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# 5.2: Reasoning and Proof

Created by: CK-12

## Inductive Reasoning

I. Section Objectives

• Recognize visual patterns and number patterns.
• Extend and generalize patterns.
• Write a counterexample to a pattern rule.

II. Problem Solving Activity-Pascal’s Triangle

• Students are going to work with a diagram of Pascal’s Triangle for this activity.
• Pascal’s Triangle is Figure02.01.01
• http://en.wikipedia.org/wiki/Pascal%27s_triangle
• Student are going to problem solve to find a rule to Pascal’s Triangle.
• Have students work in small groups.
• They can use color on the triangle to point out different patterns.
• Allow a lot of time for the students to explore the patterns of the triangle.
• Ask them to use the Wikipedia pattern to write the rule for the triangle.
• Once they have the rule, they need to write the next two rows of the triangle.
• Then demonstrate two ways that you know that your rule is accurate.
• Finally, write a conjecture and a counterexample for the rule.
• Allow time for student sharing.

III. Meeting Objectives

• Students will recognize visual patterns and number patterns.
• Students will be required to extend and generalize patterns in Pascal’s Triangle.
• Students will write conjectures and counterexamples of their rule.

IV. Notes on Assessment

• Do some independent study on Pascal’s Triangle prior to completing this activity.
• Ask leading questions if students are stuck, but refrain from offering solutions.
• Encourage students to help each other with the patterns if they are having difficulties.

## Conditional Statements

I. Section Objectives

• Recognize if- then statements.
• Identify the hypothesis and conclusion of an if- then statement.
• Write the converse, inverse and contrapositive of an if- then statement.
• Understand a biconditional statement.

• Students are going to use newspapers and magazines for this problem solving activity.
• Begin the activity by talking about how advertisers use conditional statements to lure people into purchasing their products. For example, a phone company will often offer a free phone for a cell phone plan.
• Note: If you can find one such add it would be great to bring it in for a demonstration.
• Tell students that their assignment is to use newspapers and magazines to find one such conditional advertisement.
• Then they are to take that advertisement and create a display using it to show the converse, inverse, contrapositive and biconditional statement of the advertisement.
• Students can decorate and design their display.
• Allow time for students to share their work when finished.

III. Meeting Objectives

• Students will write converse, inverse and contrapositive statements.
• Students will understand biconditional statements by writing them.
• Students will present their work to their peers.

IV. Notes on Assessment

• Be sure that the students have selected a conditional statement in an advertisement.
• Check their work for accuracy when writing each of the different statements.
• Allow time for students to share their work.
• Include creativity in student evaluations.
• Students could receive a classwork or homework grade for this assignment.

## Deductive Reasoning

I. Section Objectives

• Recognize and apply some basic rules of logic
• Understand the different parts that inductive reasoning and deductive reasoning play in logical reasoning
• Use truth tables to analyze patterns of reasoning

II. Problem Solving Activity-Write It Out

• Students are going to write statements based on Figure02.03.01 of vertical angles.
• Draw the figure on the board/overhead.
• Underneath it write the words “Vertical angles are congruent.”
• Put the students in groups.
• There are four “starters” on index cards. A starter card gives the students a beginning geometric statement. They then need to take this statement and complete it using the Law of Detachment or the Law of Syllogism.
• For example, if the starter is $\angle 1 \cong \angle 2$, then the students would need to write two other statements using the figure as a guide.
• Possible answers could be: $\angle 1$ and $\angle 2$ are vertical angles. Vertical $\angle$’s are congruent.
• Here are the three other starters: $\angle 1$ and $\angle 4$ are supplementary angles.
• $\angle 3$ and $\angle 4$ each equal $55^\circ$.
• $\angle 2$ and $\angle 3$ are adjacent angles.
• $\angle 3$ and $\angle 4$ are congruent.
• Finally, when finished, ask students to identify whether the Law of Detachment or the Law of Syllogism was at work in each set of statements.

III. Meeting Objectives

• Students will use the basic rules of logic.
• Students will understand the role of inductive and deductive reasoning.
• Students will apply this reasoning to geometric content.

IV. Notes on Assessment

• Observe students as they work.
• If groups are struggling, refer them back to their notes on previously learned material.
• Refrain from offering suggestions.
• Give feedback based on content and accuracy.

## Algebraic Properties

I. Section Objectives

• Identify and apply properties of equality
• Recognize properties of congruence “inherited” from the properties of equality
• Solve equations and cite properties that justify the steps in the solution
• Solve problems using properties of equality and congruence

II. Problem Solving Activity-Match It Up

• Students are going to create a matching game that they can then play in small groups.
• Each small group needs to create a pair for each of the properties. One card will have the name of the property on it, and the match will be a numerical example, and or a geometric example.
• Be sure that the students write out an actual example of the property and not just variables as they did in class.
• This will help them to take the lesson in the text to a new level.
• Be sure that the students have index or small cards, pens, rulers, etc.
• After the cards have all been created, have one group exchange with another group and play that team’s game.
• When finished, ask the teams to give each other feedback on the examples used.
• Here are the properties to use:
• Reflexive Property
• Symmetric Property
• Transitive Property
• Substitution Property
• Multiplication Property of Equality
• Reflexive Property of Congruence with segments and angles
• Symmetric Property of Congruence with segments and angles
• Transitive Property of Congruence with segments and angles

III. Meeting Objectives

• Students will identify and apply properties of equality.
• Students will solve equations and cite properties in their examples.
• Students solve problems using the properties.

IV. Notes on Assessment

• This activity has two parts. The first part is to observe the students as they work on creating the game.
• The second part is to watch them play it.
• Because they are switching game cards with another team, any errors will quickly come to light.
• Be sure to allow time for feedback/correction.

