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Inductive Reasoning

I. Section Objectives

  • Recognize visual patterns and number patterns
  • Extend and generalize patterns
  • Write a counterexample to a pattern rule

II. Multiple Intelligences

  • Include group work in this lesson. Rather than explaining all of the information in the lesson and then assigning group work, intersperse the group work with the lesson.
  • Begin by going over visual patterns.
  • Then have students work in pairs. Each student draws a visual pattern. Then they exchange papers with their partner. Our next step is to write a rule for the pattern they have been given, and to extend the pattern two steps.
  • Next, go over number patterns.
  • Then have the students work in pairs. Each student writes a number pattern. Then they exchange papers with their partner. Our next step is to write a rule (an equation) for the number pattern and to extend the pattern two steps.
  • Finally teach about conjectures and counterexamples.
  • Have students work with the patterns that they have previously worked with and write a conjecture and a counterexample for each pattern.
  • Multiple Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, spatial- visual, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Write all vocabulary on the board. Request that students copy this information down in their notebooks.
  • Vocabulary
    • Conjecture
    • Counterexample

IV. Alternative Assessment

  • Collect student papers.
  • Review each student’s work to assess understanding.
  • Use this to review at the beginning of the next class. You can use different student patterns in the beginning of the next class to review the previously learned material.
  • This will be especially helpful to special needs students who require a lot of review to recall previously learned concepts.

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. Multiple Intelligences

  • Students need to develop a good understanding of conditional statements and related statements in this section. Because of this, teach all of the material in the lesson and then you can use the following activity to expand student understanding.
  • Divide students into four groups. Assign each group one of the following: converse, inverse, contrapositive and biconditional.
  • Then write a conditional statement on the board/overhead.
  • Each group needs to work together to write a related statement for the given conditional statement.
  • When finished, go over the student answer as a class. Use other peers in the class to do any correcting that is needed.
  • Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, interpersonal, intrapersonal

III. Special Needs/Modifications

  • This is a difficult lesson for special needs students to understand because the language and symbols are so verbal. Many students with language based learning disabilities will find this challenging. Here is one option on how to scaffold the information in this text.
  • Alter these definitions and provide an example (in words not symbols) for each.
  • Converse- switch the hypothesis and the conclusion of the conditional statement.
  • Inverse- add not to the conditional statement to negate it.
  • Contrapositive- add not to the converse to negate it.
  • Biconditional statement- combine the conditional statement and its converse together.
  • Allow time for the students to work with these definitions and to copy them into their notebooks.

IV. Alternative Assessment

  • Use the answers from each group to assess student learning.
  • If the students are having difficulty with the activity, then after the first example add another example.
  • Repeat as necessary until the students have a good grasp of the information.

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. Multiple Intelligences

  • Teach the concepts in this lesson prior to completing the activity. This activity involves music and will assist students in a concrete example of how to write a conclusion using inductive reasoning and then use deductive reasoning to prove their conclusion.
  • Prepare a five or six music samples for students to listen to. Be sure that all of the samples are from the same genre of music, for example, all jazz. You can even make a few of them by the same artist. The students will have a couple of different options to draw a conclusion from.
  • Then allow students time to work in a pair and to write a conclusion about the samples that they have heard.
  • Have each pair exchange their conclusion with another pair. Then the students must use deductive reasoning to prove that the statement is correct.
  • Intelligences: linguistic, logical- mathematical, musical, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Begin the class with a review of the following terms. Knowledge and understanding of these terms is implied in this lesson.
    • Linear pair
    • Adjacent angles
    • Vertical angles
    • Supplementary angles
    • Complementary angles
  • Write the following new definitions on the board/overhead. Request that students copy this information down in their notebooks.
    • Law of Detachment
    • Law of Syllogism
    • Inductive Reasoning
    • Deductive Reasoning
  • Truth Tables- go over this information very slowly. Be sure that the students understand all of the symbols prior to going over the section in the textbook. You could even create a chart of symbols and explanations on the board for easier understanding.

IV. Alternative Assessment

  • Notice how students react to the music cues. Then notice the conclusions that they write about the music. Be sure that they prove their statements clearly. If they have difficulty, then provide examples or use another peer group to coach these students.

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. Multiple Intelligences

  • There is a lot of information given in this lesson. Rather than using an activity, here are some suggestions on how to differentiate all of this information so that all learners are engaged.
  • Begin lesson by writing the intention of the lesson on the board/overhead. Intention- “to combine geometric building blocks with reasoning.”
  • Throughout the lesson, refer back to this statement. Identify the geometric building blocks for the students. For example, when there is an example on angles, refer back to student notes on angles. Ask students to brainstorm a few things that they have learned about angles. Then continue with the lesson.
  • Review equality definition.
  • Write all Properties on one half of the board.
  • Write the statements of congruence on the other half of the board.
  • Show students how to combine these two together visually. Use different colored chalk or pens (on a whiteboard) to illustrate how combining these statements together can help to prove the given statement.
  • Practice this with a few examples. Encourage class participation.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • If you have many special needs students in the class, you may want to break up this lesson over two days.
  • Day one- review properties and statements of congruence. Review basics of geometry.
  • Day two- show students how to combine the two together.
  • Here are some steps for combining statements and properties.
  1. Look at whether you are working with line segments or angles. This will help you choose a statement of congruence.
  2. Choose a property that explains the given statement.
  3. Combine the statement of congruence and the property together for a final answer.

