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Unit Design
For
Geometric Relationships
Developed by
Jacqueline Isaacson
Marvin L. Winans Academy of Performing Arts
2012
Understanding by Design
Unit Design Worksheet
Unit Title: Geometric Relationships
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Subject/Course: Geometry
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Topic: Circles
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Grade(s): 10th
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Staff Name: Jacqueline Isaacson
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Stage 1 - Desired Results
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Established Goals (Common Core State Standards):
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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.
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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.
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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.
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G-C.1 Prove that all circles are similar.
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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.
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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
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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).★
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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).
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Understandings:
Students will understand
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how to recognize an angle, a circle, perpendicular lines, parallel lines, and a line segment when given a picture in a plane.
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what an inscribed circle looks like, the different parts of the circle, and how those parts relate to the shape they are inscribed with.
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what a theorem is, what we use to prove it, why it needs to be proven, and the formats used for writing proofs.
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similar circles and be able to write a proof to demonstrate their understanding.
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that the arc length intercepted by an angle is proportional to the radius.
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how algebra and geometry are related.
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how to use and apply a variety of shapes to real life.
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how to find a solution to a problem and then clearly explain why it would solve it.
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Essential Questions:
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What are the definitions of basic shapes of geometry?
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What is the relationship among an inscribed angle, radii, and chords?
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How do you prove theorems about lines and angles? ( vertical angles, transversal crosses, alt. interior angles)
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What makes a circle similar to another circle?
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Is an arc intercepted by an angle is proportional to the radius and how do we derive the formula of the sector that it creates?
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What is the use of the Pythagorean Theorem in relation to a circle and how do we use it to solve for missing parts?
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How do you measure shapes and describe their properties?
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What types of problems in life could geometry help me solve?
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Students will know:
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how to recognize an angle, a circle, perpendicular lines, parallel lines, and a line segment when given a picture in a plane.
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what an inscribed circle looks like. They should be able to tell me the different parts of the circle and how they relate to the shape they are inscribed with.
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what a theorem is and what we use to prove it. Why it needs to be proven. (to fully understand how things work together.) They will know the formats we write a proof in.
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what similar circles means and be able to write me a proof showing full understanding of why.
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how to be able to put into their own words using similarity the fact that the arc length intercepted by an angle is proportional to the radius. Then they would be able to give the formula for the area of a sector.
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that algebra relates to geometry in many ways. You want them to be able to tell me how and give examples. You want them to be able to tell me what the Pythagorean and comp. sq. does and how it relates to a circle.
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how they would/do use a variety of shapes in real life.
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how to find a solution to a problem and then clearly explain why it would solve it.
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Students will be able to:
1. describe an angle, circle, perpendicular lines, parallel lines, and line segment.
2. explain how angles, circles, perpendicular lines, parallel lines, and line segments relate.
3. show how geometric figures are used in inscribed circles.
4. identify and use geometric shapes within a circle (inscribed) to solve for missing values.
5. write a clear proof of a theorem and explain it to their peers in their own words.
6. produce mathematical problems involving similarity, arc length, and area of a sector. They will be able to explain in their own words.
7. solve for missing sides of a right triangle with algebra, the Pythagorean Theorem, and completing the square in relation to a circle.
8. explain how they use geometric shapes at home, at school, during sports, etc.
9. apply concepts of geometry to situation in real life to solve problems.
10. design situations where geometric theories would help.
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Unit Enduring Understanding:
Students will know how all geometric figures relate to circles and real life.
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Unit Question:
How do all geometric figures relate to circles and real life?
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Stage 2 - Assessment Evidence
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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.
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Key Criteria: (Rubrics, etc.)
Holistic Rubric:
Holistic Rubric for (Tests)
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4
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3
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2
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1
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Student understands the concept with no missed steps and/or confusing rules.
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Student understands the concept, but missed a step and/or is confusing learned rules.
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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.
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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
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4
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3
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2
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1
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Mathematical computation
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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.
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Few errors in mathematical computation.
