Big Idea Prior Work Future Work - Buffalo State Collegemath.buffalostate.edu/~giambrtm/MED309/SupWorksTrans.pdf · Big Idea Prior Work Future Work ... Moving Straight Ahead) Understanding

10 Looking for Pythagoras

Big Idea Prior Work Future Work

Investigating symmetry (Kaleidoscopes, Hubcaps, andMirrors); finding the equation of a circle (Shapes ofAlgebra)

Finding slopes of lines andinvestigating parallel lines(Variables and Patterns;Moving Straight Ahead)

Understanding sloperelationships ofperpendicular andparallel lines

Solving quadratic equations (Growing, Growing,Growing; Frogs, Fleas, and Painted Cubes; Say It WithSymbols)

Representing fractions asdecimals and decimals asfractions (Bits and Pieces I,II, and III)

Understandingirrational numbers asnon-terminating,non-repeatingdecimals

Exploring sampling and approximations (Growing,Growing, Growing; Samples and Populations; Frogs,Fleas, and Painted Cubes)

Understanding fractions anddecimals (Bits and Pieces I, II,and III)

Investigating rationalnumbers written asdecimals

Solving geometric and algebraic problems (Growing,Growing, Growing; Frogs, Fleas, and Painted Cubes; SayIt With Symbols; Kaleidoscopes, Hubcaps, and Mirrors)

Solving problems ingeometric and algebraiccontexts (Shapes andDesigns; Moving StraightAhead; Thinking WithMathematical Models;Covering and Surrounding)

Using thePythagoreanTheorem to solveproblems

Formulating and using symbolic rules and the syntax formanipulating symbols (Frogs, Fleas, and Painted Cubes;Say It With Symbols; Shapes of Algebra)

Formulating, reading, andinterpreting symbolic rules(Variables and Patterns;Moving Straight Ahead;Thinking With MathematicalModels; Covering andSurrounding); working withthe triangle inequality(Shapes and Designs)

Understanding the PythagoreanTheorem and how itrelates the areas ofthe squares on thesides of a righttriangle

Looking for patterns in square numbers (Frogs, Fleas,and Painted Cubes); looking for patterns in exponents(Growing, Growing, Growing)

Applying the formula forarea of a square (Coveringand Surrounding)

Understandingsquare roots aslengths of sides ofsquares

Studying transformations and symmetries of planefigures (Kaleidoscopes, Hubcaps, and Mirrors)

Measuring areas of polygonsand irregular figures (Bitsand Pieces I; Covering andSurrounding) and surfaceareas of three-dimensionalshapes (Filling and Wrapping)

Finding areas offigures drawn on acoordinate grid withwhole-numbervertices

Finding midpoints of line segments (Kaleidoscopes,Hubcaps, and Mirrors)

Measuring lengths (Shapesand Designs; Covering andSurrounding); working withcoordinates (Variables andPatterns; Moving StraightAhead; Thinking WithMathematical Models)

Calculating thedistance betweentwo points in theplane

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10

3rd Proof

10

Getting Ready

Describe the locations of these landmarks:

George Washington University

Dupont Circle

Benjamin Banneker Park

The White House

Union Station

How can you find the distance from Union Station toDupont Circle?

Find the intersection of G Street and 8th Street SE andthe intersection of G Street and 8th Street NW. How arethese locations related to the Capitol Building?

101010

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Map of Euclid

3rd Proof

N y

x�4 �2 �1�3�7 �6 �5 2 31 4 5 6 7

�3

�4

�5

�1

�2

1

2

4

5

3

City HallArt museum

Animal shelter

Greenhouse

Stadium

Gas station

Hospital Police station Cemetery



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Planning Parks

3rd Proof

N y

x�4 �2 �1�3�7 �6 �5 2 31 4 5 6 7

�3

�4

�5

�1

�2

1

2

4

5

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City Hall

(1, 1)

(4, 2)

Art museum

Animal shelter

Greenhouse

Stadium

Gas station




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Venn Diagram

3rd Proof


Finding Areas

Transparency 1.3Looking for Pythagoras

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3rd Proof

1.

5.

8.

9.

10.

6.

7.

2. 4.3.



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5 Dot-by-5 Dot Grids

3rd Proof



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16

Getting Ready

What is the side length of a square with an area of2 square units?

Is this length greater than 1? Is it greater than 2?

Is 1.5 a good estimate for � 2—?

Can you find a better estimate for � 2—?

3rd Proof



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Enclosed 5 Dot-by-5 Dot Grids

3rd Proof


Transparency 2.3B Problem SummaryLooking for Pythagoras

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3rd Proof


Transparency 2.3C Problem SummaryLooking for Pythagoras

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3rd Proof



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20

Length ofLeg 1(units)

Length ofLeg 2(units)

Area of Squareon Leg 1

(square units)

Area of Squareon Leg 1

(square units)

Area of Squareon Hypotenuse(square units)

1

1

2

1

2

3

3

1

2

2

3

3

3

4

1 1 2

3rd Proof

20



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3rd Proof

Puzzle Pieces

Puzzle Frames



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Solution

3rd Proof



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Finding Distances

3rd Proof

K

P

L

M

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24

Getting Ready

In ancient Egypt, the Nile River overflowed every year,flooding the surrounding lands and destroying propertyboundaries. As a result, the Egyptians had to remeasuretheir land every year.

