HIGH SCHOOL GEOMETRY: BURIED TREASUREUNIT OVERVIEW

Task Name: Buried Treasure

Grade: 10

Subject: Geometry

Depth of Knowledge: Level 3 (Strategic Thinking)

Task Description: Students will apply geometric constructions, postulates and theorems, and proofs to find locations on a map. Adapted from an activity from Key Curriculum’s Discovering Geometry.

Standards:

CC.9-12.G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

CC.9-12.G.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

CC.9-12.G.MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Standards for Mathematical Practice:

CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

CCSS.Math.Practice.MP4 Model with mathematics.

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

CCSS.Math.Practice.MP6 Attend to precision.

CCSS.Math.Practice.MP7 Look for and make use of structure.

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TABLE OF CONTENTS

Table of ContentsTeacher Instructions 3

Student Activity 4

Scoring Rubric 6

Performance Level Descriptions 8

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Performance Task – Teacher Instructions

Buried Treasure

To do this task, each student will need a compass and protractor, and could also use a ruler.

Before students start this task, teachers should read the following aloud with their class.

TEACHER: With this Buried Treasure Performance Task, you are expected to answer questions in complete sentences, use compass and protractor to construct geometric figures that represent the scenarios in this task, and use theorems and postulates to support your answers wherever possible.

For full credit for each question on this task, you must correctly answer the question, support your answer with appropriate theorem or postulate, and correctly construct geometric figures with compass and protractor on the attached treasure map.

Label any unnamed points as t 1 , t 2, t 3 and so on.

Are there questions?

Before you can turn in your task, you will need to get me to initial it at the very end. If you have not answered every question, I may ask you to return to the task and complete it fully.

Before students can turn in their task, check for completion. Encourage students who have not answered every question fully to go back and attempt again. Once a student has fully completed their task, initial at the end. (This will simulate the controls that the teacher has in the computer-based SBAC assessment.)

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Performance Task

Buried Treasure

Make sure to answer questions in complete sentences and explain fully. Label any unnamed points ast 1 , t 2 , t 3…

Treasure #1Pirate Albus and Bluebeard have come across Captain Coldhart burying his treasure on an island, and hope to steal it after he leaves. Pirate Albus is standing at the edge of Westend Bay at point A, and his cohort, pirate Bluebeard, is 60 paces to the north at point B. Each pirate can see Captain Coldhart off in an easterly direction burying a treasure. With his sextant, Albus measures the angle from himself to Bluebeard to the Captain (with the vertex of the angle where Albus is standing), and finds it to be85 °. With his sextant, Bluebeard measures the angle from himself to Albus to the Captain (with the vertex of the angle where Albus is standing), and finds it to be 48 °. Albus and Bluebeard mark their positions with large boulders and return to their ship, confident that they have enough information to return later and recover the treasure.

If it is possible, use your geometric tools to locate the position of the treasure on the map and mark its location. Can they recover the treasure? Explain why or why not. Support your answer with theorems and postulates as appropriate.

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Treasure #2Captain Coldhart, convinced someone in his crew stole the treasure he just buried on the island, has decided to be more careful about burying any more treasures. To hide treasure #2, he gives his trusted first mate, Dexter, two ropes, the lengths of which only the Captain knows. The Captain instructs Dexter to nail one end of the shorter rope to Hangman’s Tree (H). He’s also told to attach one end of the longer rope to Skull Rock (S). The Captain, with the ends of the two ropes in one hand, and the treasure chest tucked under one arm, walks toward the East to the point where the two ropes become taut. The Captain buries the treasure at the point where the two ropes come together, collects his ropes, and returns confidently to the ship.

If it is possible, locate the position of this second treasure on the map, if the two ropes are 100 paces and 150 paces in length. Mark the treasure location on your map. Has the Captain given himself enough information to recover the treasure? Explain why or why not. Support your answer with theorems and postulates as appropriate.

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Treasure #3After the theft from his locker of the two very important ropes that he used to bury his second treasure, and the subsequent disappearance of his trusted first mate, Dexter, Captain Coldhart is determined that no one shall find the location of his third treasure. The Captain, with his new first mate, Euclid, walks out to Deadman’s Point (D). The Captain instructs his first mate to walk inland along a straight path for a distance of 130 paces. There Euclid is to drive his sword into the ground for a marker (M), turn and face in the direction of the captain, and then turns at an arbitrary angle to the left. He continues walking for another 60 paces, stops, and waits for the Captain. The Captain, at Deadman’s Point, measures the angle formed by the lines from Euclid to himself, and from himself to the sword. The angle measures 26 °. The Captain places a boulder where he is standing (D), walks around the bay to Euclid, and instructs him to bury the treasure at this point. Alas, poor Euclid is buried with the treasure.