## Diagrams

I. Section Objectives

• Provide the diagram that goes with a problem or proof.
• Interpret a given diagram.
• Recognize what can be assumed from a diagram and what can not be
• Use standard marks for segments and angles in diagrams.

II. Problem Solving Activity-Name That Postulate!

• This is a game. The students will create the game cards and then a “Jeopardy” kind of game can be played in the large class or in small groups.
• Students are assigned the task of creating an index card with a diagram that represents each postulate.
• Students should use diagrams and also standard marks for segments and angles in their examples.
• There are eleven postulates, so if there are twenty- two students in the class, each postulate would be represented by two different diagrams. You need to assign the students the postulates to avoid too many repeats.
• Allow time for the students to create their diagrams and then use peers to check each other’s work for accuracy.
• When finished, collect the cards and play the game with the students.

III. Meeting Objectives

• Students create diagrams to better understand postulates.
• Students interpret given diagrams when playing the game.
• Students use standard marks for segments and angles when creating their game cards.

IV. Notes on Assessment

• Assessment is easier with this lesson because the students will be playing the game. You will be able to see who understands the postulates and who doesn’t.
• Also, having students check each other’s work before playing the game will definitely help to catch any errors.
• You can help add any corrections when playing the game and looking at each game card.

## Two- Column Proof

I. Section Objectives

• Draw a diagram to help set up a two- column proof.
• Identify the given information and statement to be proved in a two- column proof.
• Write a two- column proof.

II. Problem Solving Activity-Wind Generators

• Use a figure like this one of a wind generator. This is Figure 02.06.01
• www.blaineschools.org/Schools/WRMS/Tech/Zsupportin_documents/Images/wind_generator.jpg
• Here is the problem.
• “Mike Eisele did an experiment for his science project to figure out which angle of degree on a propeller of a wind generator would be the most efficient. He figured out that $75^\circ$ was the most efficient. Your task is to take this given information and write a proof to using geometric principles. We’ll call one angle of the propeller angle $1$ and the other angle $2$.”
• Show students the diagram of the wind generator. Point out the two angles that you are working with and then write this information on the board.
• On Board:

Given: $m \angle 1 & = 75^\circ\\\angle 1 & \cong \angle 2$

Prove: $m \angle 2 = 75^\circ$

III. Meeting Objectives

• Students will use a diagram to help set up a two- column proof.
• Students can draw a diagram of a wind generator and label the given angles.
• Students will write a two- column proof.

IV. Notes on Assessment

• Here is a possible answer for the given proof.
Statements Reasons
$m \angle 1 = 75$ $\angle 1 \cong \angle 2$ Given
$m \angle 1 = m \angle 2$ Definition of Congruent Angles
$75 = m \angle 2$ Substitution
$m \angle 2 = 75$ Symmetric Property
• Observe students while they work. Offer assistance when necessary.
• If you want to learn more about Mike Eisele and his experiment, see the Enrichment section of this Teacher’s Edition. The website about Mike and his experiment is www.share3.esd105.wednet.edu/mcmillend/02SciProj/ReeseC/reesec.html#Experimental

## Segment and Angle Congruence Theorems

I. Section Objectives

• Understand basic congruence properties.

II. Problem Solving Activity-Angle or Segment?

• For this problem solving activity, you will need to prepare a set of cards with angle statements and a set of cards with segment statements.
• Use numbers or letters to mark each card. Then you will know which statements the students were working with.
• Write each angle statement as reflexive, symmetric or transitive.
• Write each segment statement as reflexive, symmetric or transitive.
• Students are going to each be given a card with either an angle statement or a segment statement on it.
• Remind students to label their work with a letter or number that matches the card that they have been given.
• Then the students need to write out the property for the card and draw a diagram that illustrates the statement on the card.
• Students will need rulers, pencils and protractors for this assignment.
• All work should be accurate and measured.
• Allow a certain amount of time for this first card, when finished, ask the students to pass the card to a neighbor and repeat this assignment. You want the students to each work on two different angle cards and two different segment cards.
• This will help to secure student understanding.
• Have students share their work in small groups when finished.

III. Meeting Objectives

• Students will understand basic congruence properties.
• Students will prove basic congruence properties through diagrams and group discussions.

IV. Notes on Assessment

• Collect student work.
• Check each student’s work for accuracy and offer written feedback.

I. Section Objectives

• State theorems about special pairs of angles.
• Understand proofs of the theorems about special pairs of angles.
• Apply the theorems in problem solving.

II. Problem Solving Activity-Judges Table

• Before explaining the activity, select four students to serve as judges.
• Explain that the students are going to need to use theorems and proofs to “PROVE” each statement.
• The judges will be deciding if the students have successfully proved their statement.
• Students should work in groups of three for this assignment.
• The judges are also going to need to complete the work for all of the statements that way they know whether or not students have successfully proven their statement.
• Use Figure02.08.01- provide each group with a copy of the diagram.
• Here are some possible statements:
• $\overline{AD} \cong \overline{BC}$
• Given that $\angle 1 \cong \angle 2$, which other angles are congruent?
• $m \angle 5$ and $m \angle 6 = 90^\circ$
• You can create as many different statements as you would like.
• Allow time for the students to work and then they present their case to the judges.
• The judges accept it or decline it. If accepted, students can work on another statement. If declined, the students need to go back and try again.

III. Meeting Objectives

• Students will state theorems about special pairs of angles.
• Students will understand how to prove theorems about special pairs of angles.
• Students will apply the theorems in problem solving.

IV. Notes on Assessment

• Sit on the panel with the judges.
• Listen to the statements and offer feedback.
• The students can be given extra credit for the number of statements that they are able to prove.
• Students could also be given a classwork grade for this assignment.

## Date Created:

Feb 22, 2012

Feb 23, 2012
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