarrow \frac{2}{4} \; \left(\frac{1}{2}\right) \longleftrightarrow \frac{3}{4} \longleftrightarrow 1$

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Fraction Classifications and Structural Conversions

• Proper Fraction: A fraction where the numerator is strictly less than the denominator ($a < b$), representing a value between 0 and 1 (e.g., $\frac{3}{7}$).
• Improper Fraction: A fraction where the numerator is greater than or equal to the denominator ($a \ge b$), representing a value greater than or equal to 1 (e.g., $\frac{29}{6}$).
• Mixed Number: A combined representation featuring a whole number and a proper fraction ($W \frac{a}{b}$), representing values greater than 1 (e.g., $4 \frac{5}{6}$).

Conversion Algorithms

• Improper to Mixed: $\frac{29}{6} \implies 29 \div 6 = 4 \text{ R } 5 \implies 4\frac{5}{6}$
• Mixed to Improper: $4\frac{5}{6} \implies \frac{(4 \times 6) + 5}{6} = \frac{24 + 5}{6} = \frac{29}{6}$

Simplifying Fractions via the Greatest Common Factor (GCF)

A fraction is in simplest form (lowest terms) when its numerator and denominator share no common factors other than 1. Dividing both terms by their Greatest Common Factor (GCF) produces an equivalent fraction in simplest form:

$\frac{48}{60} \implies \text{GCF}(48, 60) = 12 \implies \frac{48 \div 12}{60 \div 12} = \frac{4}{5}$

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Comparing and Ordering Fractions

When ordering a set of fractions, calculating full common denominators is often inefficient. Students should deploy three targeted comparative strategies:

1. Benchmark Fractions ($0, \frac{1}{2}, 1$): Classify each fraction relative to $\frac{1}{2}$. For example, when comparing $\frac{4}{9}$ and $\frac{7}{12}$, recognize that $\frac{4}{9} < \frac{1}{2}$ (since half of 9 is 4.5) while $\frac{7}{12} > \frac{1}{2}$ (since half of 12 is 6). Therefore, $\frac{4}{9} < \frac{7}{12}$ immediately without finding a common denominator.
2. Cross-Multiplication Method: For two fractions $\frac{a}{b}$ and $\frac{c}{d}$, compare the cross-products $a \times d$ and $b \times c$: $\frac{5}{8} \text{ vs } \frac{7}{11} \implies (5 \times 11) \text{ vs } (8 \times 7) \implies 55 < 56 \implies \frac{5}{8} < \frac{7}{11}$
3. Least Common Denominator (LCD): When comparing three or more fractions, find the Least Common Multiple (LCM) of the denominators, convert each fraction to an equivalent form with that common denominator, and compare numerators.

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Decimal Place Value and Sub-Unit Systems

Decimal notation extends base-ten principles to values less than one. The decimal point separates whole number periods from fractional sub-units:

Test Tip: When comparing decimals of varying lengths, append placeholder trailing zeros to equalize digits after the decimal point. For example, comparing $0.4$ and $0.089$: rewrite $0.4$ as $0.400$. Since $400 \text{ thousandths} > 89 \text{ thousandths}$, $0.4 > 0.089$.

Benchmark Conversions Reference Table

Memorization of standard benchmark equivalencies eliminates time-consuming long division during testing:

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Real-World Percentage Applications

Word problems in the Mathematics Problem Solving subtest frequently assess percentage calculations in commercial environments, including discounts, sales tax, and percentage changes.

Formula Framework

• Part of a Whole: $\text{Part} = \text{Percent (in decimal form)} \times \text{Whole}$
• Discount and Sale Price: $\text{Discount} = \text{Original Price} \times \text{Discount Rate}$; $\text{Sale Price} = \text{Original Price} - \text{Discount}$
• Sales Tax and Total Cost: $\text{Tax} = \text{Sale Price} \times \text{Tax Rate}$; $\text{Total Cost} = \text{Sale Price} + \text{Tax}$
• Percent Increase / Decrease: $\text{Percent Change} = \frac{|\text{New Amount} - \text{Original Amount}|}{\text{Original Amount}} \times 100\%$

Worked Multi-Step Scenario: A winter coat with an original retail price of $80.00 is placed on clearance at a $25\%$ discount. The local sales tax rate is $7\%$. Determine the final customer cost.

1. Calculate the discount: $0.25 \times 80 = 20$, giving a $20.00 discount.
2. Determine discounted subtotal: $80.00 - $20.00 = $60.00.
3. Calculate sales tax on discounted subtotal: $0.07 \times 60 = 4.20$, giving $4.20 in tax.
4. Sum subtotal and tax: $60.00 + $4.20 = $64.20.

Common Trap: Students often calculate tax on the original $80.00 price ($0.07 \times 80 = 5.60$, yielding $5.60 in tax), yielding an incorrect distractor of $65.60.