mathematics i – unit 2

Accelerated Mathematics I Frameworks

Student Edition

Unit 2 Polygons in the Plane

1st Edition

March 31, 2008 Georgia Department of Education

One Stop Shop For Educators

Accelerated Mathematics I Unit 2 1st Edition

Georgia Department of Education Kathy Cox, State Superintendent of Schools

March 21, 2008 Copyright 2008 © All Rights Reserved

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Table of Contents

INTRODUCTION: ............................................................................................................................ 3

Robotic Gallery Guards Learning Task ............................................................................................ 8

Poor Captain Robot Learning Task ................................................................................................ 11

Triangles Learning Task ................................................................................................................. 13

Centers of Triangles Learning Task ............................................................................................... 17

3 Location Learning Tasks.............................................................................................................. 19

Cell Tower ................................................................................................................................... 19

Burn Center ................................................................................................................................. 19

Archeological Sites ..................................................................................................................... 21

Constructing with Diagonals Learning Task .................................................................................. 23

Video Game Learning Task ............................................................................................................ 28

New York Learning Task ................................................................................................................. 32

Quadrilaterals Revisited Learning Task ......................................................................................... 34

Euler’s Village Learning Task ........................................................................................................ 38





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Accelerated Mathematics 1 Unit 2 Polygons in the Plane

Student’s Edition INTRODUCTION:

Through the study of topics in the geometry strand for grades K – 8, students have acquired an extensive vocabulary of describing geometric figures in the plane, a deep understanding of the concepts of congruence and similarity for plane figures, and broad experience in using the concepts of symmetry and transformations to identify properties of geometric figures. In Grade 7, they used standard ruler and compass constructions to understand properties of lines, angles, and figures and justified these constructions. In this unit, students will apply their geometric knowledge and skills to explore angles in polygons, congruence and points of concurrency for triangles, and properties of special quadrilaterals. Students will make conjectures based on their explorations and then prove their conjectures through multiple forms of justification. In addition students will begin their student of coordinate geometry by developing the formulas for the distance between two points, the distance between a point and a line, and the midpoint of a segment and then using these formulas as they revisit quadrilaterals in the context of the coordinate plane. ENDURING UNDERSTANDINGS:

• Geometric ideas are appropriate for describing many aspects of our world. • Geometric ideas are significant in all areas of mathematics such as: algebra, trigonometry,

and analysis. • Properties of angles, triangles, quadrilaterals, and polygons are connected. • Algebraic formulas can be used to find measures of distance on the coordinate plane. • The coordinate plane allows precise communication about graphical representations. • The coordinate plane permits use of algebraic methods to obtain geometric results.

KEY STANDARDS ADDRESSED: MA1G1: Students will investigate properties of geometric figures in the coordinate plane.

a. Determine the distance between two points. b. Determine the distance between a point and a line. c. Determine the midpoint of a segment. d. Understand the distance formula as an application of the Pythagorean Theorem. e. Use the coordinate plane to investigate properties of and verify conjectures related to

triangles and quadrilaterals.

MA1G3: Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons.

a. Understand, use, and prove properties of and relationships among special quadrilaterals: parallelogram, rectangle, rhombus, square, trapezoid, and kite.

b. Find and use points of concurrency in triangles: incenter, orthocenter, circumcenter, and centroid.





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RELATED STANDARDS ADDRESSED: MA1G2. Students will understand and use the language of mathematical argument and justification.

a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirect proof as appropriate

MA1P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. MA1P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. MA1P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely.

MA1P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a

coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics.

MA1P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena.

UNIT OVERVIEW

The concepts in this unit should build on the rich background students already have. In seventh grade, students thoroughly investigated basic geometric constructions including angle bisectors, perpendicular bisectors, and parallel and perpendicular lines. This unit will build upon that knowledge by using these constructions, as they apply to triangles and polygons, to explore relationships. Students also spent time in middle school investigating what it means to be congruent. In this unit, the focus is on the minimum information necessary to conclude that





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triangles are congruent. Building on prior knowledge of quadrilaterals, students conjecture and prove or disprove properties that allow classification of quadrilaterals. Students are expected to make and appropriately justify their conclusions. That justification may be through paragraph proofs, flow proofs, two-column proofs, and other forms of communicating mathematical ideas. The emphasis of this unit is the mathematics and the communication of it, not on the rote production of two-column proofs.

