## 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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