Investigating Relationships of Area and Perimeter in Similar Polygons

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Investigating Relationships of Area and Perimeter in Similar Polygons"

Transcription

1 Investigating Relationships of Area and Perimeter in Similar Polygons Lesson Summary: This lesson investigates the relationships between the area and perimeter of similar polygons using geometry software. Key words: Perimeter, area, similar, polygons Existing Knowledge Base: Prior to this lesson, students should understand similarity of polygons (the angles are congruent and the sides are proportional). Students are also expected to have some proficiency with a dynamic geometry software package. This lesson was designed specifically for Cabri II, but could be easily adapted for use with other software packages (such as Geometer s Sketchpad). NCTM Standards : NCTM Geometry Standard: Students should be able to analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Ohio Geometry and Spatial Reasoning Standard: Students identify, classify, compare and analyze characteristics, properties, and relationships of one-, two-, and three-dimensional geometric figures and objects. Learning Objectives: Students determine that the ratio of the perimeters of similar polygons is equal to the ratio of any two corresponding sides, and that the ratio of the areas of similar polygons is equal to the square of the ratio of two corresponding sides. Materials: 1. Lab handout with detailed explanations of the activity 2. Cabri II 3. Calculator (if desired to make some calculations easier) Suggested Procedure: 1. As a motivator for the lesson, present the following problem to your students: You are purchasing index cards. A 3-inch by 5-inch pack of 100 cards costs $.98. How much should you expect a pack of inch by 6-inch cards to cost? Have the students make an initial guess and explain that this problem will be revisited after the lab. 2. Students could be grouped in pairs for this procedure. 3. Step 4 in the student lab sheet is an extension of ideas and is designed for students who finish early and need extra work to challenge them. 4. After the class has determined a method for finding the ratios of perimeter and area of similar polygons, the original motivating question can be revisited, as well as several extensions. A worksheet with sample questions (including a revisit of the original motivating question) and an extension for after the class discussion has been included with this lesson. 5. Assessment for the lesson could be determined by checking the worksheet, and also by collecting and checking the lab sheets. One suggestion is that each student should have explanations in his or her own words

2 Investigating Relationships of Perimeter and Area in Similar Polygons Group Members names: File Names: Goal: The goal of this lesson is for each group of students to determine a relationship between the ratio of the sides of similar polygons and the ratios of perimeter and area. To make sure our calculations are not affected by rounding errors, we need to change a setting in Cabri II. Under the Options menu, select Preferences. Under the Display Precision and Units menu, set the Length option to 4. Step 1: Investigating Perimeter Relationships 1. Create a triangle. Label the vertices A, B, and C. (Use the Triangle tool) 2. Create a box containing the number.5. (Use the Numerical Edit tool) 3. Create a point way outside of your triangle. 4. Using the Dilation tool (under the sixth button of the tool bar), point at the triangle so that the cursor says, Dilate this triangle. Click the mouse once. Move the cursor to the point you drew outside your triangle until the cursor says with respect to this point and click the mouse again. Move the mouse toward the number.5 until the cursor says using this factor and click the mouse again. You should get a new triangle, smaller than the original. 5. The new triangle should not overlap the old triangle. If it does overlap, drag the point you dilated it with so that they no longer overlap. 6. Label the vertices of your new triangle D, E, and F, respectively. Is triangle DEF half as big as triangle ABC? Make a prediction. (We will verify the prediction later.) 7. Determine the perimeter and side lengths for each triangle. Label them on the screen for later use. Record your results below: (Use the Distance and Length tool) AB = BC = AC = DE = EF = DF = Perimeter of triangle ABC = Perimeter of triangle DEF =

3 If two triangles are similar, what do we know about the ratios of their corresponding sides? Now calculate the ratios: DE/AB = EF/BC = DF/AB = What relationship is there between the ratios of corresponding sides and the dilation factor we used? Why does this relationship exist? Can you predict what the ratio of the perimeters will be? Make a guess and write it here: Now, calculate the ratio of the perimeters. Perimeter of DEF/Perimeter of ABC = _ What relationship is there between the ratio of the corresponding sides and the ratios of the perimeters in similar triangles? Do you think this relationship will always work, or will it change if you change the triangle? Try dragging one of the vertices of the original triangle. Notice that this affects the similar triangle. Does the relationship you wrote down above still work? _ Why or why not? Save this file so you can use it later in the lab. Let s investigate a shape other than triangles. 1. Start a new file.

