Understanding By Design - Saginaw Valley State University



Unit Design

For

Geometric Relationships

Developed by

Jacqueline Isaacson

Marvin L. Winans Academy of Performing Arts

2012

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Understanding by Design

Unit Design Worksheet

|Unit Title: Geometric Relationships |Subject/Course: Geometry |

|Topic: Circles |Grade(s): 10th |Staff Name: Jacqueline Isaacson |

|Stage 1 - Desired Results |

|Established Goals (Common Core State Standards): |

|G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance |

|along a line, and distance around a circular arc. |

| |

|G-C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles;|

|inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. |

| |

|G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles|

|are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s |

|endpoints. |

|G-C.1 Prove that all circles are similar. |

|G-C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle|

|as the constant of proportionality; derive the formula for the area of a sector. |

| |

|G-GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle |

|given by an equation |

| |

|G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★ |

| |

|G-MG.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). |

|Understandings: |Essential Questions: |

|Students will understand | |

|how to recognize an angle, a circle, perpendicular lines, parallel lines, and a |What are the definitions of basic shapes of geometry? |

|line segment when given a picture in a plane. |What is the relationship among an inscribed angle, radii, and chords? |

|what an inscribed circle looks like, the different parts of the circle, and how |How do you prove theorems about lines and angles? ( vertical angles, transversal|

|those parts relate to the shape they are inscribed with. |crosses, alt. interior angles) |

|what a theorem is, what we use to prove it, why it needs to be proven, and the |What makes a circle similar to another circle? |

|formats used for writing proofs. |Is an arc intercepted by an angle is proportional to the radius and how do we |

|similar circles and be able to write a proof to demonstrate their understanding. |derive the formula of the sector that it creates? |

|that the arc length intercepted by an angle is proportional to the radius. |What is the use of the Pythagorean Theorem in relation to a circle and how do we|

|how algebra and geometry are related. |use it to solve for missing parts? |

|how to use and apply a variety of shapes to real life. |How do you measure shapes and describe their properties? |

|how to find a solution to a problem and then clearly explain why it would solve |What types of problems in life could geometry help me solve? |

|it. | |

|Students will know: |Students will be able to: |

| | |

|how to recognize an angle, a circle, perpendicular lines, parallel lines, and a |1. describe an angle, circle, perpendicular lines, parallel lines, and line |

|line segment when given a picture in a plane. |segment. |

|what an inscribed circle looks like. They should be able to tell me the different |2. explain how angles, circles, perpendicular lines, parallel lines, and line |

|parts of the circle and how they relate to the shape they are inscribed with. |segments relate. |

|what a theorem is and what we use to prove it. Why it needs to be proven. (to |3. show how geometric figures are used in inscribed circles. |

|fully understand how things work together.) They will know the formats we write a |4. identify and use geometric shapes within a circle (inscribed) to solve for |

|proof in. |missing values. |

|what similar circles means and be able to write me a proof showing full |5. write a clear proof of a theorem and explain it to their peers in their own |

|understanding of why. |words. |

|how to be able to put into their own words using similarity the fact that the arc |6. produce mathematical problems involving similarity, arc length, and area of |

|length intercepted by an angle is proportional to the radius. Then they would be |a sector. They will be able to explain in their own words. |

|able to give the formula for the area of a sector. |7. solve for missing sides of a right triangle with algebra, the Pythagorean |

|that algebra relates to geometry in many ways. You want them to be able to tell me|Theorem, and completing the square in relation to a circle. |

|how and give examples. You want them to be able to tell me what the Pythagorean |8. explain how they use geometric shapes at home, at school, during sports, |

|and comp. sq. does and how it relates to a circle. |etc. |

|how they would/do use a variety of shapes in real life. |9. apply concepts of geometry to situation in real life to solve problems. |

|how to find a solution to a problem and then clearly explain why it would solve |10. design situations where geometric theories would help. |

|it. | |

|Unit Enduring Understanding: |Unit Question: |

| | |

|Students will know how all geometric figures relate to circles and real life. |How do all geometric figures relate to circles and real life? |