In fifth grade, students began plotting points in the first quadrant. Throughout sixth, seventh, and eighth grade they continued to progress from working in the first quadrant to using all four quadrants. Students have made scatter plots and have worked with both lines and systems of lines, including finding equations of lines, finding slopes of lines, and finding the slope of a line perpendicular to a given line. This unit offers the opportunity to use calculators, especially when computing distances. The explorations in this unit lend themselves to computing with a calculator allowing students to focus on the emerging patterns and not the arithmetic process.

In eighth grade, students discovered and used the Pythagorean Theorem. This unit allows students to extend this theorem to the coordinate plane while developing the distance formula. It includes work with distance between two points and distance between a point and a line. Students are expected to discover and use the midpoint formula. They will revisit the properties of special quadrilaterals while using slope and distance on the coordinate plane. The tasks in this unit lend themselves to extensive use of manipulatives. All students may not have had access to the same manipulatives or technology during middle school so students may need an opportunity to investigate the use of other tools. Graphing calculators and dynamic geometry software are valuable tools and should be available to all students. In addition to technological tools, MIRAs, patty paper, a compass, and a straight edge can be powerful tools for investigating concepts in geometry. Formulas and Definitions: Centroid: The point of concurrency of the medians of a triangle. Circumcenter: The point of concurrency of the perpendicular bisectors of the sides of a triangle. Distance Formula: d = 2

122

12 )()( yyxx −+− Incenter: The point of concurrency of the bisectors of the angles of a triangle. Orthocenter: The point of concurrency of the altitudes of a triangle. Sum of the measures of the interior angles of a convex polygon: 180º(n – 2).

Measure of each interior angle of a regular n-gon: nn )2(180 −o





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Midpoint Formula: ⎟⎠⎞

⎜⎝⎛ ++

2,

22121 yyxx

Exterior angle of a polygon: an angle that forms a linear pair with one of the angles of the polygon. Remote interior angles of a triangle: the two angles non-adjacent to the exterior angle. Measure of the exterior angle of a triangle: equals the sum of the measures of the two remote interior angles. Exterior Angle Inequality: an exterior angle is greater than either of the remote interior angles. Theorems: Exterior Angle Sum Theorem: If a polygon is convex, then the sum of the measure of the exterior angles, one at each vertex, is 360º. The Corollary that follows states that the measure of each exterior angle of a regular n-gon is 360º/n. If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. If two sides of one triangle are equal to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second. If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. (SSS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. (SAS) If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. (ASA) ASA leads to the AAS corollary. If two angles on a triangle are known, then the third is also known. If two triangles are congruent, then the corresponding parts of the two congruent triangles are congruent.





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Converse of the Pythagorean Theorem: If c is the measure of the longest side of a triangle, a and b are the lengths of the other two sides, and c2 = a2 + b2, then the triangle is a right triangle. If two legs of one right triangle are congruent to the corresponding legs of another right triangle, the triangles are congruent. (LL) If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding angle of another right triangle, then the triangles are congruent. (HA) If one leg and an acute angle of a right triangle are congruent to the corresponding leg and angle of another right triangle, then the triangles are congruent. (LA) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. (HL) If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If each diagonal bisects a pair of opposite angles, then the parallelogram is a rhombus. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. If three parallel lines cut off equal segments on one transversal, then they cut off equal segments on every transversal. TASKS:

The remaining content of this framework consists of student learning tasks designed to allow students to learn by investigating situations with a real-world context. The first leaning task is intended to launch the unit. Its primary focus is to explore the relationships between angles and polygons leading to a focus on triangle relationships in the next few tasks. After students become familiar with the ways in which triangle congruencies can be shown, they are then asked to use these congruencies to prove properties of quadrilaterals. The Video Game task gives students a chance to make critical connections between algebra and geometry. This is reinforced when





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quadrilaterals are revisited on the coordinate plane and when they are trying to situate homes in Euler’s Village Learning Task. The last task is designed to demonstrate the type of assessment activities students should be comfortable with by the end of the unit and gives students a chance to apply many of the ideas they have learned throughout the unit.

Robotic Gallery Guards Learning Task The Asimov Museum has contracted with a company that provides Robotic Security Squads to guard the exhibits during the hours the museum is closed. The robots are designed to patrol the hallways around the exhibits and are equipped with cameras and sensors that detect motion. Each robot is assigned to patrol the area around a specific exhibit. They are designed to maintain a consistent distance from the wall of the exhibits. Since the shape of the exhibits change over time, the museum staff must program the robots to turn the corners of the exhibit. Below, you will find a map of the museum’s current exhibits and the path to be followed by the robot. One robot is assigned to patrol each exhibit. There is one robot, Captain Robot, CR, who will patrol the entire area.

