Unittitle: Circlesperiod: 16 Days/ Block 8 Days Textbook:Core HMH (12.1 12.7)

Geometry Revised Summer 2016

UnitTitle: CirclesPeriod: 16 days/ Block 8 days Textbook:Core HMH (12.1 – 12.7)

The UbD Template, Version 2.0 Stage 1—DesiredResults
EstablishedGoals (Standards)
Priority Standards:
MAFS.912.G-C.1.1: Prove that all circles are similar.
MAFS.912.G-C.1.2: 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.
MAFS.912.G-C.1.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
MAFS.912.G-C.2.5: 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.
MAFS.912.G-CO.1.1: 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.
MAFS.912.G-GPE.1.1: 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.
MAFS.912.G-GPE.2.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
Supporting Standards:
LAFS.910.SL.2.4: Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task.
MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them. / UNDERSTANDINGS
Studentswillunderstandthat...

U1 – all circles are similar.
U2 – the central angle of a complete circle measures 360°or 2π radians.
U3 – angles and lines in circles are used in the study of optics, which has applications inscience, medicine, and technology.

/ ESSENTIAL QUESTIONS
Studentswillkeepconsidering...

Q1 – How can the study of circles help us understand connections to science and technology?

KNOW
Studentswillknow...

K1 – the precise definition of a circle, angle, perpendicular lines, parallel lines, and line segments.
K2 – the formulas for circumference, area of a circle, arc length, and area of a sector.
K3 – the properties of central angles, inscribed angles, diameters, radii, chords, tangents, and secants.
K4 – the relationships among inscribed angles, radii, and chords.
K5 – the length of the arc intercepted by an angle is proportional to the radius.
K6 – the equation of a circle.
K7 – know what makes two figures similar.
K8 – know that dilations can be combined with translations to create similar circles.

/ DO
Studentswillbe able to...

D1 – solve problems related to circles using the properties of central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents.
D2 –construct the inscribed and circumscribed circles of a triangle.
D3 – solve problems using the properties of inscribed and circumscribed circles of a triangle.
D4 – use or justify properties of angles of a quadrilateral that is inscribed in a circle.
D5 –solve problems that involve the length of the arc intercepted by an angle and the radius of a circle.
D6 – use similarity to derive the fact that the length of the arc intercepted by an angle is proportional to the radius.
D7 – derive the formula for the area of a sector and use it to solve problems.
D8 – convert angle measures between radians and degrees; define the radian measure as the constant of proportionality.
D9 – determine the center and radius of a circle given its equation in general form.
D10 – use the Pythagorean Theorem, completing the square, the coordinates of a circle’s center, and thecircle’s radius to derive the equation of a circle.
D11 – use coordinate geometry to prove simple geometric theorems algebraically.
D12 – use a sequence of transformations to prove that circles are similar.
D13 – use the measures of different parts of a circle to determine similarity.

Stage2—Evidence
Code / EvaluativeCriteria / *See attachments in CMAP
Are all desired results being appropriately assessed? / What criteria will be used in each assessment to evaluate attainment of the desired results?
Regardless of the format of the assessment, what qualities are the most important? / PERFORMANCE TASK(S):
Studentswillshowthattheyreallyunderstandbyevidenceof...
How will students demonstrate their understanding (meaning-making and transfer) through complex performance?

Journal/Student Log – Student journals from unit will be assessed for completion based on the following criteria: 8 total entries (2-3 each week) with responses to “I learned…; I still don’t understand…; I will remember…; I wonder if…; I was surprised when…; I can explain…”.(See stage 3 learning plan for exact days of journal assignment.)

Problem Solving Task (Collaborative): Two Wheels and a Belt*

Alternative Summative Assessments: *

OTHER EVIDENCE:
StudentswillshowtheyhaveachievedStage1goalsby...
What other evidence will you collect to determine whether Stage 1 goals were achieved?

