High School Geometry Days 102-112

49

STRUCTURED CURRICULUM LESSON PLAN

Day: 102 Subject: Geometry Grade Level: High School

Correlations (SG,CAS,CFS): 6A1; 7A5, 7C1, 2; 8D1; 9A3

TAP:Perform arithmetic operations involving integers, fractions, decimals, and percents, explicitly stated or within contextChoose and apply appropriate operationalprocedures and problem-solving strategies toreal-world situations

Understand number systemsUnderstand geometric properties and relationships; apply geometric concepts and formulasAnalyze and interpret data presented in charts,

graphs, tables, and other displaysDemonstrate understanding of measurement concepts and apply measurement skills

ISAT:Solve problems requiring computations with whole numbers, fractions, decimals, ratios, percents, and proportionsUnderstand and apply geometric concepts and relationshipsUse mathematical skills to estimate,

approximate, and predict outcomes and tojudge reasonableness of results

Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs

Understand and use methods of data collectionand analysis, including tables, charts, andcomparisons

Demonstrate an understanding of measurement concepts and skills

Unit Focus/Foci

Connecting Proportions and Similarity

Instructional Focus/Foci

Exploring DilationsMaterials

Graph PaperRulersColored Pencils

Educational Strategies/Instructional Procedures

Define dilation for the students.

Dilation – a similarity transformation with center c and scale factor n mapping everypoint P so that

a) if P is the center, P = P’

b) if P is not the center, then CP’ = N(CP)

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If N > 1 the dilation is an enlargementIf N < 1 the dilation is a reduction

Center at A N(Scale Factor) 2

N(ScaleFactor)E12

A C12

(AC) A (2,6) A C 2(AC) A (2,6)

A B12

(AB) B (5,3) A B 2(AB) B (6,1)

C B12

(CB) C (6,4) C B 2(CB) C (12,6)

1 1 1 11 11 11

1 1 1 11 11 11

1 1 1 11 11 11

=

= =

= =

= =

Center at Origin

N(Scale factor)12

N(ScaleFactor)

A (1,3)

B (2,2)

C (3,3)

A

B

C

1

1

1

11

11

=

=

=

=

=

=

=

2

4 12

8 8

1212

11 ( , )

( , )

( , )

Have the students work along with you on grid paper as you complete the following examplepoints for � ABCD.

B

B11

C11

B1

A=(2,6)A1=A11↓

C1 C (6,6)

(4,4)

B(4,4)

B1

A=(2,6)

A1 C1

C (6,6)

A11 C11

B11

Enlargement

Reduction

51

Explain that with a function, the denominator cannot be 0.

A (-2, 2) C (2, -2)B (2, 2) D (-2, -2)

Using a scale factor of 3 and the center of the origin, plot the points A’B’C’D’ on the same axes.

Have students note that with the center at the origin, we multiply each coordinate.

Using the same coordinate axis, use B (2,2) as the center with scale factor N = 12

dilate

ABCD and label the points A1B1C1D1.

A1B1=12

(AB)

AB = 4 units∴ =A B units1 1 2

Since B is fixed, we locate B1 at (2,2) and A1 at (0,2). Ask the students where C1 and D1 arelocated?

Answer: C1 (2,0) D1(0,0)

Remind students that if the center is not the origin, we must measure distances.

Have the students complete Dilation Activity.

Discuss their observations.

Integration with Core Subject(s)

LA: Understand explicit, factual informationUnderstand the meaning of words in context

SC: Analyze and interpret dataApply scientific method to solve problems

SS: Read and interpret maps, charts, and graphsDistinguish fact from opinion and relevant from irrelevant information

A1(-6,6) B1(6,6)

A (–2,2) B (2,2)

(0,0)

D (-2,-2) C (2,-2)

D1 (-6,-6) D11 (6,-6)

52

Connection(s)

Enrichment: Have students explore dilations used in movies such as Ghostbusters, King Kong,and Jurassic Park.

Fine Arts: Have students explore dilations used in illustrations in children’s books such as Jackand the Beanstalk.

