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1.2: Reasoning and Proof

Created by: CK-12

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 \ldots or 40, 20, 10, 5, \ldots 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 \ldots 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 of s(n) = \frac{1}{2}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\;\mathrm{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.

Symbolic Form Example
Conditional p \Rightarrow  q
Inverse :p  \Rightarrow :q
Converse q \Rightarrow p
Contrapositive :q \Rightarrow :p

Spend time reviewing example one on page 85 as a class. Stress the importance of counterexamples.

Interactive Lesson! Use the same setup as the opening activity when discussing biconditionals. Begin with a definition, such as example one on page 86. Set up your magnetic phrases in if and only if form, then illustrate to students how the biconditional can be separated into its conditional and converse.

Deductive Reasoning

Pacing: This lesson should take one class period

Goal: This lesson introduces deductive reasoning. Different than inductive reasoning, deductive reasoning begins with a generalized statement, and assuming the hypothesis is true, specific examples are deduced.

Differentiate between deductive and inductive reasoning to students by linking to the previous lessons. Deductive reasoning begins with a conjecture (hypothesis) and infers specific examples.

Stress example 5 with your students. Students can get confused with the inverse and contrapositive from the previous lesson that they make the mistake of using faulty reasoning.

When determining the truth value of p^{\land} q, students may be confused as to why the value is false if the hypothesis is false. Offer students a real life example. “If it is snowing, then it is cold.” If the hypothesis is already false, stress that it doesn’t matter the conclusion; the statement is not applicable.

Be sure the students understand the difference between \land (exclusive) \lor and (inclusive) before filling out the truth tables.

Real World Application! Show a portion of an episode of a courtroom drama scene. Ask students to apply the ideas of deductive and inductive reasoning to the lawyers. Determine which reasoning the prosecuting attorney is using. Is it different reasoning than what the defending attorney uses?

Algebraic Properties

Pacing: This lesson should take one class period

Goal: Students should have some familiarity with these properties. Here we can extend algebraic properties to geometric logic.

Fun Tip! Construct “I have, who has” cards for your class. Using the properties from this lesson (and other lessons if you have a large class), create as many cards as students in your class. The first card should read, “I have Reflexive Property of Equality. Who has the property that states if a = b, then b = a?” The next card should state, “I have the Symmetric Property of Equality. Who has…? Continue this process until the last card. The “Who has” of this card should state, “Who has the property that a equals a?” Shuffle the cards and give one to each student. Because the cards are all connected, it doesn’t matter who starts. Time the class and then challenge the students to beat their previous time. Not only does this increase listening in the classroom, but it also reinforces the properties and encourages active participation.

Teaching Strategy: Use personal whiteboards or interactive “clickers” to do a spot check. Create two or three property questions each day and begin your class with these mini-quizzes. Encourage students to create flashcards for the properties and use them for two-column proofs.

Stress to students that the properties of congruence can only be used when given congruence (\cong), not equality (=). This also hold true for the properties of equality; these properties are reserved for objects that are equivalent.

Have students list properties not mentioned in this lesson. Students may come up with the distributive property of the multiplying fractions property. Students may offer notions that are incorrect – take the time to have students learn from incorrect thoughts!

Diagrams

The best way to describe what you can and cannot assume is “Looks are deceiving.” Reiterate to students that nothing can be assumed. The picture must literally say one thousand words using notation such as tic marks, angle arcs, arrows, etc.

Additional Example! Use the following diagram and ask your students to list everything they can assume from the drawing and those things that cannot be assumed. For the latter list, ask students to list additional information needed to clarify the drawing.

Two Column Proof

Pacing: While two-column proofs will be used for the remainder of the text, this lesson should take one to two class periods

Goal: Students are introduced to the format of a two-column proof in this lesson. The purpose of two-column proofs is not only to prove geometric theorems. Organizing one’s thoughts in a logical manner allows students to become better writers and debaters.

Fun Tip! Use cut outs so students can begin to visualize two column proofs. Photocopy the following proof and cut it into sections. Shuffle the sections and place into an envelope. Give pairs of students the envelope and a sheet of paper with the given statement, the “to prove” statement, and the column separator. Have students sort through the rectangles and recreate the proof.

Given: \overline{AB} bisects \overline{DE};\overline{DE} bisects \overline{AB}

Prove: \triangle ABM \cong \triangle DCM

Reason Justification

Segment AD bisects segment BC

Segment BC bisects segment AD

Given
Segment AM \cong segment DM Midpoint Postulate
Segment BM \cong segment CM Midpoint Postulate
\angle AMB \cong \angle DMC Vertical Angles Theorem
\triangle ABM \cong \triangle DCM Side-Angle-Side Congruence Theorem

Neat Idea! Whiteboard makers also make a 2-column personal whiteboard. You could purchase these to perform a spontaneous check or you could make your own. Purchase plain white paneling from a home improvement store. Cut out the desired lengths then use electrical tape to construct your proof T-chart.

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CK.MAT.ENG.TE.1.Geometry.1.2

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