GEOMETRY - Walch · 4. Unnatural Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

GEOMETRY

SECOND EDITION

© 2007 Walch Publishing Real-Life Math: Geometryiii

How to Use This Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Geometry Around Us1. Talking Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Still Talking Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3. Natural Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4. Unnatural Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Construction and Landscaping5. Community Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6. Landscape Design I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7. Landscape Design II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8. Hanging Drywall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Design and Marketing9. Going Fishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10. The Size of Things to Come, Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11. The Size of Things to Come, Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Art 12. Quilts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13. Making Really Big Pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

14. Line Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Sports and Recreation15. Racetrack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

16. Air Supply I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

17. Air Supply II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Table of Contents

(continued)

© 2007 Walch Publishing Real-Life Math: Geometryv

The Real-Life Math series is a collection of activities designed to put math into the context ofreal-world settings. This series contains math appropriate for pre-algebra students all theway up to pre-calculus students. Problems can be used as reminders of old skills in newcontexts, as an opportunity to show how a particular skill is used, or as an enrichmentactivity for stronger students. Because this is a collection of reproducible activities, you maymake as many copies of each activity as you wish.

Please be aware that this collection does not and cannot replace teacher supervision.Although formulas are often given on the student page, this does not replace teacherinstruction on the subjects to be covered. Teaching notes include extension suggestions,some of which may involve the use of outside experts. If it is not possible to get thesepresenters to come to your classroom, it may be desirable to have individual students contact them.

We have found a significant number of real-world settings for this collection, but it is nota complete list. Let your imagination go, and use your own experience or the experience ofyour students to create similar opportunities for contextual study.

How to Use This Series

© 2007 Walch Publishing Real-Life Math: Geometryvi

OrganizationThis book is organized around six different contexts addressing geometry: Geometry AroundUs; Construction and Landscaping; Design and Marketing; Art; Sports and Recreation; andMiscellaneous. For organizational purposes, the topics covered in each activity are listed onthe teacher guide page for that activity.

Order of ActivitiesYou’ll find that the activities in this book parallel most topics taught in a typical geometrycourse. You can supplement or enrich a concept presented in your textbook with thisresource or use the activities as an introduction to a new concept.

Level of DifficultySome activities use more difficult mathematical concepts than others. It should be noted thatthe less difficult lessons, mathematically speaking, still require higher-order thinking skills.

Time ConsiderationsBecause students’ ability levels and schools’ schedules vary, time suggestions for the activitiesare not given. Before using an activity, review it and decide how much time would beappropriate for your students.

Calculators and Other TechnologyA practical way of using calculators with the activities is to consider whether or not thesituation described in the activity would warrant the use of a calculator in real life. If thesituation does, then allow students to use calculators; if it doesn’t, then don’t allow them touse calculators. In some of the activities, students can use spreadsheet, word-processing, anddesktop-publishing software.

Organizing the ClassroomThe Teaching Notes sections include suggestions on how to arrange students for theactivities. Some of the activities work best for individual student work, others are moreappropriate for students working in pairs, and some work best for groups of students.

Evaluation and AssessmentWhere appropriate, selected answers are given. However, because the lessons model real-lifesituations, exact answers cannot always be provided.

Introduction

© 2007 Walch Publishing Real-Life Math: Geometry

Geometry Around Us

Contextgeometry around us

Topicgeometric terms

OverviewIn this activity, students build confidenceabout their prior knowledge of geometry bybrainstorming about the geometric termsthey already know. Then they use thosewords to create sentences showing howthose geometric terms are used in everydayspeech.

ObjectivesStudents will be able to:

• list common geometric terms used ineveryday speech

• build confidence in their priorknowledge of geometry

• create sentences using commongeometric terms

Materials• one copy of the Activity 1 handout for

each student

• dictionary (optional)

Teaching Notes• Students should work in small groups

for this activity.

• This activity is intended as anintroductory activity to the class andworks best at the beginning of theterm. However, it can still be usedafter the course has started.

• After all the groups have generatedlists of geometric terms, have themgive you their lists so you can keep amaster list with a frequency tally ofterms on the board or overhead.

• Let students know that they don’t haveto be able to define each term; they’lldo that in Activity 2.

• Once all the geometric terms havebeen posted, have students analyze thelist and determine which terms appearmost often. There might also be somedebate about whether or not a term isactually related to geometry.

• If students are struggling withcreating their sentences, share somesentences that other groups of studentshave written as they are working.

• Students do not have to use the classes’most common words when creatingtheir sentences.

• When students are finished writingtheir sentences, post the most creativesentences around the room.

1

teacher’s page

1. Talking Geometry

(continued)

Name ______________________________________________________ Date _________________________________

© 2007 Walch Publishing Real-Life Math: Geometry23

Most interior walls in homes and offices are made of drywall. Drywall is made of gypsumand is less than half an inch thick. It is sold in sheets that measure 4 feet by 8 feet. Adrywall hanger will put up entire sheets of drywall where possible, but then will have to fillin around it with smaller pieces cut from the larger sheets.

Once the sheets are hung, a paper “tape” and plaster “mud” are used to fill in the gapsbetween the sheets and between the sheets and the ceiling (but not the gaps between thesheets and the floor, which are covered by a baseboard). Filling the gaps makes the wall lookcontinuous and smooth. The fewer taped and mudded joints, the less work the drywallhanger has.

Before a drywall hanger starts the job, he or she must decide how many 4-by-8-foot sheetsare needed to complete the job. If a wall is 16 feet long and the ceiling is 8 feet high, foursheets of drywall can just be hung next to each other. Unfortunately, few walls or rooms aredesigned to make the drywall hanger’s job that easy.

Read the problems below. In each case, find a way to complete the job using the fewestpartial pieces of drywall and the least number of joints.