Home: Have students compare the cost of a child’s meal and the amount of food it buys withthe cost of an adult meal and the amount of food it buys at their favorite restaurant.

Remediation: Work with students in small groups, taking measurements of similar figures.

Technology: Using a computer program such as Geometers Sketchpad or Cabri Geometry,have the students explore dilations:

a) using coordinatesc) examining the ratios of the perimeters and areas of two similar figuresd) comparing the ratios of the corresponding sides

Assessment

Teacher observation

Homework

Assign appropriate problems from your text.

Give students two weeks to complete a book review on Gulliver’s Travels, noting similaritiesand ratios. Include the attached worksheet on dilation activity.

Teacher Notes

Inform students that in Geometers Sketchpad and some other computer programs, the select toolcan be turned into a dilation tool.

Reserve a copy of Gulliver’s Travels on video to be shown after the unit assessment on day 117.

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Dilation Activity

1. Draw pentagon ABCDE on graph paper using the following coordinates: A(6, 9), B(0, 3),C(-12, 9), D(-9, 0), and E(3, -3).

2. Using the origin as the center and Scale Factor 2/3, find A1, B1, C1, D1, and E1.

3. Graph A1, B1, C1, D1, E1.

4. List the corresponding sides of ABCDE and A1B1C1D1E1.

5. Compare ABCDE to A1B1C1D1E1 by:a) measuring corresponding sidesb) finding slopes of corresponding sidesc) finding the measure of corresponding angles

6. Describe your observations.

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TAP:Perform arithmetic operations involving integers, fractions, decimals, and percents, explicitly stated or within contextUnderstand geometric properties and relationships; apply geometric concepts and formulasUse variables, number sentences, andequations to represent solutions and solveproblems

Demonstrate understanding of measurement concepts and apply measurement skills

ISAT:Solve problems requiring computations with whole numbers, fractions, decimals, ratios, percents, and proportionsUnderstand and apply geometric concepts and relationshipsIdentify, analyze, and solve problems using

equations, inequalities, functions, and theirgraphs

Demonstrate an understanding of measurement concepts and skills

Unit Focus/Foci



Recognizing and Using Ratios and Proportions

Materials

Copies of Similar Worksheet 1YardstickMeterstickTape measure


Have students work in groups of four to complete the following explorations:

Exploration 1

Measure the distance from your wrist to the tip of your middle finger. Then measure yourheight. Compare the length of your hand to your height.

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Exploration 2

Using a tape measure, find the circumference of your hand and neck. Find the ratio of hand toneck measurements. How do your ratios compare? How does your ratio compare to the othermembers of your group? To the other members of your class?

Explain that with a function the denominator cannot be 0.

Define a proportion as an equation that states that two ratios are equal. Remind the students thatin a proportion, the product of the extremes equals the product of the means. This is called thecross product.

extreme a c meanmean b d extreme

ad = bc(cross product)

→ ←=→ ←

Review the following examples using the property: a cb d

= if and only if ad=bc .

+3 4 9 3 4Example1:

9 3 9 312 4

3( 3) 9 49 34 4

3 9 363 3

3 27

9

x

x

x

x

x

+= =

+ = ⋅ =

+ = =

==

4 2 22 4 22 2Example 2:

6 8 6 818 24

8 32 6 126 8

2 44 3 3

22

− + − += =

− = + =

= ==

x x

x x

x

x

Have the students complete Similar Worksheet 1.


LA: Understand explicit, factual informationUnderstand the meaning of words in context

SC: Apply scientific method to solve problemsSS: Distinguish fact from opinion and relevant from irrelevant information

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Connection(s)

Enrichment: Have the students write an essay on the Golden Ratio.

Fine Arts: See Enrichment.

Home: Have parent sign homework sheet weekly.

Remediation: Teachers’ Half Dozen

Technology: Have the students explore the Golden Ratio using a drawing program such asGeometers Sketchpad or Cabri Geometry.

Assessment

Evaluate Similar Worksheet 1 using the Structured Curriculum Scoring Rubric.

