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# Proportional Reasoning

## Connected Math: Comparing and Scaling

• Advanced Topics from Math Session
• Variable Concept
• Equality Concept
• 4 Conceptions of Algebra
• Congruence Equations
• Fraction to Decimal Conversion Theorem
• Proportional Reasoning

## Investigation 4: Comparing by Finding Rates

• Rate – unit rate vs. scale up method (Pg 38, Prob
4.1 and 4.2, Pg 39-41)
• Multiple representation (Prob 4.2, Pg 41)
• Piecewise Graph & Average Rate (Prob 4.3, Pg 42)
• Reciprocal Rates (Prob 4.4, Pg 43)
• Connections to rational number product
(Connections Prob 17-20, Pg 48)
• Stacked Graph (Connections, Prob 21, Pg 49)
• Proportion to Equation (Extensions, Prob 22, Pg
49)

## Research Proportional Reasoning Among 7th Grader Students, ESM, 1998

• CMP curriculum effect on proportional
reasoning
• Overview of CMP curriculum, Pg 248
• Proportional Reasoning: math relationships
that are multiplicative in nature a/b = c/d
• Few students develop consistent conception of PR
• 3 catergories of PR: part/whole, rates, scaling
• 3 tasks for assessing PR: missing value, numerical
comparison, qualitative prediction-comparison

• Proportional Problem solution methods
• Proportional Reasoning Measure, Pg 255
• External Ratio: unit rate method, Pg 258
• Internal Ratio: compare like units, Pg 259
• Refraining from computation, Pg 250
• 6 additional strategies, Pg 260-263
• Common Multiple, Pg 260
• Building up or table, Pg 260

• Key Finding
• CMP students developed proportional thinking
through problem-based investigations that
encouraged personal construction of flexible
approaches to such tasks and were therefore
more successful in applying sensible and
effective strategies to the given task
• Unit rate method evolved as having the most
intuitive appeal

• Implications for Teaching
• Cognitive Science findings
• Context variables interfering with development of PR
include familiarity of context
• Presence of mixture, continuous quantities, presence
of integer ratios, order, and numerical complexity

• Implications for Teaching
• Research findings
• Mathematical intuition and informal knowledge systems