A1

B1

Exhibit Z

R1

Exhibit D

Entrance

Exhibit CExhibit B

Exhibit A

S T

U

VW

X

N O

P

R Q

L M

G

HI

J

K

CR

R3

Y

Z

C1

R4

L1

K1

F1

R2

E1

N1 W1

X1

G1





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1. When a robot reaches a corner, it will stop, turn through a programmed angle, and then continue its patrol. Your job is to determine the sum of the angles that R1, R2, R3, and R4 will need to turn as they patrol their area. Keep in mind the direction in which the robot is traveling and make sure it always faces forward as it moves around the exhibits.

2. What do you notice about the sum of the angles? Do you think this will always be true?

Why? 3. Determine the sum of the measures of the interior angles of any polygon. Justify your answer.

4. Looking at your results from #2 and #3, can you find a way to prove your conjecture about the

sum of the exterior angles?

5. The museum intends to create regular polygons for its next exhibition; how can the directions

for the robots be determined for a regular pentagon? Hexagon? Nonagon? N-gon?

6. A sixth exhibit was added to the museum. The robot patrolling this exhibit only makes 15º

turns. What shape is the exhibit? What makes it possible for the robot to make the same turn each time?

7. Robot 7 makes a total of 360 º during his circuit. What type of polygon does this exhibit

create?

8. The sum of the interior angles of Robot 7’s exhibit is 3,420 º. What type of polygon does the

exhibit create?





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9. What is different about the path CR travels versus the paths of R1, R2, R3 and R4 guard are convex polygons. Do your earlier generalizations hold for CR’s path?

10. A counter-clockwise turn is considered positive while a clockwise turn is negative. Investigate CR’s path using this information. How does this affect the application of your earlier generalizations on the polygon traced by CR?

11. The museum is now using Exhibit Z. Complete a set of instructions for RZ that will allow the robot to make the best circuit of this circular exhibit. Defend why you think your instructions are the best.





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Poor Captain Robot Learning Task Captain Robot’s positronic brain is misfiring and he will only take instructions to move three distances. He will no longer acknowledge angles given in directions and chooses all of them himself. He will not travel the same path twice and refuses to move at all if he cannot end up back at his starting point. 1. Captain Robot was given the following sets of instructions. Determine which instructions Captain Robot will use and which sets he will ignore. Be ready to defend your choices. Instruction Set #1: 10 cm, 5 cm, 8 cm Instruction Set #2: 7 cm, 4.4 cm, 8 cm Instruction Set #3: 10 cm, 2 cm, 8 cm Instruction Set #4: 5 cm, 5 cm, 2.8 cm Instruction Set #5: 7 cm, 5.1 cm, 1 cm 2. Determine a method that will always predict whether or not Captain Robot will move. 3. Captain Robot traveled from point A to point B to point C. His largest turn occurred at point

C and his smallest turn occurred at point A. Order the sides of the triangle from largest to smallest.

4. Is there a relationship between the lengths of the side and the measures of the angles of a

triangle? Explain why or why not.





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The museum has decided Captain Robot needs to patrol two access doors off the side of the museum. Captain Robot’s addition to his route is shown below. 5. Determine the angles Captain Robot will need to turn. Look carefully at the angles you chose

for the robot’s route. What do you notice? 6. Make a conjecture about the relationship of the exterior and interior angles of a triangle. Prove

your conjecture.





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Triangles Learning Task The students at Hometown High School decided to make large pennants for all 8 high schools in their district. The picture to the right shows typical team pennants. The Hometown High students wanted their pennants to be shaped differently than the typical isosceles triangle used for pennants and decided each pennant should be a scalene triangle. They plan to hang the final products in the gym as a welcome to all the schools who visit Hometown High. Jamie wanted to know how they could make sure that all of the pennants are congruent to each other. The students wondered if they would have to measure all six parts of every triangle to determine if they were congruent. They decided there had to be a shortcut for determining triangle congruence, but they did not know the minimum requirements needed. They decided to find the minimum requirements needed before they started making the pennants. 1. Every triangle has ____ parts, _____ sides and _____ angles. 2. First they picked out 3 sides and each person constructed a triangle using these three sides.