4 2. Create a square using (label it PQRS) and use the same method as above for creating a similar square (WXYZ, respectively), this time using a factor of.75 (Use Regular Polygon tool) Are all squares similar? 3. Determine the length of a side of each square and the perimeter of each square. Label these values. What is the ratio of the corresponding sides of the two squares? WX/PQ = What is the ratio of the perimeters of the squares? _ Does the relationship you determined for similar triangles also work for similar squares? What relationship is there between the ratio of the corresponding sides and the ratios of the perimeters in similar squares? Would you expect this to work for any similar quadrilaterals? What about other similar polygons? Why do you think this relationship exists? (Think about what you know about similarity of polygons and perimeters of polygons.) Step 2: Investigating Area Relationships Now we are ready to investigate the relationships involving ratios of sides and areas of similar polygons.

5 1. Go back to your first file (the one with the similar triangles). Determine the area of each triangle. (Use Area tool) 2. Create altitudes of your triangle and measure them. (Use Perpendicular Line tool) 3. You should have already determined the length of each side. Fill in the following information in the first row of the table below: DE AB DE/AB Altitude Of DEF Altitude of ABC Altitude DEF / Altitude ABC DEF ABC DEF / ABC At the beginning of the lab, we asked if DEF is half as big as ABC. Is it? How do we know this? 3. Now, drag a vertex of the larger triangle around so that the area of ABC is a different value. Notice that this changes the shape and area of DEF. Enter the information for this new triangle into the next row of the table. Do you notice any relationship between DE/AB, Altitude of DDEF/Altitude of DABC and DDEF/ DABC? You may need to play around with numbers a little. Try experimenting on a calculator to try to find a relationship that will work. When you come up with a relationship that works, write it below: 4. Go back to your file with the similar squares. Fill in the table below, first with your original values, and then after you drag a vertex: WX PQ WX/PQ WXYZ PQRS WXYZ/ PQRS

6 Do you notice any relationship between WX/PQ and WXYZ/ PQRS? Try checking if the relationship you wrote down in Step 3 works for similar squares. If that relationship does not work, try to come up with one that works for both triangles and squares. You may need to try relationships other than addition, subtraction, multiplication, and division. Consider using squares, square roots, cubes, cube roots, etc. When you come up with a relationship that works for both case (triangles AND squares), write it below: Step 3: Review and Conclusions Create a new file. Create a triangle and then create a similar triangle, using any dilation factor other than.5 or.75. Write down the dilation factor you used: You should be able to guess the ratio of corresponding sides. Remember to give your ratio in the form of new figure to old figure, since that is the order we have used all along. Make a guess and write it down: You should also be able to determine the ratio of perimeters of the triangles: Determine if your ratio is correct. Is it? Now, try to make a prediction for the ratio of the areas: Determine if your ratio is correct. Is it? Summarize your results below: Given the corresponding sides of similar polygons, the ratio of the perimeter is Given the corresponding sides of similar polygons, the ratio of the areas is

7 Step 4: Extension Will the relationships you discovered using triangles and squares work for other polygons? Experiment with both regular and irregular polygons. Create similar polygons and experiment with various factors of dilation. Fill in the chart below to keep track of your results. Number of Sides of Polygon Regular Or Irregular Ratio of Corresponding Sides Ratio of Perimeters Ratio of Areas Think about the following questions as you investigate: 1. Does the relationship you developed for squares and triangles still work for other regular polygons? What about irregular polygons? 2. Does it matter if my polygons are concave or convex? Write a summary of your findings in the space below:

8 Worksheet to Review Relationships of Area and Perimeter in Similar Polygons 1. Now that we have learned the relationships between corresponding sides of similar polygons and the ratios of their perimeter and area, we can revisit the original problem that motivated our discussion: You are purchasing index cards. A 3-inch by 5-inch pack of 100 cards costs $.98. How much should you expect a pack of inch by 6-inch cards to cost? What about a pack of inch by 7-inch cards? 2. Two similar heptagons have corresponding sides in a ratio of 3 inches: 7 inches. What is the ratio of their perimeters? What is the ratio of their areas? 3. Two similar pentagons have areas in the ratio of 25 cm 2 : 36 cm 2. What is the ratio of their sides? What is the ratio of their perimeters? 4. PQR has an altitude of 5.1 ft. LMN has an altitude of 6.2 ft. Given that the two triangles are similar, find the area of PQR given that the area of LMN is 9.3 ft 2 5. A regular decagon has an area of 90 square centimeters. A similar decagon has an area of 25 square centimeters. What is the ratio of the perimeters of the first decagon to the second? Extension: Hexagons ABCDEF and PQRSTU are similar. The scale factor of Hexagon ABCDEF to Hexagon PQRSTU is 14:3. The area of Hexagon ABCDEF is 24x. The area of Hexagon PQRSTU is x + 2. The perimeter of Hexagon ABCDEF is 40 + y and the perimeter of Hexagon PQRSTU is 2y-11. a) Use the scale factor to find the ratio of the areas of the hexagons. b) Use the scale factor to find the ratio of the perimeters of the hexagons. c) What relationship is there between the ratio of the areas and the ratio of the perimeters? d) Solve for x and y. Explain your work. How did you know how to set up your equation? How could you check that your answer is correct?

