Understanding By Design - Saginaw Valley State University
Unit Design
For
Geometric Relationships
Developed by
Jacqueline Isaacson
Marvin L. Winans Academy of Performing Arts
2012
[pic]
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.) |
| |
|Holistic Rubric: |
| |
| |
|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. |
| |
| |
|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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| |
| |
| |
| |
| |
| |
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| |
| |
| |
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| |
| |
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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 |
| |
|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. |
| |
| |
|Other Evidence: |
| |
|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 |
| |
|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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