Because many plots of land were rectangular, theEgyptians needed a reliable way to mark right angles.They devised a clever method involving a rope withequally spaced knots that formed 12 equal intervals.

To understand the Egyptians’ method, mark off 12 segments of the same length on a piece of rope orstring. Tape the ends of the string together to form aclosed loop. Form a right triangle with side lengths thatare whole numbers of segments.

What are the side lengths of the right triangle youformed?

Do the side lengths satisfy the relationship a2 � b2 � c2?

How do you think the Egyptians used the knotted rope?

3rd Proof



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25

The Wheel of Theodorus

3rd Proof

1

1

1

1

1

1

11

1

1

1

1

0 1 2Á2

3 4 5 6



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26

3rd Proof

Real numbers

Rational numbers

Irrational numbers

Integers

Whole numbers

Countingnumbers



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Baseball Diamond

3rd Proof



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28

Getting Ready

Triangle ABC is an equilateral triangle.

What is true about the angle measures in an equilateraltriangle?

What is true about the side lengths of an equilateraltriangle?

Line AP is a reflection line for triangle ABC.

What can you say about the measures of the followingangles? Explain.

Angle CAP Angle BAPAngle CPA Angle BPA

What can you say about line segments CP and PB?Explain.

What can you say about triangles ACP and ABP?

3rd Proof

A

P BC



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Analyzing Triangles

3rd Proof

30

60

6 units

C.


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3rd Proof

Triangle ABC

30°8 units DA B

C



Name Date Class

Labsheet 1.1Looking for Pythagoras

Maps of Euclid

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101

N y

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�3

�4

�5

�1

�2

1

2

4

5

3

City HallArt museum

Animal shelter

Greenhouse

Stadium

Gas station


N y

x�4 �2 �1�3�7 �6 �5 2 31 4 5 6 7

�3

�4

�5

�1

�2

1

2

4

5

3

City HallArt museum

Animal shelter

Greenhouse

Stadium

Gas station


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Name Date Class


Planning Parks

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102

N y

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�3

�4

�5

�1

�2

1

2

4

5

3

City Hall

(1, 1)

(4, 2)

Art museum

Animal shelter

Greenhouse

Stadium

Gas station


N y

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�3

�4

�5

�1

�2

1

2

4

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3

City Hall

(1, 1)

(4, 2)

Art museum

Animal shelter

Greenhouse

Stadium

Gas station



Name Date Class


Figures for Problem 1.3

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103

1.

5.

8.

9.

10.

6.

7.

2. 4.3.


Name Date Class

Labsheet 1ACE Exercises 15–25Looking for Pythagoras


104

15.

18.

16. 17.

20.19.

21.

24. 25.

22. 23.


Name Date Class


5 Dot-by-5 Dot Grids

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105


Name Date Class


Enclosed 5 Dot-by-5 Dot Grids


106


Name Date Class

Labsheet 3.2ALooking for Pythagoras

Puzzle Frames and Puzzle Pieces, Set AFrames

Puzzle Pieces

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107



108

Name Date Class

Labsheet 3.2BLooking for Pythagoras

Puzzle Frames and Puzzle Pieces, Set BFrames

Puzzle Pieces


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109

Name Date Class

Labsheet 3.2CLooking for Pythagoras

Puzzle Frames and Puzzle Pieces, Set CFrames

Puzzle Pieces


Name Date Class


Points on a Grid


110

K

P

L

M

Q

N


Name Date Class


The Wheel of Theodorus

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111

1

1

1

1

1

1

11

1

1

1

1

0 1 2Á2

3 4 5 6


Name Date Class


Questions A–C


112

30°8 units DA B

C


Additional Credits

Diana Bonfilio, Mairead Reddin, Michael Torocsik,nSight, Inc.

Technical Illustration

Schawk, Inc.

Cover Design

tom white.images

Team Credits

The people who made up the ConnectedMathematics 2 team—representing editorial,editorial services, design services, and productionservices—are listed below. Bold type denotes coreteam members.

Leora Adler, Judith Buice, Kerry Cashman, PatrickCulleton, Sheila DeFazio, Richard Heater, BarbaraHollingdale, Jayne Holman, Karen Holtzman, EttaJacobs, Christine Lee, Carolyn Lock, CatherineMaglio, Dotti Marshall, Rich McMahon, EveMelnechuk, Kristin Mingrone, Terri Mitchell,Marsha Novak, Irene Rubin, Donna Russo, RobinSamper, Siri Schwartzman, Nancy Smith, EmilySoltanoff, Mark Tricca, Paula Vergith, RobertaWarshaw, Helen Young

Acknowledgments 117

AC

KN

OW

LE

DG

ME

NT

S

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14 Looking for Pythagoras

Ongoing Informal Assessment

Embedded in the Student Unit Problems Use students’ work from the Problemsto check student understanding.

