# 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

• Connections with advanced topics

• 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 7^{th} 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

lead to variety of strategies

• Decimals and fractions make proportional reasoning more

difficult

• Divide large by small bug – use variety of units to avoid

• Students must create understanding through small group

and full class discourse

• Justification of reasoning essential to developing

understanding, assessment must include this

• Unfamiliar context and large numbers decrease success