1. You need to hang drywall on a wall that is 20 feet long and 10 feet high. How shouldyou position the 4-by-8-foot sheets to limit the number of joints and the number ofsheets used? Describe your plan below. Be sure to list the number of sheets of drywallused and the total length of joints filled.

2. You need to hang drywall on a wall that is 13 feet long and 11 feet high. How shouldyou position the sheets to limit the number of joints and the number of sheets used?Describe your plan. List the number of sheets of drywall used and the total length ofjoints filled. Then draw a scale model of your plan.

8. Hanging Drywall

Name ______________________________________________________ Date _________________________________


12. QuiltsQuilts are a traditional American form of art that have been around for centuries. Quilts aretypically constructed by designing a block pattern or template on a grid. Each individualblock is then joined with other blocks to form the overall pattern. Follow the steps below tocreate your own quilt design.

Use the books your teacher provides or search the Internet for a web site that shows quiltdesigns. While you are looking over the designs, consider how the patterns work. Payparticular attention to the symmetry, rotation, reflection, or translations that might bepresent in the design. Choose one quilt pattern to study in greater detail.

Using the quilt pattern you chose, describe its pattern in terms of symmetry, rotation,reflection, or translations. Then draw the basic design in the grid that follows.

(continued)

Name ______________________________________________________ Date _________________________________


12. QuiltsGrid Design

Now it is your turn to come up with a design of your own.

Step 1: On a sheet of grid paper, outline a square. The square will form the foundation ofyour quilt design.

Step 2: Make a pattern inside the square. For now, keep the design simple. For example, youmay want to create a design using symmetrical triangles.

Step 3: Draw the reflection of your design.

Step 4: Make six to eight copies of the design and its reflection.

Step 5: Using the squares you made, create different designs using rotations, reflections, andtranslations. When you are satisfied with a pattern, tape or glue it onto a piece ofposter board.

Write a brief description of how your design works.

Problem Solving with Your Design

1. Suppose you wanted to make your quilt fit on a queen-size bed. How much materialof each color would you need? A standard-sized queen mattress measures 80 � 60 inches.

2. Suppose you wanted to make your quilt fit on a king-size bed. How much material ofeach color would you need? A standard-size king mattress measures 84 � 72 inches.

Name ______________________________________________________ Date _________________________________


16. Air Supply IDivers have to plan for many different things before they actually get in the water. One ofthe most important things they have to plan for is how much air they will need for a scubadive. As such, dive supervisors learn how such things as changes in pressure, temperature,and breathing rates affect the available volume of air.

Imagine you are a dive supervisor planning a scuba dive. First, you will explore therelationship between volume and pressure, as described by Boyle’s law. Second, you willapply what you have learned to determine the duration of the air supply for a scuba dive.

Boyle’s Law

One of the first gas laws that divers learn about is Boyle’s law. Boyle’s law explains therelationship between pressure and volume: At constant temperature, the absolute pressureand the volume of a given mass of gas are inversely proportional. As you go deeper in thewater, the pressure increases, and the volume of gas decreases; or you could say theopposite—as the pressure decreases, the volume increases. Understanding this relationship isimportant because as a diver goes deeper, less air is available. For example, suppose you had a1-gallon milk container at the surface of the water. This milk container is under 1 atmosphere (atm) of pressure. 1 atm is equal to 14.7 pounds per square inch, or psi. If youinverted the milk container and took it to a depth of 33 feet, it would be under 2 atm ofpressure (29.4 psi). Every 33 feet of water is equal to 1 atm of pressure. Because there is nowtwice the absolute pressure on the container, the volume is decreased to one-half gallon. Ifyou took the container to a depth of 66 feet, then the pressure would be equal to 3 atm andwould compress the volume to one-third gallon.

Boyle’s law can be expressed as:

Boyle’s law

P1V1 = P2V2

where

P1 = initial pressure

V1 = initial volume

P2 = final pressure

V2 = final volume

(continued)

EDUCATIONWALCH

Daily Warm-UpsGEOMETRYNCTM Standards

Jillian Gregory

iiiTable of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v About the CD-ROM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viPart 1: Two- and Three-Dimensional Shapes . . . . . . . . . . . . . . . . . . . . . . . 1Part 2: Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Part 3: Transformations and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 127Part 4: Visualization, Spatial Reasoning, and Geometric Modeling . . . 138Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Daily Warm-Ups: Geometry, NCTM Standards

IntroductionThe first few minutes of each class period are critical, as they set the tone for the entire lesson. Set your class on the right path with these warm-up problems, prompts, and brain benders. You can avoid wasted time by engaging your students from the minute they step in the classroom.

The warm-ups are organized in four parts based on the Geometry standards and expectations from the National Council of Teachers of Mathematics Principles and Standards for School Mathematics.

The standards for grades 9–12 include the following:• Analyze characteristics and properties of two- and three-dimensional geometric shapes and

develop mathematical arguments about geometric relationships.• Specify locations and describe spatial relationships using coordinate geometry and other

representational systems.• Apply transformations and use symmetry to analyze mathematical situations.• Use visualization, spatial reasoning, and geometric modeling to solve problems.Each warm-up can be classified under more than one standard and several expectations within each

standard. For ease of use, the warm-ups have been categorized under the standard that best represents the goal of the problem.

As a teacher, you can pick and choose the warm-ups you need each day based on your lesson plan. Therefore, the order in which you use the problems is at your discretion.

These warm-ups are a springboard to an engaging learning environment that will guarantee future success for your students.

v


About the CD-ROMDaily Warm-Ups: Geometry, NCTM Standards is provided in two convenient formats: an easy-to-use reproducible book and a ready-to-print PDF on a companion CD-ROM. You can photocopy or print activities as needed, or project them on a large screen via your computer.