| | |

|Stage 2 - Assessment Evidence |

|Performance Tasks: |

| |

|Goal: Your task is to show your knowledge of geometric shapes relating to circles by creating a blueprint of a new performing arts school and proposing the new |

|building to board members. |

| |

|Role: You are part of a team of architects with a strict budget. You need to create a blueprint and proposal of a new creative building. Since it is a performing |

|arts school your design should be very avant-garde and utilize unusual techniques in building structure. Your clients prefer organic shapes and circles to promote |

|a calming, creative atmosphere. Your job is to create a school building that is sound in structure, creative with the use of circles, within the budget, and |

|propose this plan to a board of school members. |

| |

|Audience: Your target audience is the school board, principal, parents, and high school students. |

| |

|Situation: The challenge involves creating an 18” x 20” blueprint of the new school and a name for the school. The design must include circles and have a secure |

|structure. Then write a clear proposal explaining why you used the structure you used with the strict budget of $5,000. |

| |

|Product: One 18” x 20” blueprint of the new school. One 4 page essay explaining the blueprint, budget usage, and structure of the school. One 5 minute speech on |

|why your school is the one to choose for the project. |

| |

|Standard: Your blue print, paper, and speech will be judged with a rubric. |

| |

|Key Criteria: (Rubrics, etc.) |

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|Holistic Rubric: |

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|Holistic Rubric for (Tests) |

|4 |

|3 |

|2 |

|1 |

| |

| |

|Student understands the concept with no missed steps and/or confusing rules. |

|Student understands the concept, but missed a step and/or is confusing learned rules. |

|Student work has a lot of errors showing the student has not fully grasped the concept. They are mixing up concepts or rules, and work lacks clarity. |

|Student attempted the problem, but clearly does not understand the concept. |

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| |

|Building A Structure : Analysis of Created Product: Blue Print |

|Teacher Name: |

| |

|Student Name:     ________________________________________ |

| |

|CATEGORY |

|4 |

|3 |

|2 |

|1 |

| |

|Mathematical computation |

|All instances with geometric computation are correct. Example: If a triangle is inscribed, then all angles depicted are correct. All measurements of perimeter and |

|area are correct, etc. |

|Few errors in mathematical computation. |

|At least 10 errors in mathematical computation. |

|More than 10 errors in mathematical computation. |

| |

|Mathematical usage of tools |

|All drawn geometric shapes have been drawn using geometric construction techniques and tools (compass and straight edge). Shows a clear understanding of how to |

|construct geometric figures. |

|80% of drawn geometric shapes have been drawn using geometric construction techniques and tools (compass and straight edge). Shows a clear understanding of how to |

|construct geometric figures. |

|70% of drawn geometric shapes have been drawn using geometric construction techniques and tools (compass and straight edge). Shows a surface level understanding of|

|how to construct geometric figures. |

|60% or below of drawn geometric shapes have been drawn using geometric construction techniques and tools (compass and straight edge). Shows a shallow understanding|

|of geometric construction. |

| |

|Mathematical usage of geometry on blueprint and in paper. |

|Plans for the structure show a strong understanding of how geometric figures relate to a circle in the correct manner. All mathematical analysis is correct and |

|makes sense. All measurements are correct. |

|Plans for the structure show an understanding of how geometric figures relate to a circle in the correct manner. 80% of mathematical analysis is correct and makes |

|sense. Most measurements are correct. |

|Plans for the structure show little understanding of how geometric figures relate to a circle in the correct manner. 70% of mathematical analysis is correct and |

|makes sense. Many measurements are incorrect. |

|Plans for the structure show very little to no understanding of how geometric figures relate to a circle in the correct manner. 69% or below of mathematical |

|analysis is correct and makes sense. |

| |

|Information Gathering |

|Accurate information taken from several sources in a systematic manner. |

|Accurate information taken from a couple of sources in a systematic manner. |

|Accurate information taken from a couple of sources but not systematically. |

|Information taken from only one source and/or information not accurate. |

| |

| |

| |

|Computations: |

| |

|Teacher Name: |

| |

|Student Name:     ________________________________________ |

| |

|CATEGORY |

|4 |

|3 |

|2 |

|1 |

| |

|Mathematical Concepts |

|Explanation shows complete understanding of the mathematical concepts used to solve the problem(s). |

|Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s). |

|Explanation shows some understanding of the mathematical concepts needed to solve the problem(s). |

|Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written. |

| |

|Mathematical Reasoning |

|Uses complex and refined mathematical reasoning. |

|Uses effective mathematical reasoning. |

|Some evidence of mathematical reasoning. |

|Little evidence of mathematical reasoning. |

| |

|Mathematical Errors |

|90-100% of the steps and solutions have no mathematical errors. |

|Almost all (85-89%) of the steps and solutions have no mathematical errors. |

|Most (75-84%) of the steps and solutions have no mathematical errors. |

|More than 75% of the steps and solutions have mathematical errors. |

| |

|Mathematical Terminology and Notation |

|Correct terminology and notation are always used, making it easy to understand what was done. |

|Correct terminology and notation are usually used, making it fairly easy to understand what was done. |

|Correct terminology and notation are used, but it is sometimes not easy to understand what was done. |

|There is little use, or a lot of inappropriate use, of terminology and notation. |

| |

|Strategy/ |

|Procedures |

|Typically uses an efficient and effective strategy to solve the problem(s). |

|Typically uses an effective strategy to solve the problem(s). |

|Sometimes uses an effective strategy to solve problems, but does not do it consistently. |

|Rarely uses an effective strategy to solve problems. |

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|Research: |

| |

|Teacher Name: |

| |

|Student Name:     ________________________________________ |

| |

|CATEGORY |

|4 |

|3 |

|2 |

|1 |

| |

|Research Packet |

|Research packet is complete with all parts filled in. |

|Research packet is missing a few parts but have been completed since. |

|Research packet is vague and missing parts. |

|Research is inaccurate or unfinished. |

| |

|Research in packet |

|Research is accurate, clear, and follows format of packet. |

|Research is mostly complete, but has a few inaccuracies. |

|Student used unreliable websites to find information or research is inaccurate. |

|Student’s research is incomplete and unreliable either by using improper websites or making things up. |

| |

| |

|Oral Presentation Rubric : GRASPS Rubric: |

|Architecture Project |

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|Teacher Name: |

| |

|Student Name:     ________________________________________ |

| |

|CATEGORY |

|4 |

|3 |

|2 |

|1 |

| |

|Vocabulary |

|Uses vocabulary appropriate for the audience. Extends audience vocabulary by defining words that might be new to most of the audience. |

|Uses vocabulary appropriate for the audience. Includes 1-2 words that might be new to most of the audience, but does not define them. |

|Uses vocabulary appropriate for the audience. Does not include any vocabulary that might be new to the audience. |

|Uses several (5 or more) words or phrases that are not understood by the audience. |

| |

|Time-Limit |

|Presentation is 4-5 minutes long. |

|Presentation is 3 minutes long. |

|Presentation is 2 minutes long. |

|Presentation is less than 2 minutes OR more than 5 minutes. |

| |

|Content |

|Shows a full understanding of the topic, both geometric and economic. |

|Shows a good understanding of the topic, both geometric and economic. |

|Shows a good understanding of parts of the topic, both geometric and economic. |

|Does not seem to understand the topic very well. |

| |

|Evaluates Peers |

|Fills out peer evaluation completely and always gives scores based on the presentation rather than other factors (e.g., person is a close friend). |

|Fills out almost all of the peer evaluation and always gives scores based on the presentation rather than other factors (e.g., person is a close friend). |

|Fills out most of the peer evaluation and always gives scores based on the presentation rather than other factors (e.g., person is a close friend). |

| |

|Fills out most of the peer evaluation but scoring appears to be biased. |

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|Other Evidence: |

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|Journals, rough drafts, vocabulary activity, quizzes, concept maps, observations, daily assignments, textbook, worksheets, prompts on backing up mathematical |

|thought. |

|BEFORE |DURING |AFTER |

| | | |

|Pre-quizzes on vocabulary |Vocabulary log |Posttest GRASPS |

|The teacher will test the students on vocabulary to |Students will keep a running vocabulary log for |GRASPS project on architecture. |

