GEOMETRY A. Process Strands: Mathematical Reasoning Problem Solving, Communication, connections, and use of tools MAJOR CONCEPTS

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1 GEOMETRY A Geometry is for students who have successfully completed Algebra IA. This course develops logical thinking and the proof of theorems, using axioms and postulates. Specific topics will include the properties and relationships of points, lines and planes, congruence, parallelism and geometric models. Content Strand: Geometry and Measurement Process Strands: Mathematical Reasoning Problem Solving, Communication, connections, and use of tools Major Stems: MAJOR CONCEPTS Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems Applies the concepts of congruence by solving problems Applies concepts of similarity by solving problems Solves problems involving perimeter, circumference, or area Uses units of measurement appropriately and consistently when solving problems across content strands; makes conversions within or across systems and makes decisions concerning an appropriate degree of accuracy in problem situations Solves problems on and off the coordinate plane

2 DISCIPLINE COURSE CONTENT COMPETENCIES 1. Students will demonstrate that geometry contains a symbolic language for communicating ideas. 2. Students will apply logical and mathematical reasoning effectively to arrive at appropriate solutions to problems. 3. Students will apply the concepts of measurement in multiple dimensions. 4. Students will solve problems using the characteristics and properties of geometric figures. 5. Students will define and describe physical entities and attributes in the real world using spatial relationships. 6. Students will apply geometric patterns and relationships to represent information, interpret data, and make predictions. 7. Students will demonstrate that geometry connects to the other strands of mathematics and to other disciplines. DISCIPLINE COURSE PROCESS COMPETENCIES 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will communicate their understanding of mathematics. 4. Students will create and use representations to communicate mathematical ideas and to solve problems. 5. Students will recognize, explore, and develop mathematical connections GEOMETRY A STEM 1 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

3 Topics Competencies Knowledge/Skills Process Skills Create formal proofs for angles, lines, circles, distance, midpoint, and polygons 1. Students will demonstrate that geometry contains a symbolic language for communicating ideas. 2. Students will apply logical and mathematical reasoning effectively to arrive at appropriate solutions to problems. 3. Students will solve problems using the characteristics and properties of geometric figures. 4. Students will apply geometric patterns and relationships to represent information, interpret data, and make predictions. 5. Students will demonstrate that geometry connects to the other strands of mathematics and to other disciplines. 1. Create a 2 column or flow proof 2. Use deductive reasoning to establish or refute conjectures 3. Establish precise descriptions of conditions that characterize a class of objects. 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will communicate their understanding of mathematics. 4. Students will create and use representations to communicate mathematical ideas and to solve problems. 5. Students will recognize, explore, and develop mathematical connections Sample Performance Assessment SPA #1 Topics in SPA#1 Discipline Process Skills Addressed in SPA#1 Given: quadrilateral ABCD, with AB CD, BC AD Prove: ABCD is a parallelogram Auxiliary lines, Reflexive property, Triangle Congruence Postulates, Definition of Congruent triangles, and Parallel Line properties 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will communicate their understanding of mathematics.

4 Discipline Competencies Addressed in SPA#1 4. Students will create and use representations to communicate mathematical ideas and to solve problems. 5. Students will recognize, explore, and develop mathematical connections 1. Students will apply logical and mathematical reasoning effectively to arrive at appropriate solutions to problems. 2. Students will solve problems using the characteristics and properties of geometric figures. 3. Students will demonstrate that geometry connects to the other strands of mathematics and to other disciplines. SPA#1 Rubric Level 4 Level 3 Level 2 Level 1 GEOMETRY A-STEM 2 Applies the concepts of congruence by solving problems

5 Topics Competencies Reflections, translations, or rotations on or off coordinate plane Congruent polygons and triangles 1. Students will demonstrate that geometry contains a symbolic language for communicating ideas. 2. Students will apply the concepts of measurement in multiple dimensions 3. Students will solve problems using the characteristics and properties of geometric figures. Knowledge/Skills Process Skills 1. Students will study congruent figures by way of rigid transformations on and off the coordinate plane 2. Students will understand congruence by way of translations, reflections, and rotations 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will create and use representations to communicate mathematical ideas and to solve problems. 4. Students will recognize, explore, and develop mathematical connections.