The depth and breadth of the collection gives you the opportunity to pick and choose specific skills and concepts that correspond to your curriculum and instruction. The activities address all of the NCTM Standards for Geometry. Use the table of contents and the information on the title pages for each part to help you select appropriate tasks.

Suggestions for use:• Choose an activity to project or print out and assign.• Select a series of activities. Print the selection to create practice packets for learners who need

help with specific skills or concepts.

vi


Part 1: Two- and Three-Dimensional ShapesNational Council of Teachers of Mathematics: “Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.”

Expectations

• Analyze properties and determine attributes of two- and three-dimensional objects.• Explore relationships (including congruence and similarity) among classes of two- and three-dimensional

geometric objects, make and test conjectures about them, and solve problems involving them.• Establish the validity of geometric conjectures using deduction, prove theorems, and critique

arguments made by others.

• Use trigonometric relationships to determine lengths and angle measures.

Daily Warm-Ups: Geometry, NCTM Standards© 2008 Walch Education

6

Vertex of an AngleAn angle is a plane figure formed by two rays that share a common endpoint. In the figure below, ray A and ray C share a common endpoint. The vertex of an angle is the point at which the two sides of the angle meet. The vertex is point B. This angle is written in notation form as ∠ABC or ∠CBA .

It is important to write the points on the angle in the appropriate order. Since B is the vertex in the angle on the left, it must be written in between points A and C.

The vertex of each angle on the right is B. The angles can be written as ∠ABC or ∠CBA and ∠CBD or ∠DBC .

Name all the angles in each figure below.

A

B C45˚

C

B A35˚

D

55˚

A

B C45˚

C

B A35˚

D

55˚

1.

2.

FD

E

R

S

T

J

LK

BC

A

S

T

Q

J

MK

D

EC

D

C

B

FD

E

R

S

T

J

LK

BC

A

S

T

Q

J

MK

D

EC

D

C

B

FD

E

R

S

T

J

LK

BC

A

S

T

Q

J

MK

D

EC

D

C

B

FD

E

R

S

T

J

LK

BC

A

S

T

Q

J

MK

D

EC

D

C

B

3.

4.


96

Properties of PolyhedronsA polyhedron is a solid bounded by plane regions. The faces of the solid are polygons. The faces meet at common segments called edges. The edges have endpoints known as vertices of the polyhedron.

A regular polyhedron is a convex polyhedron whose faces are congruent regular polygons. The polygons are arranged in such a way that adjacent faces form congruent dihedral angles. Below are some examples of polyhedrons.

dodecahedron octahedron

Give the specific name for each regular polyhedron below.

1. 2. 3.

Part 2: Coordinate GeometryNational Council of Teachers of Mathematics: “Specify locations and describe spatial relationships using coordinate geometry and other representational systems.”

Expectations

• Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations.

• Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.

Daily Warm-Ups: Geometry, NCTM Standards © 2008 Walch Education

111

PointsA point has location, but not size. A point is represented by a dot and an uppercase letter. Points can be located in the Cartesian coordinate system and are written as (x, y). Look at the example below. The ordered pair for point A is (2, 4).

Example

Remember, the order of the coordinates matters. For example, point (2, 4) is not the same as point (4, 2).

Plot the following points on the grid below: A(4, 3), B(6, 6), C(−1, 5), D(−2, −4)

A (2, 4), )

y

x

y

x

A (2, 4), )

y

x

y

x


126

Graphing SolidsUse the isometric dot area below to sketch each solid.

1. a triangular prism with 3-4-5 right triangles as bases and a height of 6 units

2. a cube with an edge length of 5 units

Part 3: Transformations and SymmetryNational Council of Teachers of Mathematics: “Apply transformations and use symmetry to analyze mathematical situations.”

Expectations

• Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.

• Use various representations to help understand the effects of simple transformations and their compositions.

Daily Warm-Ups: Geometry, NCTM Standards © 2008 Walch Education

127

Line SymmetryAn isosceles triangle has a vertical line of symmetry. The letter B, however, has a horizontal line of symmetry.

For each figure, find the number of lines of symmetry. If possible, draw the lines of symmetry on the figure.

1. 3. 5.

2. 4. 6.


136

Adding VectorsThe sum of two vectors is called the resultant. Vectors can be added geometrically. Look at the example below.

Example

• Add the vectors on the right using the triangle method.

• Connect the initial point of the first vector with the terminal point of the second vector.

• Construct the sum of the two vectors from the initial point of the second vector with the terminal point of the first vector.

Find the sum of the two vectors.

1. 2. 3.

Part 4: Visualization, Spatial Reasoning, and Geometric ModelingNational Council of Teachers of Mathematics: “Use visualization, spatial reasoning, and geometric modeling to solve problems.”

Expectations

• Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.• Visualize three-dimensional objects from different perspectives and analyze their cross sections.• Use vertex-edge graphs to model and solve problems.• Use geometric models to gain insights into, and answer questions in, other areas of mathematics.• Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest

such as art and architecture.

Expeditions in Your ClassroomGeometry

Nora Priest

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Project Skills Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Project Putt-Putt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Ripping Rooms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Fashionistas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

At the Scene of the Crime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Protectors of the Realm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Superhero Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Thinking Outside the Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Director’s View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

This Is Air Traffic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

The Great Geometry Race . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Geometry Project Assessment Rubric . . . . . . . . . . . . . . . . . . . . . . 207

Contents

Expeditions in Your Classroom: Geometry ©2007 Walch Publishing iiiiii

Introduction

We all remember a project we did in school, often with more vivid recall than we can summon forentire courses or years. And for good reason. Projects command attention. They force students tograpple with new information, skills, and technologies in ways that embed learning in memory.They contextualize education and help students truly understand why “I need to know that.”

This book contains ten projects designed to leave a lasting mark. These projects provide studentswith authentic tasks involving real problems, real products, and real people, and use themes thathook young people. At the same time, they have teachers thoroughly in mind.