|see what they know prior to starting the unit. |personal reference. | |

| | |Student Review |

|Brainstorming |Questioning |Students will re-teach parts of the unit as partners |

|The students will brainstorm ideas for their building |The teacher will ask questions as the students work |to the class to help promote learning. |

|before beginning the GRASPS project. This will promote|and learn. | |

|critical thinking and problem solving. | | |

| |Daily assignments | |

|Think/pair/ share |Daily book and worksheet assignments. | |

|Students will think, pair, and share about problems. | | |

|The teacher will give them feedback on what they will |Drawings | |

|be learning. |Students will practice their constructions with small| |

| |assignments similar to the GRASPS assignment to check| |

| |for understanding. | |

| | | |

| |Observations | |

| |The teacher will monitor the students’ in-class work | |

| |and check for understanding. | |

| | | |

| |Quizzes | |

| |The teacher will retest the students on vocabulary | |

| |and information to see if they learned the terms and | |

| |information that they had not learned before. | |

|Describe the assessment/s and state the prompt if applicable. □ F X S | | |

| | | |

|What type of scoring tools will be used for evaluation? | | |

|X Analytic rubric □ Checklist | | |

|X Holistic rubric □ Answer Key | | |

|X Criterion rubric □ Other | | |

| | | |

|Student Self-Assessment and Reflection: |

| |

|Students will write a few short journal entries to reflect on their work and demonstrate understanding. |

|They will be asked to write about: |

|How do you feel about the unit so far? |

|What is the easiest concept for you to understand, what is the most difficult? |

|Explain a concept in your own words. |

|How does what we learned today relate to real life? |

|How did you do this week? |

| |

|At the end of the project, students will reflect on how much work they have done. They will be expected to give other groups a grade for their work as well. |

|Students will ask themselves: |

|What parts of the project did I contribute to? |

|What could I have done more on? |

|What did I learn from this project? |

|How did the circles make creating the building more difficult? |

|How did the budget make creating the building more difficult? |

|What other subjects did I have to use to complete this project? |

|How will this project and unit help me later on in life? |

|Did performing in front of an audience cause me to pay attention to the information more closely? |

|Did working with a group of peers help me to understand concepts I didn’t understand before? |

| |

|Group project peer analysis questionnaire: |

|What parts did I contribute to, what parts did the others do? |

|Who did not do anything? |

|Who helped you the most when it came to understanding the mathematical analysis? |

|As a group, how did you think you did? Rate your group on a scale of 1-10, with one being the lowest and ten being the highest. |

|Stage 3 - Learning Plan |

|Differentiated Instruction: |

| |

|Level C – 90 points (All activities are required) |

| |

|1. Unit vocabulary list (10 points) |

|2. “Do Now” unit bell work tracker (2 points a day – 30 points) |

|3. Daily homework problem sets (5 points a day – 50 total points) |

| |

|Level B – 25 points (Activity one is required, choose one of the next two) |

| |

|1. Use a compass and protractor to discover how to make inscribed shapes and complete a worksheet. (10 points) |

|2. Create a series of 5 mini textbooks showing a variety of proofs of inscribed shapes, angles, similar circles, and constructions of all the geometric shapes. |

|Must explain in own words. ( 15 points) |

|3. Create/complete work packet with a partner. (15 points) |

| |

|Level A – 56 points |

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|GRASPS project: create a blueprint for a new school with a given budget |

|Research packet with filled in questions and essays ( 8 points) |

|Written notes showing all work of all mathematical computation (20 points) |

|Fully rendered blueprint (16 points) |

|Oral presentation (12 points) |

|Learning Activities: |

| |

|The end of the unit project asks students to answer the question of, where are we going? They will have to rethink what they have learned and re organize the |

|information to create a blueprint of a new school with a given budget which will deepen understanding of geometry in relation to real life. |

|Students will use geometric tools to understand the construction of geometric objects and why we compute the way we do. They will use a variety of worksheets, |

|inquiry based group work, and choices to tailor to their needs. |

|The teacher will hook students’ interest by giving them choices and projects that use new tools and ideas they have not used before. Students will either create a |

