engaging mathematics: teks-based activities … · 2010-09-27 · teks-based activities geometry...

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TEKS-BasedTEKS-BasedActivitiesActivities

GeometryGeometry

ISBN: 978-1-934950-66-1

www.theansweris4.net

Engaging Mathematics:TEKS-Based Activities

Geometry

Teacher Edition

Product ID: 407-1604

Creating solutions that are the worldwide standard for

educational excellence®

Revolutionizing education to inspire and advance future

generations®

SAMPLE

Table of Contents

Introduction ......................................................................................................... i–vii Overview of Materials ............................................................................................ i Texas Essential Knowledge and Skills (TEKS) Alignment Chart .......................... iv

Geometric Structure .......................................................................................... 1–43 Points, Lines, and Planes, Activity 1 ..................................................................... 2 Points, Lines, and Planes, Activity 2 ..................................................................... 4 Points, Lines, and Planes, Activity 3 ..................................................................... 8 Inductive Reasoning ........................................................................................... 12 Deductive Reasoning, Activity 1 ......................................................................... 14 Deductive Reasoning, Activity 2 ......................................................................... 16 Logical Reasoning .............................................................................................. 20 Non-Euclidean Geometries ................................................................................ 24 Conversions Between Measurement Systems ................................................... 26 One- and Two-Dimensional Coordinate Systems, Activity 1 .............................. 28 One- and Two-Dimensional Coordinate Systems, Activity 2 .............................. 30 One- and Two-Dimensional Coordinate Systems, Activity 3 .............................. 32 Angle Relationships, Activity 1 ............................................................................ 34 Angle Relationships, Activity 2 ............................................................................ 36 Angle Relationships, Activity 3 ............................................................................ 38 Angle Relationships, Activity 4 ............................................................................ 40 Angle Relationships, Activity 5 ............................................................................ 42

Transformations ............................................................................................... 44–71 Translations ........................................................................................................ 44 Reflections .......................................................................................................... 46 Rotations, Activity 1 ............................................................................................ 48 Rotations, Activity 2 ............................................................................................ 50 Dilations Activity ................................................................................................. 52 Transformation Compositions, Activity 1 ............................................................ 56 Transformation Compositions, Activity 2 ............................................................ 58 Investigating Polygons, Activity 1 ....................................................................... 60 Investigating Polygons, Activity 2 ....................................................................... 62 Investigating Polygons, Activity 3 ....................................................................... 66 Investigating Polygons, Activity 4 ....................................................................... 68 Tessellations ....................................................................................................... 70

Triangles and Congruence ............................................................................ 72–103 Classifying Triangles .......................................................................................... 72 Triangle Angle Sum Theorem, Activity 1 ............................................................ 76 Triangle Angle Sum Theorem, Activity 2 ............................................................ 78 Exterior Angles ................................................................................................... 80 Isosceles and Equilateral Triangles .................................................................... 84 Corresponding Parts of Congruent Triangles ..................................................... 88 Triangle Congruence, Activity 1 .......................................................................... 90 Triangle Congruence, Activity 2 .......................................................................... 92 Triangle Congruence, Activity 3 .......................................................................... 94

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Perpendicular Bisectors and Angle Bisectors of Triangles .................................. 96 Median .............................................................................................................. 100 Midsegments of Triangles ................................................................................. 102

Triangle Inequalities and Right Triangles .................................................. 104–135 Triangle Side and Angle Inequalities ................................................................. 104 Triangle Inequality Theorem ............................................................................. 106 Pythagorean Theorem Models .......................................................................... 110 Using the Pythagorean Theorem ...................................................................... 112 Pythagorean Triples .......................................................................................... 116 Converse of the Pythagorean Theorem ............................................................ 118 Applications of the Pythagorean Theorem ........................................................ 122 Special Right Triangles, Activity 1 ..................................................................... 124 Special Right Triangles, Activity 2 ..................................................................... 126 Special Right Triangles, Activity 3 ..................................................................... 128 Special Right Triangles, Activity 4 ..................................................................... 130 Special Right Triangles, Activity 5 ..................................................................... 134

Polygons ....................................................................................................... 136–199 Parallelograms, Activity 1 .................................................................................. 136 Parallelograms, Activity 2 .................................................................................. 140 Parallelograms, Activity 3 .................................................................................. 142 Rectangles, Activity 1 ........................................................................................ 146 Rectangles, Activity 2 ........................................................................................ 148 Rectangles, Activity 3 ........................................................................................ 150 Rhombi, Activity 1 ............................................................................................. 152 Rhombi, Activity 2 ............................................................................................. 154 Rhombi, Activity 3 ............................................................................................. 158 Comparing Squares and Rhombi ...................................................................... 160 Squares, Activity 1 ............................................................................................ 164 Squares, Activity 2 ............................................................................................ 166 Trapezoids, Activity 1 ........................................................................................ 168 Trapezoids, Activity 2 ........................................................................................ 170 Trapezoids, Activity 3 ........................................................................................ 174 Kites, Activity 1.................................................................................................. 176 Kites, Activity 2.................................................................................................. 178 Quadrilaterals ................................................................................................... 182 Polygons, Activity 1 ........................................................................................... 184 Polygons, Activity 2 ........................................................................................... 186 Polygons, Activity 3 ........................................................................................... 190 Polygons, Activity 4 ........................................................................................... 192 Perimeter and Area ........................................................................................... 194 Area, Activity 1 .................................................................................................. 196 Area, Activity 2 .................................................................................................. 198