Construct a triangle with sides of 3 inches, 4 inches, and 6 inches. Compare your triangle to other students’ triangles. Are any of the triangles congruent? Are three sides enough to guarantee congruent triangles? Explain.

3. Next the class decided to use only 2 sides and one angle. They choose sides of 5 inches and 7

inches with an angle of 38°. Using these measures, construct a triangle and compare it to other students’ triangles. Are any of the triangles congruent?

4. Joel and Cory ended up with different triangles. Joel argued that Cory put her angle in the

wrong place. Joel constructed his triangle with the angle between the two sides. Cory constructed her sides first then constructed her angle at the end of the 7 in. side not touching the 5 in. side. Everybody quickly agreed that these two triangles were different. They all tried Cory’s method, what happened? Which method, Joel’s or Cory’s will always produce the same triangle?

5. Now the class decided to try only 1 side and two angles. They chose a side of 7 in. and angles

of 35° and 57°. Construct and compare triangles. What generalization can be made?





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6. Jim noticed that Sasha drew her conclusion given two angles and the included side. He wondered if the results would be the same if you were given any two angles and one side. What do you think?

7. The last situation the class decided to try was to use three angles. They chose angles of 20°,

40°, and 120°. How do you think that worked out? Construct a triangle using these three angles and compare with others. Can they prove two triangles are congruent using the three corresponding angles? Explain why or why not.

8. Summarize the results using the chart below. Discuss what is meant by the common

abbreviations and how they would help to remember the triangle situations you have just explored.

Common

Abbreviation Explanation Abbreviation Significance

SSS

SAS

SSA

ASA

AAS

AAA

9. SSS, SAS, and ASA are generally accepted as postulates. Look up the definition of a

postulate. Discuss the need for postulates in this case. Why can’t we just use SSS as a single postulate? Could we use less than the three given or could we choose a different set as postulates?





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10. PROVE that AAS is always true. If we can prove a statement is always true, what do you we call it?

11. The methods listed in the table, which can be used for proving two triangles congruent, require three parts of one triangle to be congruent to three corresponding parts of another triangle. Nakita thought she could summarize the results but she wanted to try one more experiment. She wondered if the methods might be a bit shorter for right triangles since it always has one angle of 90o. She said: “I remember the Pythagorean Theorem for finding the length of a side of a right triangle. Could this help? My father is a carpenter and he always tells me that he can determine if a corner is square if it makes a 3 – 4 – 5 triangle.” Nakita chose to create a triangle with a hypotenuse of 6 inches and a leg of 4 inches. Does her conjecture work? Why or why not?

12. What if Nakita had chosen 6 inches and 4 inches to be the length of the legs? Does her

conjecture work? Why or why not? 13. What are the minimum parts needed to justify that the two right triangles are congruent?

Using the list that you already made, consider whether these could be shortened if you knew one angle was a right angle. Create a list of ways to prove congruence for right triangles only.

14. Once it is known that two triangles are congruent, what can be said about the parts of the

triangles? Write a statement relating the parts of congruent triangles. 15. Construct an equilateral triangle. Find the point, that when connected to the vertices of the

triangle, divides the triangle into 3 congruent isosceles triangles.





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Congruent triangles can be used to solve problems encountered in everyday life. The next two situations are examples of these types of problems. 16. In order to construct a new bridge, to replace the current bridge, an engineer needed to

determine the distance across a river, without swimming to the other side. The engineer noticed a tree on the other side of the river and suddenly had an idea. She drew a quick sketch and was able to use this to determine the distance. Her sketch is to the right. How was she able to use this to determine the length of the new bridge? You do not have to find the distance; just explain what she had to do to find the distance.

17. A landscape architect needed to determine the distance

across a pond. Why can’t he measure this directly? He drew the following sketch as an indirect method of measuring the distance. He stretched a string from point J to point N and found the midpoint of this string, point L. He then made stretched a string from M to K making sure it had same center. He found the length of MN was 43 feet and the length of segment LK is 19 feet. Find the distance across the pond. Justify your answer.





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Centers of Triangles Learning Task A developer plans to build an amusement park but wants to locate it within easy access of the three largest towns in the area as shown on the map below. The developer has to decide on the best location and is working with the ABC Construction Company to minimize costs wherever possible. No matter where the amusement park is located, roads will have to be built for access directly to the towns or to the existing highways.