ACE exercises Use ACE exercises for homeworkassignments to assess student understanding.

Mathematical Reflections Have students summarize their learning at the end of eachInvestigation.

Looking Back and Looking Ahead At the end of the unit, use the first two sections to allow students to show what they know about the unit.

Additional ResourcesTeacher’s Guide Use the Check for Understandingfeature of some Summaries and the probingquestions that appear in the Launch, Explore, orSummarize sections of all Investigations to checkstudent understanding.

Summary Transparencies Use these transparenciesto focus class attention on a summary check forunderstanding.

Self AssessmentNotebook Check Students use this tool toorganize and check their notebooks before givingthem to their teacher. Located in AssessmentResources.

Self Assessment At the end of the unit, studentsreflect on and provide examples of what theylearned. Located in Assessment Resources.

Formal Assessment Choose the assessment materials that areappropriate for your students.

Additional ResourcesMultiple-Choice Items Use these items for homework, review, a quiz, or add them to theUnit Test.

Question Bank Choose from these questions forhomework, review, or replacements for Quiz,Check Up, or Unit Test questions.

Additional Practice Choose practice exercises for each investigation for homework, review, or formal assessments.

ExamView CD-ROM Create practice sheets, reviewquizzes, and tests with this dynamic software. Giveonline tests and receive student progress reports.(All test items available in Spanish.)

Spanish Assessment ResourcesIncludes Partner Quizzes, Check Ups, Unit Test,Multiple-Choice Items, Question Bank, NotebookCheck, and Self Assessment. Plus, the ExamViewCD-ROM has all test items in Spanish.

IndividualSkills, rich problemsThe UnitUnit Test

GroupRich problemsInvest. 3Partner Quiz

IndividualSkillsInvest. 2Check Up

StudentWork

FocusFor UseAfter

Assessment

G3d4 Using the Pythagorean Theorem

G3d3 The Pythagorean Theorem

✔✔N2d2 Squaring Off

A2c1 Coordinate Grids

Local TestSAT10ITBS

Terra Nova

CAT6 CTBSNAEPInvestigation

Correlation to Standardized Tests

NAEP National Assessmentof Educational Progress

CAT6/Terra Nova California Achievement Test, 6th Ed.CTBS/Terra Nova Comprehensive Test of Basic Skills

ITBS Iowa Test of Basic Skills, Form MSAT10 Stanford Achievement Test, 10th Ed.

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Name ____________________________________________ Date ____________ Class ____________

Partner QuizLooking for Pythagoras

PDF

1. The three line segments are drawn on centimeter dot paper.

a. Find the length of each segment to the nearest ten-thousandth of a centimeter.

b. Could these line segments be arranged to form a triangle?

If no, explain why not.

If yes, answer this question: Could they form a right triangle? Explain your reasoning.

2. Use the diagram below to answer parts (a)–(f). Show all the work you do to find your solutions.

a. What is the area of square ABCD?

B

A

C

D

CMP08_sn_142-151_LP.qxd 4/24/06 2:21 PM Page 149

150


PDF

b. What is the length of line segment AB?

c. What is the distance from A to C?

d. What is the area of triangle ABC?

e. Explain how the area of square ABCD compares to the area of triangle ABC.

f. Explain how the perimeter of square ABCD compares to the perimeter of triangle ABC.

Name ____________________________________________ Date ____________ Class ____________

Partner Quiz (continued)

Looking for Pythagoras


151

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Investigation 1

3. The gardener wants to brace the gate on the left below by adding a diagonalstrip of wood between the two horizontal strips (as shown on the right).

The section of gate at theleft is the same as the section of gateon the right.The section of gate onthe right (with the diagonal strip ofwood) is what the gardener wantsthe gate to look like.

a. How long should the diagonal strip of wood be? (Do not worry abouttrimming the ends to make a perfect fit.) Show all the work you do to findyour solution.

Think about what you know. Youknow the length of the two boards (5 ft)and the distance between them (3 ft). Howcan you find the length of the diagonal?

b. A standard tape measure is marked in feet and inches. If your answer forpart (a) is written only in feet, rewrite it in feet and inches.

What is your answer from part (a)?

How many feet are in your answer?

How many inches are in your answer? There are 12 inches in 1 foot. Thedecimal places that follow a whole numberare out of 10, so you need to figure outhow many inches are represented by thedecimal number.For example, if the answer is 4.6, that is 4 feet and some inches. To find the number of inches, figure out how manyinches 0.6 is. That means = .

Multiply 6 � 12 to get 72. Then divide 72by 10 to get 7.2. So the answer is 4 feetand 7.2 inches.

HINT

HINT

HINT

Name ____________________________________________ Date ____________ Class ____________



PDF

3 ft

5 ft

610

x12


154


Check-Up1. a. If AB is a side of the square, the other

points could be (3, 2) and (2, �1) or(�4, 1) and (�3, 4).If AB is a diagonal of the square, theother points are (�2, 2) and (1,1).

b. If AB is a side, .If AB is a diagonal, .

c. If AB is a side, 10 square units.If AB is a diagonal, 5 square units.