The high-school curriculum is packed, and, as teachers well know, a project can quickly take on alife of its own. Expeditions in Your Classroom provides activities and materials that scaffold studenttasks, set clear criteria for final products, and offer assessment tools and a detailed outline of projectsteps so that teachers can focus energy on instruction rather than project management.

About Project-Based LearningIn Real Learning, Real Work1, Adria Steinberg describes the qualities of powerful projects: the six A’s.

AuthenticityStudents solve problems and questions that are meaningful and real. People outside school wallstackle the same challenges. What students create and do has value beyond school.

Academic Rigor Students encounter challenging material and learn critical skills, knowledge, and habits of mindessential for success in one or more disciplines.

Applied LearningStudents put their knowledge and skills to work in hands-on ways, and learn how to organize andmanage themselves along the way.

Active ExplorationStudents go into the field. They investigate and communicate their discoveries.

Adult RelationshipsStudents connect with adults with relevant expertise. They observe them, work with them, and getsupport and feedback.

AssessmentStudents play an active role in defining their goals and assessing their progress. Adults around themgive them ongoing and varied opportunities to demonstrate progress.

1Steinberg, Adria. Real Learning, Real Work (Transforming Teaching). New York, NY: Routledge, 1998.

Expeditions in Your Classroom: Geometry ©2007 Walch Publishingiivv

Introduction

Expeditions in Your Classroom: Geometry ©2007 Walch Publishing

Project Format and MaterialsEach project contains the following materials:

Teacher Pages• Overview: information on project learning goals, prior knowledge or experience needed by

students, time needed for the project, and team formation information• Suggested Steps: a day-by-day view of how to deliver project activities• Project Management Tips and Notes: suggestions for how to handle possible issues or

information on project options and variations• Extension Activities: suggested activities for extending the project or exploring related areas • NCTE/IRA Standards Connection: a list of standards students will address through the

project• Answer Key: answers for Before You Go and Skill Check questions (Many answers will vary,

and therefore, have been omitted from the answer keys.)

Student Pages• Expedition Overview: a description of the project challenge, learning objectives, key

vocabulary terms, materials needed, and web resources students use for project activities• Before You Go: lead-in activities designed to review fundamental skills or knowledge needed

for the project• Off You Go: activities that support the core project, including guidelines and instructions for

final products or presentations• Expedition Tools: handouts and worksheets associated with project activities• Check Yourself: two assessment tools that students use to check skill development (practice

problems or questions) and evaluate their project performance overall

A Geometry Project Assessment Rubric is also included and can be used with any project.

vv

Projects challenge students to flex more than one mental muscle at a time and integrate skills theyoften see dissected and covered in discrete math book chapters. Each project in this book has a coreskill focus, but also gives students an opportunity to practice other skills. Use this chart as areference to help you find the best project for your needs.

C = Core skill

X = Other skills covered (sometimes optional)

Expeditions in Your Classroom: Geometry ©2007 Walch Publishing

Project Skills Chart

vvii

Project Page

Project Putt-Putt

1 X X C C X X X X

RippingRooms

18 X X C C X C X

Fashionistas 40 X C X X C

At the Sceneof the Crime

64 X X C C C

Protectors ofthe Realm

95 X X X C C

SuperheroChallenge

113 X X X X C C

ThinkingOutside the Box

130 X X X X X C X

Director’sView

151 X X X C C

This Is AirTraffic Control

168 X X X X C C

The GreatGeometryRace

195 X X X X X X X X X X

Mea

sure

men

tR

atio

an

d p

rop

ort

ion

Sca

le d

raw

ing

Cla

ssif

yin

g p

oly

go

ns

Tria

ng

le a

nd

an

gle

mea

sure

men

tsP

oly

go

nm

easu

rem

ents

Cir

cle

geo

met

ryTh

ree-

dim

ensi

onal

shap

es a

nd v

isua

lizat

ion

Tran

sfo

rmat

ion

sC

oo

rdin

ate

syst

ems

Cal

cula

tin

g s

lop

eG

eom

etri

c m

od

elin

gG

rap

h t

heo

ryB

asic

tri

go

no

met

ry

Expeditions in Your Classroom: Geometry ©2007 Walch Publishing 6655

Teacher

page

At the Scene of the Crime

• scale: the ratio between the size of something and a representation of it• triangulation: a method of calculating the location of an object using known

measurements of two other objects; creating a triangle from three objects and usingside and angle measurements to calculate an unknown measurement

Suggested StepsPreparation

• Gather any materials and props you may be providing for students.• Provide administrators and colleagues with an overview of the project. Confirm any

areas of the school, types of crime, and so forth that are off-limits or inappropriate.• Write and send a note home to parents explaining the project. • Arrange for students to have access to computers and the Internet.

Day 1

1. Form student teams, provide an overview of the project, and show students the project materials.

2. Assign students to work independently or as a team to complete Before You Go: CrimeScene Reconstruction Crash Course.

3. Provide students with computer time and Internet access to research their responsesusing the project web resources. Have them complete the activity for homework if needed.

Homework

Have students complete Before You Go: Crime Scene Reconstruction Crash Course as needed.

Day 2

1. Assign Before You Go: X Marks the Spot (Crime Scene Coordinates). Have studentswork in small groups to find measurements.

2. Allocate specific objects in the room, or let each group select two or three objects.

3. As an extension of the project and the math skills covered, show students how todetermine paths of trajectory or calculate the angle of impact from splattered orshattered evidence. Consider using the web resources on the following page.

Expeditions in Your Classroom: Geometry ©2007 Walch Publishing6666

Teacher

page


• Criminologywww.usoe.k12.ut.us/stc/ccc/7-8/Lessons%5C7%20Math%5CScientific%5CCriminology.htm

• Gizmos & Gadgets—Bullet Trajectory Rodswww.csigizmos.com/products/sceneaccessories/bullettrajectory.html

4. Before the end of class, provide a preview of Activity 1: Design a Crime.

5. Discuss limits you have. Emphasize that students should strive for funny, mysterious,or baffling crimes—not scary, gory, or disgusting ones.