IV. Alternative Assessment

  • Verbally and visually check- in with students to be sure that they are following the lesson. If necessary, go back over previously learned information so that you are sure to have everyone following along.

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. Multiple Intelligences

  • Complete this activity half-way through the lesson. Once you have gone over the definitions of the eleven postulates, divide the students into eleven groups.
  • Assign each group a postulate.
  • Request that each group design a diagram that best proves their given postulate.
  • When finished, have each group share and justify their diagram.
  • Request that they explain how the diagram illustrates the proof.
  • Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, interpersonal, intrapersonal.
  • When finished, go back to the text and work on the section where we combine the postulates, properties and statements of congruence together.

III. Special Needs/Modifications

  • Review all properties from the previous lesson.
  • Review the statements of congruence.
  • Review the steps for combining the two together.
  • Write all of the postulates on the board/overhead. Request that the students make a three column chart for each.
Postulate Definition example

IV. Alternative Assessment

  • Work with the student groups to understand proving each postulate.
  • Assessment is done through observation and verbal questions.
  • Provide students with plenty of “think time” so that you receive the most accurate response.

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. Multiple Intelligences

  • To work through this lesson, allow students to work in pairs and discuss their work. There are logical- mathematical, visual, linguistic, interpersonal and intrapersonal aspects to working with a peer.
  • Since many students have a difficult time with proofs, one suggestion is to provide students with a fill- in the blank proof before working on the exercises at the end of the section.
  • You can use the proof on page 101- 102 to do this or write one of your own.
  • Provide the students with a series of statements on the board. The students fill- in the reasons.
  • Provide students with a two- column proof where some of the statements are blank and some of the reasons are blank. Where there are blank statements, the students will need to use the reason to write a statement. Where the reason is blank, the students will need to use the statement to write the reason.
  • This will assist the students in interacting with the information in the proof and discussing it will help with retention.
  • Now students should be able to complete number fifteen in the exercises which requires that they write a two- column proof without any assistance.

III. Special Needs/Modifications

  • Be sure that the students have a current list of postulates, properties and vocabulary where they can access it easily.
  • Here are some helpful hints for students in working on two- column proofs.
  1. Draw a diagram to better understand the vocabulary in the given. For example, if you are working on proving that points are collinear, then draw a diagram of the collinear points. Then look at what statements and reasons you can write about it.
  2. Look at the vocabulary in the given. Example 2 has the word “bisects” in it. Therefore, you will need a statement and reason that explains bisects.

Example 2 also has a congruent symbol in it. Therefore, you will need a statement and a reason that addresses congruency.

IV. Alternative Assessment

  • Give a homework assignment where students write their own two- column proof based on a common given.
  • The next day review the assignment and answers with the students. In small groups, have them write one “best” proof for the given.

Segment and Angle Congruence Theorems

I. Section Objectives

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

II. Multiple Intelligences

  • The best way to address different learning styles in this lesson is to use diagrams and to teach this lesson as a class discussion.
  • The students will need to break down the concepts provided to gain an excellent understanding of the material.
  • Prior to teaching the lesson, write the intention on the board or overhead. This will assist all visual learners and help special needs students too.
  • “To prove congruence properties, we turn congruence statements into number statements, and use properties of equality.
  • Here are some steps to write on the board:
  1. Take the given and notice whether you are working with segments or angles.
  2. Think of converting to measurement. For example \overline{AB} \cong \overline{AB}, as a statement, we can say that AB = AB. We are working with the measurement or length of the segment here. We have changed this to numbers. With angles, change to show the measurement of the angles is equal.

III. Special Needs/Modifications

  • Be sure that students have a page of notes out that explain the properties of equality.
  • Review each of the properties and what each means.
  • Show students how to draw a diagram to illustrate a given statement. A picture often helps special needs students.
  • Explain that postulates don’t need to be proven.
  • Explain that theorems need to be proven.

IV. Alternative Assessment

  • Use an observation checklist to observe students as they work.
  • Notice who is having difficulty and who isn’t. Be sure to make a note of those students so that they can be offered assistance or a peer tutor.
  • When possible, use peers to help explain concepts. The student teaching and the student learning both benefit greatly.

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. Multiple Intelligences

  • Use the following activity to assist students in understanding theorems about special pairs of angles.
  • Teach the content in the lesson. Be sure that the students have a good understanding of the concepts in the lesson. Then divide students into groups.
  • Each group must make a diagram that illustrates each of the following theorems. They need to write a two column proof to show how the diagram illustrates the theorem.
  • When all have finished, allow students time to share their work with the class.
  • If time does not allow for students to work on all four of the theorems, assign each group a different theorem to work with.
  • Students can complete their work using colored pencils, rulers and large chart paper.
  • Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Review the basic definitions of right angles, supplementary angles, complementary angles, and vertical angles.
  • Draw an example of each on the board/overhead.
  • Explain to students how to move from the basic definition of each angle relationship to the theorem.
  • Write each of the theorems on the board/overhead and request that students copy this information in their notebooks.

IV. Alternative Assessment

  • Examine the work that the students completed on their chart paper. Notice which groups were more accurate than others.
  • Pay attention to which groups expressed their reasoning well verbally.
  • Notice which groups expresses their reasoning well in the diagram.
  • If a letter grade is needed, assign the same assignment for homework and then grade student work the following day. This will give you an excellent understanding of which students have a good grasp of the concepts and who still needs more practice.

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