|series of 5 mini textbooks with visual aids that explain the main unit points or complete a work packet with a partner. |

|At the end of the unit, students will present their GRASPS project, reflect on their project, and peer evaluate the projects. |

| |

| |

|Students will be equipped with materials and supplies to complete each learning activity. |

|Essential Vocabulary |

|Angles: two rays that share one end-point. |

|Area: any particular extent of space or surface; part. |

|Arc: any unbroken part of the circumference of a circle or other curved line. |

|Bisector: a line or plane that bisects  an angle or line segment. |

|Center: the middle point, as the point within a circle or sphere equally distant from all points of the circumference or surface. |

|Chord: the line segment between two points on a given curve. |

|Circle: a closed plane curve consisting of all points at a given distance from a point within it called the center. Congruent: equal parts. |

|Diameter: a straight line passing through the center of a circle or sphere and meeting the circumference or surface at each end. |

|Equidistant: equally distant. |

|Interior Angles: an angle formed between parallel lines by a third line that intersects them. |

|Inscribe: to draw or delineate (one figure) within another figure so that the inner lies entirely within the boundary of the outer, touching it at as many |

|points as possible. |

|Line Segment: a finite section of a line. |

|Parallel Line: extending in the same direction, equidistant at all points, and never converging or diverging. |

|Perimeter: the border or outer boundary of a two-dimensional figure. |

|Perpendicular Line: meeting a given line or surface at right angles. |

|Proportion: comparative relation between things or magnitudes as to size, quantity, number, etc.; ratio. |

|Pythagorean Theorem: the theorem that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. |

|Radius: a straight line extending from the center of a circle or sphere to the circumference or surface. |

|Segment: a part cut off from a figure, especially a circular or spherical one, by a line or plane, as a part of a circular area contained by an arc and its |

|chord or by two parallel lines or planes. |

|Similar: having the same shape; having corresponding sides proportional and corresponding angles equal. |

|Vertical Angle: one of two opposite and equal angles formed by the intersection of two lines. |

|Sequencing the Learning |

|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

| | | | | |

|Pre-quiz: |Identifying Basic geometric |Intro to circles |Manipulation of/ similarity of |Manipulation of/ similarity of |

|Identify what basic |shapes and how they relate to | |circles |circles |

|vocabularies are. Identify what|each other. |Level C: daily assignment | | |

|the basic geometric shapes are.| | |Level C: daily assignment |Quiz over week information. |

| |Level C: daily assignment | | | |

|Complete basic math skills i.e.| | | |Level C: Packet for the week |

|multiplication, division, and | | | | |

|solving for variables | | | | |

| | | | | |

|Level C: daily assignment | | | | |

|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

| | | | | |

|Basic proofing knowledge by |Proofing of similar circles |Level B: activity day |Inscribed circles |Quiz |

|brainstorming in pairs | | | | |

| |Think-pair-share activity |Inquiry based work sheet by |Level C: daily assignment |Level C: Packet for the week |

|Level C: daily assignment | |drawing constructions | | |

| |Level C: daily assignment | | | |

| | |Level C: daily assignment | | |

|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

| | | | | |

|Pythagorean theorem with |Research on relation to real |Level B: choice assignment |Level B: choice assignment |Level C: Packet for the week |

|inscribed and non triangles |life day. How to research | | | |

| |effectively. |Work day |Work day |Level B: choice assignment due |

|Level C: daily assignment | | | | |

| |Level C: daily assignment | | | |

|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

| | | | | |

|Level A |Level A |Level A |Level A |Level A |

| | | | | |

|GRASPS work day |GRASPS work day |GRASPS work day |GRASPS work day |GRASPS work day |

| | | | | |

| |Research packet due | |Mathematical computation work | |

| | | |packet due | |

|MONDAY |TUESDAY |WEDNESDAY |THURSDAY |FRIDAY |

| | | | | |

|Level A |Presentations |Presentations |Geometric relationships test | |

| | | | | |

|GRASPS work day |Blueprints due | |Unit vocabulary packets due | |

| | | | | |

|Complete peer evaluations | | | | |

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