The Framework

The RTF is a visual and structural approach to teaching mathematical reasoning that emphasizes understanding relationships over memorized procedures. By giving students a consistent way to see the structure of any problem, RTF builds reasoning that transfers from arithmetic to algebra and beyond.

The relational triangle is the central visual tool of the framework, representing relationships between the whole and parts in math problems. It is used across addition, subtraction, multiplication, division, fractions, proportions, and algebra — making it a single anchor for the entire mathematical journey.

Before solving any problem, students ask:

• What is known? Identify known values and place them in the triangle.
• What is unknown? Identify what to find and mark it with a variable.
• How are the quantities related? Determine the operation: combine or undo.

These questions activate the structural reasoning process and prevent answer-getting shortcuts.

A three-stage problem-solving process that students apply consistently across all math contexts.

Applicable from simple to multi-step algebraic problems, including fractions and proportions.

Multi-step problems involve linking multiple relational triangles via the cascade effect:

• The unknown of one triangle becomes the whole of the next.
• Solve step by step, following the structure without pre-planning.
• The cascade approach simplifies complex problems by chaining relational models.

RTF extends to complex math by interpreting fractions as scaling operators rather than static values:

• Fractions are viewed as transformations; e.g., (1/3)x = 9 is interpreted as how many one-thirds add up to 9.
• Rational numbers are seen as scaling factors; e.g., 0.5 × 20 = 10.

The structure remains the same — only the interpretation of the relationships changes.

RTF helps diagnose errors based on structural misunderstandings:

• Placing the whole in a part position.
• Using the wrong operation (e.g., addition instead of multiplication).
• Skipping the model phase and jumping to calculation.
• Correct answers achieved through luck without structural reasoning.
• Cascade errors, such as mistransferring the result to the next triangle.
• Confusing reciprocal rules in fractions.

Errors are categorized as position, operation, or process mistakes — guiding targeted correction.

RTF is a comprehensive, visual, and structural approach to teaching mathematical reasoning that emphasizes understanding relationships, diagnosing errors structurally, and applying a consistent problem-solving cycle across all levels of math.