2. 14.5 square units.Possible strategy: Subdivide the figure andadd the smaller areas:1 � 6 � 1.5 � 2 � 1 � 1.5 � 1.5 � 14.5square units.

3. 7 square units.Possible strategy: Enclose the triangle in asquare, and subtract the area of the threeright triangles from the area of the square:16 � (2 � 3 � 4) � 7 square units.

4. a, b. The areas of the squares on sides PR and PQ are both 13 square units,so PR and PQ both have length

. The area of the third square is 26 square units, so RQ has length

.

c. ( )( ) � 6.5 square units

Partner Quiz1. a.

b. The three segments will form a triangle because the sum of the twoshorter lengths is greater than thelongest length: 2 � 2.2361 � 3.6056.However, they do not form a righttriangle because they do not fit thePythagorean Theorem—that is,22 � ( )2 ( )2."13"5

Á5 < 2.2361

2.0000

Á13 < 3.6056

"13"1312

xR

Q

P

y

�2 4 62�4�6

�4

�6

6

4

"26

"13

1.5

1.5

6

1

2

11.5

x

y

�2

�2 A

B

4�4

�4

4

"5"10

Looking for Pythagoras Assessment Answers

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2. a. 25 square units (Students’ methods will vary.)

b. , or 5.c. (approx. 7.07).d. 12.5 square units.e. The area of the square ABCD is twice

the area of the triangle ABC.f. The perimeter of the square is

4 � 5 � 20. The perimeter of thetriangle is 5 � 5 � � 17. Theperimeter of the square is slightlygreater than the perimeter of thetriangle. In particular, the relationshipbetween the areas of the two figuresdoes not hold for the perimeters of thetwo figures.

3. a. Since 32 � 52 � 34, the length of the diagonal strip should be � 5.83 ft.

b. To convert 0.83 ft to inches, multiply by 12. The result is 9.96, so, to thenearest inch, the strip should be about5 ft 10 in long.

Multiple-Choice Items1. A 2. H 3. A 4. G5. D 6. G 7. A

Unit Test1. a. Figure ACES is a parallelogram.

b. The lengths are as follows: AS, ;SE, ; CE, ; AC, .

c. Opposite sides are the same length.d. 13 square units

2.

3. area � 6 square units,perimeter � 2 � 2 � 13.4 units

4. area � 10 square units,perimeter � 12 � 17.0 units

5. a, b.

c. must be greater than 1 but lessthan 2, as 12 � 1 and 22 � 4.So, � must be greater than 2 and therefore greater than .

6. There are two possible positions (labeledT and T’):

7. a. Since this is a 30-60-90 triangle, thelength of the shorter leg is half thelength of the hypotenuse. The wire willbe attached to the ground 7 ft from thebuilding.

b. Using the Pythagorean Theorem,since 142 � 72 � 147, the wire will beattached � 12.12 ft up the side of the building.

Question Bank1. Kathy can claim that since the process will

continue to “bring down a 0” each time,the string of remainders will cycle througha series of numbers. In the division that isshown, the remainders (with 0s attached)are 10, 30, 20, 60, 40, and 50. They willcontinue to display this pattern, causingthe digits in the decimal to repeat thepattern 142857. (Kathy’s calculatorincluded a 3 after the second 4 due torounding.)

"147

S

T

T’

"4"2"2

"2

Á4

Á2

"2

"5"20

157Á3 Á11 Á17 Á36

Á5

1.5

1 5 63 42

"26"13"26

"13

"34

"50

"50"25

Looking for Pythagoras Assessment Answers (continued)


156


2. a. (2, 4)b. Answers will vary. Students may talk

about drawing parallel lines (from thedefinition of a parallelogram), onethrough point A parallel to segmentCE and one through point E parallel tosegment AC. These lines meet at thefourth vertex of the parallelogram.Students may also reason using whatthey know about slopes of parallellines. For example, the line through

points C and E has slope , so the line

through points A and S must also have

slope . Moving from point A up 1 and

to the right 4, point S is located at (2, 4).c. Placing a vertex at point (4, 0) will

form a second parallelogram.3. a.

b. The sum of the areas of the two smallersquares is equal to the area of thelargest square.

4. The length of segment AB is .Students might draw a square withsegment AB as a side, or they might drawa right triangle with segment AB as thehypotenuse.

5. a. Since 62 � 32 � 100, the length ofsegment AB is � 10.

b. Since 52 � 32 � 34, the length ofsegment CD is � 5.83.

6. Possible answer: students may think aboutthis problem is by finding two perfectsquares that have a sum of 13 (only 4 and9 work) and using their square roots (2and 3) as the vertical and horizontaldistances from point X. There are threepossible locations of the treasure asmarked on this map.