Days 3 and 4

1. Introduce Activity 1: Design a Crime and crime scene criteria.

2. Provide time for teams to meet to plan their crimes. Remind students that you needone legible Design a Crime Worksheet per team. Suggest that each team appoints arecorder.

3. Collect worksheets as teams finish.

4. Inform students of the day the crime scene investigations will take place. Instruct them to begin assembling materials they will need. Note that your final approval isstill pending.

5. Review proposed crimes. If time allows, begin your review in class.

6. Make any notes, requests, or comments to support students’ understanding. Note anycalculations or math that might be unfamiliar to students or pose stumbling blocks.

7. Mark worksheets as “approved” or “needs more work.”

Day 5

1. Return Design a Crime Worksheets to students. Highlight any important points fromyour review.

2. Assign which teams will swap crime scenes. Explain that each team will investigate theother team’s scene.

3. Let each team know the location they will investigate or have teams meet quickly torelay this information. No other details of the crime should be discussed.



Teacher

page

4. Review the next day’s activities. State the following: • Two activities will take place during one class—the crime scene setup AND the

scene investigation.• Students must be ready at the start of class, with the materials they need for both

activities ready to go.

Note: Give students at least one day of advance notice to assemble materials and props.Use the in-between time to review problem-solving strategies, any potential stumblingblocks in math that you discovered in your review of the crime scenes, and so forth.Alternatively, give students time to prepare their witness or fine-tune their setup plan.

Day 6 (Crime Day)

1. Define the tasks and time limits for students: 10 minutes to set up and 30 minutes toinvestigate the other team’s scene.

2. Tell each team to appoint a timekeeper. Dispatch teams to set up their scenes. Teamsmay opt to send only one or two members.

3. After 10 minutes, announce the switch. Dispatch teams to investigate scenes.

4. Call an end to the scene investigations after 30 minutes.

5. Ask one or two members of each team to collect evidence and restore crime scenelocations to their original state. Or, members of the investigating teams can do this.

Day 7

1. Review the requirements of Activity 3: Crime Scene Report and the Crime SceneReport Template.

2. Assign the due date of the report. Specify if class time will be used or if students mustcomplete the report as homework.

3. Let teams meet to decide how they will complete assignments. Make suggestions as needed.

Report Due Date

1. Direct teams to review their partner team’s report.

2. Tell reviewing teams to write their comments on the final page.

3. Discuss, debrief, and collect reports.

Final Day

1. Have students complete the Skill Check problems.

2. Check and review answers.

3. Have students complete the Self-Assessment and Reflection worksheet and submit it (optional).

Project Management Tips and Notes• The crime scene setup/investigation activities are best suited to an extended or block

period. Alternatively, direct students to select specific crime scene locations of yourchoosing that can remain undisturbed from one day to the next so setup can be done in advance.

• Some students may struggle to develop crime scene sketches. Show students a floorplan example. Remind them that they can use basic shapes and symbols to representitems. Explain some of the symbols commonly used (for doors, windows, and so forth).

• To constrain the project, construct one scene of your own design for all teams toinvestigate. If you do, ask students to brainstorm and submit ideas for you to use.

Suggested Assessment Use the Geometry Project Assessment Rubric or the following point system:

Team and class participation 10 points

Crime scene design 10 points

Crime scene investigation 15 points

Crime scene report 60 points

Project self-assessment 5 points

Extension Activities • Have students present their findings as crime scene investigators testifying in court.• Create crime scene floor plans using a computer-aided design (CAD) tool.• Design an interdisciplinary project with science classes (physics for projectiles, biology

for forensics, and so forth).


At the Scene of the CrimeTeacher

page

At the Scene of the CrimeOff You Go

Activity 1: Design a Crime

Directions1. With your team, review the crime scene criteria below and brainstorm crime ideas.

2. Choose your best idea and formulate a detailed, step-by-step plan and story line foryour crime. What crime occurred and how?

3. Record the details of your plan on the Design a Crime Worksheet.

4. Prepare your witness or witnesses.

5. Following your teacher’s instructions, stage your crime at the location you selected.


Goal: To brainstorm, detail, and stage a “crime” (with your team)to be investigated by another team

Materials: pencil, paper, ruler, measuring tape or string, “evidence” props

Tools: Design a Crime Worksheet

Crime Scene Criteria

❑ Your crime should be clever and creative. However, this is an educationaladventure, not meant to be gory or to spread panic. Use common sense. Yourcrime should be fictitious. Obviously, you will not do anything illegal.

❑ Your crime must occur in school locations approved by your teacher.

❑ Provide at least one witness—a member of your team who is prepared toanswer questions posed by the investigating team that might help them solve the crime.

❑ Leave a minimum of three items of evidence at the scene that relate directlyto the perpetrator or perpetration of the crime.

❑ Evidence must be located on multiple planes (floor, wall, ceiling, and soforth) in order to challenge investigators’ skills in coordinate math,measurement, and three-dimensional drawing.

❑ You must be able to set up your crime scene in 10 minutes or less.

At the Scene of the CrimeExpedition Tool

Design a Crime Worksheet

Your team name: __________________________________________________________

Your crime scene location: ___________________________________________________

1. What will the crime scene location look like? Attach a detailed sketch that showsimportant features and objects (walls, doors, windows, furniture, and so forth) and theirdimensions. Include accurate labels and measurement information. Your drawing doesnot need to be done to scale.

2. What is the crime and how will it happen? Write a step-by-step description of how thecrime will occur.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

3. Make a list of the evidence perpetrators will leave behind. What, if anything, about thelocation will change as a result of the crime? You can provide location informationrelative to a baseline, another object in the room, or another piece of evidence.

