<meta http-equiv="refresh" content="1; url=/nojavascript/"> Reasoning and Proof | CK-12 Foundation

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.
• 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.
• 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.

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.

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.

Date Created:

Feb 22, 2012

Feb 23, 2012
You can only attach files to None which belong to you
If you would like to associate files with this None, please make a copy first.