Similar Polygons. Copy both triangles onto tracing paper. Measure and record the sides of each triangle. Cut out both triangles.

Similar Polygons. Copy both triangles onto tracing paper. Measure and record the sides of each triangle. Cut out both triangles. -7 Similar Polygons MAIN IDEA Identify similar polygons and find missing measures of similar polygons. New Vocabulary polygon similar corresponding parts congruent scale factor Math Online glencoe.com

More information

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid

Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,

More information

Analysis in Geometry. By Danielle Long. Grade Level: 8 th. Time: 5-50 minute periods. Technology used: Geometer s sketchpad Geoboards NLVM website

Analysis in Geometry. By Danielle Long. Grade Level: 8 th. Time: 5-50 minute periods. Technology used: Geometer s sketchpad Geoboards NLVM website Analysis in Geometry By Danielle Long Grade Level: 8 th Time: 5-50 minute periods Technology used: Geometer s sketchpad Geoboards NLVM website 1 NCTM Standards Addressed Problem Solving Geometry Algebra

More information

A convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon.

A convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon. hapter 7 Polygons A polygon can be described by two conditions: 1. No two segments with a common endpoint are collinear. 2. Each segment intersects exactly two other segments, but only on the endpoints.

More information

POLYGONS

POLYGONS POLYGONS 8.1.1 8.1.5 After studying triangles and quadrilaterals, students now extend their study to all polygons. A polygon is a closed, two-dimensional figure made of three or more nonintersecting straight

More information

Exploring Geometric Figures Using Cabri Geometry II

Exploring Geometric Figures Using Cabri Geometry II Exploring Geometric Figures Using Cabri Geometry II Regular Polygons Developed by: Charles Bannister. Chambly County High School Linda Carre.. Chambly County High School Manon Charlebois Vaudreuil Catholic

More information

How do changes in dimensions of similar geometric figures affect the perimeters and the areas of the figures? ACTIVITY: Creating Similar Figures

How do changes in dimensions of similar geometric figures affect the perimeters and the areas of the figures? ACTIVITY: Creating Similar Figures .6 Perimeters and Areas of Similar Figures How do changes in dimensions of similar geometric figures affect the perimeters and the areas of the figures? ACTIVITY: Creating Similar Figures Work with a partner.

More information

Exploring Trigonometric Ratios

Exploring Trigonometric Ratios Exploring Trigonometric Ratios Lesson Summary: Students will explore the trigonometric ratios by constructing a table of values and exploring the table as the shape of the triangle changes. The lab is

More information

Tessellations. A tessellation is created when a shape is repeated over and over again to cover the plane without any overlaps or gaps.

Tessellations. A tessellation is created when a shape is repeated over and over again to cover the plane without any overlaps or gaps. Tessellations Katherine Sheu A tessellation is created when a shape is repeated over and over again to cover the plane without any overlaps or gaps. 1. The picture below can be extended to a tessellation

More information

Lesson 6: Polygons and Angles

Lesson 6: Polygons and Angles Lesson 6: Polygons and Angles Selected Content Standards Benchmark Assessed: G.4 Using inductive reasoning to predict, discover, and apply geometric properties and relationships (e.g., patty paper constructions,

More information

Five-Minute Check (over Lesson 1 5) CCSS Then/Now New Vocabulary Key Concepts: Polygons Example 1: Name and Classify Polygons Key Concepts:

Five-Minute Check (over Lesson 1 5) CCSS Then/Now New Vocabulary Key Concepts: Polygons Example 1: Name and Classify Polygons Key Concepts: Five-Minute Check (over Lesson 1 5) CCSS Then/Now New Vocabulary Key Concepts: Polygons Example 1: Name and Classify Polygons Key Concepts: Perimeter, Circumference, and Area Example 2: Find Perimeter

More information

Pythagorean Theorem. Inquiry Based Unit Plan

Pythagorean Theorem. Inquiry Based Unit Plan Pythagorean Theorem Inquiry Based Unit Plan By: Renee Carey Grade: 8 Time: 5 days Tools: Geoboards, Calculators, Computers (Geometer s Sketchpad), Overhead projector, Pythagorean squares and triangle manipulatives,

More information

10.1: Areas of Parallelograms and Triangles

10.1: Areas of Parallelograms and Triangles 10.1: Areas of Parallelograms and Triangles Important Vocabulary: By the end of this lesson, you should be able to define these terms: Base of a Parallelogram, Altitude of a Parallelogram, Height of a

More information

Polygons are figures created from segments that do not intersect at any points other than their endpoints.