Mark the location of evidence on your crime scene sketch.


Item Description Location Measurements

(continued )


4. Is there anything else important to know about the scene (for example, lightingconditions or angle of lighting, security, time of day, used or vacant, and so forth)?

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

5. What will your witness(es) see and when? Invent a story for each witness and mark thelocation or vantage point of each on your crime scene sketch.

• Who is the witness?

• When was he or she at the scene of the crime?

• Why was he or she at the scene of the crime?

• What are his or her physical characteristics?

• What was his or her vantage point or field of view?

• What did/could the witness see? What didn’t the witness see and why?

• What else does the witness know or not know about the crime scene?

6. List any materials or props will you need to stage your crime.

______________________ ______________________

______________________ ______________________

______________________ ______________________

______________________ ______________________


At the Scene of the CrimeOff You Go

Activity 2: Investigate the Scene

Directions1. Organize your team and assemble the materials you need to document the scene. All

team members should take notes. Assign specific roles and duties to each person; forexample, there should be a lead investigator, a sketch artist, a photographer, anevidence recorder, a witness interviewer, and so forth. You may need to play more than one role.

2. Go to the scene of the crime. As a team, determine your approach.

• What are your initial impressions? Do you have any hunches or theories?• What are the boundaries of the crime scene? • How will you do your walk-through?• How will you get the measurements you need?

3. Use the Crime Scene Checklist and the Witness Statement tools to help you conductyour search, collect data, and document the scene. Remember, according to the FBI,physical evidence cannot be overdocumented!

4. Interview witnesses at any stage of your crime scene search. Witnesses may beinterviewed once, and for no more than five minutes. You may interview only onewitness at a time.

5. When your investigation is complete, compare notes as a team and do a final survey of the scene.


Goal: To investigate and document another team’s crime scene

Materials: notebook, sketch paper, pencils, digital camera, ruler,measuring tape or string, compass, protractor

Tools: Crime Scene Checklist, Evidence Inventory, Witness Statement


Crime Scene ChecklistUse this checklist to help you collect the information you need during your crime scene search.

❑ Investigator names

❑ Type of crime

❑ Crime scene location

❑ Date and time arrived at scene

❑ Scene conditions

❑ Search approach and methods used

❑ Description of the crime scene and search findings

❑ Measurements (scene area and boundaries; location and size of evidence; other relevant measurements such as victim’s height, shoe size, size of windows and doors,and so forth)

❑ List of sketches (overview and side-view shots required; others optional as needed)

❑ List of photos (optional)

❑ List of witnesses interviewed/witness statements

❑ Evidence inventory




Evidence InventoryList key items of evidence discovered at the scene.

Item Description Location Measurements


Witness StatementWitness name: ____________________________________________________________

Interviewed by: ___________________________________________________________

Relationship to victim or other witnesses: _______________________________________

Profile (age, occupation, relevant physical characteristics, and so forth):

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

Statement:

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________


planea flat surface that stretches indefinitely in two dimensions

Wordstrips! 180 Geometry Facts & Figures • Lines & Angles © 2008 Walch Publishing

1

planea flat surface that stretches indefinitely in two dimensions


1

midpointthe point on a line segment that divides it into two congruent segments

X YM

5 5


10

M is the midpoint of XY, since XM MY.≅

midpointthe point on a line segment that divides it into two congruent segments

X YM

5 5


10

M is the midpoint of XY, since XM MY.≅

equiangulara polygon in which all of the interior angles are congruent

L P 108˚

108˚

108˚

108˚

108˚

M O

N

110˚65˚

90˚135˚

130˚

QR

S

TU

Wordstrips! 180 Geometry Facts & Figures • Polygons © 2008 Walch Publishing

48

Equiangular Not equiangular

equiangulara polygon in which all of the interior angles are congruent

L P 108˚

108˚

108˚

108˚

108˚

M O

N

110˚65˚

90˚135˚

130˚

QR

S

TU

Wordstrips! 180 Geometry Facts & Figures • Polygons © 2008 Walch Publishing

48

Equiangular Not equiangular

central angle ofa circleand angle whose vertex is the center of the circle

#98

C

AB

Wordstrips! 180 Geometry Facts & Figures • Circles © 2008 Walch Publishing

98

central angle ofa circleand angle whose vertex is the center of the circle

#98

C

AB

Wordstrips! 180 Geometry Facts & Figures • Circles © 2008 Walch Publishing

98

networksa set of nodes (locations) connected by links (arcs between places) used to represent, analyze, and solve problems

Wordstrips! 180 Geometry Facts & Figures • Reasoning & Proof © 2008 Walch Publishing

180

Each node is linked to all other nodes.

networksa set of nodes (locations) connected by links (arcs between places) used to represent, analyze, and solve problems

Wordstrips! 180 Geometry Facts & Figures • Reasoning & Proof © 2008 Walch Publishing

180

Each node is linked to all other nodes.

Hands-On MathGeometry

byPam Meader and Judy Storer

illustrated by Jennifer DeCristoforo

Contents

To the Teacher .................................................... iv

Topic: Ratio

1. The Golden Ratio .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. How Accurate Were Ancient Measuring Devices? .. . . . . . . . . . . . . . . . . . . . . . . . . 7

3. Unit Fractions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Topic: Area

4. Area/Percent Scavenger Hunt .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5. Discovering Area of a Circle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6. Circumference vs. Area of a Circle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7. Area of a Parallelogram and a Trapezoid .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8. Surface Area of a Sphere .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Topic: Angles

9. What’s My Angle? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10. How Many Degrees? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11. Properties of Parallel Lines .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

12. Angles Inscribed in Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13. Sums of Angles in a Triangle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Topic: Triangles

14. Similar Triangles .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

15. Pythagorean Theorem ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

16. Median Line of a Triangle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Topic: Quadrilaterals

17. Investigating Special Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

18. Diagonals and Special Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

iv

To the Teacher

Since the early 1970s, we have been teaching math to learners of all ages, from young children to adults, who represent many different cultures and socioeconomic backgrounds. We believe that all learners

can

do math by first overcoming any math anxiety and then by participating in meaningful, cooperative learning activities that relate to various learning modalities (e.g., auditory, visual, kinesthetic, and tactile) of each learner.