Similarity and Trigonometry ........................................................................ 200–229 Similar Figures, Activity 1 .................................................................................. 200 Similar Figures, Activity 2 .................................................................................. 202 Similar Figures, Activity 3 .................................................................................. 204

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Similar Figures, Activity 4 ................................................................................. 206 Similar Triangles, Activity 1 .............................................................................. 210 Similar Triangles, Activity 2 .............................................................................. 214 Similar Triangles, Activity 3 .............................................................................. 216 Similar Triangles, Activity 4 .............................................................................. 218 Right Triangle Proportions, Activity 1 ................................................................ 220 Right Triangle Proportions, Activity 2 ................................................................ 222 Right Triangle Proportions, Activity 3 ................................................................ 224 Right Triangle Proportions, Activity 4 ................................................................ 228

Surface Area and Volume ........................................................................... 230–271 Components of Solids, Activity 1 ...................................................................... 230 Components of Solids, Activity 2 ...................................................................... 234 Components of Solids, Activity 3 ...................................................................... 236 Perspectives of Solids, Activity 1 ...................................................................... 238 Perspectives of Solids, Activity 2 ...................................................................... 242 Surface Area, Activity 1 .................................................................................... 246 Surface Area, Activity 2 .................................................................................... 248 Surface Area, Activity 3 .................................................................................... 250 Volume, Activity 1 ............................................................................................. 254 Volume, Activity 2 ............................................................................................. 256 Volume, Activity 3 ............................................................................................. 258 Similar Solids and Dimensional Changes, Activity 1 ......................................... 260 Similar Solids and Dimensional Changes, Activity 2 ......................................... 262 Similar Solids and Dimensional Changes, Activity 3 ......................................... 264 Similar Solids and Dimensional Changes, Activity 4 ......................................... 266 Similar Solids and Dimensional Changes, Activity 5 ......................................... 270

Circles ........................................................................................................... 272–311 Angles and Arcs Within a Circle, Activity 1 ....................................................... 272 Angles and Arcs Within a Circle, Activity 2 ....................................................... 276 Angles and Arcs Within a Circle, Activity 3 ....................................................... 278 Angles and Arcs Within a Circle, Activity 4 ....................................................... 280 Angles and Arcs Within a Circle, Activity 5 ....................................................... 282 Circumference, Area, and Parts of Circles, Activity 1 ....................................... 284 Circumference, Area, and Parts of Circles, Activity 2 ....................................... 286 Circumference, Area, and Parts of Circles, Activity 3 ....................................... 288 Circumference, Area, and Parts of Circles, Activity 4 ....................................... 290 Circumference, Area, and Parts of Circles, Activity 5 ....................................... 292 Geometric Probability ....................................................................................... 296 Angle Relationships–Chords, Secants, Tangents, Activity 1 ............................ 298 Angle Relationships–Chords, Secants, Tangents, Activity 2 ............................ 300 Angle Relationships–Chords, Secants, Tangents, Activity 3 ............................ 304 Segment Relationships, Activity 1 .................................................................... 306 Segment Relationships, Activity 2 .................................................................... 308 Segment Relationships, Activity 3 .................................................................... 310

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Engaging Mathematics: Geometry TEKS-Based Activities

© Region 4 Education Service Center All rights reserved.

Activity Objective The student will determine the number of handshakes required for everyone in a given group to shake hands with each other exactly once.

Materials • Smiles All Around • Highlighters—pink and yellow • Straightedge (optional)

Facilitation Questions • What strategies could you use to help you determine the solution to each

problem? Possible answers: I could act out the handshakes with people in my class. I could name each smiley face and make an organized list. I could draw a line to represent each handshake.

• What strategy could you use to be certain you have not counted a line (or handshake) twice? Possible answer: I could use a different color or style of line as I draw handshakes for each person. If there is already a line drawn, then I do not need to draw it again.

• If Person 1 shook hands with Person 3, does Person 3 need to shake hands with Person 1? Why or why not? No, because the two people have already met and shaken hands.

Answers

Number of People

Number of Perimeter Handshakes (pink)

Number of Interior Handshakes (yellow)

Total Number of Handshakes

3 3 0 3

4 4 2 6

5 5 5 10

6 6 9 15

Investigating Polygons, Activity 3

The intent of this

activity is to lay the foundation for

determining the total number of diagonals in a

polygon.

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Student Name: ________________________________________ Date: ________________

Smiles All Around How many handshakes are necessary for each group of people so that each person has shaken the hand of every other person with no repeated handshakes? Use the pictures below to help you determine the number of handshakes and record your data in the table at the bottom of the page.