Polygons are figures created from segments that do not intersect at any points other than their endpoints. Unit #5 Lesson #1: Polygons and Their Angles. Polygons are figures created from segments that do not intersect at any points other than their endpoints. A polygon is convex if all of the interior angles

More information

The Area of a Triangle Using Its Semi-perimeter and the Radius of the In-circle: An Algebraic and Geometric Approach

The Area of a Triangle Using Its Semi-perimeter and the Radius of the In-circle: An Algebraic and Geometric Approach The Area of a Triangle Using Its Semi-perimeter and the Radius of the In-circle: An Algebraic and Geometric Approach Lesson Summary: This lesson is for more advanced geometry students. In this lesson,

More information

Geometry: 1-1 Day 1 Points, Lines and Planes

Geometry: 1-1 Day 1 Points, Lines and Planes Geometry: 1-1 Day 1 Points, Lines and Planes What are the Undefined Terms? The Undefined Terms are: What is a Point? How is a point named? Example: What is a Line? A line is named two ways. What are the

More information

Properties of Special Parallelograms

Properties of Special Parallelograms Properties of Special Parallelograms Lab Summary: This lab consists of four activities that lead students through the construction of a parallelogram, a rectangle, a square, and a rhombus. Students then

More information

Unit 3: Triangle Bisectors and Quadrilaterals

Unit 3: Triangle Bisectors and Quadrilaterals Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties

More information

6-1 Properties and Attributes of Polygons

6-1 Properties and Attributes of Polygons 6-1 Properties and Attributes of Polygons Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up 1. A? is a three-sided polygon. triangle 2. A? is a four-sided polygon. quadrilateral Evaluate each expression

More information

The Elementary School Math Project. Mirror, Mirror

The Elementary School Math Project. Mirror, Mirror The Elementary School Math Project Mirror, Mirror Math Grows Up (Geometry/Spatial Sense) Objective Students will use spatial reasoning and problem solving strategies to determine which regular polygons

More information

Unit 8. Quadrilaterals. Academic Geometry Spring Name Teacher Period

Unit 8. Quadrilaterals. Academic Geometry Spring Name Teacher Period Unit 8 Quadrilaterals Academic Geometry Spring 2014 Name Teacher Period 1 2 3 Unit 8 at a glance Quadrilaterals This unit focuses on revisiting prior knowledge of polygons and extends to formulate, test,

More information

Exploring Geometric Mean

Exploring Geometric Mean Exploring Geometric Mean Lesson Summary: The students will explore the Geometric Mean through the use of Cabrii II software or TI 92 Calculators and inquiry based activities. Keywords: Geometric Mean,

More information

Discovering Math: Exploring Geometry Teacher s Guide

Discovering Math: Exploring Geometry Teacher s Guide Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional

More information

Chapter 3. Chapter 3 Opener. Section 3.1. Big Ideas Math Blue Worked-Out Solutions. Try It Yourself (p. 101) So, the value of x is 112.

Chapter 3. Chapter 3 Opener. Section 3.1. Big Ideas Math Blue Worked-Out Solutions. Try It Yourself (p. 101) So, the value of x is 112. Chapter 3 Opener Try It Yourself (p. 101) 1. The angles are vertical. x + 8 120 x 112 o, the value of x is 112. 2. The angles are adjacent. ( x ) + 3 + 43 90 x + 46 90 x 44 o, the value of x is 44. 3.

More information

Inscribed Right Triangles

Inscribed Right Triangles This lesson introduces students to the properties of inscribed right triangles. The properties are: 1. If a right triangle is inscribed in a circle, then its hypotenuse is a diameter of the 2. If one side

More information

Activities Grades 3 5 ROTATING STRING SHAPES. Make multi-sided shapes with string. [45 minutes]

Activities Grades 3 5 ROTATING STRING SHAPES. Make multi-sided shapes with string. [45 minutes] Activities Grades 3 5 www.exploratorium.edu/geometryplayground/activities ROTATING STRING SHAPES Make multi-sided shapes with string. [45 minutes] Materials: String, about 2.5 meters (8 feet) long, tied

More information

We are going to investigate what happens when we draw the three angle bisectors of a triangle using Geometer s Sketchpad.

We are going to investigate what happens when we draw the three angle bisectors of a triangle using Geometer s Sketchpad. Krystin Wright Geometer s Sketchpad Assignment Name Date We are going to investigate what happens when we draw the three angle bisectors of a triangle using Geometer s Sketchpad. First, open up Geometer

More information

Investigation: Area of a Parallelogram

Investigation: Area of a Parallelogram Investigation: Area of a Parallelogram In this investigation, you will discover properties of the area of a parallelogram. You will make a table that shows the relationships of the segments, angles, and

More information

The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons.

The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons. Interior Angles of Polygons The angle sum property of triangles can help determine the sum of the measures of interior angles of other polygons. The sum of the measures of the interior angles of a triangle

More information

Characteristics of Solid Figures Face Edge Vertex (Vertices) A shape is characterized by its number of... Faces, Edges, and Vertices Faces - 6 Edges - 12 A CUBE has... Vertices - 8 Faces of a Prism The

More information

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT!