With the release of the NCTM Standards in 1988 and in 2000 the NCTM Principles and Standards for School Mathematics, teaching mathematics has become more student-driven and hands-on. Emphasis on deductive and inductive reasoning through a discovery process enables a student to truly

understand

mathematics. We feel the labs presented in this book address these concerns. First, our labs address students’ various learning styles. We provide hands-on activities of measuring, constructions, etc., which address the kinesthetic learner. We give opportunities to write and communicate ideas and to visualize concepts, thus including the visual learner. Finally, we furnish opportunities for group discussions and talking through problems, enabling the auditory student to be involved.

The study of Euclidean geometry lends itself to discovery of theorems through hands-on applications. Students who derive their own meaning for various theorems will own them and will understand what the theorems mean.

We hope you enjoy trying these activities with your students. We believe learning should be learner-centered, not teacher-driven. As quoted in the NCTM Principles and Standards, “students should conduct . . . explorations which will allow them to develop a deeper understanding of important geometric ideas. . . .”

—

Pam and Judy

t

Triangles

75

TeacherPage

Similar Triangles

Procedure1. This lab works best outdoors during a sunny day. If you don’t have these conditions,

then you could use a lamp with a bright light to cast shadows.

2. Sometimes building shadows may be hard to measure. Students must measure the perpendicular distance of the shadow. If your site has trees or flagpoles, they may work better.

3. When students measure their heights, the calculated heights may not match their exact heights. Discuss why this may happen. Some students may question where to measure the shadow, from the back of the student’s foot or in the front. Have students experiment to see which gives the more accurate measure. Make sure the students set up their proportions correctly and in the correct order.

Learning OutcomesStudents will be able to

• solve for unknown heights using proportions.

• measure various lengths with appropriate measuring tools.

Overview Student groups will calculate the height of buildings, flagpoles, or trees by measuring the objects’ shadows and solving a proportion formula. Students will also calculate each other’s heights using the same method.

Time Requirements30 minutes

Group SizePairs or groups of three

Materials• rulers

• tape measures

• lab sheets

• calculators

ACTIVITY 14

Name Date

© 2001 J. Weston Walch, Publisher 76 Hands-On Math: Geometry

Similar Triangles

Part OneIn this lab, we are going to try to figure out the height of a

building or flagpole using the process of similar figures and proportions.

1. Go outside and measure the shadow of a building or flagpole in feet.

Measure of shadow of building or flagpole: ___________________________________

2. Now place a ruler perpendicular to the ground and measure its shadow.

Measure of shadow of ruler in inches: ________________________________________

3. Change the measure of the ruler’s shadow to feet by dividing by 12.

Measure of ruler’s shadow in feet: ___________________________________________

4. You are now ready to calculate the height of the building or flagpole.

What is your calculated building height? _____________________________________

Part Two1. Have your partner measure the length of your shadow and try to calculate

your height.

2. Use the proportion of the length of the ruler to its shadow as the other part of the proportion. Does your calculated height come close to your actual height? ________

ACTIVITY 14

Proportion: Height of building =

Height of ruler

Length of building’s shadow Length of ruler’s shadow

Fill in with your measurements:

Height of building _______=

Height of ruler _______

Length of shadow _______ Length of shadow _______

shadow shadow

ruler

77

TeacherPage

Pythagorean Theorem

Procedure

Part One1. Pass out a 3" × 4" × 5" right triangle and 25 one-inch square tiles

2. Have groups build squares on each of the legs of the right triangle. If they do it correctly, all 25 tiles will be used, with 9 on the 3-inch side and 16 on the 4-inch side.

3. Have groups take the tiles and build a “square” on the hypotenuse using as many of the 25 tiles as they need. They will see that all the tiles will be used on the hypotenuse. This should show students that the sum of the squares of the legs equals the square formed on the hypotenuse.

Part Two1. In this activity, the students are using the converse of the Pythagorean theorem. That

is, if the squares of two sides of a triangle equal the square of the hypotenuse, then the triangle must be a right triangle.

2. Using the 6-8-10-inch string, the groups will locate right triangles in the room and use the string to prove the triangles are right triangles. To do this, students will put the

Learning OutcomeStudents will be able to apply the Pythagorean theorem to real-life situations.

Overview Students will use a variety of hands-on methods to learn about, test, and calculate with the Pythagorean theorem.

Time Requirements40–60 minutes

Group SizePairs

MaterialsFor each group

• 25 one-inch-square tiles

• 3" × 4" × 5" right triangles

• 26-inch piece of string

• centimeter rulers

• centimeter dot-matrix paper

ACTIVITY 15

78 Hands-On Math: GeometryTeacher

Page

6-inch length on one leg of the triangle and the 8-inch length on the other leg. The remaining 10-inch length should form the hypotenuse.

3. The groups should also try an object that doesn’t have a right angle to show that the three pieces will not fit together.

Part Three1. Using dot-matrix paper students make several right triangles.

2. They then measure the legs and put the measurements into the Pythagorean formula to determine the length of the hypotenuse.

3. Teams then measure the hypotenuse with a ruler to verify the results. This activity gives students practice with the Pythagorean theorem with visual results to verify their work.

Part FourIn this activity, the student will begin to see a spiral form.

