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Unit 2 Triangle similarity October 1, 2012

Transcript

Unit 2

Triangle similarityOctober 1, 2012

DO NOW

• NO D.E.A.R TODAY for 3rd block• Triangle A’B’C’ is a translation image of

triangle ABC. What is the rule of the translation?

A’B’C’ ordered points are: A’ (-2, 5)B’ ( 1, 2) and C’ ( 3, 4)ABC ordered points are:A( 3, - 1) B (6, -4) and C ( 8, -2)

Activating/launch

How do you remember the names? Think cycles!

Monomial, Binomial, TrinomialThere are special names for polynomials with 1, 2 or 3 terms:

Balance – what is balance? What can we balance? Why balance?

What motive is there for understanding balance, especially in a math class?

What is a polynomial?

made up of terms that are only added, subtracted or multiplied.

what are polynomials.htm

Matching Expressions and Words

P-4

Add four to n, then multiply by two.

Add two to n, then multiply by four.

Multiply n by four, then add two

4n + 2

4(n +2)

2(n + 4)

We do:

YOU DO

• Each student will receive a copy of the assessment task Interpreting Expressions and a mini-whiteboard, pen, and eraser.

• Each pair of students will need a copy of Card set A: Expressions, Card Set B: Words, a glue stick, a felt-tipped pen, and a large sheet of paper or card for making a poster.

Wrap up• Summary: How are theorems/rules useful in everyday

school subjects and life outside of school? • Where are ratios found in everyday life?• Homework: Assigned 10/1 due 10/5• Devise an area for your pet, to play basketball, soccer,

football, to dance, or act. Come up with a blueprint, label the coordinates of your area, (create on graph paper) how would you come up with the scale factor for the actual area, how would you calculate perimeter of the area if the units are in feet? In addition, you want to put a wall/fence around your area, the fencing or wall is $3 per yard, how much will it cost to fence the entire area?

• 15 minutes D.E.A.R

10 – 2 - 121. DO NOW – what is a ratio? What is scale factor?

And what is the image of P (3, - 4) reflected across line x = - 1

2. Launch – video 3. Discuss rotations, theorems, reflections, glide

reflections4. Symmetry 5. Summary: How are theorems/rules useful in

everyday school subjects and life outside of school?

rotations

• Pg. 603

• Reflections• Pg. 612

• Glide reflectionsWhat is the image of

triangle TEX for glide translation where translate is (x, y – 5) and line reflection is x = 0

Stations – ratios of rectangles

• Small groups of 3 or 4• Answer question on the graph paper• Turn in, this is an assignment • 9-4 symmetry (form K)• similar polygons preview

practice

• Assessment• Skill based task– worksheet

• Practice– Problem task

Each set of partners/threes has to create a poster and an example for

their specific rule• Theorems 6 – 9 Proofs using coordinate

geometry• Properties of Parallelograms pg. 369A• Special Parallelograms pg. 396 “special

parallelograms” 6-13, 6-14• Coordinate geometry pg. 369B

• Why should we use variables as coordinates when writing a coordinate proof?

• Polygon Angle sum theorem pg. 371• Corollary to the polygon angle sum thm pg. 372• Polygon exterior angle sum thm pg. 373• Theorem 6-3, 6-4, 6-5, 6-6 pg. 378 – 384

parallelogram• Theorem 6-7 pg. 385• Theorem 6-8, 6-9, 6-10 pg. 388• Theorem 6-11, 6-12 pg. 390• Theorem 6 -16, 6 -17, 6- 18 pg. 404• Theorem 6 – 19, 6-20, 6-21 pg. 410• Theorem 6-23 pg. 415

10-2-12

• Academic Vocabulary: center of dilation, line segment

Skills to master– Given a center & scale factor, experiment to

visually see that when performing dilations the line segment, the pre-image, the segment which becomes the image is longer or shorter based on the ratio given by the scale factor

Additional skills to master

• Given a line segment, a point not on the line segment, and a dilation factor, construct a dilation of the original segment

• Recognize that the length of the resulting image is the length of the original segment multiplied by the scale factor & that the original & dilated image are parallel to each other

Activities

• We do: 9-5 enrichment questions and regular 9-5 questions

• Homework: 9 -6 compositions of reflections due Wednesday Oct 3rd

assessment

• Skill based– Create a dilation of

segment AB through C, with a scale factor of 2 to create segment EF. Find the lengths of EF, AC, BC, CE, and CF

• Problem Task– Locate the center of

dilation and scale factor in the following pair of triangles

– Need ordered pairs for triangles TRS, T’R’S’

Wrap up

• How is scale factor calculated? How is it applied to create images from transformations?

10-3-12

• Do Now• Launch• Instructions & Review• Quiz

Think about a Plan

• Similar polygons 7-2

Teacher Created Argumentation Tasks (W1-MP3&6)•Your classmate provides the following solutions to the problems below. In complete sentences, identify and explain the error in each explanation, and tell me how you would help your classmate reach an accurate conclusion. (From www.pearsonsuccessnet.com – Chapter 7: Find the Errors! for sections 7.2-7.3)

Skill-based task•Solve the following proportion

• Prentice Hall Geometry Textbook1.Pg. 437 # 27 – 32

The triangles below are

H.O.T ?