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! FINDING THE DISTANCE BETWEEN TWO POINTS DISTANCE FORMULA- (x₂-x₁)²+(y₂-y₁)² Find the distance between the points ( -3,2) and

More information

Geometry. Geometry is the study of shapes and sizes. The next few pages will review some basic geometry facts. Enjoy the short lesson on geometry.

Geometry. Geometry is the study of shapes and sizes. The next few pages will review some basic geometry facts. Enjoy the short lesson on geometry. Geometry Introduction: We live in a world of shapes and figures. Objects around us have length, width and height. They also occupy space. On the job, many times people make decision about what they know

More information

Activity Set 4. Trainer Guide

Activity Set 4. Trainer Guide Geometry and Measurement of Plane Figures Activity Set 4 Trainer Guide Int_PGe_04_TG GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set #4 NGSSS 3.G.3.1 NGSSS 3.G.3.3 NGSSS 4.G.5.1 NGSSS 5.G.3.1 Amazing

More information

CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY

CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY Specific Expectations Addressed in the Chapter Explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that

More information

Remaining Fractions Two-Step Equations

Remaining Fractions Two-Step Equations Remaining Fractions Two-Step Equations Lesson 61 61 Remaining Fractions If a whole has been divided into parts and we know the size of one part, then we can figure out the size of the other parts. What

More information

Grade Level: High School

Grade Level: High School Lesson I: Triangles- Exterior Angle Theorem KEY WORDS: Triangles, exterior-angle theorem, and remote interior angles. Grade Level: High School SUMMARY: With this investigation students will discover the

More information

Isosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors

Isosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A

More information

Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

More information

The Polygon Angle-Sum Theorems

The Polygon Angle-Sum Theorems 6-1 The Polygon Angle-Sum Theorems Common Core State Standards G-SRT.B.5 Use congruence... criteria to solve problems and prove relationships in geometric figures. MP 1, MP 3 Objectives To find the sum

More information

3 rd Six Weeks

3 rd Six Weeks Geometry 3 rd Six Weeks 014-015 MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY Nov 9 10 11 1 13 6-1 Angle Measures in Polygons Class: Wksht #1 6- Properties of Parallelograms Class: Wksht # 6-3 Proving Parallelograms

More information

LESSON PLAN #1: Discover a Relationship

LESSON PLAN #1: Discover a Relationship LESSON PLAN #1: Discover a Relationship Name Alessandro Sarra Date 4/14/03 Content Area Math A Unit Topic Coordinate Geometry Today s Lesson Sum of the Interior Angles of a Polygon Grade Level 9 NYS Mathematics,

More information

TImath.com. Geometry. Angle Bisectors in a Triangle

TImath.com. Geometry. Angle Bisectors in a Triangle Angle Bisectors in a Triangle ID: 8889 Time required 40 minutes Activity Overview In this activity, students will explore the relationships between an angle bisector and segments in a triangle. They will

More information

Perimeter and area formulas for common geometric figures:

Perimeter and area formulas for common geometric figures: Lesson 10.1 10.: Perimeter and Area of Common Geometric Figures Focused Learning Target: I will be able to Solve problems involving perimeter and area of common geometric figures. Compute areas of rectangles,

More information

Greater Nanticoke Area School District Math Standards: Grade 6

Greater Nanticoke Area School District Math Standards: Grade 6 Greater Nanticoke Area School District Math Standards: Grade 6 Standard 2.1 Numbers, Number Systems and Number Relationships CS2.1.8A. Represent and use numbers in equivalent forms 43. Recognize place

More information

Geometer s Sketchpad. Discovering the incenter of a triangle

Geometer s Sketchpad. Discovering the incenter of a triangle Geometer s Sketchpad Discovering the incenter of a triangle Name: Date: 1.) Open Geometer s Sketchpad (GSP 4.02) by double clicking the icon in the Start menu. The icon looks like this: 2.) Once the program

More information

PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

More information

EXPLORING GEOMETRIC FIGURES Grade 10 6-Day Lesson Plan

EXPLORING GEOMETRIC FIGURES Grade 10 6-Day Lesson Plan 1 EXPLORING GEOMETRIC FIGURES Grade 10 6-Day Lesson Plan Tangrams Geoboards Equation Grapher Green Globs AlphaShapes Protractor Compass Created by Sandra Metzler 2 OVERALL OBJECTIVES 1- Students will increase

More information

Heron s Formula. Key Words: Triangle, area, Heron s formula, angle bisectors, incenter

Heron s Formula. Key Words: Triangle, area, Heron s formula, angle bisectors, incenter Heron s Formula Lesson Summary: Students will investigate the Heron s formula for finding the area of a triangle. The lab has students find the area using three different methods: Heron s, the basic formula,

