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 , then the students would need to write two other statements using the figure as a guide.
- Possible answers could be: and are vertical angles. Vertical ’s are congruent.
- Here are the three other starters: and are supplementary angles.
- and each equal .
- and are adjacent angles.
- and 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 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 and the other angle .”
- 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:
Prove:
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 |
---|---|
Given | |
Definition of Congruent Angles | |
Substitution | |
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:
- Given that , which other angles are congruent?
- and
- 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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Feb 22, 2012Last Modified:
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