7. Celia’s sketch shows that one of thesmaller triangles has a hypotenuse oflength (because the smaller square hasan area of 5). The larger triangle has ahypotenuse that is twice as long as that ofthe smaller triangle (or 2 times ) and ithas length (because the larger squarehas an area of 20).So, � 2 � .

Á5

Á5

"5"20

"20"5

"5

X

"34

"100

"17

13

9

4

14

14



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8. slope of line a � , slope of line b � ,

slope of line c � , slope of line

d � � �1

9. a.

b. Answers may vary. Students may saythat all of the points are included inone of the line segments, so all thepossible slopes have been accountedfor. They may also point out that all the combinations of using a 0, 1, 2, or 3 in both the numerator and thedenominator of the fractions for theslopes are included (when looking atequivalent fractions such as � �

� 1).10. a. Point A is at (0, 1); point B is at (2, 2).

b. The slope of line AB is .

c. y � x � 1

11. a, b.

c. Two pairs of line segments areoverlying: the segments with slope 1 or

, and the segments with slope or .

The slopes are equal, so the linesegments lie on the same path.

12. a. Since this is a 30-60-90 triangle, thelength of the shorter leg is half thelength of the hypotenuse. The wire will be attached to the ground 7 ft fromthe building.

b. Using the Pythagorean Theorem,since 142 � 72 � 147, the wire will beattached � 12.12 ft up the side of the building.

"147

24

12

22

A

12

12

11

22

33

no

slo

pe

A J

F G

IH

B C D E32

23

12 1

3

� 221

� 003

� 331 � 13

3

233

32

213

12



16


1. A and B are two vertices of a square.

a. What could the coordinates of the other two vertices be? Add the points tothe grid, and label their coordinates.

b. What is the area of the square you have identified?

c. What is the side length of the square you have identified?

In 2 and 3, find the area of the polygon. Show all work you do.

2. 3.

x

y

2

�2

�2 OA

B

42�4

�4

4

Name ____________________________________________ Date ____________ Class ____________

Check-UpLooking for Pythagoras

Investigation 2for use after

8CMP06_AR_LP_016-032.qxd 3/21/06 4:58 PM Page 16

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4. a. On one of the grids below, identify the point named by each coordinatepair. Connect points P, Q, and R to make a closed figure. (There are twogrids in case you need a clean one for one of the parts below.)

P (�1, �2) Q (2, �4) R (1, 1)

b. Find the lengths of the sides PQ, QR, and PR by using squares with sidePQ, side QR and side PR. Show all your work.

c. What is the area of figure PQR?

x

y

�2 4 62�4�6

�4

�2

�6

6

4

2

O

x

y

�2 4 62�4�6

�4

�2

�6

6

4

2

O

Name ____________________________________________ Date ____________ Class ____________

Check-Up (continued)




20


1. Without using a calculator, choose the number that could be a representationof on a calculator.

A. 5.2915026 B. 4.8989794 C. 5.9160797 D. 4.1231056

2. A surveyor uses the following diagram to determine the distance between twodocks, A and B, on opposite sides of a lake. Which is the closest estimate of thedistance between the two docks?

F. 2 miles G. 1 mile H. 2.3 miles J. 3 miles

3. Which one of the diagrams below could be used to solve the followingproblem: Justine rides her bike 3 miles to the east and then 10 miles to thesouth. How far is she from her starting point?

A. B. C. D. 3

10

3 103

10

3

10

3 mi2 mi

A B

"28

Name ____________________________________________ Date ____________ Class ____________

Multiple-Choice ItemsLooking for Pythagoras


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4. Between which two consecutive whole numbers does lie?

F. 5 and 6 G. 6 and 7 H. 7 and 8 J. 4 and 5

5. In triangle ABC below, side AB has a length of 6 cm and side BC has a lengthof 10 cm. What is the length of side AC?

A. 4 cm B. 6 cm C. 11.66 cm D. 8 cm

6. A square board has an area of 5 square feet. To the nearest tenth of a foot,what is the length of one side of the board?

F. 2.5 ft G. 2.2 ft H. 5 ft J. 1.3 ft

7. What is the length in centimeters of the hypotenuse of the right trianglebelow?

A. B. C. 8 � 2 D. 22 � 82"2 1 8"22 1 82

2 cm

8 cm

6 cmA

C

B

10 cm

"42

Name ____________________________________________ Date ____________ Class ____________

Multiple-Choice Items (continued)



22


Place a ✓ next to each item you have completed.

Notebook Organization

Problems and Mathematical Reflections are labeled and dated.

Work is neat and easy to find and follow.

Vocabulary

All words are listed. All words are defined or described.

Assessments

Check-Up

Partner Quiz

Unit Test

Assignments

Name ____________________________________________ Date ____________ Class ____________

Notebook ChecklistLooking for Pythagoras


18


1. The three line segments below are drawn on centimeter dot paper.

a. Find the length of each segment to the nearest ten-thousandth of acentimeter.

b. Could these line segments be arranged to form a triangle? If no, explainwhy not. If yes, answer this question: Could they form a right triangle?Explain your reasoning.

