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3.2: 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. Cross- curricular- Music

  • Prepare several different examples of repetitive music.
  • For example, rap, classical, folk song, childrens song.
  • You want a few with a refrain or a clear consistent pattern.
  • You will need to prepare this ahead of time.
  • You want the students to develop a rule for each selection.
  • Brainstorm a list of possible pattern rules and decide on one.
  • Then write this one on the board.
  • Ask the students to come up with a counterexample to the pattern rule.
  • You can do this first one as a whole class so that the students understand the idea.
  • Then break students off into groups.
  • Have them listen to each of the other selections and write a rule and a counter example for each.
  • When finished, allow time for students to share their work.

III. Technology Integration

  • Have the students complete a google search on patterns in nature.
  • Students are going to select one or more examples of patterns in nature.
  • Ask them to write a rule for each pattern.
  • Then write a counter example for each pattern.
  • Finally allow time for students to share their work.

IV. Notes on Assessment

  • Assessment is done through class sharing.
  • Do the students understand how to write a pattern rule?
  • Are the rules accurate?
  • Do the students understand how to write a counterexample?
  • Provide feedback as needed.

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. Cross- curricular-Literature

  • Provide students with a copy of the poem “The Road Not Taken” by Robert Frost.
  • Read the poem with the class.
  • Discuss the meaning of the poem and the thoughts behind it.
  • Then tell the students that they are going to change the poem to be written in all conditional statements.
  • They can reword it if they wish.
  • Allow time for the students to work on this in small groups.
  • When they have finished, ask them if the meaning of the poem has changed with their conditional statements.
  • Ask them how conditional statements can impact different statements.
  • Allow time for the students to read their poems.

III. Technology Integration

  • Have the students use the following website to investigate conditional statements further.
  • www2.edc.org/makingmath/mathtools/conditional/conditional.asp
  • Ask the students to use the diagrams to write three different conditional statements.
  • Allow time for the students to share their work in small groups.

IV. Notes on Assessment

  • You will hear how well the students understand conditional statements by listening to their poems.
  • Provide feedback/correction as needed.

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. Cross- curricular-Mount Everest

  • Begin this activity with a discussion about deductive and inductive reasoning.
  • Review these concepts so that the students are not confused when working on this activity.
  • Have students research through books or technology facts about people who have climbed Mt. Everest.
  • Ask the students to make a list of at least ten facts about people who have climbed Everest.
  • Then write this on the board, “If you have climbed Mt. Everest, then you….”
  • Tell the students that they are to write at least five different statements using deductive reasoning to complete this statement.
  • When finished, have students share their work in small groups.
  • Ask each group to assess whether or not the students have successfully written statements using deductive reasoning.

III. Technology Integration

  • Complete a websearch on Aristotle.
  • Ask students to do some research about Aristotle and how he developed the concept of logic.
  • Students can write a short essay about this or apply it to a real life example.
  • Collect student work.

IV. Notes on Assessment

  • Collect student statements.
  • Assess them for accuracy.
  • Provide students with feedback/correction as needed.
  • When working on the technology integration, ask the students to share what they have discovered about Aristotle and logic.
  • This can become a lively discussion about how the actions of someone in the past impacts the way we work today- draw a connection to the judicial system.

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. Cross- curricular- Scale Design

  • This activity involves students exploring the concept of equality.
  • Bring in several different scales for students to work with.
  • Then prepare an assortment of items for students to work with. For example, apples, bananas, bags of flour, bags of rice, oranges, etc. You can use non food items too.
  • Students need to come up with collections of items that demonstrate equality.
  • For example, apples and oranges- can you put so many apples to equal so many oranges?
  • Have students make a list of the items that equal other items.
  • Then ask students to use the properties from the chapter and write a reflexive statement, a symmetric statement and a transitive statement about two of their equal statements.
  • Finally, allow time for the students to share their work.

III. Technology Integration

  • Have students explore how properties apply to circuits.
  • Use the following website for this exploration.
  • www.allaboutcircuits.com/vol_4/chpt_7/4.html
  • Begin a discussion about the information. Create a list of important facts on the board.
  • You can also have the students do this in small groups.
  • Students can then create their own diagrams to demonstrate how the circuit works and how algebraic properties impact circuits.
  • Allow time for the students to share their diagrams.

IV. Notes on Assessment

  • Observe students as they work.
  • Then collect all student statements and diagrams.
  • Check student work for accuracy.
  • Provide feedback/correction as needed.