6 Sample Performance Assessment SPA #1 Topics in SPA#1 Discipline Process Skills Addressed in SPA#1 Discipline Competencies Addressed in SPA#1 Congruency, Triangle congruence shortcuts (SSS, SAS, AAS, HL, ASA), Transformations 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will create and use representations to communicate mathematical ideas and to solve problems. 4. Students will recognize, explore, and develop mathematical connections. 1. Students will demonstrate that geometry contains a symbolic language for communicating ideas. 2. Students will apply the concepts of measurement in multiple dimensions 3. Students will solve problems using the characteristics and properties of geometric figures. SPA#1 Rubric for Geometry A

7 Level 4 Student will use sophisticated mathematical reasoning, provide strong supportive arguments and include examples and/or counterexamples Level 3 Student uses sound mathematical reasoning and includes supportive arguments Level 2 Student uses somewhat appropriate mathematical reasoning, but includes incomplete or faulty arguments Level 1 Student uses limited mathematical reasoning and includes no arguments GEOMETRY A STEM 3 Applies concepts of similarity by solving problems Topics Competencies Knowledge/Skills Process Skills Apply concepts of similarity to define the trigonometric functions as ratios of sides of right triangles 1. Students will demonstrate that geometry contains a symbolic language for communicating ideas. 2. Students will apply logical and mathematical reasoning effectively to arrive at appropriate solutions to problems. 3. Students will apply the concepts of measurement in multiple dimensions. 4. Students will solve problems using the characteristics and properties of geometric figures. Trig. Functions (sine, cosine, tangent), Inverse functions, Special right triangles (45,45,90 and 30,60,90), Law of Sines, Law of cosines, Anle of elevation and depression 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 3. Students will use mathematical reasoning and proof. 4. Students will communicate their understanding of mathematics 5. Students will recognize, explore, and develop mathematical connections Sample Performance An archeologist uncovers the remains of a square based Assessment SPA #1 Egyptian pyramid. The base is intact and measures 130

8 Topics in SPA#1 Discipline Process Skills Addressed in SPA#1 Discipline Competencies Addressed in SPA#1 meters on each side. The top of the pyramid has eroded away, but what remains of each face of the pyramid forms a 65 degree angle with the ground. What was the original height of the pyramid? Trig functions, pyramid definition and properties. 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will recognize, explore, and develop mathematical connections 1. Students will apply logical and mathematical reasoning effectively to arrive at appropriate solutions to problems. 2. Students will apply the concepts of measurement in multiple dimensions. 3. Students will solve problems using the characteristics and properties of geometric figures. SPA#1 Rubric for Geometry A Level 4 Student will create an effective diagram with given info clearly stated. Student shows thorough understanding of the problems mathematical ideas and processes. Students use multiple strategies that lead to correct answer. Level 3 Student creates accurate diagram. Student shows basic understanding of the problems mathematical ides and processes. Uses appropriate strategies that lead to correct answer. Level 2 Student creates inappropriate and/or unclear diagram. Student indicates a partial understanding of the problem. Selects some appropriate strategies that lead to partially correct answer. Level 1 Students does not create diagram. Student shows little understanding of the problem. Uses poor or inappropriate strategies that lead to incorrect answer.

9 GEOMETRY A STEM 4 Solves problems involving perimeter, circumference, or area

10 Topics Competencies Knowledge/Skills Perimeter, circumference, and arc length of twodimensional figures as well as surface area and volume of three-dimensional figures 1. Students will understand that geometry contains a symbolic language for communicating ideas. 2. Students will understand that logical and mathematical reasoning can be used effectively to problem-solve in order to arrive at appropriate solutions. 3. Students will understand that concepts of measurement can be applied in multiple dimensions. 4. Students will understand that the characteristics and properties of geometric figures can be used to solve problems. 5. Students will understand that spatial relationships can be used to define and describe physical entities and attributes in the real world 6. Students will understand that geometric patterns and relationships can be used to represent information, interpret data, and make predictions. 7. Students will understand that geometry connects to the other strands of mathematics and to other disciplines 1. The types of measurements to be used for perimeter, circumference, area, and volume. 2. Which formulas relating to perimeter, circumference, area, or volume is appropriate to use for a given shape. 3. Howe different formulas relating to area and volume were developed. 4. The properties of different polygons, circles, and solids. 5. The connection between area of plane figures and surface area of some solids. Process Skills 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will communicate their understanding of mathematics 4. Students will create and use representations to communicate mathematical ideas and to solve C.A.C.E.S. (User Friendly) Generic problems. Course Competencies/Assessment MODEL 5. Students will recognize, explore, and develop mathematical connections