Homework


Teacher Notes

Prepare copies of Similar Worksheet 2A.

Answers to Similar Worksheet 1:

1. 12

2. 32

13. 12 or12.5

24. 8

5. 23

6. 4 or 6

7. 4

E8.

R9. 50 and130

10.7.2days

Bonus: 2:48 pm

==

=

== −= = −=

=+

° °

x

x

x

x

x

x x

x

re

r

57

Similar Worksheet 1

Solve each proportion:

1. 12

623x +

= 2. x

x845+

= 3. xx+−

=42 3

34

4. 4

16xx= 5.

x x− = +25

34

6. 23

162

xx

=+

7. xx

xx

−+

= −+

31

26

8. Solve for er

r:

Ee

R= +

9. Find the measure of two supplementary angles if their measures are in a ratio of 5:13.

10. Carlos can install 5 car stereos in 2 days. How long does it take him to install 18 stereos?

Bonus: Carlos works from 9:00am to 5:00pm. He eats lunch from 12:00 to 1:00pm. At whattime will he complete installation of the 18th stereo?

58




TAP:Perform arithmetic operations involving integers, fractions, decimals, and percents, explicitly stated or within contextUnderstand geometric properties and relationships; apply geometric concepts and formulasUse variables, number sentences, and equations to represent solutions and solve problemsDemonstrate understanding of measurement concepts and apply measurement skills

ISAT:Solve problems requiring computations with whole numbers, fractions, decimals, ratios, percents, and proportionsUnderstand and apply geometric concepts and relationshipsIdentify, analyze, and solve problems using equations, inequalities, functions, and their graphsDemonstrate an understanding of measurement concepts and skills

Unit Focus/Foci



Identifying and Solving Problems Involving Similar Figures

Materials

Copies of Similar Worksheet 2A


Review last night’s homework and answer any questions the students may have.

Discuss the phrase if and only if with the students. Have the students note that the conditionalstatement and its converse must be true. (The hypothesis leads to the conclusion and theconclusion leads to the hypothesis.)

Review the definition of similar polygons and discuss the following examples:

Similar polygons: two polygons are similar if and only if their corresponding angles arecongruent ( ≅ ) and the measure of their corresponding sides is proportional.

59

Ex. 1 Ex. 2

Inform students that the symbol for similar is ~.

Have the students complete Similar Worksheet 2A.


LA: Understand exp licit, factual informationUnderstand the meaning of words in contextApply information presented in the text to a new or different situation


Connection(s)

Enrichment: Have students find pictures of household articles that are similar.




Technology: Have students explore Similar Polygons using a drawing program such asGeometers Sketchpad or Cabri Geometry.

Assessment


Homework


8

6

10

4

3

5

5.0

5.0

2.5

7.52

2.0

1

3

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Teacher Notes

Inform students that a quiz will be given on Day 110.

Prepare copies of Similar Worksheet 2B.

Answers to Similar Worksheet 2A:1. Polygons are similar2. Polygons are not similar3. MN and QR, MP and QT, NO and RS, PO and TS4. ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠M and Q N and R P and T Oand S; ; ;5. 336. 157. 88. 68°

61

Similar Worksheet 2A

Determine whether each pair of polygons is similar.

1.

Trapezoid MNOP∼ Trapezoid QRST

3. List all pairs of corresponding sides.

4. List all pairs of corresponding angles.

5. Find the length of TS .

6. Find the length of QT .

7. Find the length of NO .

8. Find m∠ T

5cm

4cm

6cm

7.5cm

8cm

16cm

32cm

24cm

M N

OP 22cm

112°16cm Q

R

T

S

X

Y 24cm

12cm

62






Unit Focus/Foci



Identifying and Using Similar Triangles to Solve Problems

Materials

Copies of Similar Worksheet 2BProtractorsRulers or straightedges


Review last night’s homework with the students and answer any questions the students mayhave.