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At least 10 errors in mathematical computation.
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More than 10 errors in mathematical computation.
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Mathematical usage of tools
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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.
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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.
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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.
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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.
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Mathematical usage of geometry on blueprint and in paper.
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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.
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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.
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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.
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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.
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Information Gathering
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Accurate information taken from several sources in a systematic manner.
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Accurate information taken from a couple of sources in a systematic manner.
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Accurate information taken from a couple of sources but not systematically.
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Information taken from only one source and/or information not accurate.
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Computations:
Teacher Name:
Student Name: ________________________________________
CATEGORY
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4
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3
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2
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1
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Mathematical Concepts
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Explanation shows complete understanding of the mathematical concepts used to solve the problem(s).
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Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s).
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Explanation shows some understanding of the mathematical concepts needed to solve the problem(s).
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Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written.
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Mathematical Reasoning
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Uses complex and refined mathematical reasoning.
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Uses effective mathematical reasoning.
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Some evidence of mathematical reasoning.
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Little evidence of mathematical reasoning.
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Mathematical Errors
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90-100% of the steps and solutions have no mathematical errors.
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Almost all (85-89%) of the steps and solutions have no mathematical errors.
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Most (75-84%) of the steps and solutions have no mathematical errors.
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More than 75% of the steps and solutions have mathematical errors.
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Mathematical Terminology and Notation
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Correct terminology and notation are always used, making it easy to understand what was done.
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Correct terminology and notation are usually used, making it fairly easy to understand what was done.
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Correct terminology and notation are used, but it is sometimes not easy to understand what was done.
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There is little use, or a lot of inappropriate use, of terminology and notation.
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Strategy/
Procedures
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Typically uses an efficient and effective strategy to solve the problem(s).
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Typically uses an effective strategy to solve the problem(s).
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Sometimes uses an effective strategy to solve problems, but does not do it consistently.
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Rarely uses an effective strategy to solve problems.
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Research:
Teacher Name:
Student Name: ________________________________________
CATEGORY
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4
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3
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2
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1
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Research Packet
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Research packet is complete with all parts filled in.
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Research packet is missing a few parts but have been completed since.
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Research packet is vague and missing parts.
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Research is inaccurate or unfinished.
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Research in packet
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Research is accurate, clear, and follows format of packet.
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Research is mostly complete, but has a few inaccuracies.
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Student used unreliable websites to find information or research is inaccurate.
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Student’s research is incomplete and unreliable either by using improper websites or making things up.
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Oral Presentation Rubric : GRASPS Rubric:
Architecture Project
Teacher Name:
Student Name: ________________________________________
CATEGORY
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4
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3
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2
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1
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Vocabulary
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Uses vocabulary appropriate for the audience. Extends audience vocabulary by defining words that might be new to most of the audience.
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Uses vocabulary appropriate for the audience. Includes 1-2 words that might be new to most of the audience, but does not define them.
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Uses vocabulary appropriate for the audience. Does not include any vocabulary that might be new to the audience.
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Uses several (5 or more) words or phrases that are not understood by the audience.
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Time-Limit
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Presentation is 4-5 minutes long.
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Presentation is 3 minutes long.
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Presentation is 2 minutes long.
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Presentation is less than 2 minutes OR more than 5 minutes.
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Content
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Shows a full understanding of the topic, both geometric and economic.
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Shows a good understanding of the topic, both geometric and economic.
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Shows a good understanding of parts of the topic, both geometric and economic.
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Does not seem to understand the topic very well.
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Evaluates Peers
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Fills out peer evaluation completely and always gives scores based on the presentation rather than other factors (e.g., person is a close friend).
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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).
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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).
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Fills out most of the peer evaluation but scoring appears to be biased.
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Other Evidence:
Journals, rough drafts, vocabulary activity, quizzes, concept maps, observations, daily assignments, textbook, worksheets, prompts on backing up mathematical thought.