2. Use the diagram at the right to answer parts (a)–(f) on this page and the next page. Show all the work you do to find your solutions.

a. What is the area of square ABCD?

b. What is the length of line segment AB?

c. What is the distance from A to C?

B

A

C

D

Name ____________________________________________ Date ____________ Class ____________

Partner QuizLooking for Pythagoras



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d. What is the area of triangle ABC?

e. Explain how the area of square ABCD compares to the area of triangleABC.

f. Explain how the perimeter of square ABCD compares to the perimeter oftriangle ABC.

3. The gardener wants to brace the gate on the left by adding a diagonal strip ofwood between the two horizontal strips.

a. How long should the diagonal strip of wood be? (Do not worry abouttrimming the ends to make a perfect fit.) Show all work you do to find yoursolution.

b. A standard tape measure is marked in feet and inches. If your answer forpart (a) is written only in feet, rewrite it in feet and inches.

3 ft

5 ft

Name ____________________________________________ Date ____________ Class ____________





1. Kathy heard her teacher say that every fraction can be written as a decimal that either repeats or terminates.

She tried on her calculator to confirm this, and it gave

her an answer of 2.142857143. She was not sure whetherthis decimal is repeating or terminating, so she workedthe problem out by hand as shown. When she reached this point in her division, she said, “Now I see that thisdecimal will repeat.” What is Kathy’s evidence that this is a repeating decimal?

2. a. The points A, C, and E are labeled on the grid. Place a fourth vertex S toform parallelogram ACES. Give the coordinates of vertex S.

b. Explain how you decided where to put vertex S. What facts about theparallelogram helped you to decide on the position?

c. Is there another way to make a parallelogram using points A, C, and E anda fourth vertex? If so, locate the fourth vertex. If not, explain why not.

x

y

2

�2

�2 O 4 62�4�6

�4

�6

6

4A

CE

157

7�15.000000000 14 10 7 30 28 20 14 60 56 40 35 50 49 10

2.1428571

Question BankLooking for Pythagoras

28



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3. a. Find the area of each square below.

b. Describe any relationship you notice in your answer to part (a).

4. Without using a ruler, find the length of segment AB.

5. Use the Pythagorean Theorem to find the length of each line segment. Showall work you do to find your solutions.

a. What is the length of segment AB?

b. What is the length of segment CD?

C

D

B

A

B

A

Question Bank (continued)



30




6. Redbeard’s treasure is buried at one of the grid points shown below at adistance of from the X. Where could the treasure be? Explain how youlocated a possible position.

7. Celia’s brother says his algebra teacher said that is the same as 2 ? ,but this does not seem correct to him. Celia made this sketch to explain whythis is indeed true. Explain how Celia used her sketch to help her brotherunderstand.

8. Find the slopes of all the line segments on the grid.

slope of line a: _____________

slope of line b: _____________

slope of line c: _____________

slope of line d: _____________c

a

b

d

"5"20

X

"13


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9. a. Starting at point A, use a ruler to draw as many line segments as you can,all with different slopes. Each line segment must start at A and end atanother grid point. Give the slope of each line segment.

b. How do you know you have drawn all the possible line segments?

10.

a. Give the coordinates of points A and B.

b. What is the slope of line AB?

c. What is an equation of line AB?

x

y

2

�2

�2 O 4 62�6

�4

�6

6

4

AB

A


32


11. a. Starting at point A, use a ruler to draw and label line segments on the grid

below with slopes 1, , , , and . (The first line segment is drawn as anexample.)

b. Using a different color, draw and label more line segments, all starting at

point A, with slopes of , , , , and .

c. Are any of the line segments you drew in part (b) the same as the linesegments you drew in part (a)? Explain why this does or does not happen.

12. Wire is strung between a building and the ground, making a 30-60-90 triangleas shown. The wire is attached to the ground 14 feet from the building.

a. How long must the wire be?

b. How far up the side of the building is the wire attached?

14 ft

30�

60�

25

24

23

22

21

A

15

14

13

12




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Mathematical Ideas

After studying the mathematics in Looking for Pythagoras:

1. a. I learned these things about side lengths and areas of right triangles and squares:

b. Here are page numbers of notebook entries that give evidence of what Ihave learned, along with descriptions of what each entry shows:

2. a. The mathematical ideas that I am still struggling with:

b. This is why I think these ideas are difficult for me:

c. Here are page numbers of notebook entries that give evidence of what Iam struggling with, along with descriptions of what each entry shows:

Class Participation

I contributed to the classroom discussion and understanding of Looking forPythagoras when I . . . (Give examples.)

Name ____________________________________________ Date ____________ Class ____________

Self AssessmentLooking for Pythagoras


24


Learning Environment

Rate the learning activities using this scale:

1 I consistently struggled to understand the mathematics and I’m still notsure that I know it.

2 I struggled somewhat but now I understand more than I did.

3 I had to work, but I feel confident that I understand now.

4 I understood everything pretty easily and I feel confident that I knowthe mathematics in these problems.

5 Everything came easily. I knew most of the mathematics before we did this.

Learning Activities:

Problems from the Investigations

ACE Homework Assignments

Mathematical Reflections

Check-Up

Partner Quiz

Unit Test

Check any of the following that you feel are the most helpful in adding to thesuccess of your learning.