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. Cross- curricular-Airports

  • Begin this activity by reviewing each of the eleven postulates in the chapter.
  • Make a list of them and their characteristics on the board.
  • Then move on to the activity.
  • Use a copy of the map of the runway at O’Hare International Airport. This is Figure02.05.01.
  • www.en.wikipedia.org/wiki/O'Hare_International_Airport
  • Ask the students to use colored pencils to find an example of each of the eleven postulates.
  • They need to use a color to highlight each example.
  • Then they can use this color as an indicator and write a description of HOW the example illustrates the postulate.
  • Do this for all of the eleven postulates.
  • When students are finished, allow time for them to share their work.

III. Technology Integration

  • Have the students do a search for different housing floor plans.
  • They can use the following website for the search or another of their own choosing.
  • www.thehousedesigners.com/
  • Then ask the students to make a list of how the different postulates apply to housing floor plans.
  • Would it be possible for houses to be designed without these postulates?
  • Conduct a class discussion on this topic.

IV. Notes on Assessment

  • Collect the airport maps and student notes.
  • Check them for accuracy.
  • Did the students follow the directions?
  • Is each of the eleven postulates represented?
  • Were the students able to write a written explanation of how the postulate is shown in the map?
  • Provide feedback/correction as needed.

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. Cross-curricular- Cooking

  • In this activity, the students are going to need to prove the following statement.
  • “You must have eggs to make a chocolate cake.”
  • Assign half of the class the job of proving that this is a true statement.
  • Assign the other half of the class the job of disproving the statement.
  • This can branch off into technology as well.
  • If students have access to computers, they can search recipes and cake information on line.
  • Some students will break right off and talk about dairy- free or vegan cakes.
  • This is great because students can talk about that, but they will need to prove it.
  • Tell students that they need at least four different statements.
  • Tell students that they will need to use resources to back up their statements.
  • Allow students time to work.
  • When finished, allow them time to share their arguments.
  • The class can assess whether they successfully proved it or not.
  • You may want to do this first in small groups.
  • Have each group select the best proof.
  • Then have a whole class debate using the best proofs.
  • Ask the students to share what worked or was challenging about this assignment.
  • Students may figure out that they can be very specific in their proof.

III. Technology Integration

  • Incorporate technology into the above activity by allowing students computer access to do their recipe/cooking searches.

IV. Notes on Assessment

  • Assess student work through the debates and discussions.
  • Collect students work and read through their proofs.
  • This is a GREAT class for demonstrating how challenging it can be to prove or disprove something.
  • For fun, you could serve chocolate cake when finished.

Segment and Angle Congruence Theorems

I. Section Objectives

  • Understand basic congruence properties.
  • Prove theorems about congruence.

II. Cross- curricular-Roller Coasters

  • Use the following image from Wikipedia for the first part of this lesson.
  • This is Figure02.07.01
  • www.en.wikipedia.org/wiki/File:Wooden_roller_coaster_txgi.jpg
  • Review the segment and angle congruence theorems from the lesson in the text.
  • Make a list of them on the board.
  • Then distribute this image to the students.
  • Students are going to work in pairs or small groups.
  • They need to use the image to explain why segment and angle congruence theorems are important to roller coaster design.
  • Allow time for the students to work on this.
  • This is a written explanation and should include the definitions from the text applied in a real life context.
  • Allow time for the students to share their work when finished.

III. Technology Integration

  • Have students complete a websearch of roller coasters.
  • Ask each student to select one that best uses the segment and angle congruence theorems.
  • Then conduct a large class discussion on this.
  • Be sure that the students see how the theorems apply in real life.
  • If segments and angles weren’t congruent, how would this impact the operations of the roller coaster?

IV. Notes on Assessment

  • Assessment is completed through class discussion.
  • Observe students as they work and listen to their ideas in the discussion.
  • Are the students connecting the theorems to the design?
  • Help them to make the connections.

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. Cross- curricular-Theorems in Art

  • Students are going to use art to prove the different theorems.
  • Use the following image from this website for this activity. This is Figure 02.08.01.
  • www.prestonsteed.com/Sale_pages/Right_Angles/Right.Angles.html
  • Then ask the students to come up with an example of each of the following theorems in this painting.
  • 1. Right Angle Theorem
  • 2. Supplements of the Same Angle Theorem
  • 3. Complements of the Same Angle Theorem
  • 4. Vertical Angles Theorem
  • Have students discuss their findings in small groups.
  • Allow time for sharing in the large group as well.

III. Technology Integration

  • Students can use the power point presentation in this website to explore different angles and their relationships.
  • www.learninginhand.com/lessonplans/angles.html
  • The activities themselves are an excellent integration of technology.

IV. Notes on Assessment

  • Assess student work through observation.
  • Walk around and listen to students as they discuss the painting.
  • Sit in and interject thoughts are ideas.
  • Are the students connecting the theorems to the painting?
  • Are they discovering the angle relationships?
  • Offer feedback/suggestions as needed.

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