Using a protractor and a straightedge, have the students draw two triangles so that each trianglehas angles of 45° and 65°. Now measure the third angle of each triangle; it should measure 70°.Why? Measure the length of the sides of the triangles. Compute the ratios of the correspondingsides. Are the triangles similar? Explain.

63

Review the postulate and theorems involving similarity:

Angle-Angle Similarity (AA)-If two angles of one triangle are congruent to two angles ofanother triangle, then the triangles are similar.

Side-Side Similarity (SSS)-If the measures of the corresponding sides of two triangles areproportional, then the triangles are similar.

Side-Angle Side Similarity (SAS)-If the measures of two sides of a triangle are proportional tothe measures of two corresponding sides of another triangle and the included angles arecongruent, then the triangles are similar.

Have students complete Similar Worksheet 2B.


LA: Understand explicit, factual informationUnderstand the meaning of words in contextApply information presented in the text to a new or different situation

SC: Apply scientific method to solve problemsSS Distinguish fact from opinion and relevant from irrelevant information

Connection(s)

Enrichment: Have students create a collage using pictures of similar household articles.

3 4

5

6 8

10

2 34 6

64




Technology: Search the Internet for uses of similarity.

Assessment

Evaluate the in-class assignment using the Structured Curriculum Scoring Rubric.

Homework


Teacher Notes

Prepare the Mid-Unit Assessment. Prepare copies of Similar Worksheet 3.

Answers to Similar Worksheet 2B:

1. AA2. AA3. Similar4. Similar by SSS or AA5. a) J1(-1, -1) b) 1 1 1JKL J K L bydefinition∆ ∆∼

K1(3, -1) L1(3, 2) c) The rating of similitude is 2:16. See student work. A possible solution:

Statements Reasons1. AB//ED 1. Given2. ABE BED∠ ≅ ∠ 2. Alt. Int. angles3. ACB DCE∠ ≅ ∠ 3. Vertical angles4. ABC DEC∆ ∆∼ 4. AA

A B

C

E D

AB//EDProve ∆ ABC∼∆DEC

65

Similar Worksheet 2B

Explain why the triangles are similar.

1. 2.

Determine whether the figures in questions 3 and 4 are similar and give the reason why or whynot.

3. A rhombus with 4 cm sides containing a 50° angle and a rhombus with 8cm sides containinga 130° angle

4. An equilateral triangle with 5cm sides and an equilateral triangle with 7cm sides.

5. Draw JKL∆ with vertices J(-2, -2), K(6, -2), and L(6, 4). Draw its image J1 K1 L1 under the

dilation with center (0, 0) and scale factor 12

.

a) What are the vertices of J1 K1 L1?b) Is 1 1 1JKL J K L∆ ∆∼ ?c) What is the ratio of similitude?

6.

A B

C

D E

A B

C

E D

Given: AB//EDProve: ∆ABC∼∆DEC

66






Unit Focus/Foci



Solving Problems Using Proportions and Similarity

Materials

Copies of Similar Worksheet 3


Quick Quiz: On a sheet of paper, write all the ways you know to test for similarity of triangles.What are the requirements for each? Are there any short cuts? After the quiz, discuss theanswers with the students. Allow them to answer orally.

Review last night’s homework with the students and answer any questions the students mayhave.

Have students complete Similar Worksheet 3 and discuss the results.

67




Connection(s)

Enrichment:

Fine Arts:



Technology:

Assessment


Homework

Study for Mid-Unit Assessment.

Teacher Notes

Remind students that the Mid-Unit Assessment will be given at the next class meeting.Remind students that the book reviews are due in six days.Prepare copies of the Mid-Unit Assessment.

Answers to Similar Worksheet 3:

1. a. ∆ABC ~∆EFG AA

b. ∆ABC ~ ∆EFG

c. ∆ABC ~ ∆EFG SSS

2. AB = 27 BC = 15

3. MT = 11.2

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4.