I-330

I-320

I-310

Crazytown

Busytown

Lazytown

1. Just by looking at the map, choose the location that you think will be best for building the

amusement park. Explain your thinking. 2. Now you will use some mathematical concepts to help you choose a location for the tower.

In the previous lesson, you learned how to construct medians and altitudes of triangles. In 7th grade, you learned how to construct angle bisectors and perpendicular bisectors. Investigate the problem above by constructing the following:

a) all 3 medians of the triangle b) all 3 altitudes of the triangle c) all 3 angle bisectors of the triangle d) all 3 perpendicular bisectors of the triangle

You have four different kinds of tools at your disposal- patty paper, MIRA, compass and straight edge, and Geometer’s Sketch Pad. Use a different tool for each of your constructions.





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3. Choose a location for the amusement park based on the work you did in part 2. Explain why you chose this point.

4. How close is the point you chose in part 3, based on mathematics, to the point you chose by

observation? You have now discovered that each set of segments resulting from the constructions above always has a point of intersection. These four points of intersection are called the points of concurrency of a triangle.

The intersection point of the medians is called the centroid of the triangle. The intersection point of the angle bisectors is called the incenter of the triangle. The intersection point of the perpendicular bisectors is called the circumcenter of the triangle. The intersection point of the altitudes is called the orthocenter of the triangle.

5. Can you give a reasonable guess as to why the specific names were given to each point of

concurrency? 6. Which triangle center did you recommend for the location of the amusement park? 7. The president of the company building the park is concerned about the cost of building roads

from the towns to the park. What recommendation would you give him? Write a memo to the president explaining your recommendation.





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3 Location Learning Tasks

Cell Tower A cell service operator plans to build an additional tower so that more of the southern part of Georgia has stronger service. People have complained that they are losing service, so the operator wants to remedy the situation before they lose customers. The service provider looked at the map of Georgia below and decided that the three cities: Albany, Valdosta, or Waycross, were good candidates for the tower. However, some of the planners argued that the cell tower would provide a more powerful signal within the entire area if it were placed somewhere between those three cities. Help the service operator decide on the best location for the cell tower. Compose a memo to the president of the cell company justifying your final choice for the location of the tower. Use appropriate mathematical vocabulary and reasoning in your justification.

Burn Center Now, you can apply your new found knowledge to a more important situation: Several groups have decided to fund an acute trauma hospital located where it would give the most people of Georgia the most access. Using one of the sets of cities indentified below, or your choices of three other cities, and the map from the earlier question, decide where the new trauma hospital should be located. Be sure to mathematically justify your answer. City set one: Bainbridge, Savannah, and Dalton City set two: Thomasville, Sylvania, and Calhoun City set three: Valdosta, Augusta, and Marietta





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Archeological Sites

“Ceramic vessels are common artifacts found by archaeologists on Native American sites occupied after 1000 BC. This date marks when most southeastern Indians began making clay pottery. It marks, too, the beginning of other broad cultural changes.

Population had grown enough that the territory each group used to get seasonally available foods got smaller. People started staying much of the year in villages, many of which were located near rivers or along major tributaries. To supplement the food they hunted and gathered people began growing crops. Pottery became a key element in this village-based lifestyle. Once people became settled and less nomadic, pottery vessels became practical, everyday tools used for cooking and storage.

Pottery’s relative abundance at archaeological sites is due to the fact that it is very durable. It can last for thousands of years, whether buried in the ground or lying on the surface. For today’s archaeologists, pottery holds clues to past life. Archaeologists and other historians have learned through their research that pottery styles are distinctive to particular groups of people, and the styles change over time. This knowledge, which is refined as more research is done, helps archaeologists determine how old a site is, which group of people lived there, and what interactions they might have had with people living in other far and near places. Archaeologists also study vessel shapes and sizes to infer whether the pots were used for cooking, serving, or food storage.

Indeed, archaeologists can understand much about how a group of people lived by studying vessel sizes and shapes. The storage capacity of vessels, for example, allows calculation of how much stored food people had, and, from that, estimates are possible of how many people lived at a site. Functions of different sizes of pottery can also be determined. A small-necked vessel probably stored liquids or very small seeds, rather than large seeds. Large open vessels, such as bowls, probably weren’t used for storage, since they would be difficult to seal from moisture, rodents, and insects.

