Course 3 Unit 1 - Reasoning and Proof

In Courses 1 and 2, students have come to understand mathematics as an active science of exploring, discovering, explaining, and applying patterns in quantity and change, shape and motion, and data and change. In Course 3, informal mathematical explanations and reasoning are extended to more formal arguments and proofs.

Unit Overview
This first unit in Course 3 signals the new expectations of more formal arguments by considering reasoning strategies used in games, with numbers, and with area and reasoning with if-then statements in Lesson 1. Once an initial understanding of desirable characteristics of a mathematical argument is developed, students are asked in Lesson 2 to construct arguments (deductive reasoning) based on geometric assumptions. In Lesson 3, students use inductive reasoning (observed patterns for a number of cases) to prove a general relationship for all cases and prove that an equation or inequality is true for all or most values of the variables. In Lesson 3, students reason statistically. They learn how to design a good experiment to compare two different treatments and how to use randomization to produce a sampling distribution in order to decide if one treatment is more effective than another treatment. They also learn the difference between sample surveys, experiments, and observational studies. As synthesis, students compare algebraic, geometric, and statistical reasoning.

Objectives of the Unit
  • Recognize the differences between, as well as the complementary nature of, inductive and deductive reasoning
  • Develop some facility in analyzing and producing deductive arguments in everyday contexts and in geometric, algebraic, and statistical contexts
  • Know and be able to use the relations among the angles formed when two lines intersect, including the special case of perpendicular lines
  • Know and be able to use the necessary and sufficient conditions for two lines to be parallel
  • Use symbolic notation to represent numerical patterns and relationships and use rules for transforming algebraic expressions and equations to prove those facts
  • Distinguish between sample surveys, experiments, and observational studies; know the characteristics of a well-designed experiment
  • Use statistical reasoning to decide whether one treatment causes a better result than a second treatment

Sample Overview
The sample student material below is from Lesson 1, "Reasoning Strategies." In this investigation, students examine arguments presented in differing forms and using differing reasoning strategies. They are asked to judge these arguments on the basis of how well the arguments convince them that the asserted proposition is true. From these experiences, they are asked to identify characteristics of arguments that make them convincing and logically correct. This approach to developing reasoning abilities is recommended in the Common Core State Standards for Mathematics within the Standards for Mathematical Practice number 3: Construct viable arguments and critique the reasoning of others. (Alignment of Core-Plus Mathematics with the Common Core State Standards)

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.

View the Unit Table of Contents and Sample Lesson Material
You will need the free Adobe Acrobat Reader software to view and print the sample material.

How the Geometry and Trigonometry Strand Continues
In Course 3 Unit 3, Similarity and Congruence, students will extend their ability to reason formally in geometric settings. Deductive reasoning is used to prove theorems concerning similar and congruent triangles and their application to special quadrilaterals, and necessary and sufficient conditions for parallelograms. Circular functions (sine and cosine), are used to model periodic change in Unit 6, Circles and Circular Functions.
     In Course 4: Preparation for Calculus, geometry and algebra become increasingly intertwined. Students develop understanding of two-dimensional vectors and their application and the use of parametric equations in modeling linear, circular, and other nonlinear motion. In addition, students intending to pursue programs in the mathematical, physical, and biological sciences, or engineering extend their ability to visualize and represent three-dimensional surfaces using contours, cross sections, and reliefs; and to visualize and sketch surfaces and conic sections defined by algebraic equations. They also extend their understanding of, and ability to reason with, trigonometric functions to prove or disprove trigonometric identities and to solve trigonometric equations. They also geometrically represent complex numbers and apply complex number operations to find powers and roots of complex numbers expressed in trigonometric form. (See the CPMP Courses 1-4 descriptions.)

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