Given: ∆BCE ~ ∆DEF

Prove: ∆BCE ~ ∆ACF

Statements Reasons

1. ∆BCE ~∆DEF 1. Given

2. ∠E ≅ ∠F 2. Definition of similar triangles3. ∠∠C ≅ ∠∠C 3. Reflexive property

4. ∆BCE ~ ∆ACF 4. AA

F

ED

A B C

69

Similar Worksheet 3

1. Use the given information to determine if ∆ ABC ∼ EFG and explain why they are similar.

a) ∠ ≅ ∠ ∠ ≅ ∠A E C F,

b) ∠ ≅ ∠ =B GACEF

ABEG

,

c) ABEG

CBFG

CAFE

= =

2.

3. MAH BUG

FindMT

∆ ∆∼

4.

A B

C F

E G

D

CBA

BD Bisects ADC

AD 45 CD 25

AC 42

Find AB and BC

∠

= =

=

M

28

A

T

H35

U B

G

10

25

S

F

ED

A B C

Given : BCE DEF∆ ∆∼Prove: BCE ACF∆ ∆∼

70






Unit Focus/Foci



Using Proportions and Similar Triangles to Solve Problems

Materials

Copies of Mid-Unit Assessment


Have students complete the Mid-Unit Assessment.




71

Connection(s)

Enrichment: Have students create a scale drawing of their favorite room, including furniture.


Home: See Enrichment.


Technology: See Enrichment.

Assessment

Evaluate the Mid-Unit Assessment using the Structured Curriculum Scoring Rubric.

Homework

Have students work on their book reviews.

Teacher Notes

Answers to Mid-Unit Assessment:

1. False2. True3. True4. True5. IJ = 22.5m JK = 19.5m6. GM = 114mm GO = 152mm7. (-4, 24) or (16, 24)8. Ratio of Similarity 2:1 J K L1 1 11 1 3 1 3 2( , ) ( , ) ( , )− − −

72

9. Given:

NQNO

NPNM

=

Prove: NQNO

QPOM

=

Statements Reasons

1. NQNO

NPNM

=1. Given

2. ∠ ≅ ∠N N 2. Reflexive

3. ∆NPQ ∼ ∆NMO 3. SAS

4. NQNO

QPOM

= 4. Definition of ∼ triangles

10. Given: AB ⊥ BC BD ⊥ AC

Prove:

Statement Reasons1. AB BC and BD AC⊥ ⊥ 1. Given2. ∠ ∠ABC and ADBareright angles 2. Definition of ⊥ lines3. ∠ ≅ ∠ABC ADB 3. All right angles are ≅4. ∠ ≅ ∠A A 4. Reflexive5. ∆ABC ∼ ∆ADB 5. AA

B

CDA

M P N

Q

O

73

Mid-Unit Assessment

Answer questions 1-4 true or false. Explain your answer.1. All isosceles right triangles are ∼.2. All isosceles right triangles are ∼.3. All equilateral triangles are ∼.4. If the vertex angles of two isosceles triangles are congruent, the triangles are ∼.

Solve each of the following:5. Polygon ABCDE ∼ Polygon HIJKL. If BC = 13m. CD = 15m. IJ = 5x + 2 and JK = 7x – 2.

Find IJ and JK.

6.

7. ∆ABC with vertices A(2, 3), B(2, 6), and C(4, 6) is similar to ∆XYZ with the vertices X(6, 9), Y(6, 24), and Z. Give two possible coordinates for the vertex.

8. Draw ∆JKL with the vertices J(-2, -2) K(6, -2), and L(6, 4). Draw its image J K L1 1 1 under

the dilation center (0, 0) with scale factor 12

.

E

G

D

MT

∆GET ∼ ∆ GDMGM = 4x + 2GD = 5x + 12GE = 32mmGT = 24mmFind the length of GM and GD.

2:1 Similarity Ratio

J ( 1, 1)

K (3, 1)

L (3, 2)

1

1

1

− −

−

74

9.

Given: NQNO

NPNM

=

Prove: NQNO

QDOM

=

10.

