wvde.us · Web viewand discuss the deductive process as a chain of conditional statements that together produce a convincing argument. Distribute to students a 2.03 Flow Proof Story

Lesson PlanLesson 2: Getting Ready for Proof #2

Mathematics High School Math II

Unit Name: Unit 5: Similarity, Right Triangle Trigonometry, and Proof

Lesson Plan Number & Title: Lesson 2: Getting Ready for Proof #2

Grade Level: High School Math II

Lesson Overview: Students will continue to hone their reasoning skills, to justify conjectures with convincing arguments, and to assemble and connect the language of geometry developed in Math I for use in constructing more formal proofs. This lesson is designed for 45 – 90 minutes depending on the sophistication of the understanding of the students.

Focus/Driving Question: How can convincing arguments be formulated to support conjectures using conditional statements and valid reasons?

West Virginia College- and Career-Readiness Standards:

M.2HS.42 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Implementation may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.2HS.C.3. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.

.Manage the Lesson:

This lesson is the second of two lessons that will provide the scaffolding for the understanding of formal proof. Up to this point, the work of geometry has been primarily one of conjecture based upon investigations. Methods of rigorous proof will be addressed later in the unit in the context of specific theorems. A variety of approaches to understanding these theorems will afford accessibility for all students, with increased demand placed upon the STEM student who will be progressing to Math III STEM.

Academic Vocabulary Development:

Vocabulary addressed in this unit will include primarily the vocabulary developed in Math I, with an emphasis on how these terms, along with the postulates and theorems will become the building blocks for creating credible proofs. Students should refer to the foldables and glossaries compiled in Math I.

Launch/Introduction:

To encourage students to organize their thinking, launch the lesson with the Digit Place Game. Using a document camera or the chalkboard draw four columns and label them Step, Guess, Digits, and Place. The object of the game is to determine a secret number in as few guesses as possible. The teacher has selected the secret two-digit number. (Later a three-digit number will be used.) The class will try to figure out what the secret number is by considering some specific clues. No digits can be repeated and no

number can begin with 0. As a student makes a guess, the teacher will record the guess in the second column (the first column will list the step number as 1). The teacher will respond with two values. The first value will be recorded in the Digits column and will stand for the number of correct digits in the guess. The second value will be recorded in the Place column and will stand for the number of correct digits which are located in the correct place. The next student who makes a guess will use these responses to refine his/her guess for Step 2. Students should be asked to explain why they have made a particular guess and how they used the information from the previous guess. The game continues until the secret number is discovered. Distribute to students 2.01 Analyzing the Digit Place Game for students to discuss with a partner. Facilitate the exchange of ideas, making sure that students are discussing the choices and justifying their reasoning. Students may play the game with a partner using a three-digit number or the game can be continued as a whole class.

Investigate/Explore:

Ask students what it means to them when someone says, “Prove it!” Discuss how proof could be defined. (Proof is a convincing argument that something is true; extend that idea to mathematics as a rigorous mathematical argument which unequivocally demonstrates the truth of a given statement.)

Read to the class a 2.02 Sherlock Holmes Passage from Arthur Conan Doyle’s short story The Adventure of the Dancing Men and discuss the deductive process as a chain of conditional statements that together produce a convincing argument. Distribute to students a 2.03 Flow Proof Story as a quick way of demonstrating to students that a flow proof is an ordered series of statements.

Ask students to recall their work with solving equations and how they justified the steps of solution using the properties of equality. Distribute 2.04 Getting Ready for Proof to provide a quick review of the properties of equality and how they are used to justify reasoning.

In his set of books, collectively called Elements, Euclid defined many geometric terms with a goal of creating a common language about geometry that would enable mathematicians to communicate their discoveries. Likewise, students need to develop that common language so that their classmates will be able to understand their arguments. In addition to the properties of equality, students will need to recall vocabulary and basic concepts that were addressed in Math I. The “common notions” of Euclid will be regarded as basic assumptions. These include:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Prior to class draw on a piece of patty paper a large capital L with the horizontal segment of length 5 centimeters and the vertical segment of length 8 centimeters. Without showing the “L” to the students, describe it as follows: “My shape has a horizontal segment and a vertical segment that together look like a capital L.” Ask the students to draw the figure on white boards, if available, and then display them to

the class. Ask students why the shapes are not the same, and what other information would allow them to draw the figure more accurately. Most will realize that they need to know more about the length of the horizontal and vertical segments. Tell students that the horizontal segment is of length 5 and the vertical segment is of length 8. Ask if this information is sufficient. Most students will recognize that units of measurement are also needed. Tell students then that the horizontal segment is of length 5 centimeters and the vertical segment is of length 8 centimeters. Now ask students if they can draw the figure. Students will then know that they need a ruler marked in centimeters to recreate the figure that the teacher has drawn. After they’ve drawn the figure, place your figure on patty paper atop some of the students’ figures. This short activity should emphasize to students that size is an important attribute used to describe any figure and using the same measurement units is essential to the description. Introduce Postulate 1, the Ruler Postulate, which will prove critical to the work developed with dilations. In addition to Euclid’s assumptions, Postulate 1, the Ruler Postulate, “Given a ray and a positive number, there exists exactly one point on the ray whose distance from the endpoint of the ray is the number,” will also be accepted without proof.