If it is possible, locate the position of this third treasure on the map and mark it. Has the Captain given himself enough information to locate the treasure? Explain why or why not. Support your answer with theorems and postulates as appropriate. If it is not possible, explain how many possible locations for treasure there are; find and mark them on the map.

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Before turning this in, you must have your teacher initial here______.5

HIGH SCHOOL GEOMETRY: BURIED TREASURERUBRIC

High School Geometry: Buried TreasureBuried Treasure Rubric

Possible Responses Points Section Points

1. a)Correctly state that they can recover the treasureb) Support their answer with angle-side-angle, and explain that they are given two angles and the side in between them.c) Correctly construct and label a triangle which locates X on the map, which accurately represents the situation.Partial Credit

State the postulate but do not support their answer. Correctly describes the triangle, but states the wrong postulate or theorem. Construct the triangle correctly, but do not label it, or labels it in a way that does

not represent the situation. Construction is incomplete, but what they have drawn is correct. Triangle is constructed incorrectly, but there are correct aspects to the

construction.

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2

(1)(1)

(1)

(1)(1)

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2. a) Correctly state that he can recover the treasureb) Support their answer with side-side-side, and explain that they are given the 3 distances, and will get the same triangle regardless of whether they are given any angles.c) Correctly construct the intersection which locates the treasure, and accurately represents the situation.Partial Credit

State the postulate but do not support their answer. Correctly describes the triangle, but states the wrong postulate or theorem. Construct the intersection correctly, but do not label it in a way that represents

the situation. Construction is incomplete, but what they have drawn is correct. Intersection is constructed incorrectly, but there are correct aspects to the

construction.

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2

(1)(1)

(1)

(1)(1)

5

3. a) Correctly state that he cannot recover the treasureb) Support their answer with SSA (non-theorem), or that the other two angles are unknown, and there are two possible locations, or describe the situation and correctly state that the remaining side and angles could vary.c) Correctly construct and label the triangle, and give the two locations of the treasure, and accurately represent the situation.

Partial Credit Just state that SSA is not valid, but don’t explain in context. Incorrectly answers the question, and supports it with SAS. Answers the question correctly, but does not specify that there are two possible

spots for the treasure. Triangles are correctly constructed, but are not labeled in a way that represents

the situation. Only one triangle is drawn, the triangle is constructed incorrectly, but there are

correct aspects to it (for example, only one side or one angle)

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3

(1)(1)(1)

(2)

(½ )

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Total Points 16

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High School Geometry: Buried Treasure

Rubric

Performance Level Descriptions and Cut ScoresPerformance is reported at four levels: 1 through 4, with 4 as the highest. When scores fall between cut scores, teachers must review performance level descriptions in relation to the performance task overall, and use their discretion to assign a performance level holistically.

Level 1: Demonstrates Minimal Success (0-3 points)The student’s response shows few of the elements of performance that the task demands as defined by the CCSSM. The student’s work shows a minimal attempt and lack of coherence. The student fails to use appropriate tools strategically, such as a compass, protractor, postulates and theorems. The student is unable to make sense of the problem and apply geometric concepts in this modeling situation.

Level 2: Performance Below Standards (4-8 points)The student’s response shows some of the elements of performance that the task demands as defined by the CCSSM. The student might ignore or fail to address some of the constraints of the problem. The student may occasionally make sense of quantities or relationships in the problem. The student attempts to use appropriate tools, such as a compass, protractor, postulates and theorems, with limited success. The student may recognize some characteristics of triangles, but has trouble generalizing or applying geometric methods in this modeling situation.

Level 3: Performance at Standard (9-12 points)For most of the task, the student’s response shows the main elements of performance that the task demands as defined by the CCSSM with few minor errors or omissions. The student explains the problem and identifies constraints. The student makes sense of quantities and their relationships in the modeling situation. The student uses appropriate tools, such as a compass, protractor, postulates and theorems. The student might discern patterns or structures and make connections between representations. The student is able to make sense of the problem and apply geometric concepts to this modeling situation.

Level 4: Achieves Standards at a High Level (13-16 points)The student’s response meets the demands of nearly all of the tasks as defined by the CCSSM and is organized in a coherent way. The communication is clear and precise. The body of works looks at the overall situation of the problem and process, while attempting to the details. The student routinely interprets the mathematical results, applies geometric concepts in the context of the situation, reflects on whether the results make sense and uses all appropriate tools strategically.

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