M P N

Q

O

B

CDA

Given AB BCBD AC

ove ABC ADB

:

Pr :

⊥⊥

≈∆ ∆

75






Unit Focus/Foci



Using Proportional Parts of Triangles to Solve Problems

Materials

Copies of Similar Worksheet 4CompassesRulers


Review with the students theorems and corollaries involving parallel lines and proportional parts.

Example 1: If a line is parallel to one side of a triangle and intersects the other two sides in twodistinct points, then it separates these sides into segments of proportional lengths.

76

In figure at the left,BD AE . Given that BA = 4,BC = 2, and DE = 6. What is the length of CD ?(Answer: CD = 3)

Example 2: A segment whose endpoints are the midpoints of two sides of a triangle is parallel tothe third side of the triangle, and its length is one-half the length of the third side.

Example 3: If three or more parallel lines intersect two transversals, then they cut off thetransversals proportionally.

Given: ∠ ≅ ∠ ≅ ∠1 2 3AB = 9, BC = 15 and DE = 11What is the length of EF ?

(Answer: 1813

18 33or . )

Have the students work in pairs to complete the following:1.

ARSBC with R the midpoint of AB andS the midpoint of AC

R S AB = 16, BC = 30, AC = 34Find AR , RS, and AS .

B C (Answer: AR = 8, RS=15, AS =17)

A

B

C

D

E

6

2

4

D

E

B

C

4A

3

10

A

B

C

D

E

F

1

2

3

77

2. UV TSRU=4

UT = 7

VS = 16 Find RV (Answer 917

)

3.

lmnAB = 3 BC = 5DE x= − 4 EF x=

Find DE and EF (Answer: DE = 6 and EF = 10)

Discuss the solutions with the students.

Have student complete Similar Worksheet 4.




Connection(s)

Enrichment: Have students read a Paul Bunyon tale and compare the larger-then-life figures toaverage figures.

Fine Arts: Have students write a story similar to a Paul Bunyon tale and include illustrations.



Technology: Have students research the tallest human on the Internet.

R

ST

U V7

4

16

l m n

A B C

D E F

78

Assessment

Teacher observation

Homework


Teacher Notes

79

Similar Worksheet 4

1. Use the following steps to separate into four congruent parts.

A. Given AB and point C not on AB , draw AC .

B. Place the compass point at A; construct an arc intersecting ACuuur

at D. Using the samecompass setting, construct DE , EF , FG , and GC so that AD DE EF FG≅ ≅ ≅ .

C. Draw GB . Construct lines through E, F, and G that are parallel to GB .

2. Explain why AD=DE=EF=FG .

A B

C

A B

C

D

E

F

G

80






Unit Focus/Foci



Recognizing and Using Similar Triangles

Materials


Have the students sit in groups of four.

On the chalkboard or overhead, draw two similar triangles whose sides have a common ratio of 3to 4. Have the students make conjectures about the relationship of the perimeters and areas ofthe two triangles.

Have the students calculate the perimeter and area of each triangle and make a conjecture aboutthe corresponding sides to the perimeter and area.

Draw two similar triangles whose sides have a common ratio of 2 to 3 and calculate theperimeter and area of each triangle; make a conjecture about the corresponding sides to theperimeter and area.

Have the members of the group discuss their results and prepare a report.

81

Randomly select a group to report their conjectures to the class and discuss the results.

Have students note that the perimeters have the same ratios as their corresponding sides, and theareas have the square of the ratio.

Have each group explore four quadrilaterals and discuss their observations. Does the conjecturestill hold? Ask the class to make a conjecture about the volume of similar figures. The volumeshave the cube of the ratio of the corresponding sides.

Draw two similar polygons on the chalkboard and have the students name the correspondingparts. (Students sometimes have a difficult time naming the corresponding parts.)




Connection(s)

Enrichment: See Fine Arts.

Fine Arts: Have students write a paragraph describing any larger than life character they haveread about, excluding Paul Bunyon and Gulliver.

Home: Have parent sign homework record weekly.


Technology: Have students research larger than life characters on the Internet.

Assessment

Teacher observation

Homework


Teacher Notes

Select a review format to be used at the next class meeting.

