Connecting with Mathematics

What is Mathematical Investigation?
Dissections and Area
Linearity and Proportional Reasoning
Pythagoras and Cousins
Pascal's Revenge: Combinatorial Algebra
Games and Number Theory


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The Linearity and Proportional Reasoning module explores the gamut of proportional thinking from its elementary origins through generalizations in several directions. The first session looks at proportional reasoning through fractions and ratios, as well as multiplication. The second session presents linear interpolation and linear combinations. The third session explores functions for which f(a+b)=f(a)+f(b) and addresses the common misconception that any function has this property. In the fourth session, functions of two variables emerge from old-fashioned and new style word problems, and their graphs are investigated. The final session takes another look at linear combinations using perspectives from both middle school and higher level mathematics (through an accessible introduction to the basis for a vector space).

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