Quiz – Lesson Quizzes 12-2*(embedded in 12-2PowerPoint)
Quiz – Lesson Quizzes 12-6* (embedded in 12-6 PowerPoint)
Unit Test: Circles*(severalFSA-item types incorporated; NOTE: see Alternative Summative Assessments above)

Stage 3—LearningPlan
Code
(Transfer, Meaning, Acquisition) / *See attachments in CMAP
PRE-ASSESSMENT
What pre-assessments will you use to check student’s prior knowledge, skill levels, and potential misconceptions?
What’s the goal for (or type of) each learning event? / LEARNING EVENTS
Studentsuccessattransfer,meaning,andacquisitiondependsupon...
Are all three types of goals (acquisition, meaning, and transfer) addressed in the learning plan? Does the learning plan reflect principles of learning and best practices? Is there tight alignment with Stages 1 and 2? Is the plan likely to be engaging and effective for all students?
NOTE ON HOMEWORK: Anytime homework has been assigned on the previous day, solutions from the previous day’s assignment will be displayed as students enter so they can check their answers. For the first 5 minutes of class and in groups, students will discuss and share with others how they achieved the correct homework solutions. Groups without questions may research information on that day’s lesson topic (written on board). Teacher will go over any remaining questions.
NOTE ON MATH NATION LESSON(S): Due to copy/printing constraints, teacher will either project appropriate pages from the Math Nation Geometry Section 8 Circles Study Guide*and have students write pertinent info and examples on their own paper, or teacher will print so that 4 pages of material from study guide are on one sheet of paper. Teacher may choose to print only the “Beat the Test”pages (in a reduced format) for exit slip purposes.

Day 1:Student discussion with a shoulder partner will take place for 3 minutes contrasting and comparing a 2-D circle (on paper) to a 3-D sphere (e.g., beach ball). The goal is that students will discover the definition of a circle. Students will take notes and be guided through examples from 12-1 PowerPoint* on lines that intersect circles in order to have precise definitions and related theorems. After slides 3 and 9, teacher will instruct students to turn to their shoulder partner and, for 1 minute each, discuss what they learned and how they could use that information to solve problems. Assign Lesson 12-1 Practice* as homework (formative assessment).

Day 2:See NOTE ON HOMEWORKbefore Day 1.Students will take notes and be guided through examples from 12-2 PowerPoint* on arcs and chords in order to have precise definitions and related theorems. Incorporate a class discussion about how existing knowledge of the terminology will translate to student understanding and discovery of meaning and naming conventions within a circle. Guided examples are embedded in PowerPoint and labeled as “Example 1, etc.”Independent practice is embedded in PowerPoint and labeled as “Check It Out!” with example numbers.Lesson Quiz 12-2 (#1-4) will serve as a checkpoint of today’s lesson and is embedded in last 3 slides of PowerPoint (formative assessment). Assign a journal entry (covering last 2 days) to be written as homework (see Stage 2 Performance Task for instruction on questions to be answered in journal).

/ PROGRESS MONITORING
How will you monitor students’ progress toward acquisition, meaning, and transfer, during lesson events?
What are potential rough spots and student misunderstandings?
How will students get the feedback they need?
Stage 3—LearningPlan
What’s the goal for (or type of) each learning event? / LEARNING EVENTS

Day 3: See NOTE ON MATH NATION LESSON(S) before Day 1. Utilize pgs.5-11 from Math Nation Geometry Section 8 Circles Study Guide*(these pages cover circumference and arc length). Instruction and guided practice will be provided on pgs.5-9. Use thumbs up/down throughout to check for understanding (formative assessment). Prior to each independent practice, teacher will instruct students to turn to their shoulder partner and, for 1 minute each, discuss what they learned and how they could use that information to solve problems. Students will then complete the “Try It!” at the top of pg.8 and on pg.9 as independent practice. Once complete, teacher will display answers for the problems and students will discuss how they achieved correct solutions or what they need to do differently. “Beat the Test” #1 & 2 (pgs.10 & 11) will be completed individually as an exit slip and turned in before the end of class (formative assessment). Assign a journal entry to be written as homework (see Stage 2 Performance Task for instruction on questions to be answered in journal).
Day 4: Students will be placed in groups by teacher of approximately 3-4 students each. Problem solving task “Two Wheels and a Belt”* (higher order question) will be assigned to each group (collaboration). Task downloaded from CPALMS will either need to be edited or students will need to be informed that arc cos is the same as inverse cosine (cosˉ¹). Teacher will use classroom observations while checking on the work within each group and will adjust group and class instruction on task as needed based on those observations. Groups will complete the task and turn it in before the end of class (performance task).

