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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

• 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

Proportional Reasoning Among 7th Grader
Students, ESM, 1998

• CMP curriculum effect on proportional
 • 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
  • 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