❏ Working on my own in class.

❏ Discussing a problem with a partner.

❏ Working in a small group of 3 or 4 people.

❏ Discussing a problem as a whole class.

❏ Individual or group presentation to the whole class.

❏ Hearing how other people solved the problem.

❏ Summarizing the mathematics as a class and taking notes.

❏ Completing homework assignments.

Name ____________________________________________ Date ____________ Class ____________

Self Assessment (continued)



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1. Draw line segments to connect points A, C, E, and S, in that order.

a. What shape is figure ACES?

b. Without using a ruler, find the lengths of line segments AS, SE, CE, andAC.

AS ____ SE ____ CE ____ AC ____

c. How do the lengths of the sides compare?

d. What is the area of the figure?

2. Arrange the following numbers on a number line.

, , , , , 1.5,

10 5 63 42�1

"11"5"36"17157"3

S

A

O

E

C

x

y

4

2

3

32 4 5 6 71

1

6

5

Name ____________________________________________ Date ____________ Class ____________

Unit TestLooking for Pythagoras


26


Name ____________________________________________ Date ____________ Class ____________

Unit Test (continued)


For Exercises 3–4, find the perimeter and area of the figure.

3.

4.

5. a. On the dot grid below, draw and label a line segment with length .

b. Draw and label a line segment with length .

c. Which is greater, � or ? Explain how you know.

6. Label a grid dot below with the letter T so that the length of ST is .

S

"10

"4"2"2

"4

"2

Area

Perimeter

Area

Perimeter


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Name ____________________________________________ Date ____________ Class ____________

Unit Test (continued)


7. A 14-foot piece of wire is strung between a building and the ground, making a30-60-90 triangle as shown.

a. How far straight out from the base of the building is the wire attached tothe ground?

b. How far up the side of the building is the wire attached?

14 ft

30�

60�


113

Materials

PACING:

Mathematical Goals

Launch

Explore

Summarize

Materials

Materials

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Name Date Class

Dot PaperLooking for Pythagoras

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99


Name Date Class

Centimeter Grid PaperLooking for Pythagoras


100


Glossary 115

GL

OS

SA

RY

conjecture A guess about a pattern or relationshipbased on observations.

hypotenuse The side of a right triangle that isopposite the right angle. The hypotenuse is thelongest side of a right triangle. In the triangle below,the side labeled c is the hypotenuse.

irrational number A number that cannot bewritten as a fraction with a numerator and adenominator that are integers. The decimalrepresentation of an irrational number never endsand never shows a repeating pattern of a fixed number of digits. The numbers , , , and pare examples of irrational numbers.

legs The sides of a right triangle that are adjacent tothe right angle. In the triangle above, the sideslabeled a and b are the legs.

perpendicular Forming a right angle. For example,the sides of a right triangle that form the right angleare perpendicular.

Pythagorean Theorem A statement about therelationship among the lengths of the sides of a righttriangle. The theorem states that if a and b are thelengths of the legs of a right triangle and c is the length of the hypotenuse, then .a2 1 b2 5 c2

"5"3"2

b

ac

rational number A number that can be written as afraction with a numerator and a denominator thatare integers. The decimal representation of a rationalnumber either ends or repeats. Examples of rational

numbers are , , 7, 0.2, and 0.191919. . . .

real numbers The set of all rational numbers andall irrational numbers. The number line representsthe set of real numbers.

repeating decimal A decimal with a pattern of afixed number of digits that repeats forever, such as0.3333333. . . and 0.73737373. . . . Repeating decimalsare rational numbers.

square root If , then s is the square root ofA. For example,-3 and 3 are square roots of 9because 3 ? 3 5 9 and -3 ? -3 5 9. The symbolis used to denote the positive square root. So, wewrite 5 3. The positive square root of a numberis the side length of a square that has that number asits area. So, you can draw a segment of length bydrawing a square with an area of 5, and the sidelength of the square will be .

terminating decimal A decimal that ends, orterminates, such as 0.5 or 0.125. Terminatingdecimals are rational numbers.

"5

"5

"9

"5

A 5 s2

7891

12

H

C

T

I

L

P

S

R

8cmp06te_LPGL.qxd 2/9/06 9:42 AM Page 115

142


PDF

Name ____________________________________________ Date ____________ Class ____________

1ACE Exercises 1–6Looking for Pythagoras

Investigation 1

For Exercises 1–6, use the map below.

1. Give the coordinates of each landmark.

a. art museum b. hospital c. greenhouse

2. What is the shortest driving distance from the animal shelterto the stadium?

Remember that a car can driveonly on roads.