Day 5: Each student will utilizeLesson 12-3 (pgs.503-512) from Core Geometry consumable*, one sheet of plain paper, a compass, and a pair of scissors. Students will complete the exploration, example, and reflectionson pg.503 & 504 in small groups(collaboration)leading to the derivation of the area ofsector of a circle at the top of pg.505. Independent practice will be provided on pg.508 #3-7; teacher will go over solutions after completion by students. Higher Order QuestionFSA Practice Exit Slip* (students will turn in at end of class) is #6 (pg.13) from FSA Practice Test* (formative assessment). Differentiation Strategy – Give students the option of choosing 9 problems as their homework assignment from pgs.511 & 512: students choose any 6 questions from pg.511 #1-9, choose 1 question from pg.512 #1-3, and choose any 2 questions from pg.512 #4-7.

Day 6: See NOTE ON HOMEWORKbefore Day 1. Technology – Lab Day (pre-scheduled) constructing inscribed and circumscribed circles. Each student will need one handout per construction*, a compass, a straightedge, and a pencil. Each student will view the videos* ( and ( guiding them through the constructions so that they can start and stop them at their own pace. After both constructions are complete by all students, students will trade papers with the person next to them and use peer assessment to check each other’s constructions and discuss what they learned and/or correct any mistakes (formative assessment). Assign a journal entry to be written as homework (see Stage 2 Performance Task for instruction on questions to be answered in journal).
Day 7: See NOTE ON MATH NATION LESSON(S)before Day 1. Utilize pgs.43-52 from Math Nation Geometry Section 8 Circles Study Guide* (these pages cover inscribed angles). Instruction and guided practice will be provided throughout the study guide, including the “Let’s Practice!” sections. Use thumbs up/down to check for understanding (formative assessment). Prior to each independent practice, teacher will instruct students to turn to their shoulder partner and, for 1 minute each, discuss what they learned and how they could use that information to solve problems. Students will then complete the “Try It!” on pgs.46 & 48 as independent practice. Once complete, teacher will display answers for the problems and students will discuss how they achieved correct solutions or what they need to do differently. “Beat the Test” #1-3 (pgs.50-52) will be completed as an exit slip and turned in before the end of class (formative assessment). Assign a journal entry to be written as homework (see Stage 2 Performance Task for instruction on questions to be answered in journal).

Day 8: See NOTE ON MATH NATION LESSON(S) before Day 1. Utilize pgs.53-61 from Math Nation Geometry Section 8 Circles Study Guide* (these pages cover inscribed polygons). Instruction and guided practice will be provided throughout the study guide, including the “Let’s Practice!” sections. Use thumbs up/down to check for understanding (formative assessment). Prior to each independent practice, teacher will instruct students to turn to their shoulder partner and, for 1 minute each, discuss what they learned and how they could use that information to solve problems. Students will then complete the “Try It!” on pgs.55 & 59 as independent practice. Once complete, teacher will display answers for the problems and students will discuss how they achieved correct solutions or what they need to do differently. “Beat the Test” #1 & 2 (pgs.60 & 61) will be completed as an exit slip and turned in before the end of class (formative assessment). Differentiation Strategy – Give students the option of choosing 9 problems as their homework assignment from Lesson 12-4 Practice* (taken from pgs.824-826 in the HMH hardback textbook): students choose any 3 questions from #12-16, choose 2 questions from #17-20, choose 1 question from #21 & 22, choose 1 question from #26 & 27, and choose 2 question from #33-35).Introduce Alternative Summative Assignment* options.
Day 9:See NOTE ON HOMEWORK before Day 1. Each student will utilizeLesson 12-5 (pgs.519-522) from Core Geometry consumable*, a compass, and a straightedge. Students will complete the exploration, proof, and reflections on pg.519 & 520 in small groups(collaboration). Teacher will instruct students to turn to their shoulder partner and, for 1 minute each, discuss what they learned and how they could use that information to solve problems. Students will write Theorem 12-5-3 from the HMH Geometry hardback textbook pg.832 on their consumable pages for lesson 12-5 and teacher will guide students through the practice examples at the bottom of pg.520. Independent practice/classwork assignment will be assigned from pgs.521 (1-13) & 522 (1-5). Teacher will walk around providing one-on-one help. This assignment can be completed for homework as time warrants (formative assessment).
Day 10:See NOTE ON HOMEWORK before Day 1. Students will write notes and be guided through examples from 12-6 PowerPoint* on segment relationships in circles in order to have precise definitions and related theorems. Guided examples are embedded in PowerPoint and labeled as “Example 1, etc.” Prior to each independent practice, teacher will instruct students to turn to their shoulder partner and, for 1 minute each, discuss what they learned and how they could use that information to solve problems. Independent practice is embedded in PowerPoint and labeled as “Check It Out!” with examples numbers. Lesson Quiz 12-6 (#1-4) will serve as a checkpoint of today’s lesson and is embedded in last 2 slides of PowerPoint. Assign a journal entry (covering last 2 days) to be written as homework (see Stage 2 Performance Task for instruction on questions to be answered in journal).
Day 11: See NOTE ON MATH NATION LESSON(S) before Day 1. Utilize pgs.20-25 from Math Nation Geometry Section 8 Circles Study Guide* (these pages cover circles in the coordinate plane-part 1). Instruction and guided practice will be provided throughout the study guide, including the “Let’s Practice!” sections. Use thumbs up/down to check for understanding (formative assessment). Prior to each independent practice, teacher will instruct students to turn to their shoulder partner and, for 1 minute each, discuss what they learned and how they could use that information to solve problems. Students will then complete the “Try It!” on pgs.24 & 25 as independent practice. Once complete, teacher will display answers for the problems and students will discuss how they achieved correct solutions or what they need to do differently. Assign a journal entry to be written as homework (see Stage 2 Performance Task for instruction on questions to be answered in journal).

