3.2 Decimals & Order of Operations

3.2 Decimals & Order of Operations

Quick Summary: Decimal notation extends the base-ten place value system to fractional parts using powers of ten ($10^{-1}, 10^{-2}, 10^{-3}$). Master exact decimal algorithms, converting between fractions and decimals, identifying terminating vs. repeating decimals, and applying the strict Order of Operations (PEMDAS/GEMS) to complex multi-step numerical expressions.

Decimals and the Order of Operations represent core computational skills tested on Praxis 5003. Elementary teachers must possess complete mastery of positional place value, place-alignment rules, quotient calculations, fraction-decimal equivalences, and grouping hierarchies.

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Decimal Place Value & Base-Ten Structure

The decimal system is a positional base-ten system where each place value represents a power of $10$. Positions to the left of the decimal point represent non-negative powers of ten ($10^0 = 1, 10^1 = 10, 10^2 = 100$), while positions to the right represent negative powers of ten (fractional parts):

$$\text{Tenths } (10^{-1} = 0.1), \quad \text{Hundredths } (10^{-2} = 0.01), \quad \text{Thousandths } (10^{-3} = 0.001), \quad \text{Ten-Thousandths } (10^{-4} = 0.0001)$$

Reading and Writing Decimals

When reading a decimal, read the whole number part, say "and" for the decimal point, and read the decimal part as a whole number followed by the place value of the last digit.

• Example: $43.085$ is read as "forty-three and eighty-five thousandths."
• Expanded Notation: $43.085 = (4 \times 10^1) + (3 \times 10^0) + (0 \times 10^{-1}) + (8 \times 10^{-2}) + (5 \times 10^{-3})$

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Fundamental Decimal Algorithms

1. Addition and Subtraction

Align decimal points vertically so that digits of equal place value are directly above each other. Annex trailing zeros to equalise decimal lengths before computing.

$$\text{Evaluate: } 45.2 - 18.735$$ $$\begin{array}{rl} 45.200 & \text{(Annex two trailing zeros)} \\ - 18.735 & \text{(Align decimal points vertically)} \\ \hline 26.465 & \end{array}$$

2. Multiplication

Multiply numbers as if they were whole integers, ignoring decimal points initially. Count the total number of decimal places across both factors. Place the decimal point in the final product so it has that same total number of decimal places.

$$\text{Evaluate: } 3.14 \times 0.25$$

• Factor 1: $3.14$ ($2$ decimal places)
• Factor 2: $0.25$ ($2$ decimal places)
• Total decimal places required = $2 + 2 = 4$ places.
• Integer product: $314 \times 25 = 7850$.
• Placing decimal point 4 places from right: $0.7850 = 0.785$.

3. Division

To divide by a decimal divisor:

1. Shift the decimal point in the divisor to the right until it becomes a whole number.
2. Shift the decimal point in the dividend to the right by the exact same number of places.
3. Place the decimal point in the quotient directly above the new decimal point in the dividend.
4. Perform standard long division.

$$\text{Evaluate: } 14.04 \div 0.36$$

• Shift divisor decimal point 2 places right: $0.36 \to 36$.
• Shift dividend decimal point 2 places right: $14.04 \to 1404$.
• Compute whole number division: $1404 \div 36 = 39$.

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Converting Between Fractions & Decimals

Decimal to Fraction

Write the decimal as a fraction with the numerator as the digits and the denominator as the corresponding power of $10$, then simplify to lowest terms.

• Example: $0.375 = \frac{375}{1000} = \frac{375 \div 125}{1000 \div 125} = \frac{3}{8}$

Fraction to Decimal

Divide the numerator by the denominator using long division ($a \div b$).

Terminating vs. Repeating Decimals

A rational number in simplest form $\frac{a}{b}$ produces either a terminating decimal (ends after a finite number of digits) or a repeating decimal (has a repeating block of digits denoted by a vinculum bar $\bar{x}$).

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Order of Operations (PEMDAS / GEMS)

When evaluating expressions with multiple operations, mathematicians follow a standardized order of operations to ensure a single, unambiguous result.

The GEMS Framework

While many candidates recall PEMDAS (Please Excuse My Dear Aunt Sally), modern mathematics instruction uses GEMS to avoid common order-of-operation misconceptions:

1. G – Grouping Symbols: Evaluate expressions inside grouping symbols from innermost to outermost: Parentheses (), Brackets [], Braces {}, Absolute Value bars |x|, and horizontal Fraction Bars $\frac{\text{Numerator}}{\text{Denominator}}$.
2. E – Exponents: Evaluate all exponential terms, powers, and roots.
3. M – Multiply / Divide: Perform all multiplication and division strictly from left to right as they appear in the expression.
4. S – Subtract / Add: Perform all addition and subtraction strictly from left to right as they appear in the expression.

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Step-by-Step Worked Exemplar

$$\text{Evaluate: } 18 - [3^2 + 2 \times (14 - 4 \times 2)] \div 5$$

1. Innermost Grouping Symbol $(14 - 4 \times 2)$:

   • Inside parentheses, perform multiplication before subtraction: $4 \times 2 = 8$.
   • Perform subtraction: $14 - 8 = 6$.
   • Expression becomes: $18 - [3^2 + 2 \times 6] \div 5$

2. Outer Grouping Symbol $[3^2 + 2 \times 6]$:

   • Evaluate exponent: $3^2 = 9$.
   • Evaluate multiplication: $2 \times 6 = 12$.
   • Evaluate addition: $9 + 12 = 21$.
   • Expression becomes: $18 - 21 \div 5$

3. Division before Subtraction:

   • Compute division: $21 \div 5 = 4.2$.

4. Final Subtraction:

   • Compute subtraction: $18 - 4.2 = 13.8$.

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Praxis Traps & Common Teacher Misconceptions

• Evaluating PEMDAS Literally: Executing all multiplications before divisions regardless of position is a frequent error. $12 \div 3 \times 4 = 16$, NOT $1$.
• Ignoring Fraction Bar Groupings: The fraction bar acts as an implicit grouping symbol. In $\frac{10 + 6}{2 + 2}$, evaluate numerator ($16$) and denominator ($4$) fully before dividing: $16 \div 4 = 4$.
• Misplacing Zeros in Decimal Multiplication: When multiplying $0.04 \times 0.02$, $4 \times 2 = 8$. With 4 total decimal places, annex leading zeros: $0.0008$.