Because ancient pots are usually found broken into hundreds of pieces (sherds), archaeologists find it tedious and often impossible to glue them back together. One way they get an idea of how large a pot was is by calculating its original circumference. Rim sherds are used to do this because they often indicate how large the vessel opening was.” Source: http://rla.unc.edu/lessons/Lesson/L206/L206.htm





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Archeologists have different ways to determine the size of a vessel based on the rim sherd. Your job is to determine at least one method for calculating the diameter of the original circumference of a rim sherd using your knowledge of triangles. Use the drawing below, or the rim sherd your teacher has provided. After you have discovered a method that will always work, write a set of directions an archeologist could follow.





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Constructing with Diagonals Learning Task 1. Construct two segments of different length that are perpendicular bisectors of each other.

Connect the four end points to form a quadrilateral. What names can be used to describe the quadrilaterals formed using these constraints?

2. Repeat #1 with two congruent segments. Connect the four end points to form a quadrilateral.

What names can be used to describe the quadrilaterals formed using these constraints?

3. Construct two segments that bisect each other but are not perpendicular. Connect the four end

points to form a quadrilateral. What names can be used to describe the quadrilaterals formed using these constraints?

4. What if the two segments in #3 above are congruent in length? What type of quadrilateral is

formed? What names can be used to describe the quadrilaterals formed using these constraints?

5. Draw a segment and mark the midpoint. Now construct a segment that is perpendicular to the

first segment at the midpoint but is not bisected by the original segment. Connect the four end points to form a quadrilateral. What names can be used to describe the quadrilaterals formed using these constraints?

6. In the above constructions you have been discovering the properties of the diagonals of each

member of the quadrilateral family. Stop and look at each construction. Summarize any observations you can make about the special quadrilaterals you constructed. If there are any quadrilaterals that have not been constructed yet, investigate any special properties of their diagonals.





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7. Complete the chart below by identifying the quadrilateral(s) for which the given condition is

necessary. Conditions Quadrilateral(s) Explain your reasoning

Diagonals are perpendicular.

Diagonals are perpendicular and only one diagonal is bisected.

Diagonals are congruent and intersect but are not perpendicular.

Diagonals bisect each other.

Diagonals are perpendicular and bisect each other.

Diagonals are congruent and bisect each other.

Diagonals are congruent, perpendicular and bisect each other

8. As you add more conditions to describe the diagonals, how does it change the types of

quadrilaterals possible? Why does this make sense?





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9. Name each of the figures below using as many names as possible and state as many properties

as you can about each figure.

F

ED

C

BA

Figure Names Properties A

B

C

D

E

F





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10. Identify the properties that are always true for the given quadrilateral by placing an X in the appropriate box.

Property Parallelogram Rectangle Rhombus Square Isosceles Trapezoid

Kite

Opposite sides are parallel

Only one pair of opposite sides is parallel

Opposite sides are congruent

Only one pair of opposite sides is congruent

Opposite angles are congruent

Only one pair of opposite angles is congruent

Each diagonal forms 2 ≅triangles

Diagonals bisect each other

Diagonals are perpendicular.

Diagonals are Congruent

Diagonals bisect vertex angles

All ∠s are right s ∠

All sides are congruent

Two pairs of consecutive sides are congruent





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11. Using the properties in the table above, list the minimum conditions necessary to prove that a quadrilateral is:

a. a parallelogram

b. a rectangle

c. a rhombus

d. a square

e. a kite

f. an isosceles trapezoid





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Video Game Learning Task John and Mary are fond of playing retro style video games on hand held game machines. They are currently playing a game on a device that has a screen that is 2 inches high and four inches wide. At the start, John's token starts ½ inch from the left edge and half way between the top and bottom of the screen. Mary's token starts out at the extreme top of the screen and exactly at the midpoint of the top edge.

1" 2" 3" 4"

1"

2"

0.1"

0.1"

Mary

John

Starting Position

As the game starts, John's token moves directly to the right at a speed of 1 inch per second. For example, John’s token moves 0.1inches in 0.1 seconds, 2 inches in 2 seconds, etc. Mary's token moves directly downward at a speed of 0.8 inches per second.

Mary

Mary - afteronesecond

1" 2" 3" 4"

1"

2"

0.1"

0 .1"

John - afteronesecond

John

After One Second





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Let time be denoted in this manner: t = 1 means the positions of the tiles after one second

1. Discuss the movements of both tokens relative to each other.

2. Find the distance between John and Mary’s tokens at times t = 0, t = ¼, t = 1/2, t = 1.

If Mary's token gets closer than ¼ inch to John’s token, then Mary's token will destroy John’s, and Mary will get 10,000 points. However, if John presses button A when the tokens are less than 1/2 inch apart and more than ¼ inch apart, then John’s token destroys Mary's, and John gets 10,000 points.

