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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.

II. Problem Solving Activity-Advertisements

  • 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 recognize if- then statements in advertisements.
  • 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
    • Addition Property of Equality
    • 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.
  • Prove theorems about congruence.

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.

Proofs about Angle Pairs

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.

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