82






Unit Focus/Foci



Reviewing Proportions and Similarity

Materials


Review Unit Seven using a format from Appendix D.


LA: Apply information presented in the text to a new or different situationSC: Apply scientific method to solve problemsSS: Distinguish fact from opinion and relevant from irrelevant information

83

Connection(s)

Enrichment:

Fine Arts:

Home: Have a family member quiz you on the materials in Unit Seven.


Technology:

Assessment

Teacher observation

Homework

Have students study for the Unit Seven Assessment.

Teacher Notes

Prepare copies of the Unit Seven Assessment.

84






Unit Focus/Foci



Assessing Proportions and Similarity

Materials

Copies of Unit Seven Assessment


Administer the Unit Seven Assessment.


LA: Apply information presented in the text to a new or different situationSC: Apply scientific method to solve problemsSS: Distinguish fact from opinion and relevant from irrelevant information

Connection(s)

Enrichment: Have students write a children’s story using proportions.


85

Home: See Enrichment.


Technology:

Assessment

Evaluate the Unit Seven Assessment using the Structured Curriculum Scoring Rubric.

Homework

Have students complete the Gulliver’s Travels Book Review.

Remind students that the Gulliver’s Travels Book Review is due at the next class meeting.

Teacher Notes

Answers to Unit Seven Assessment:

1. T2. F3. F4. T5. T6. 6 27. 2 108. x = 1.59. 2 7

10. x = 187

11. x = 8, y = 4 312. AA or SSS13. AA14. AA15. 350 miles16. 27.5 ft17. 36°, 48°, and 96°18. K(10, 0)19. K(12.5, 0)20. See student work.

86

Unit Seven Assessment

Answer 1-5 True or False.

1. If two angles of one triangle are congruent to two angles of another, the triangles aresimilar.

2. An equation stating two proportions are equal is a ratio.

3. Two or more parallel lines divide transversals into equal parts.

4. A ratio is the comparison of two quantities.

5. Polygons are similar if their corresponding sides are in proportion.

Find the geometric mean of:

6. 4 and 18

7. 2 and 20

Solve for x

8. 3

1 5x

x=

+

9. 2 4

6 2x

x− =

+

10. ABC ADE∆ ∆∼

7 9

2 x

A

B C

D E

87

11. Solve for x and y.

Explain why each pair of triangles is similar.

12.

13. 14.

15. On a map, l inch represents 250 miles. Chicago is 1.4inches from St. Louis. How manymiles are there between Chicago and St. Louis?

16. Susan is 5ft 6in tall and casts a shadow of 3ft. Susan is standing next to a tree that cast ashadow of 15ft. How tall is the tree?

x y

4

12

B

CA

D

F

E

M A

K

C H

M N

C R

AMN CR

Explainwhy.MAN CAR∆ ∆

P

∼

88

17. The measures of the angles of a triangle are in the ratio 3: 4: 8. Find the measure of eachangle.

Find coordinates for K so that HIJ HKL∆ ∼ .

18. H(0, 0), I(0, 3), J(6, 0), L(0, 5)

19. H(0, 0), I(0, 8), J(10, 0), L(0, 10)

20. Given: B is the midpoint of AD . C is the midpoint of AE . Prove ABC ADE∆ ∆∼ .

D

B

AC E

89






Unit Focus/Foci



Viewing Proportional Parts and Similarity

Materials

Video of Gulliver’s TravelsTelevisionVCR


Collect Gulliver’s Travels book reviews.

Have students view the video of Gulliver’s Travels. Discuss Gulliver in relation to theBrobdingnagians and Lilliputians.

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Connection(s)

Enrichment: Have students prepare a brief report describing fractals including illustrations.

Fine Arts: Find pictures in newspapers and magazines of fractals in nature.



Technology: Have students research fractals on the Internet.

Assessment

Evaluate Gulliver’s Travels book review using the Structured Curriculum Scoring Rubric.

Homework

Have students select three pieces of their work for their portfolios and explain why each piecewas selected.