As a review of Math I geometric concepts, students can play a Jeopardy game that incorporates many of the terms that will be utilized as justifications in proofs in this unit. Suggested questions can be found at 2.05 PowerPoint Jeopardy Game. Students can play the game using polleverywhere.com or simply white boards. They can record their answers on a 2.06 Jeopardy Game Sheet, if desired. The Transformation Graphic Organizer that was used in Lesson 8 of Unit 5 in Math I could prove useful in reviewing the transformations and their properties.

Summarize/Debrief:

Discuss with students that getting all of these ideas in place will be advantageous as they seek to justify theorems in this unit. Emphasize that it is important for them to be comfortable with all of the concepts learned in Math I so they can use them to draw conclusions in this course.

Materials:

2.01 Analyzing the Digit Place Game2.02 Sherlock Holmes Passage2.03 Flow Proof Story2.04 Getting Ready for Proof2.05 PowerPoint Jeopardy Game2.06 Jeopardy Game Sheet2.07 Journal Writing on the Digit Place Game

Career Connection:

Although logical reasoning skills are advantageous in most professions, they are particularly needed in a variety of STEM careers from computer science to engineering. Additionally, they would be an asset to careers in the Health Sciences.

Lesson Reflection:

Students will be asked to play the Digit Place Game with their families, and then complete a 2.07 Journal Writing on the Digit Place Game describing the experience.

In lesson 1, teachers were provided with a guide to aid them in reflecting upon the lesson as they seek to improve their practice. Certainly, it may not be feasible to formally complete such a reflection after every lesson, but hopefully the questions can generate some ideas for contemplation.

ANALYZING THE DIGIT PLACE GAME

Suppose you are playing the Digit Place Game and the first guess is the following:

a) Could the answer be 31? Explain to your partner why or why not.

b) Could the answer be 15? Explain why or why not.

c) Which would give you more information, a guess of 37 or a guess of 76? Justify your choice. What information would your guess provide?

Now suppose in a second game, the following two guesses have been made with the following results:

a) Explain why we know that the solution to this problem is in the 20's.

b) Are there any numbers in the 20's that can't be a solution? Explain to your partner which number(s) and why.

c) Why would 51 be a better third guess than 20? Write an explanation of your reasoning.

STEP GUESS DIGITS PLACE

1 13 1 0

STEP GUESS DIGITS PLACE

1 34 0 0

2 42 1 0

Arthur Conan DoyleThe Adventure Of The Dancing MenHolmes had been seated for some hours in silence with his long, thin back curved over achemical vessel in which he was brewing a particularly malodorous product. His head was sunkupon his breast, and he looked from my point of view like a strange, lank bird, with dull grayplumage and a black top-knot.

"So, Watson," said he, suddenly, "you do not propose to invest in South African securities?"

I gave a start of astonishment. Accustomed as I was to Holmes's curious faculties, this suddenintrusion into my most intimate thoughts was utterly inexplicable.

"How on earth do you know that?" I asked.

He wheeled round upon his stool, with a steaming test-tube in his hand, and a gleam ofamusement in his deep-set eyes.

"Now, Watson, confess yourself utterly taken aback," said he.

"I am."

"I ought to make you sign a paper to that effect."

"Why?"

"Because in five minutes you will say that it is all so absurdly simple."

"I am sure that I shall say nothing of the kind."

"You see, my dear Watson" -- he propped his test-tube in the rack, and began to lecture with theair of a professor addressing his class --"it is not really difficult to construct a series ofinferences, each dependent upon its predecessor and each simple in itself. If, after doing so, onesimply knocks out all the central inferences and presents one's audience with the starting-pointand the conclusion, one may produce a startling, though possibly a meretricious, effect. Now, itwas not really difficult, by an inspection of the groove between your left forefinger and thumb, tofeel sure that you did NOT propose to invest your small capital in the gold fields."

"I see no connection."

"Very likely not; but I can quickly show you a close connection. Here are the missing links of thevery simple chain: 1. You had chalk between your left finger and thumb when you returned fromthe club last night. 2. You put chalk there when you play billiards, to steady the cue. 3. Younever play billiards except with Thurston. 4. You told me, four weeks ago, that Thurston had anoption on some South African property which would expire in a month, and which he desired youto share with him. 5. Your check book is locked in my drawer, and you have not asked for thekey. 6. You do not propose to invest your money in this manner."

"How absurdly simple!" I cried.

"Quite so!" said he, a little nettled. "Every problem becomes very childish when once it isexplained to you. Here is an unexplained one. See what you can make of that, friend Watson."He tossed a sheet of paper upon the table, and turned once more to his chemical analysis.

http://www.eastoftheweb.com/short-stories/UBooks/AdveDanc.shtml

What is a transformation? Define the following in your own words:

Translation:

Reflection:

Rotation:

Dilation:

How are the image and the preimage related?

When is prime notation used?List the properties that are preserved for each of the above transformations?Translation:

Reflection:

Rotation:

Dilation (only the dilation considered in class):

What is an isometry?

What is a rigid and a non-rigid motion?

Which of the transformations are rigid and which are non-rigid motions?

JOURNAL WRITING

Last night you played the “Digit Place” game at home. Describe each of the following:

how each game progressed who played the game how many times you played the game how easy or hard it was to explain the directions strategies used by both players any discussion related to the game that occurred during or after

the game