3.2 Fractions and Decimals

Fractions Represent Parts of a Whole

Fractions are among the most-missed topics in elementary math, which is exactly why the ParaPro tests both your fluency and your ability to teach them. A fraction names a part of a whole or a part of a set. The numerator (top) counts how many parts you have; the denominator (bottom) tells how many equal parts make the whole.

Three categories appear on the test:

• Proper fraction — numerator < denominator, so the value is less than 1 (¾).
• Improper fraction — numerator ≥ denominator, so the value is 1 or more (5/4).
• Mixed number — a whole number plus a proper fraction (1¼).

Converting Mixed Numbers and Improper Fractions

Mixed → improper: multiply the whole number by the denominator, add the numerator, and keep the denominator.

$2\frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{11}{4}$

Improper → mixed: divide the numerator by the denominator; the quotient is the whole number and the remainder is the new numerator.

$\frac{11}{4} = 11 \div 4 = 2 \text{ R } 3 = 2\frac{3}{4}$

Equivalent Fractions and Simplifying

Multiplying or dividing numerator and denominator by the same number gives an equivalent fraction: 1/2 = 2/4 = 4/8. To simplify, divide both by their greatest common factor — 6/12 ÷ 6/6 = 1/2.

Operations with Fractions

Adding and subtracting require a common denominator. If the denominators already match, combine the numerators and keep the denominator. If they differ, rewrite each fraction over the least common denominator first.

$\frac{3}{8} + \frac{2}{8} = \frac{5}{8} \qquad \frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Multiplying is the easiest case — multiply across the tops and across the bottoms, then simplify. No common denominator is needed.

$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$

Dividing means multiplying by the reciprocal (flip the second fraction, then multiply).

$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

Worked Example: A recipe needs ⅔ cup of flour, but you are making half a batch. How much flour? Half of ⅔ is ½ × ⅔ = 2/6 = ⅓ cup. This is the kind of real-world fraction problem the ParaPro favors.

Decimals and Conversions

Decimals are just fractions with denominators that are powers of ten, so the two forms convert cleanly.

Fraction → decimal: divide the numerator by the denominator.

$\frac{3}{4} = 3 \div 4 = 0.75$

Decimal → fraction: write the decimal over the matching power of ten, then simplify. 0.75 = 75/100 = ¾; 0.6 = 6/10 = ⅗.

Benchmark Equivalents to Memorize

Knowing these by heart saves time and powers mental-math estimates.

Comparing and Ordering

To compare fractions, convert them to a common denominator or to decimals, then compare. To order 3/5, 1/2, and 7/10, rewrite over 10: 6/10, 5/10, 7/10 → ordered least to greatest that is 1/2, 3/5, 7/10.

Comparing decimals is a common student stumbling block. Line up the decimal points and compare place by place, padding with zeros if needed. Many students think 0.45 > 0.5 because "45 is bigger than 5" — rewriting 0.5 as 0.50 makes it clear that 0.50 > 0.45. The ParaPro frequently asks you to diagnose exactly this misconception.

Worked Example: Order 0.7, 0.07, and 0.77 from least to greatest. Pad to two places: 0.70, 0.07, 0.77. Compare: 0.07 < 0.70 < 0.77, so the order is 0.07, 0.7, 0.77.

Helping Students with Fractions

Fractions are abstract, so application answers reward concrete models before procedures:

1. Area models (shaded circles or rectangles) and fraction bars show that ½ and 2/4 cover the same space.
2. Number lines place fractions between whole numbers, reinforcing that ¾ is close to 1.
3. Money and food make fractions familiar — a quarter is ¼ of a dollar; a slice is a fraction of a pizza.
4. Equivalence first: students should see that 1/2 = 2/4 with manipulatives before they trust the rule.
5. Address the classic error directly: "more pieces means smaller pieces," so 1/8 < 1/4 even though 8 > 4.

Good scaffolding moves from manipulatives, to pictures, to symbols — never the reverse.

Recap

Fractions name parts of a whole; add and subtract only over a common denominator, multiply straight across, and divide by the reciprocal; fractions and decimals convert through division and powers of ten; benchmark equivalents speed everything up; and effective fraction teaching is visual, equivalence-first, and grounded in money or food.