

Course 3, Unit 1  Reasoning and Proof
Overview
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. Geometric definitions
and relationships from Courses 1 and 2 that students will use in
this unit are summarized in Key
Geometric Ideas from Courses 1 and 2 (297 KB).
Local Deductive Systems
In CorePlus Mathematics, the intent is not to organize geometry
as a complete deductive mathematical system. Rather, our approach is
to enable students to experience the connectedness of geometry: how from
a few undefined terms and some basic assumptions (postulates) they can
deduce important properties of geometric figures.
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 1216.)

Principles of logical reasoning: Affirming the Hypothesis
and Chaining Implications (See pages 1014.)

Line reflection assumptions and properties: Inductive and
deductive reasoning in transformational geometry (See pages 1316.)

Relations among angles formed by two intersecting lines or by
two parallel lines and a transversal: Although introduced in
middle school, theorems are now formally proved. (See pages 3039.)

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 7480 and 8991.)

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.)
