Curriculum Overview: Arithmetic Foundations

This curriculum is designed to provide students with the absolute numerical fluency required for the Digital SAT and higher-level mathematics. It bridges the gap between basic calculation and algebraic manipulation by focusing on number sense, operational order, and proportional reasoning.

## Prerequisites

Before beginning this curriculum, students should possess the following foundational skills:

• Basic Arithmetic Proficiency: Comfort with addition, subtraction, multiplication, and division of whole numbers.
• Numerical Recognition: Ability to identify and order positive whole numbers on a number line.
• Elementary Logic: A basic understanding of "greater than" and "less than" relationships.

## Module Breakdown

The curriculum is structured into five progressive modules, moving from raw operations to complex real-world applications.

## Learning Objectives per Module

Module 1: Operational Logic

• PEMDAS Mastery: Evaluate complex expressions by correctly sequencing Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction.
• Integer Manipulation: Predict outcomes of operations involving positive and negative integers without calculator assistance.

Module 2: Number Theory

• Factors & Multiples: Differentiate between the divisors (factors) and products (multiples) of a number.
• GCF & LCM: Calculate the Greatest Common Factor and Least Common Multiple to simplify fractions and find common denominators.

Module 3: Exponents and Roots

• Simplification: Apply exponent rules for multiplying, dividing, and raising powers to a power.
• Radical Forms: Break down square and cube roots into simplest radical form.
• Fractional Exponents: Convert between radical expressions and fractional exponents ($x^{1/2} = \sqrt{x}$).

Module 4: Fractions, Decimals, and Percentages

• Fluent Translation: Convert values between forms to match SAT answer choices (e.g., $0.$75 \leftrightarrow 3/4 \leftrightarrow 75$%$).
• Percent Change: Calculate percentage increase or decrease over time using the formula: $\text{Percent Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100$

Module 5: Applied Proportions

• Unit Conversions: Execute multi-step conversions (speed, weight, distance) using dimensional analysis.
• Proportion Solving: Set up and cross-multiply ratios to solve for unknown variables in word problems.

## Success Metrics

To demonstrate mastery of the Arithmetic Foundations curriculum, students must meet the following benchmarks:

• Non-Calculator Accuracy: Achieve 90% or higher accuracy on integer and PEMDAS drills without using digital tools.
• Fluency Speed: Convert common fractions to decimals (e.g., 1/8 to 0.125) in under 3 seconds.
• Expression Translation: Successfully translate a 3-sentence word problem into a single solvable arithmetic equation.
• Error Analysis: Identify the specific "trap" reason (e.g., order of operations error vs. sign error) in incorrect practice problems.

[!IMPORTANT] Mastery of these foundations is the single greatest predictor of success in the SAT Algebra and Advanced Math domains.

## Real-World Application

Arithmetic is not merely a classroom exercise; it is the language of practical logic.

• Financial Literacy: Understanding percent change and interest rates is essential for managing personal loans, credit cards, and investments.
• Engineering & Architecture: Unit conversions and proportional scaling are required to translate blueprints into physical structures.
• Culinary Arts & Chemistry: Scaling recipes or chemical solutions relies heavily on ratios and fractional manipulation.

▶Click to see how this leads to Algebra

Once you can manipulate integers and follow PEMDAS, Algebra simply replaces known numbers with variables ($x, y$). If you can solve $2 + 3 = 5$, you are one step away from solving $2 + x = 5$.