More information

Archimedes Calculation of Pi Barbara J. Meidinger Helen Douglas. Part I: Lesson Plans

Archimedes Calculation of Pi Barbara J. Meidinger Helen Douglas. Part I: Lesson Plans Archimedes Calculation of Pi Barbara J. Meidinger Helen Douglas Part I: Lesson Plans Day One: Introduction Target Learners: Second semester geometry students Student Materials: Scientific or graphing calculators

More information

TImath.com. Geometry. Triangle Sides & Angles

TImath.com. Geometry. Triangle Sides & Angles Triangle Sides & Angles ID: 8792 Time required 40 minutes Activity Overview In this activity, students will explore side and angle relationships in a triangle. First, students will discover where the longest

More information

Study Guide and Review

Study Guide and Review Fill in the blank in each sentence with the vocabulary term that best completes the sentence. 1. A is a flat surface made up of points that extends infinitely in all directions. A plane is a flat surface

More information

Examples: 1. Write the angles in order from 2. Write the sides in order from

Examples: 1. Write the angles in order from 2. Write the sides in order from Lesson 1 Triangle Inequalities 17. I can apply the triangle inequalities theorems When considering triangles, two basic questions arise: Can any three sides form a triangle? What is the relationship between

More information

Author(s): Hope Phillips

Author(s): Hope Phillips Title: Fish Aquarium Math *a multi-day lesson Real-World Connection: Grade: 5 Author(s): Hope Phillips BIG Idea: Volume Designing and building aquariums includes mathematical concepts including, but not

More information

Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

More information

Which shapes make floor tilings?

Which shapes make floor tilings? Which shapes make floor tilings? Suppose you are trying to tile your bathroom floor. You are allowed to pick only one shape and size of tile. The tile has to be a regular polygon (meaning all the same

More information

Classifying Shapes and Figures Grade Four

Classifying Shapes and Figures Grade Four Ohio Standards Connection Geometry and Spatial Sense Benchmark E Use attributes to describe, classify and sketch plane figures and build solid objects Indicator 2 Describe, classify, compare and model

More information

Geometry Vocabulary Booklet

Geometry Vocabulary Booklet Geometry Vocabulary Booklet Geometry Vocabulary Word Everyday Expression Example Acute An angle less than 90 degrees. Adjacent Lying next to each other. Array Numbers, letter or shapes arranged in a rectangular

More information

Winter 2016 Math 213 Final Exam. Points Possible. Subtotal 100. Total 100

Winter 2016 Math 213 Final Exam. Points Possible. Subtotal 100. Total 100 Winter 2016 Math 213 Final Exam Name Instructions: Show ALL work. Simplify wherever possible. Clearly indicate your final answer. Problem Number Points Possible Score 1 25 2 25 3 25 4 25 Subtotal 100 Extra

More information

Three-Dimensional Figures or Space Figures. Rectangular Prism Cylinder Cone Sphere. Two-Dimensional Figures or Plane Figures

Three-Dimensional Figures or Space Figures. Rectangular Prism Cylinder Cone Sphere. Two-Dimensional Figures or Plane Figures SHAPE NAMES Three-Dimensional Figures or Space Figures Rectangular Prism Cylinder Cone Sphere Two-Dimensional Figures or Plane Figures Square Rectangle Triangle Circle Name each shape. [triangle] [cone]

More information

Tessellating with Regular Polygons

Tessellating with Regular Polygons Tessellating with Regular Polygons You ve probably seen a floor tiled with square tiles. Squares make good tiles because they can cover a surface without any gaps or overlapping. This kind of tiling is

More information

Target To know the properties of a rectangle

Target To know the properties of a rectangle Target To know the properties of a rectangle (1) A rectangle is a 3-D shape. (2) A rectangle is the same as an oblong. (3) A rectangle is a quadrilateral. (4) Rectangles have four equal sides. (5) Rectangles

More information

6.1 Ratios, Proportions, and the Geometric Mean

6.1 Ratios, Proportions, and the Geometric Mean 6.1 Ratios, Proportions, and the Geometric Mean Obj.: Solve problems by writing and solving proportions. Key Vocabulary Ratio - If a and b are two numbers or quantities and b 0, then the ratio of a to

More information

Review of Ratio and Proportion Ratio- a comparison of two quantities. the ratio of p and q is p q

Review of Ratio and Proportion Ratio- a comparison of two quantities. the ratio of p and q is p q Review of Ratio and Proportion Ratio- a comparison of two quantities. the ratio of p and q is p q or p:q or p to q The ratio of to is EX1/ Find the ratio of shaded boxes to unshaded. EX2/ Find the ratio