3. Find a time at which John can press the button and earn 10,000 points. Draw the

configuration at this time.

4. Compare your answers with your group. What did you discover?

5. Estimate the longest amount of time John could wait before pressing the button.

6. Drawing pictures gives an estimate of the critical time, but inside the video game, everything is done with numbers. Describe in words the mathematical concepts needed in order for this video game to work.





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Inside the computer game the distance between John and Mary’s token are computed using a mathematical formula based on the coordinates of the tokens. Our goal now is to develop this formula. To help us think about the distance between the tokens in our video game, it may help us to look first at a one-dimensional situation. Let’s look at how you determine distance between two locations on a number line:

7. What is the distance between 5 and 7? 7 and 5? -1 and 6? 5 and -3?

8. Can you find a formula for the distance between two points, a and b, on a number line? Now that you can find the distance on a number line, let’s look at finding distance on the coordinate plane:

9. Using graph paper, find the distance between:

a. (1, 0) and (4, 4) b. (-1, 1) and (11, 6) c. (-1, 2) and (2, -6)

10. Find the distance between points (a, b) and (c, d) shown below.

(c, d)

(a, b)





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11. Using your solutions from 16, write a formula to find the distance between point (x1, y1) and point (x2, y2). Will your formula work for any pair of points? Why or why not?

Let’s revisit the video game. Draw a diagram of the game on a coordinate grid placing the bottom left corner at the origin and place John and Mary’s tokens at the starting positions.

12. Write an ordered pair for John’s token and an ordered pair for Mary’s token when t = 0, when t = ½, when t = 1 ½, and when t = 2 and find the distance between the tokens at each of these times.

13. Write an ordered pair for John and Mary’s tokens at any time t.

14. Write an equation for the distance between John and Mary’s tokens at any time t.

15. Using a graphing utility, graph the equation you derived for the distance between the two tokens.

16. What does the graph look like? What are the characteristics of this graph?

17. What do the variables represent?

18. Recall that when John and Mary are between ¼ and ½ inches apart, John may press the button to earn 10,000 points. What interval of time represents John’s window of opportunity to score points?





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New York Learning Task Emily works at a building located on the corner of 9th Avenue and 61st Street in New York City. Her brother, Gregory, is in town on business. He is staying at a hotel at the corner of 9th Avenue and 43st Street. The streets of New York City were laid out in a rectangular pattern. In this part of town, Avenues run in a North-South direction and they are numbered from east to west, in other words the further east you go, the lower the number. That means the Avenues east of 9th Ave are 8th Ave, 7th Ave, etc. Streets run in an east-west direction. They increase in number as you proceed north. So, north of 41st Street is 42nd Street, then 43rd Street, etc. The distance between the avenues is the same as the distance between the streets. All the blocks are approximately the same size.

1. Gregory called Emily at work, and they agree to meet for lunch. They agree to meet at a

corner half way between Emily’s work and Gregory’s hotel. Where should they meet? Justify your answer using a diagram.

2. After lunch, Emily has the afternoon off and wants to show her brother her apartment. Her

apartment is three blocks north and four blocks west of the hotel. At what intersection is her apartment building located?

3. Gregory walks back to his hotel for an afternoon business meeting. He and Emily are going

to meet for dinner. They decide to be fair and will meet half way.

How far is it from Emily’s apartment to Gregory’s hotel? Where should they meet for dinner? How far are they going to walk to meet? Is their walking distance ½ the distance from Emily’s apartment and Gregory’s hotel? Why

or why not?





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On appropriate graph paper, plot the points A = (0, 0), B = (6, 0), and C = (4, 12). 4. Find the point midway between A and B. This point is called the midpoint of the segment

AB. Find the distance from A to midpoint and the distance from midpoint to B. What do you notice?

5. Find the midpoint of segment BC. Check using distances. 6. Find the midpoint of a segment whose endpoints are (x1, y1) and (x2, y2). Would this formula

work for any endpoints? Why or why not?