11 Sample Performance Assessment SPA #1 Topics in SPA#1 Discipline Process Skills Addressed in SPA#1 Discipline Competencies Addressed in SPA#1 An engineer is in the process of designing the new bridge that will span the Lamprey River. To her it seems fitting that the arch of the bridge will be the arc of a circle. For aesthetic reasons the arc should measure 60 degrees. The bridge will span 180 meters. Calculate the length of the arc. Properties of circles ( circumference, arc length, chords, central angle, radii), triangle sum, equilateral triangle properties. 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will create and use representations to communicate mathematical ideas and to solve problems. 4. Students will recognize, explore, and develop mathematical connections 1. Students will understand that logical and mathematical reasoning can be used effectively to problem-solve in order to arrive at appropriate solutions. 2. Students will understand that concepts of measurement can be applied in multiple dimensions. 3. Students will understand that the characteristics and properties of geometric figures can be used to solve problems. 4. Students will understand that geometry connects to the other strands of mathematics and to other disciplines SPA#1 Rubric for Geometry A

12 Level 4 The student presents an accurate diagram of the problem with findings written in and calculated on the side. The students shows a mastering of the 7 topics and there connections. The problem has an answer that makes physical and mental sense. Level 3 The student presents a diagram of the problem with findings written in and calculated on the side. The students shows knowledge of the 7 topics with limited errors. The problem has an answer that makes physical and mental sense. Level 2 The student show written calculations and connections to some of the 7 topics with errors in some. The problem has an answer that corresponds to the mental math with no physical connection to compare. Level 1 The students show some of the 7 topics and there calculations with major errors. No answer is derived at.

13 GEOMETRY A STEM 5 Uses units of measurement appropriately and consistently when solving problems across content strands; makes conversions within or across systems and makes decisions concerning an appropriate degree of accuracy in problem situations. Topics Competencies Knowledge/Skills Process Skills Measurement/conversion strategies; scales; accuracy and approximate error 1. Students will understand that logical and mathematical reasoning can be used effectively to problem-solve in order to arrive at appropriate solutions. 2. Students will understand that concepts of measurement can be applied in multiple dimension 3. Students will understand that geometric patterns and relationships can be used to represent information, interpret data, and make predictions. 4. Students will understand that geometry connects to the other strands of mathematics and to other disciplines 1. Dimensional analysis 2. Analyze precision, accuracy, and approximate error in measurement situations. 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 3. Students will use mathematical reasoning and proof. 4. Students will communicate their understanding of mathematics 5. Students will create and use representations to communicate mathematical ideas and to solve problems. 6. Students will recognize, explore, and develop mathematical connections

14 Sample Performance Assessment SPA #1 Topics in SPA#1 Discipline Process Skills Addressed in SPA#1 You are a marine biologist in charge of installing a rectangular fish tank for exotic fish at the National Zoo. The standard size fish tanks have a height of 12 ft and width of 6ft, with varying lengths of 40, 20, 10, and 5 feet. In order for the fish to survive, the tank must hold at least 10,000 gallons of water (1 gallon of water = cu ft). Find the volume in cubic feet and gallons for each tank and determine which tank will be suitable for your requirements. Write your findings in a report that includes the following and addresses the actual exhibit dimensions described in the extension: the volume of all of the standard tanks in cubic feet and in gallons a diagram (including dimensions) of the standard fish tank that will provide the fish with the optimal environment for survival; an explanation of what happens to the volume when the length is changed and height or width remain the same. Extension: The actual length of the exhibit space is 17½ ft. You must decide if ordering a custom length tank that is 17½ ft long will meet the specifications needed for a safe environment for the fish (10,000 gallons of water). Show whether or not a tank with dimensions 17½ ft. x 12 x 6 will suit your requirements. If not, find the dimensions of a fish tank that meets the necessary specifications. If possible, find the dimensions of a fish tank that gives you exactly 10,000 gallons of water. Measurement changes, Units, Attribute: radius/ height/ surface area/ volume, Investigating changes i.e. linear dimensions impact on a solid object s surface area and volume, and Investigating linear scale factors of similar figures or solids and its use in problem solving area and volume scenarios 1. Students will understand that a variety of problemsolving strategies can be used to investigate everyday as well as increasingly complex mathematical situations. 2. Students will understand that exploring, justifying and synthesizing mathematical conjectures are part of systemic reasoning which is common to all content areas and a defining feature of mathematics. 3. Students will understand that actively exploring, investigating, describing, and explaining mathematical ideas promotes communication which leads to a greater comprehension of mathematical concepts. 4. Students will understand that representing ideas and connecting the representations lies at the heart of understanding mathematics. 5. [Students will understand that progress is made by asking relevant questions, conducting careful investigations evaluating the validity of results and

