4.1 Number Operations: Integers, Decimals, Fractions, and Percents

4.1 Number Operations: Integers, Decimals, Fractions, and Percents

As a paraprofessional, you will frequently assist students with core math concepts. The GACE Paraprofessional Assessment evaluates both your personal computational skills and your ability to support instruction. Because calculators are prohibited on this exam, a deep, conceptual understanding of place value, number operations, and arithmetic rules is essential. This section covers the fundamental mathematical building blocks you must master to succeed on the exam and effectively guide students in the classroom.

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Place Value: The Foundation of Numbers

Our numbering system is a base-10 system: the position of a digit determines its value, with each place ten times larger than the one to its right. Digits to the left of the decimal point represent whole numbers (ones, tens, hundreds, thousands), while digits to the right represent fractional parts (tenths, hundredths, thousandths).

• Thousands: $1,000$s place
• Hundreds: $100$s place
• Tens: $10$s place
• Ones: $1$s place
• Decimal Point
• Tenths: $\frac{1}{10}$ or $0.1$ place
• Hundredths: $\frac{1}{100}$ or $0.01$ place
• Thousandths: $\frac{1}{1,000}$ or $0.001$ place

Classroom Application

Students often struggle to compare decimals, incorrectly believing $0.452$ is larger than $0.5$ because "452 is larger than 5." Paraprofessionals can guide them to pad decimals with trailing zeros ($0.5$ becomes $0.500$) to compare place values directly, showing that 5 tenths is greater than 4 tenths ($0.5 > 0.452$).

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Order of Operations (PEMDAS)

When evaluating expressions with multiple operations, you must follow the standard order of operations to arrive at the correct, unique solution. The acronym PEMDAS serves as a helpful mnemonic:

1. Parentheses (and other grouping symbols like brackets or fraction bars)
2. Exponents (powers and roots)
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

[!IMPORTANT] A common trap on the GACE exam is treating multiplication as always preceding division, or addition as always preceding subtraction. Multiplication and division are equal in priority. You must perform them in the order they appear from left to right. The same applies to addition and subtraction.

Example:

Evaluate $12 - 3 \times 2 + 8 \div 4$

• First, perform multiplication and division from left to right:
  • $3 \times 2 = 6$
  • $8 \div 4 = 2$
  • The expression becomes: $12 - 6 + 2$
• Next, perform addition and subtraction from left to right:
  • $12 - 6 = 6$
  • $6 + 2 = 8$
• Correct Answer: $8$ (An incorrect order might yield $12 - 6 + 2 = 12 - 8 = 4$, which is wrong!)

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Arithmetic with Integers (Signed Numbers)

Integers include all positive and negative whole numbers, and zero.

• Addition (Same Signs): Add their values and keep the sign: $-5 + (-3) = -8$.
• Addition (Different Signs): Subtract the smaller value from the larger and keep the sign of the larger absolute value: $-7 + 10 = 3$ and $4 + (-9) = -5$.
• Subtracting Negatives: Change the double negative to addition: $6 - (-2) = 6 + 2 = 8$ and $-4 - (-3) = -4 + 3 = -1$.
• Multiplication/Division: Like signs yield positive results ($(-4) \times (-5) = 20$ and $(-15) \div (-3) = 5$); unlike signs yield negative results ($(-6) \times 3 = -18$ and $24 \div (-8) = -3$).

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Fractions, Decimals, and Percents

Paraprofessionals frequently help students transition between fractions, decimals, and percents. You should be comfortable converting these fluidly.

• Fractions: Add or subtract by finding a common denominator: $\frac{1}{3} + \frac{2}{5} = \frac{5}{15} + \frac{6}{15} = \frac{11}{15}$. Multiply straight across: $\frac{2}{3} \times \frac{4}{7} = \frac{8}{21}$. Divide by multiplying by the reciprocal: $\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8} = 1\frac{1}{8}$.
• Conversions: To change a fraction to a decimal, divide numerator by denominator (\frac{5}{8} = 0.625). Convert a decimal to a percent by shifting the decimal point two places right ($0.625 = 62.5%$) and a percent to a decimal by shifting it two places left ($4.5% = 0.045$). Convert a percent to a fraction by placing it over $100$ and simplifying ($35% = \frac{35}{100} = \frac{7}{20}$).

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Prime Numbers, Factors, and Multiples

• Primes & Composites: A prime number is a whole number greater than $1$ with exactly two factors: $1$ and itself (e.g., $2, 3, 5, 7, 11$). Number $2$ is the only even prime; $1$ is neither prime nor composite. Composite numbers have more than two factors (e.g., $4, 6, 8, 9$).
• Factors & GCF: Factors are numbers multiplied to get a product. The factors of $12$ are $1, 2, 3, 4, 6,$ and $12$. The Greatest Common Factor (GCF) is the largest factor shared by numbers (e.g., the GCF of $12$ and $18$ is $6$).
• Multiples & LCM: Multiples are products of a number and integers. Multiples of $4$ are $4, 8, 12, 16, \dots$. The Least Common Multiple (LCM) is the smallest multiple shared by numbers (e.g., the LCM of $4$ and $6$ is $12$).

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Calculator-Free Exam Strategies

Since calculators are prohibited, use these strategies to maintain speed and accuracy:

1. Estimate First: Round numbers to get a ballpark figure. If your estimate is around $50$, and the choices are $5.2$, $49.8$, $502$, and $5,020$, the correct answer is clearly $49.8$.
2. Use Scratch Paper: Carefully align decimal columns during addition/subtraction. Write out step-by-step intermediate calculations for PEMDAS to avoid sign errors.
3. Use Divisibility Rules: A number is divisible by $3$ if the sum of its digits is divisible by $3$ (e.g., $147$ because $1+4+7=12$). A number is divisible by $4$ if its last two digits are divisible by $4$ (e.g., $1,024$ because of $24$ is).