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Quadrilaterals Revisited Learning Task

1. Determine the type of quadrilateral created using each set of coordinates given below. Examine the diagonals of each quadrilateral. Identify any special features of the diagonals and/or the triangles created by the diagonals. Justify all answers using your knowledge of algebra and geometry.

a) A = (-3, -1), B = (-1, 2), C = (4, 2), and D = (2, -1).

b) E = (1, 2), F = (2, 5), G = (4, 3) and H = (5, 6).

c) P = (4, 1), W = (-2, 3), M = (2,-5), and K = (-6, -4). 2. Plot points A = (1, 0), B = (-1, 2), and C = (2, 5).

a. Find coordinates of a fourth point D that would make ABCD a rectangle. Justify that

ABCD is a rectangle.

b. Find coordinates of a fourth point D that would make ABCD a parallelogram that is not a rectangle. Justify that ABCD is a parallelogram but is not a rectangle.

In the 17th century a French mathematician named Rene Descartes combined algebraic and geometric knowledge to create a branch of mathematics known as analytic geometry. This process usually involves the placement of geometric figures on the coordinate plane. It is more commonly referred to as coordinate geometry. You used coordinate geometry in problems 1 and 2 in this task. 3. What algebraic ideas did you use in problems 1 and 2? Using this algebraic knowledge, what

previously learned geometric theorems did you prove to be true in these examples?





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In problems 1 and 2, you made conjectures about specific quadrilaterals. In each case, you could support your conjecture using your knowledge of slope and distance. In each of these cases, your ‘proof’ is only valid for that specific example. But coordinate geometry can be used as a valid proof of geometric theorems. Using variables for coordinates when placing the a figure on the coordinate plane allows you to prove conjectures for generalized cases. 4. Look below at the square placed on the coordinate plane without specific numerical vertices.

Using the given coordinates and your knowledge of squares, determine the coordinates of vertices C and D.

5. When placing a geometric figure on the coordinate plane certain decisions can be made to

ease mathematical calculations. Typically, figures are placed in the 1st quadrant, the origin is used as a vertex or midpoint, and, when possible, other vertices are placed on the axes. Why would these choices make calculations easier?





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6. Find the missing coordinates of the vertices in the figures below.

(___,___ )(0, 0)

(___,___ ) (m, n)H G

E F

(c,0 )(0, 0)

(___,___ )(a,b)YZ

W X

a. EFGH is a rectangle.

b. MNOP is a parallelogram.

(___,___ )(b,c)

(0, 0) (a,0)

OP

M N

c. WXYZ is an isosceles trapezoid.

d. QRST is a rhombus. (Be very careful here. Use only a, b and 0 in your coordinates.)

(___,___ )

(___,___ )(a,b)

(0, 0)

ST

Q R

7. Use the figures above to prove at least two of the following:

a. The diagonals of a parallelogram bisect each other.

b. The diagonals of an isosceles trapezoid are congruent.

c. The diagonals of all rectangles are congruent.





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8. Look at the figure below. If you want to prove the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other, why would the given coordinates make the algebraic calculations easier? Use the figure to prove the given statement.

(2a,0)

(2d, 2e)(2b,2c)

(0, 0)

L

H

K

J

9. Prove that in any triangle, the line segment joining the midpoints of two sides of the triangle is

parallel to the third side. Think carefully about how you want to place your triangles and how you want to name the vertices.





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Euler’s Village Learning Task You would like to build a house close to the village of Euler. There is a beautiful town square in the village, and the road you would like to build your house on begins right at the town square. The road follows an approximately north east direction as you leave town and continues for 3,000 feet. It passes right by a large shade tree located approximately 200 yards east and 300 yards north of the town square. There is a stretch of the road, between 300 and 1200 yards to the east of town, which currently has no houses. This stretch of road is where you would like to locate your house. Building restrictions require all houses sit parallel to the road. All water supplies are linked to town wells and the closest well to this part of the road is 500 yards east and 1200 yards north of the town square. 1. How far from the well would it be if the house was located on the road 300 yards east of

town? 500 yards east of town? 1,000 yards east of town? 1,200 yards east of town? (For the sake of calculations, do not consider how far the house is from the road, just use the road to make calculations)

2. The cost of the piping leading from the well to the house is a major concern. Where should

you locate your house in order to have the shortest distance to the well? Justify your answer mathematically.

3. If the cost of laying pipes is $22.5 per linear yard, how much will it cost to connect your

house to the well? 4. The builder of your house is impressed by your calculations and wants to use the same

method for placing other houses. Describe the method you used. Would you want him to place the other houses in the same manner?

5. Write a formula that the builder could use to find the cost of laying pipes to any house along

this road. How would you have to change your formula for another road?

mathematics i – unit 2

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