1.2 Reasoning and Proof
Inductive Reasoning
Pacing:This lesson should take one class period
Goal:This lesson introduces students to inductive reasoning. Inductive reasoning applies easily to algebraic patterns,
integrating algebra with geometry.
A great way to start this lesson is to further expand upon inductive reasoning. Inductive reasoning uses patterns to
make generalizations. Simply put, inductive reasoning takes repeated specific examples and extends it to a general
conjecture.
Begin by writing an arithmetic or a geometric sequence such as 1, 4 , 7 , 10 ...or 40, 20 , 10 , 5 ,...Ask students to
recognize the pattern and write the generalization in words (this also lends itself to exposure to sequences and series,
a topic usually found in Advanced Algebra).
Challenge:Offer students this type of pattern: 14, 10 , 15 , 11 , 16 , 12 , 17 ...The pattern here is to subtract four then
add five.
Take the opportunity to further discuss the triangular numbers, as seen on page 76. The triangular numbers are
formed such that the dots form a triangle and also a numerical pattern ofs(n) =^12 n^2 −n.
In examples 1 and 2 on page 76, relate the even and odd numbers to a symbolic pattern. For example, even numbers
can be represented by the expression 2n, while all odd numbers can be represented by 2n+1.
Real Life Connection!Apply the idea of counterexample to real life situations. Begin by devising a statement, such
as, “If the sun is shining, then you can wear shorts.” While this is true for warm weather states such as Florida and
California, for those living in the Midwest or Northern states, it is quite common to be sunny and 12 degrees! Have
students create their own statements and encourage other students to find counterexamples.
Conditional Statements
Pacing:This lesson should take two class periods
Goal:This lesson introduces the all-important conditional statements. Students will gain an understanding of how
converses, inverses, and contrapositives are formed from a conditional and further explore truth values of each of
these statements.
The first portion of this lesson may be best taught using direct instruction and several visual aids. Design phrases
you can laminate, such as “you are sixteen” and “you can drive.” Adhere magnets to the back of the phrases (to stick
to the white board), or you can use a SMART board. Begin by writing the words “IF” and “THEN,” giving ample
space to place your phrases. When discussing each type of conditional, show students how each is constructed by
rearranging your phrases, yet leaving the words “IF” and “THEN” intact.
Have students create a chart listing the type of conditional, its symbolic form and an example. This allows students
an easy reference sheet when trying to decipher between converse, conditional, contrapositive, and inverse.
1.2. Reasoning and Proof