Day 12: See NOTE ON MATH NATION LESSON(S) before Day 1. Utilize pgs.26-29 from Math Nation Geometry Section 8 Circles Study Guide* (these pages cover circles in the coordinate plane-part 2). Teacher will need to remind students how to complete the square and go through a couple of examples of their choosing. Instruction and guided practice will be provided throughout the study guide, including the “Let’s Practice!” sections. Use thumbs up/down to check for understanding (formative assessment). Prior to each independent practice, teacher will instruct students to turn to their shoulder partner and, for 1 minute each, discuss what they learned and how they could use that information to solve problems. Students will then complete the “Try It!” at the bottom of page 27 as independent practice. Once complete, teacher will display answers for the problems and students will discuss how they achieved correct solutions or what they need to do differently. “Beat the Test” #1 & 2 (pgs.28 & 29) will be completed as an exit slip and turned in before the end of class (formative assessment). Assign a journal entry to be written as homework (see Stage 2 Performance Task for instruction on questions to be answered in journal).
Day 13: See NOTE ON MATH NATION LESSON(S) before Day 1. Utilize pgs.30-36 from Math Nation Geometry Section 8 Circles Study Guide* (these pages cover circle transformations). Students will complete the study guide in small groups(collaboration) while the teacher walks around checking on progress and answering any questions. Once complete, teacher will display answers for the problems and students will discuss in their groups how they achieved correct solutions or what they need to do differently. “Beat the Test” #1 & 2 (pgs.3536) will be completed as an exit slip and turned in before the end of class (formative assessment). Assign a journal entry to be written as homework (see Stage 2 Performance Task for instruction on questions to be answered in journal).
Day 14: Students will turn in their completed journals (see Performance Tasks in stage 2 evidence).See NOTE ON MATH NATION LESSON(S) before Day 1. Utilize pgs.37-42 from Math Nation Geometry Section 8 Circles Study Guide* (these pages cover radians and degree). Students will complete the study guide in small groups(collaboration) while the teacher walks around checking on progress and answering any questions. Once complete, teacher will display answers for the problems and students will discuss in their groups how they achieved correct solutions or what they need to do differently. “Beat the Test” #1 (pg.42) will be completed as an exit slip and turned in before the end of class (formative assessment).
Day 15: Summative Unit Review
Day 16: Other Evidence – Administer Summative Unit 10 Circles Test* and collect Alternative Summative Assessments.

/ PROGRESS MONITORING