Wrap up

G-SRT.2 Ratios, Proportions, Similar Polygons

• What you will discover:1. Determine if 2 figures are similar using properties of

transformations2. Determine if 2 triangles are similar, given their angle

measures and side lengths3. Calculate scale factor4. Given 2 similar triangles determine angle measures

and side lengthsRatio & proportions worksheet

Congruence

• 2 triangles are congruent if and only if corresponding pairs of sides & corresponding pairs of angles are congruent

• We will be using what we know about proving angles congruent

AP: Skill based task

• Complete the following statements: – Given triangle QXR congruent to triangle NYC– A) line segment QX congruent to line segment ___– B) Angle Y congruent to angle ____

– Explain your answer/rationale, use your notes from class

Problem Task

• If each angle in one triangle is congruent to its corresponding angle in another triangle, are the two triangles congruent? Explain

Coordinates to prove simple geometric theorems algebraically

• Given a triangle, use slopes to verify that length and height are perpendicular

• Explore perimeter of a variety of polygons• Textbook 1.8 Perimeter, Circumference, Area • Google earth • Unit 2 G.GPE.7 Worksheet 1

Essential questions

1) How do you factor a trinomial? 2) How is scale factor calculated? How is it applied to create images from transformations?3) What are the characteristics of similar polygons? Similar triangles? What is AA Similarity Theorem?4) How do you prove the Pythagorean Theorem? How can the Pythagorean theorem prove distance formula?

Essential Question

• 5) How can coordinates be used to calculate area of triangles? What about the perimeter of other polygons?

Perform Arithmetic Operations on Polynomials

• Interpreting algebraic expressions• Create algebraic operations with polynomials

Skill-based taskSolve the following equations:

1.Solve the following equationsa. x2 + 9x = 36b. 3x2 + 8x + 2 = 0c. 4x2 – 25 = 0

1.A square has a side length of (x – 2). Write the equations that represent the perimeter and area of the square.2.A rectangle has an area of 8 square yards and is represented by the equation (10x^2 + 5x).

a. Find the equations that represent the dimensions of the rug.b. Solve for x.

3.The width of a rectangle is six less than two times its length. a. Write the equation that represents the width.b. What equation represents the perimeter of this rectangle?c. What equation represents the area of this rectangle?d. What is the smallest possible length of the rectangle?

Problem Task1)Jake determines that the area of his square is represented by the equation (x^2 + 81). Is he correct? If so, explain why. If not, how would you explain to him why he is incorrect?

2) The area of Jane’s rectangular dining room table is represented by the equation (x^2 + 8x + 15). If the width is represented by the equation (x + 3), write an equation to represent the perimeter of the table.3) The perimeter of a rectangular playground is represented by the equation (6x + 6).

If the length is (x + 4), what equation represents the width?Using your answer from part a, what is the area of this playground?

4) The sides of a triangle are represented by the equations 2x+1, 4x, and 5x – 5. If this triangle is dilated by a scale factor of 2x, what will be the perimeter?

Problem Task (H.O.T. ?) http://illustrativemathematics.org/illustrations/603 – “Are they similar” activity Jan uses an overhead projector to enlarge a picture 5 in. high and 7 in. wide. She projects the picture on a blackboard 4 ft 2 in. high and 12 ft wide. What are the dimensions of the largest picture that can be projected on the blackboard?

Pythagorean Theorem

10-5-12

Understand and Apply Pythagorean Theorem

• Know that in a right triangle• a^2 + b^2 = c^2 (Pythagorean Thm)

• Explore various proofs of the Pythagorean Thm

• Students find examples of right triangles in your own personal environment

ResourcesTextbook Correlation: 8.1 The Pythagorean Theorem and its Converse“Proofs of the Pythagorean Theorem” Activityhttp://map.mathshell.org/materials/download.php?fileid=804MARS Tasks (HS): E04: Proofs Of The Pythagorean TheoremE08: Pythagorean TriplesMARS Problem Solving Lesson (HS): Proofs of the Pythagorean TheoremTexas Instrument 8.G.6 LessonsTexas Instrument 8.G.7 LessonsCMP2 Resources

Skill-based TaskSolve for x in each problem below

5. If the height of a cone is 10 meters and the radius is 6 meters, what is the slant height?

Problem TaskProve the Pythagorean Theorem: a² + b² = c².Explain the logical reasoning behind a proof of the Pythagorean Theorem. (Why did it make sense to prove the theorem using this method?)Investigate the historical context of one of the proofs of the Pythagorean Theorem and present the proof in context to the class.TVs are measured along their diagonal to find their dimension. How does a 52-inch HD (wide-screen) TV compare to a traditional 52-inch (full screen) TV? A 65 ft. ladder is propped against a building and reaches a point 33 ft. high. If the base of the ladder slides 7ft away from the building, what is the new height reached by the ladder?

Theorems

• Derive distance formula using Pythagorean Thm

• Derive midpoint formula using Pythagorean Thm

• Overlap a map with coordinate grid and use the Pythagorean Thm to find the distance between two locations

Skill Based

• Using the Pythagorean Theorem, find the distance between (4, 2) and (7, 10).

• Worksheet from Pennsylvania Department of Education Website – “FOG – Unit 2 – 8.G.8 Worksheet 1”

Problem Task• Using the Pythagorean Theorem to prove the distance

formula.• (Teachers: For assistance, review pg. 52 of the

textbook and the following website. Select the link that says “The distance between any two points” -- http://www.themathpage.com/alg/pythagorean-distance.htm )

• • List 3 coordinate pairs that are 5 units away from the

origin in the first quadrant. Describe how to find the points and justify your reasoning. (Note: Points on the axes are not in the quadrant.)

• Skill-based task• Calculate the area of triangle ABC with

altitude CD, given • A(-4, -2), B(8, 7), C(1, 8) and D(4, 4)

• Problem Task• Find the perimeter of a real-world shape using a

coordinate grid and Google Earth.• Jack is building a play area for his dog. On the

blueprint, the coordinates are located at (1, 6) (4, 2) (6, 10) (8, 2) and (11, 6).– What is the perimeter of the play area if the units are

in feet?• If fencing is $3 per yard, how much will it cost to

fence the entire area?

Problem TaskLocate the center of dilation and scale factor in the following pair of triangles.

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