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BEFORE
Pre-quizzes on vocabulary
The teacher will test the students on vocabulary to see what they know prior to starting the unit.
Brainstorming
The students will brainstorm ideas for their building before beginning the GRASPS project. This will promote critical thinking and problem solving.
Think/pair/ share
Students will think, pair, and share about problems. The teacher will give them feedback on what they will be learning.
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DURING
Vocabulary log
Students will keep a running vocabulary log for personal reference.
Questioning
The teacher will ask questions as the students work and learn.
Daily assignments
Daily book and worksheet assignments.
Drawings
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.
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AFTER
Posttest GRASPS
GRASPS project on architecture.
Student Review
Students will re-teach parts of the unit as partners to the class to help promote learning.
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Describe the assessment/s and state the prompt if applicable. □ F X S
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What type of scoring tools will be used for evaluation?
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X Analytic rubric □ Checklist
X Holistic rubric □ Answer Key
X Criterion rubric □ Other
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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:
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How do you feel about the unit so far?
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What is the easiest concept for you to understand, what is the most difficult?
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Explain a concept in your own words.
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How does what we learned today relate to real life?
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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:
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What parts of the project did I contribute to?
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What could I have done more on?
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What did I learn from this project?
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How did the circles make creating the building more difficult?
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How did the budget make creating the building more difficult?
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What other subjects did I have to use to complete this project?
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How will this project and unit help me later on in life?
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Did performing in front of an audience cause me to pay attention to the information more closely?
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Did working with a group of peers help me to understand concepts I didn’t understand before?
Group project peer analysis questionnaire:
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What parts did I contribute to, what parts did the others do?
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Who did not do anything?
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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.
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Stage 3 - Learning Plan
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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
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Research packet with filled in questions and essays ( 8 points)
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Written notes showing all work of all mathematical computation (20 points)
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Fully rendered blueprint (16 points)
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Oral presentation (12 points)
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Learning Activities:
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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.
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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.
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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.
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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.
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Essential Vocabulary
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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.
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Sequencing the Learning
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MONDAY
Pre-quiz:
Identify what basic vocabularies are. Identify what the basic geometric shapes are.
Complete basic math skills i.e. multiplication, division, and solving for variables
Level C: daily assignment
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TUESDAY
Identifying Basic geometric shapes and how they relate to each other.
Level C: daily assignment
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WEDNESDAY
Intro to circles
Level C: daily assignment
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THURSDAY
Manipulation of/ similarity of circles
Level C: daily assignment
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FRIDAY
Manipulation of/ similarity of circles
Quiz over week information.
Level C: Packet for the week
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MONDAY
Basic proofing knowledge by brainstorming in pairs
Level C: daily assignment
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TUESDAY
Proofing of similar circles
Think-pair-share activity
Level C: daily assignment
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WEDNESDAY
Level B: activity day
Inquiry based work sheet by drawing constructions
Level C: daily assignment
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THURSDAY
Inscribed circles
Level C: daily assignment
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FRIDAY
Quiz
Level C: Packet for the week
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MONDAY
Pythagorean theorem with inscribed and non triangles
Level C: daily assignment
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TUESDAY
Research on relation to real life day. How to research effectively.
Level C: daily assignment
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WEDNESDAY
Level B: choice assignment
Work day
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THURSDAY
Level B: choice assignment
Work day
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FRIDAY
Level C: Packet for the week
Level B: choice assignment due
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MONDAY
Level A
GRASPS work day
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TUESDAY
Level A
GRASPS work day
Research packet due
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WEDNESDAY
Level A
GRASPS work day
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THURSDAY
Level A
GRASPS work day
Mathematical computation work packet due
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FRIDAY
Level A
GRASPS work day
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MONDAY
Level A
GRASPS work day
Complete peer evaluations
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TUESDAY
Presentations
Blueprints due
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WEDNESDAY
Presentations
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THURSDAY
Geometric relationships test
Unit vocabulary packets due
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FRIDAY
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UBD Unit Design Worksheet / Saginaw Valley State University
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