3. What is the shortest driving distance from the hospital to thegas station?

HINT

N y

x

�4 �2 �1�3�7 �6 �5 2 31 4 5 6 7

�3

�4

�5

�1

�2

1

2

4

5

3

City HallArt museum

Animal shelter

Greenhouse

Stadium

Gas station



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4. What are the coordinates (x, y) of a point halfway from City Hall to thehospital if you travel by taxi?

Is there more than one answer to this question?

Explain.

5. What are the coordinates of a point halfway from City Hall to thehospital if you travel by helicopter?

How can a helicopter traveldifferently than a car? Refer back toProblem 1.1.

Is there more than one answer to this question?

Explain.

6. a. Which landmarks on the map are 7 blocks from City Hall by car?

Check to see if there is more thanone answer to this question.

b. Give precise driving directions from City Hall to each landmark youlisted in part (a).

HINT

HINT

Name ____________________________________________ Date ____________ Class ____________

1ACE Exercises 1–6 (continued)


Investigation 1

PDF


144


PDF

The points (0, 0) and (3, 2) are two vertices of a polygon with integer coordinates.

8. What could the other two vertices be if the polygon is a square?

9. Suppose the polygon is a non-rectangular parallelogram. What could the other two vertices be?

10. What could the other vertex be if the polygon is a right triangle?

2 4

2

4

–2

O

y

x–4 –2

–4

2 4

2

4

–2

O

y

x–4 –2

–4

2 4

2

4

–2

O

y

x–4 –2

–4

Name ____________________________________________ Date ____________ Class ____________


Investigation 1


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41. Put the following set of numbers in order on a number line.

2.3 2 5 2 4

4 –2.3 –2 – 2.09

Write all the numbers in the sameform, such as all in decimal form, or all infraction form.

HINT

Name ____________________________________________ Date ____________ Class ____________

2ACE Exercise 41Looking for Pythagoras

Investigation 2

PDF

42

52

14

14

42

√_

√_

√_

2 531 40-1-2-3–3 –2 –1 0 1 2 3 4 5


146


PDF

For Exercises 8–11, use the map. Find the flying distance in blocks between the2 landmarks without using a ruler.

8. Greenhouse and stadium

flying distance:

Explain how you found this answer.

Name ____________________________________________ Date ____________ Class ____________


Investigation 3

N y

x

�4 �2 �1�3�7 �6 �5 2 31 4 5 6 7

�3

�4

�5

�1

�2

1

2

4

5

3

City HallArt museum

Animal shelter

Greenhouse

Stadium

Gas station


Remember what flying distancerefers to. It is not the same as the drivingdistance. Look it up if you do notremember.

HINT


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9. Police station and art museum

flying distance:


10. Greenhouse and hospital

flying distance:


11. City Hall and gas station

flying distance:


Name ____________________________________________ Date ____________ Class ____________

3ACE Exercises 8–11 (continued)


Investigation 3

PDF


26


For Exercises 1–3, refer to the map on the following page.

1. Which landmarks are 5 blocks apart by car?

2. The taxi stand is 5 blocks by car from the hospital and 5 blocks by car from thepolice station. Give the coordinates of the taxi stand.

3. The airport is halfway between City Hall and the hospital by helicopter. Givethe coordinates of the airport.

4. Let a right triangle with vertices at (0, 0), (1, 0) and (0, 1) be the unit formeasuring area in the following questions.

a. Draw a square with vertices (0, 1), (1, 0), (0,�1), and (�1, 0).What is the area of this square in the triangle units described above?

b. Draw a square around the square you made in part (a) with two of the vertices at (1, 1) and (�1, 1). What are the other two vertices? What is the area of this square in triangle units?

c. Draw the square of the next size. One of its vertices is (0, �2). What are theother three vertices? What is the area of this square in triangle units?

d. What are the four vertices of the square of the next size? What is its area intriangle units?

e. What do you notice about the areas of the squares, as the squares get larger?

Name ____________________________________________ Date ____________ Class ____________

Additional PracticeLooking for Pythagoras

Investigation 1

Ox

y

3

2

1

�1

�2

�3

�2�3 �1 1 2 3

8CMP06_PW_LP_026-044.qxd 3/10/06 8:43 PM Page 26

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Name ____________________________________________ Date ____________ Class ____________

Additional Practice (continued)


Investigation 1


28


Investigation 1

For Exercises 5–10, use the given lengths to find the area of each figure. Showyour calculations. Record which formulas you can use as part of your reasoning.

5. 6.

7. 8.

9. 10.

6 cm 3 cm

3 cm

4 cm3 cm 3 cm

4 cm

5 cm

4 cm

5 cm

4 cm

5 cm

4 cm

9 cm

4 cm

Name ____________________________________________ Date ____________ Class ____________




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Investigation 1

Name ____________________________________________ Date ____________ Class ____________



For Exercises 11–14, find the area of the figure. Explain our reasoning.

11. 12.

13. 14.


Documents

Big Idea Prior Work Future Work - Buffalo State Collegemath.buffalostate.edu/~giambrtm/MED309/SupWorksTrans.pdf · Big Idea Prior Work Future Work ... Moving Straight Ahead) Understanding