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Course 3, Unit 1 - Reasoning and Proof

This unit signals the new expectations in Course 3. The unit provides a logical foundation for proof. The informal mathematical explanations in Courses 1 and 2 are extended to more formal arguments and proofs. Students begin to develop an understanding of mathematical reasoning in geometric, algebraic, and statistical contexts.

Key Ideas from Course 3, Unit 1

  • Inductive and deductive reasoning strategies: Inductive reasoning is used to discover general patterns or principles based on evidence from experiments or several cases. Deductive reasoning involves reasoning from facts, definitions, and accepted properties to conclusion using principles of logic. Inductive reasoning is often used to develop conjectures which may provide statements that can be proved for all or many cases using deductive reasoning. (See pages 12-14.)

  • Principles of logical reasoning: Affirming the Hypothesis and Chaining Implications (See pages 10-14.)

  • Relations among angles formed by two intersecting lines or by two parallel lines and a transversal: (See pages 35-39.)

  • Rules for transforming algebraic expressions and equations: After recognizing patterns and representing the patterns in algebraic notation, the relationships (properties) are proved by writing chains of steps that involve replacing an expression by one which is equivalent to it. Transforming an equation or an inequality involves applying operations to both sides of the equation. Properties used include addition, subtraction, multiplication, and division of both sides of an equation, using the laws of exponents, rearranging terms in an expression, and using the distributive property. (Algebraic Properties and Properties of Equality are formalized on page 64.)

  • Design of experiments, sample surveys, and observational studies: This includes the role of randomization, control groups, and blinding in experiments. (See pages 74-80 and 89-91.)

  • Randomization test: A method to determine whether a difference between two treatment groups can be reasonably attributed to the random assignment or whether you should believe that the treatments caused the difference. (See page 85 for a description of the method.)

  • Statistical significance: Results from an experiment are called "statistically significant" when it is unreasonable to attribute the results solely to the random assignment of treatments to subjects. For the randomization test applied in this unit, if the difference of the means from the actual experiment is in the outer 5% of the distribution generated by a randomization test, conclude that the results are statistically significant. (See page 85.)

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