The key is to keep one of the legs constant with a length of one. The other leg is the hypotenuse of the previous right triangle.

8-inch

6-inch

10-inch

6-inch 8-inch

10-inch(doesn’t fit)

11

1

1

1

1

1

1

Name Date


Pythagorean Theorem

Part OneYour teacher will give you a picture of a right triangle and

some 1-inch tiles. Count out 25 tiles. Determine how long the “legs” (the sides of the triangle that form a right angle) are in tiles.

One leg is __________ tiles long.

The other leg is __________ tiles long.

Now form squares on each of these legs with the tiles. Example: If the leg were 6 tiles long, the square would be 6 tiles by 6 tiles. Fill in this square with the remaining tiles. If you do this correctly, you will use all 25 tiles.

Now look at the “slanted side” of the triangle. The slanted side is called the hypotenuse. What do you estimate the length of the hypotenuse to be in tiles?

Estimated length is __________ tiles.

Now use your tiles to see how long the hypotenuse is.

The hypotenuse is __________ tiles long.

Following the same procedure as above, use as many tiles as you need to make a square on the hypotenuse and fill it in with the tiles.

What do you notice?

_________________________________________________________________________

_________________________________________________________________________

Can you make a rule about the legs and hypotenuse of a right triangle from what you have observed? Please state it below.

_________________________________________________________________________

_________________________________________________________________________

The Pythagorean theorem states that if a triangle is a right triangle, then:

a2 + b2 = c2

Explain in words what this means:

_________________________________________________________________________

_________________________________________________________________________

ACTIVITY 15

(continued)

Name Date


Pythagorean Theorem (continued)

Part TwoThe converse of the Pythagorean theorem states that if a triangle has sides of length

a, b, and c, and a2+b2=c2, then the triangle is a right triangle. This means that you can deter-mine whether a triangle has a right angle by testing the sides in the formula.

The converse is used in laying pipe, constructing houses, or anything that requires formation of a right angle.

Take a 26" piece of string, measure 6 inches, and place a knot at the 6" mark. From that knot, measure 8 inches and knot again. The remaining section should be 10", so cut off any leftover string.

Could 6, 8, and 10 be the measures of the sides of a right triangle? _______________

How do you know? ________________________________________________________

_________________________________________________________________________

Using this string, find places around the room that appear to form right angles and use this measuring device to check them. List below the angles you examined, and whether they were right angles.

_________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

Now find an angle that is not a right angle, and test to see that the 6-8-10 string will not work.

ACTIVITY 15

(continued)

Name Date



Part ThreeOn your matrix paper, draw a right triangle by connecting dots up and down or right

and left. The legs can be any length you want.

Then connect the diagonal (the hypotenuse).

ACTIVITY 15

(continued)

Name Date



Test the lengths with the Pythagorean theorem. In this example, the triangle drawn has the lengths of 4 and 5 cm. Plug these lengths into the Pythagorean formula to find the length of the hypotenuse.

a2 + b2 = c2

as 42 + 52 = c2

16 + 25 = c2

41 = c2

6.4 = c

Now measure the hypotenuse with your centimeter ruler to verify this solution.

Try several examples on the dot-matrix paper to demonstrate the Pythagorean theorem.

Part FourIn nature, the spiral shell is formed by connected right triangles

in which one leg stays one unit long, while the other leg increases to the length of the previous hypotenuse.

Look at the diagram on the right. Try creating this spiral on your grid paper.

See if you or any of your classmates can find a shell, or a photograph of a shell, that illustrates this.

ACTIVITY 15

41 = c2

83

TeacherPage

Median Line of a Triangle

ProcedureBefore the lab, review the use of a compass with the class. Have them practice drawing

circles first. Then introduce/review how you manipulate the compass to bisect line segments.

You may also need to review how to use a protractor to record angle measurements.

1. After reviewing how to bisect a line, have the students bisect line segments MN and MP.

2. Next have the students connect the two midpoints found and identify these points as S and T, respectively.

3. When they measure ∠MST and ∠N, they will find that they are equal.

4. The same will occur with ∠P and ∠MTS. They will be equal.

Learning OutcomesStudents will be able to

• determine the midpoints of two sides of a triangle using a compass.

• discover that a segment whose endpoints are the midpoints of two sides of a trian-gle is parallel to the third side of the triangle, and its length is one half the length of the third side.

Overview Through construction and measurement, students will discover properties of the line that connects the midpoints of two sides of a triangle.

Time Requirements30–45 minutes

Group SizePairs

Materials• compass

• protractor

• lab sheet

• ruler

ACTIVITY 16

84 Hands-On Math: GeometryTeacher

Page

5. Line segments ST and NP are parallel because if corresponding angles on the same side of a transversal are equal, the lines are parallel.

6. The students should notice that the measure of ST is half the measure of NP.

7. Students will discover that this will hold true for any triangle regardless of the shape.

Name Date


Median Line of a Triangle1. Use a compass to bisect line segments MN and

MP.

2. Label the midpoint of MN as S and the midpoint of MP as T, then draw line segment ST.

3. Use a protractor and measure ∠MST and ∠N. ∠MST = ________________ ∠N = ________________

4. Next measure ∠P and ∠MTS. ∠P = _______ ∠MTS _________

5. Compare the angle measures. What relationship does this suggest for line segments ST and NP? ______________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

Why? ____________________________________________________________________

_________________________________________________________________________

6. Measure ST and NP. How do these values compare?

_________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

ACTIVITY 16

(continued)

M

N P

Name Date


Median Line of a Triangle (continued)

7. Repeat the above steps for the obtuse and right triangles below.

Do you get the same results? Explain.

_________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

ACTIVITY 16

Obtuse triangle Right triangle

GEOMETRY - Walch · 4. Unnatural Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Documents

Transcript of GEOMETRY - Walch · 4. Unnatural Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....