15 Discipline Competencies Addressed in SPA#1 SPA#1 Rubric developing models to explain what has been found.] 6. [Students will understand that appropriate representations and mathematical language is used to present ideas clearly and logically for a given situation.] 7. Students will understand that mathematical reasoning and proof help to formalize mathematical arguments that influence the structure of mathematics. 1. Students will understand that geometry contains a symbolic language for communicating ideas. 2. Students will understand that logical and mathematical reasoning can be used effectively to problem-solve in order to arrive at appropriate solutions and to develop mathematical arguments about geometric relationships. 3. Students will understand that concepts of measurement can be applied in multiple dimensions and understand measurable attributes of objects and the units, systems and processes of measurement. 4. Students will understand that the characteristics and properties of geometric figures can be used to solve problems. 6. Students will understand that geometric patterns and relationships can be used to represent information, interpret data, and make predictions.

16 Level 4 The Student will: Reproduce rectangular prisms to represent the fish tank and label its dimensions in general terms and with the specific examples; Compute the volume of all the various prisms and a few additional examples and report the findings in a table; Accurately convert cubic feet into gallons for all provided prisms and correctly modify the table accordingly Provide an analysis and generalization with proof that shows the volume changes at the same rate as the length when the other two dimensions remain constant Create a plan for a 17½ foot tank and provide a clear logical explanation why the tank of this size will not hold 10,000 gallons of water; Summarize their tank calculations and use a report to show and analyze how the change in length changed the volume and identify which standard tank would work and provide at least one suggestion on how to modify the problem to meet the 10,000 gallon requirement. The Student will: Reproduce a rectangular prism to represent the fish tank and label its dimensions; Compute the volume of the given various prisms and report the findings in a table; Convert cubic feet into gallons for all entries and correctly modify the table accordingly; Provide an analysis that shows the volume changes at the same rate as the length when the other two dimensions remain constant Create a plan for a 17½ foot tank and correctly explain that the tank of this size will not hold 10,000 gallons of water; Summarize their tank calculations and use a report to show and analyze how the change in length changed the volume. Level 2 The Student will: Reproduce a rectangular prism to represent the fish tank; Compute most of the volumes of the various prisms and report the findings in a table; Convert cubic feet correctly for most of the prisms into gallons and modify the table accordingly Provide an analysis that shows the volume changes at the same rate as the length; Create a plan for a 17½ foot tank and state that the tank of this size will not hold 10,000 gallons of water; Summarize their tank calculations and use a report to show how the change in length changed the Level 1 The Student will: Reproduce a rectangle to represent the fish tank; Compute correctly the volume of two of the prisms; Convert cubic feet into gallons for two entries and modify the table accordingly; State that the volume changes but does not provide a coherent explanation. Create a plan for a 17½ foot tank guess that the tank of this size will not hold 10,000 gallons of water; Summarize their tank calculations.

17 GEOMETRY A STEM 6 Solves problems on and off the coordinate plane Topics Distance, midpoint, perpendicular and parallel lines, and slope Competencies 1. Students will understand that geometry contains a symbolic language for communicating ideas. 2. Students will understand that logical and mathematical reasoning can be used effectively to problem-solve in order to arrive at appropriate solutions. 3. Students will understand that concepts of measurement can be applied in multiple dimensions. 4. Students will understand that the characteristics and properties of geometric figures can be used to solve problems. 5. Students will understand that spatial relationships can be used to define and describe physical entities and attributes in the real world. 6. Students will understand that geometric patterns and relationships can be used to represent information, interpret data, and make predictions. Students will understand that geometry connects to the other strands of mathematics and to other disciplines Knowledge/Skills Process Skills 1. Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content. 2. Students will use mathematical reasoning and proof. 3. Students will communicate their understanding of mathematics. 4. Students will create and use representations to communicate mathematical ideas and to solve problems. 5. Students will recognize, explore, and develop mathematical connections