• Highlight the handshakes along the perimeter of the group in pink. • Highlight the handshakes in the interior of the group yellow.

Communicating About Mathematics Describe any patterns you notice in the table above.

Group A Group C

Group B Group D

Number of People

Number of Perimeter Handshakes (pink)

Number of Interior Handshakes (yellow)

Total Number of Handshakes

3

4

5

6

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Activity Objective The student will find the centroid of three triangles.

Materials The Three of Us Patty paper Straightedge

Facilitation Questions How could you fold your patty paper to find the midpoint of a side of a triangle?

I could line up the two end points of the triangle and fold a crease to mark the midpoint of the side.

How could you use your ruler to draw an obtuse triangle? I could use the ruler to draw two rays that form an angle that is greater than 90, and I could then connect two points on the rays to make an obtuse triangle.

Answers

Possible answers:

Median

Students will work in

groups of three to

complete this

activity.

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The Three of Us Determine which member of your group is Student A, Student B, and Student C. Use your patty paper and follow the instructions below.

Student A Student B Student C • Draw an obtuse triangle

on your patty paper. • Choose one side and fold

the side in order to find the midpoint.

• Use your ruler to draw a line connecting the midpoint and the opposite vertex.

• Repeat the process for all three sides.

• Draw an acute triangle on your patty paper.

• Choose one side and fold the side in order to find the midpoint.



• Draw a right triangle on your patty paper.

• Choose one side and fold the side in order to find the midpoint.



Sketch a picture of each triangle in your group.

Communicating About Mathematics What do you observe about the lines drawn?

_______________________________________________________________

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Activity Objective The student will solve problems involving properties of squares.

Materials • Round Robin: Squares • Graphing calculator

Facilitation Questions • What equation could you write to help you determine the side length of the

square? I could use the Pythagorean Theorem and write a a ( )2 2 28 2+ = where a represents the side length of the square.

• What type of triangle is formed by the diagonal of the square? A 45˚-45˚-90˚ triangle is formed.

• How could you use what you know about 45˚-45˚-90˚ triangles to help you determine the side length?

In a 45˚-45˚-90˚ triangle, the hypotenuse is 2 times the length of the leg. If I take the

diagonal and divide it by 2 , the result will be the side length of the square.

Answers

1. 3 2 inches

2. 64 square inches

3. 24 inches

4. 5x 2 inches

Squares, Activity 2

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Round Robin: Squares • Pass your paper to the person seated at your right and solve the first problem on the paper

you have received. You may work with your group to solve the problem.

• Upon completion, pass the papers to the right again and solve the second problem. Again, you may work together to solve the problem.

• Continue this process for the remaining problems.

1. A square has side lengths that

measure 3 inches. Find the length of the diagonal. Write your answer in simplest radical form.

3. The diagonal of a square measures

6 2 inches. Find the perimeter of the square.

2. The diagonal of a square measures

8 2 inches. Find the area of the square.

4. A square has side lengths that measure 5x inches. Find the length of the diagonal. Write your answer in simplest radical form.

Communicating About Mathematics Describe how you determined the side length of the square when you were given the length of the diagonal.

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Activity Objective The student will create a regular polygon by rotating a triangle. Students will find the area of the triangle and the resulting polygon.

Materials • Spinning Triangles • Spinners—1 set of spinners, copied on card stock, per group of

3 students • Scissors • Ruler • Calculator (optional)

Facilitation Question • How many triangles are drawn to form the polygon?

Answers will vary depending on the triangle. To form a polygon using Triangle A, five triangles are drawn. To form a polygon using Triangle B, six triangles are drawn. To form a polygon using Triangle C, eight triangles are drawn.

Answers Possible answers:

Triangle Polygon Formed Area of Triangle

(cm2) Area of Polygon

(cm2)

A pentagon ≈ 7.7 ≈ 38

B hexagon ≈ 7 ≈ 42

C octagon ≈ 5.7 ≈ 46

Polygons, Activity 2

Place

students in groups of three for

this activity.

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Spinning Triangles Each person in your group should choose one of the isosceles triangles from Spinners. • Measure the base and height of your isosceles triangle to the nearest tenth of a centimeter.

Record the dimensions on your triangle. • Cut out your triangle. • Align the vertex angle of your isosceles triangle with the point below. • Trace the triangle around the point below as many times as possible without overlapping

triangles. • Then work with your group to complete the table.

Triangle Polygon Formed Area of Triangle

(cm2) Area of Polygon

(cm2)

A

B

C

Communicating About Mathematics If another isosceles triangle were provided and the measure of the vertex angle was 30˚, what type of polygon would be formed using the procedure above? Justify your answer.

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Spinners Cut along the bold dotted lines. Four sets of spinners are provided.

Triangle C Triangle B Triangle A




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engaging mathematics: teks-based activities … · 2010-09-27 · teks-based activities geometry...

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