More information

CHAPTER 6. Polygons, Quadrilaterals, and Special Parallelograms

CHAPTER 6. Polygons, Quadrilaterals, and Special Parallelograms CHAPTER 6 Polygons, Quadrilaterals, and Special Parallelograms Table of Contents DAY 1: (Ch. 6-1) SWBAT: Find measures of interior and exterior angles of polygons Pgs: 1-7 HW: Pgs: 8-10 DAY 2: (6-2) Pgs:

More information

TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry

TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry ASLE Expectation: Focus Objective: Level: Strand: AR04MGE040408 R.4.G.8 Analyze characteristics and properties

More information

Geometer s Sketch Pad Instructions Basic Constructions

Geometer s Sketch Pad Instructions Basic Constructions Geometer s Sketch Pad Instructions Basic Constructions Tool Menu Basic constructions will use the Selection Arrow Tool, Point Tool, Compass Tool, and Straightedge Tool. Familiarize yourself with the location

More information

Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to:

Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to: Pre-Algebra Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 Click the mouse button

More information

6-1 Angles of Polygons

6-1 Angles of Polygons Find the sum of the measures of the interior angles of each convex polygon. 1. decagon A decagon has ten sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures.

More information

Pythagorean Theorem Differentiated Instruction for Use in an Inclusion Classroom

Pythagorean Theorem Differentiated Instruction for Use in an Inclusion Classroom Pythagorean Theorem Differentiated Instruction for Use in an Inclusion Classroom Grade Level: Seven Time Span: Four Days Tools: Calculators, The Proofs of Pythagoras, GSP, Internet Colleen Parker Objectives

More information

Surface Area and Volume

Surface Area and Volume 1 Area Surface Area and Volume 8 th Grade 10 days by Jackie Gerwitz-Dunn and Linda Kelly 2 What do you want the students to understand at the end of this lesson? The students should be able to distinguish

More information

The Pythagorean Packet Everything Pythagorean Theorem

The Pythagorean Packet Everything Pythagorean Theorem Name Date The Pythagorean Packet Everything Pythagorean Theorem Directions: Fill in each blank for the right triangle by using the words in the Vocab Bo. A Right Triangle These sides are called the of

More information

First Six Weeks (29 Days)

First Six Weeks (29 Days) Mathematics Scope & Sequence Grade 6 Revised: May 2010 Topic Title Unit 1: Compare and Order, Estimate, Exponents, and Order of Operations Lesson 1-1: Comparing and Ordering Whole Numbers Lesson 1-2: Estimating

More information

Geometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources:

Geometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources: Geometry Honors: Extending 2 Dimensions into 3 Dimensions Semester 2, Unit 5: Activity 30 Resources: SpringBoard- Geometry Online Resources: Geometry Springboard Text Unit Overview In this unit students

More information

ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE

ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE Keywords: Central angle An angle whose vertex is the center of a circle and whose sides contain the radii of the circle Arc Two points on a circle

More information

Activity Set 3. Trainer Guide

Activity Set 3. Trainer Guide Geometry and Measurement of Plane Figures Activity Set 3 Trainer Guide GEOMETRY AND MEASUREMENT OF PLANE FIGURES Activity Set 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development

More information

Geometry. Unit 6. Quadrilaterals. Unit 6

Geometry. Unit 6. Quadrilaterals. Unit 6 Geometry Quadrilaterals Properties of Polygons Formed by three or more consecutive segments. The segments form the sides of the polygon. Each side intersects two other sides at its endpoints. The intersections

More information

Geometry Notes Chapter 12. Name: Period:

Geometry Notes Chapter 12. Name: Period: Geometry Notes Chapter 1 Name: Period: Vocabulary Match each term on the left with a definition on the right. 1. image A. a mapping of a figure from its original position to a new position. preimage B.

More information

2. Sketch and label two different isosceles triangles with perimeter 4a + b. 3. Sketch an isosceles acute triangle with base AC and vertex angle B.

2. Sketch and label two different isosceles triangles with perimeter 4a + b. 3. Sketch an isosceles acute triangle with base AC and vertex angle B. Section 1.5 Triangles Notes Goal of the lesson: Explore the properties of triangles using Geometer s Sketchpad Define and classify triangles and their related parts Practice writing more definitions Learn

More information

Geometry Chapter 9 Extending Perimeter, Circumference, and Area

Geometry Chapter 9 Extending Perimeter, Circumference, and Area Geometry Chapter 9 Extending Perimeter, Circumference, and Area Lesson 1 Developing Formulas for Triangles and Quadrilaterals Learning Target (LT-1) Solve problems involving the perimeter and area of triangles

More information

Quadrilaterals GETTING READY FOR INSTRUCTION

Quadrilaterals GETTING READY FOR INSTRUCTION Quadrilaterals / Mathematics Unit: 11 Lesson: 01 Duration: 7 days Lesson Synopsis: In this lesson students explore properties of quadrilaterals in a variety of ways including concrete modeling, patty paper