18 Sample Performance Assessment SPA #1 Topics in SPA#1 Discipline Process Skills Addressed in SPA#1 Your assignment is to make a map of your neighborhood and show the distance to several of your favorite places. Place your map on a coordinate grid and let your home be located at the origin. Below are the things you are going to include on your map. You know your school is 6 blocks due east of your home and your best friend, JoJo s house is 8 blocks due north of school. State Avenue runs diagonally through town. Your house is located at the corner of First Street and State Avenue and JoJo lives at A Street and State Avenue. The Pizza Shoppe, located on State Avenue, is halfway between your house and JoJo s house. The Community Center is located on Maine Avenue, another street that runs diagonally across town and is parallel to your street, State Avenue. The Community Center is three blocks due west and 1 block due north of your home. 1) Use graph paper to make your map. Be sure to label the scale and axes. For streets that run north/south or east/west, let one square on the grid represent one city block. Write the coordinates of the school, your best friend s house, the Pizza Shoppe, and the Community Center. 2) Find the distance between the following locations: a. Your house and JoJo s b. Your house and the Pizza Shoppe c. Your house and the community center 3) Use your knowledge of coordinate geometry to verify that State Avenue and the street where the community center is located will not intersect. 4) Write a summary which includes all calculations, the map, and explanations about the formulas, calculations or other methods used to complete the assignment. Distance Formula, Slope, Midpoints, and Rectangular coordinates 1. Students will understand that a variety of problem-solving strategies can be used to investigate everyday as well as increasingly complex mathematical situations. 2. Students will understand that exploring, justifying and synthesizing mathematical conjectures are part of systemic reasoning which is common to all content areas and a defining feature of mathematics. 3. Students will understand that actively exploring, investigating, describing, and explaining mathematical ideas promotes communication which leads to a greater comprehension of mathematical concepts. 4. Students will understand that mathematical connections will help them become aware of the usefulness of mathematics, serve to bridge the concrete and the abstract, and enable deeper understanding of important ideas. 5. Students will understand that representing ideas and connecting the representations lies at the heart of understanding mathematics. 7. [Students will understand that appropriate representations and mathematical language is used to present ideas clearly and logically for a given situation.]

19 Geometry Competencies Addressed in SPA#1 1. Students will understand that geometry contains a symbolic language for communicating ideas. 2. Students will understand that logical and mathematical reasoning can be used effectively to problem-solve in order to arrive at appropriate solutions and to develop mathematical arguments about geometric relationships. 3. Students will understand that concepts of measurement can be applied in multiple dimensions and understand measurable attributes of objects and the units, systems and processes of measurement. 4. Students will understand that the characteristics and properties of geometric figures can be used to solve problems. 5. Students will understand that spatial relationships can be used to specify location and define or describe physical entities and attributes in the real world. 6. Students will understand that geometric patterns and relationships can be used to represent information, interpret data, and make predictions. 7. Students will understand that geometry connects to the other strands of mathematics and to other disciplines.

20 Level 4 The student will: Create a map in the coordinate plane that accurately includes the points from the written descriptions representing home, school, JoJo s, and the Pizza Shoppe and segments representing the streets rather than lines; Determine lengths of segments on a line given the coordinates by the absolute value of the difference between the two coordinates and apply this concept to horizontal and vertical distances on the coordinate plane; Use the distance and the Pythagorean Theorem (right triangle) to correctly compute the distance between home and JoJo s and between home and the community center and verify their distances; Compute the slope of State Avenue and Maine Avenue to be 4/3 by the slope formula or the definition of slope (rise over run) and include an illustration; Use the slope of the diagonal streets and provide proof to verify that Maine and State Avenues will not intersect with equal slopes streets must be parallel; Level 3 The student will: Create a map in the coordinate plane that includes the points representing home, school, JoJo s, and the Pizza Shoppe and segments representing the streets rather than lines; Determine lengths of segments on a line given the coordinates by the absolute value of the difference between the two coordinates and apply this concept to horizontal and vertical distances on the coordinate plane; Use the distance or the Pythagorean Theorem (right triangle) to correctly compute the distance between home and JoJo s and between home and the community center; Compute the slope for Maine and State Avenue, verify they have the same slope and include an illustration; Use the slope of the diagonal streets to conclude that that Maine and State Avenues will not intersect with equal slopes streets must be parallel; Summarize their work and support their conclusions with the mathematical representations and calculations. Level 2 The student will: Create points on a coordinate plane that represents most of the points correctly; Determine lengths of segments on a line given by counting the graph paper units or using a graph paper ruler to measure the lengths; Use the distance or the Pythagorean Theorem (right triangle) to compute the distance between home and JoJo s or between home and the community center, Identify the slope for Maine and State Avenue, with an illustration; Use the slope of the diagonal streets to conclude that that Maine and State Avenues will not intersect; Summarize their work and draw their conclusions with a few calculation errors. Level 1 The student will: Create a map in the coordinate plane that includes two of the points correctly; Determine lengths of segments on a line by counting; Use the distance or the Pythagorean Theorem (right triangle) to compute the distance between home and JoJo s; State that Maine and State Avenues have the same slope but cannot verify with calculations; Use the slope of the diagonal streets to conclude that that Maine and State Avenues will intersect with equal slopes; Summarize their work and provide at most two correct calculations. Summarize their work and support their conclusions with proof, mathematical representations and accurate calculations.

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