2.5 Fraction, Decimal, and Percent Models and Equivalence

Why Equivalence Matters on Praxis 5164

ETS Category I.B expects middle school mathematics teachers to move flexibly among fractions, decimals, and percents, and to represent the same rational number with multiple models. On Praxis 5164, equivalence is not a memorized chart—it is evidence that a candidate understands place value, proportional reasoning, and how to diagnose student misconceptions.

A percent is a special ratio "out of 100." The number 35% means 35/100 = 0.35. Because percent, decimal, and fraction forms describe the same quantity, converting among them is scaling by powers of 10 (or rewriting a ratio), not changing the underlying value.

Converting Among Fractions, Decimals, and Percents

Use these reliable pathways:

Worked example. Convert 3/8 to a decimal and a percent.

• 3 ÷ 8 = 0.375 (terminating decimal).
• 0.375 × 100 = 37.5%.
• Check: 37.5/100 = 375/1000 = 3/8 after dividing numerator and denominator by 125.

Worked example. Convert 0.06 to a fraction and a percent.

• 0.06 = 6/100 = 3/50 in lowest terms.
• 0.06 = 6%.

Worked example. Convert 45% to a fraction and a decimal.

• 45% = 45/100 = 9/20.
• 45% = 0.45.

Representing Repeating Decimals as Fractions

Every repeating decimal is rational and therefore equal to a fraction of integers. Praxis items often ask you to recover that fraction algebraically.

Example: single repeating digit. Let x = 0.3̅ = 0.333…

• Multiply by 10: 10x = 3.333…
• Subtract: 10x − x = 3.333… − 0.333… ⇒ 9x = 3 ⇒ x = 1/3.

Example: two repeating digits. Let x = 0.45̅ = 0.454545…

• Multiply by 100 (two repeating digits): 100x = 45.4545…
• Subtract: 100x − x = 45 ⇒ 99x = 45 ⇒ x = 45/99 = 5/11.

Example in the style of 0.583̅ = 7/12. Express 0.583̅ (bar over the final digit 3 only) as a fraction. Interpret 0.583̅ as 0.58333…

Let x = 0.58333…

• Then 100x = 58.333… and 1000x = 583.333…
• Subtract: 1000x − 100x = 583.333… − 58.333… ⇒ 900x = 525
• x = 525/900 = 7/12 after dividing by 75.

Indeed 7 ÷ 12 = 0.58333…, so 0.583̅ = 7/12. This is the exact algebraic pattern ETS expects: multiply by powers of 10 so the repeating tails cancel, then simplify.

Models That Make Equivalence Visible

Praxis teaching-scenario items frequently ask which model best supports a concept. Match the model to the idea:

1. Area models (grids, circles, rectangles): shade part of a whole. A 10×10 grid is especially powerful because each small square is 1%, so shading 25 squares shows 25% = 25/100 = 1/4 = 0.25 at once.
2. Base-10 blocks: a flat can represent 1, a rod a tenth, a unit a hundredth (or rescale so a flat is 1 and rods are tenths). These make decimal place value concrete: 0.3 is three rods; 0.03 is three small cubes.
3. Set models: a collection of discrete objects (12 counters) where a subset represents the fraction (3 of 12 = 1/4). Useful when the "whole" is a set, not a continuous region.
4. Colored rods / fraction bars: equal-length rods partitioned into different equal parts (halves, thirds, sixths) support comparing fractions and seeing equivalence such as 2/4 = 1/2 by length matching.

Teaching tip: A 10×10 percent grid converts percent ↔ decimal ↔ fraction more transparently than a pie chart, because the denominator 100 is built into the model.

Classroom Teaching Scenario: Model Misconceptions

A teacher asks students to shade 0.4 on a 10×10 hundredths grid and also to shade 2/5 on a separate rectangle divided into fifths, then decide whether the two shadings represent the same amount.

• Student A shades 4 small squares on the hundredths grid (thinking "0.4 means 4 hundredths") and shades 2 of 5 parts on the fifths rectangle. The student concludes the amounts are different because the pictures "look different."
• Student B shades 40 small squares (correct for 0.4 = 40/100) but argues that 2/5 cannot equal 0.4 because "fifths are bigger pieces than hundredths."

Diagnosis: Student A has a place-value error—confusing tenths with hundredths. Student B correctly shades the decimal but does not yet understand that different-sized unit fractions can compose equal totals. The instructional move is not to assert "they are equal," but to convert both to a common representation: redraw so both are on a hundredths grid (2/5 = 40/100), or use colored rods of equal total length labeled 2/5 and 4/10.

Praxis Traps

• Treating longer decimal digit strings as larger values (0.125 vs 0.13) without aligning place value.
• Writing a repeating decimal's "bar length" incorrectly when converting (multiplying by 10 instead of 100 for two repeating digits).
• Claiming a non-terminating decimal must be irrational—repeating non-terminating decimals are rational.
• Choosing a set model when the problem is about continuous area (or vice versa), which can hide the intended equivalence.
• Converting percent to decimal by moving the point the wrong direction (45% → 45.0 instead of 0.45).

Mastery for Praxis 5164 means you can convert accurately, justify the conversion with a model, and explain why a student's model does or does not show equivalence.