More information

E XPLORING QUADRILATERALS

E XPLORING QUADRILATERALS E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this

More information

Geometry Chapter 9 Extending Perimeter, Circumference, and Area

Geometry Chapter 9 Extending Perimeter, Circumference, and Area Geometry Chapter 9 Extending Perimeter, Circumference, and Area Lesson 1 Developing Formulas for Triangles and Quadrilaterals Learning Targets LT9-1: Solve problems involving the perimeter and area of

More information

Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479

Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479 Ohio Proficiency Test for Mathematics, New Graduation Test, (Grade 10) Mathematics Competencies Competency in mathematics includes understanding of mathematical concepts, facility with mathematical skills,

More information

3 rd 5 th Grade Math Core Curriculum Anna McDonald School

3 rd 5 th Grade Math Core Curriculum Anna McDonald School 3 rd 5 th Grade Math Core Curriculum Anna McDonald School Our core math curriculum is only as strong and reliable as its implementation. Ensuring the goals of our curriculum are met in each classroom,

More information

Inscribed (Cyclic) Quadrilaterals and Parallelograms

Inscribed (Cyclic) Quadrilaterals and Parallelograms Lesson Summary: This lesson introduces students to the properties and relationships of inscribed quadrilaterals and parallelograms. Inscribed quadrilaterals are also called cyclic quadrilaterals. These

More information

Exterior Angles of Polygons

Exterior Angles of Polygons easures, hape & pace EXEMPLAR 14: Exterior Angles of Polygons Objective: To explore the angle sum of the exterior angles of polygons Key Stage: 3 Learning Unit: Angle related with Lines and Rectilinear

More information

Area of Parallelograms and Triangles

Area of Parallelograms and Triangles Context: Grade 7 Mathematics. Area of Parallelograms and Triangles Objective: The student shall derive the formulas for the area of the triangle and the parallelogram and apply these formulas to determine

More information

Ch 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and Angles [and Triangles]

Ch 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and Angles [and Triangles] h 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and ngles [and Triangles] Warm up: Directions: Draw the following as accurately as possible. Pay attention to any problems you may be having.

More information

Tutorial 1: The Freehand Tools

Tutorial 1: The Freehand Tools UNC Charlotte Tutorial 1: The Freehand Tools In this tutorial you ll learn how to draw and construct geometric figures using Sketchpad s freehand construction tools. You ll also learn how to undo your

More information

Running head: A GEOMETRIC INTRODUCTION 1

Running head: A GEOMETRIC INTRODUCTION 1 Running head: A GEOMETRIC INTRODUCTION A Geometric Introduction to Mathematical Induction Problems using the Sums of Consecutive Natural Numbers, the Sums of Squares, and the Sums of Cubes of Natural Numbers

More information

Surface Area and Volume Nets to Prisms

Surface Area and Volume Nets to Prisms Surface Area and Volume Nets to Prisms Michael Fauteux Rosamaria Zapata CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version

More information

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:

More information

A Different Look at Trapezoid Area Prerequisite Knowledge

A Different Look at Trapezoid Area Prerequisite Knowledge Prerequisite Knowledge Conditional statement an if-then statement (If A, then B) Converse the two parts of the conditional statement are reversed (If B, then A) Parallel lines are lines in the same plane

More information

A. 32 cu ft B. 49 cu ft C. 57 cu ft D. 1,145 cu ft. F. 96 sq in. G. 136 sq in. H. 192 sq in. J. 272 sq in. 5 in

A. 32 cu ft B. 49 cu ft C. 57 cu ft D. 1,145 cu ft. F. 96 sq in. G. 136 sq in. H. 192 sq in. J. 272 sq in. 5 in 7.5 The student will a) describe volume and surface area of cylinders; b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c) describe how changing

More information

Identifying Triangles 5.5

Identifying Triangles 5.5 Identifying Triangles 5.5 Name Date Directions: Identify the name of each triangle below. If the triangle has more than one name, use all names. 1. 5. 2. 6. 3. 7. 4. 8. 47 Answer Key Pages 19 and 20 Name

More information

Geometers Sketchpad Worksheet: Investigating Inscribed and Central Angles

Geometers Sketchpad Worksheet: Investigating Inscribed and Central Angles Geometers Sketchpad Worksheet: Investigating Inscribed and Central Angles In this activity, you will be investigating central angles of a circle and comparing the measurement of the arc to the measurement

More information

SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.

More information

Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids

Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?

More information

BASIC GEOMETRY GLOSSARY

BASIC GEOMETRY GLOSSARY BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that

More information

G3-33 Building Pyramids

G3-33 Building Pyramids G3-33 Building Pyramids Goal: Students will build skeletons of pyramids and describe properties of pyramids. Prior Knowledge Required: Polygons: triangles, quadrilaterals, pentagons, hexagons Vocabulary:

More information