Teacher Notes

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Unit Seven Additional Activity 1Pentagrams


Exploring Pentagrams

MaterialsPlain paperRulersProtractors


Have students work in pairs.

1. Rotate one point around another 5 times, 72° each time.2. Draw the sides and diagonals of this regular pentagon. Label all points of intersection.3. Measure several lengths and find as many ratios as possible, always taking the ratio of the

larger to the smaller.4. List your observations and discuss them with your partner.

Teacher Notes

See Mathematics Teacher, November 1996, “Segments in a Pentagram.”

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Unit Seven Additional Activity 2Fractals

Instructional Focus/FociExploring Fractals

Materials

Plain paperRulersColored pencils


Remind students that fractals are patterns that are self-similar. Small parts of the figure aresimilar to the whole figure. A fractal must be described by steps that can be repeated.

Have students work in cooperative groups of three. Have each group create the Sierpinskitriangle or Koch snowflakes.

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Sierpinski Triangle

1. Draw an equilateral triangle.2. Find the midpoint of each side.3. Connect the midpoints, dividing the original triangle into 4 congruent triangles.4. Shade the middle triangle.5. Repeat the process in each unshaded triangle.

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Koch Snowflake

1. Draw an equilateral triangle.2. Divide each side into 3 congruent segments.3. Draw an equilateral triangle on each side, using the middle segment as the base.4. Erase the middle segment on each side.5. Repeat the process on each equilateral triangle.

Have each group design their own fractal.

Have students create a graphing calculator or computer program to create their fractals. Havestudents explore fractals on the Internet.

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Unit Seven Additional Activity 3Exploring Spirals of Squares with Geometers Sketchpad

We want to learn to write Sketchpad script to make spirals composed of squares, with an arbitrary ratiobetween the lengths of successive squares. The main difficulty is in figuring out exactly what rotationswe want to do and in getting the order of the “givens” right for the script. (It goes the opposite way fromwhat you might think.)

Begin by making a square, using either ruler and compass constructions or 90° rotations (the latter isquicker and doesn’t leave any hiddens). Label the vertices A,B,C,D as shown (the “Relabel” option onthe “Display” menu is useful). Make two line segments p and q, and mark both (under “Transform”) andmeasure (under “Measure”) their ratio, which we will use as the expansion ratio.

We want to make the spiral go counterclockwise as it winds outward from its center. To do this, we willrotate the square 90° clockwise about its corner A, expand it by the ratio p/q, and clean up the hiddens.Then, if we are making a script for this, we have to figure out how to select appropriate pieces of thesquare in the right order so that we can repeat the process to make a spiral with a lot of pieces.

Therefore, open a “New script” and hit “Record.” Select point A as the center of rotation. Now selectpoints B,C,D, which we will rotate and dilate into the new square. We don’t need to select A because it isthe center of rotation. We don’t select any segments because it is easier to redraw them after we have thenew vertex points.

Choose Rotation from the Transform menu. The angle is –90° about A because we are rotatingclockwise. You should see two new points below your first square. Both are selected; in addition, Bappears to be selected, though that is really the rotated image of D. Now do “Dilation by marked ratio;”you get three more points down and to the right. Hide the first two points. If you are clever, you mayfigure out how also to hide the image of D, which is underneath B, but it is not necessary. In my figure Ihave labeled the three new points B’, C’, and D’, though you don’t need to label them in your sketch.Take the Segment tool and draw segments AB’, B’C’, C’D’, and D’A, so now you have a square. Thetricky thing is to figure out how to end the script by selecting the new points in the right order so that youcan call the script again. So look in your Script window, and read what the “Givens” are. They seem tocome out in different orders; you may see:

Object DPoint AObject CObject BSegment pSegment q

The next step is going to be a –90° rotation about point C’ instead of A. So whatever order A, B, C, Dappear in the givens, we now want to select the new points by the mapping:

A → C’ B → D’ C → A’ D → B’

High School Geometry Days 102-112

Documents

Transcript of High School Geometry Days 102-112