Foundations of Geometry – Unit 5 – Circles
U9-
BY THE END OF THIS UNIT:
CORE CONTENT
Cluster Title: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.Standard: 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Concepts and Skills to Master
· Accurately communicate parts of a circle using appropriate mathematical language
· Understand why the ratio of circumference to diameter can be expressed as pi, Π
· Identify major and minor arcs and semicircles
· Compute the circumference and area of a circle
· Solve mathematical and real-life problems involving circles
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Formulas for finding the circumference and area of the circle
· The value of pi and how to leave answers to problems in terms of pi (Ex: exact circumference)
· Understand the relationship between the radius and diameter of a circle
Academic Vocabulary
circle, center, diameter, radius, chord, semicircle, major arc, minor arc, central angle, circumference, pi, exact circumference
Suggested Instructional Strategies
· Assess student knowledge of this standard using a pretest reviewing lessons 1-8 and 10-6. Modify core content based on student results.
· Use Lesson 1-8 and/or 10-6 from the Foundations of Geometry or Geometry book as a supplemental tool if needed.
· Have students complete the Circumference Ratio Geometry Activity as an exploration. (See Problem Task)
· Tell students that the circumference of a circle can be thought of as the perimeter of a circle in order to investigate the meaning pi. / Resources
· Textbook Correlation Online Teacher Resource Center: www.pearsonsuccessnet.com
1-8 Perimeter, Circumference, and Area (Use Content Relevant to Circles only)
10-6 Circles and Arcs (Use 10-6 ELL Support and 10-6 Activities, Games, and Puzzles)
· Online Practice: Have students review the parts of the circle by sketching the drawings from this site. Then, read the review questions and answer the exercises for immediate feedback. http://www.mathgoodies.com/lessons/vol2/geometry.html
· Patty Paper Geometry Book by Michael Serra
Finding the Center of a Circle p.105 or p.115 or use this link (p.105 only): http://www.pflugervilleisd.net/curriculum/math/documents/Circles_properties.pdf
Sample Formative Assessment Tasks
Skill-based task
Circumference and Area
Find the circumference and area of each circle.
Leave your answer in terms of π.
Algebra
Find the value of the variable.
Word Problem
A Ferris wheel has a 50-m radius. How many kilometers will a passenger travel during a ride if the wheel makes 10 revolutions? Round your answer to the nearest tenth of a kilometer. / Problem Task
Geometry Activity
Objective: Discover the special relationship that exists between the circumference of a circle and its diameter.
I. Gather Data and Analyze - Collect ten round objects.
A. Measure the circumference and diameter of each object using a millimeter measuring tape. Record the measures in a table like the one below.
B. Compute the value of to the nearest hundredth for each object. Record the result in the fourth column of the table.
(Note to teacher: Each ratio should be near 3.1)
Object / Circumference = C / diameter = d /
1.
2.
3.
…
10.
II. Make a Conjecture
Question: What seems to be the relationship between the circumference and the diameter of the circle?
(Note to teacher: Student answers should be C ≈ 3.14d)
CORE CONTENT
Cluster Title: Experiment with transformations in the plane (Note: Do not allow cluster title to mislead you. G.CO.1 is a part of the Congruence domain and has broad meaning.)Standard: 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. (Note: Arc length is covered in G.C.5 in the Geometry Curriculum Guide.)
Concepts and Skills to Master
· Find the measure of a central angle and the measure of its intercepted arc
· Compute the circumference of a circle in terms of pi to compute the distance of an arc
· Calculate distances along circular paths or part of a circle’s circumference (i.e. arc length - a concept further developed in Geometry)
SUPPORTS FOR TEACHERS
Critical Background KnowledgeFormula for computing the circumference of a circle; Exact Circumference (leave your answer in terms of pi); Congruent circles have congruent radii
Academic Vocabulary
circle, center, diameter, radius, congruent circles, central angle, semicircle, minor arc, major arc, adjacent arcs, intercepted arc, circumference, pi, concentric circles, arc length, congruent arcs, exact circumference
Suggested Instructional Strategies
· Be sure to highlight for students that an arc’s measure in a circle is the same value as the size of its central angle.
· Explain to students that as it relates to standard G.C.5, the length of an arc can be found by multiplying the ratio of the arc’s measure to 360 degrees by the circle’s circumference.
· Optional: Although the proportion below will be taught in Geometry, you may choose to differentiate by using it for struggling students. u u
· Students often confuse arc measure with arc length. Explain to them that an arc’s measure is expressed in degrees but an arc’s length is measured in units. CONTINUED NEXT PAGE…
· IMPORTANT!!! Only calculate distances of circular paths that are simple fractional parts of a circle’s circumference (e.g. ). Allow students to demonstrate further mastery by doing this for circles of various radii but DO NOT move to more complex fractional parts of a circle to develop computing arc length. This concept will be further taught Geometry.
· The concepts and skills to master in this standard of the unit can be introduced to students at a level 1 or a level 2 competency in order to better promote conceptual mastery in Geometry. / Resources
· Textbook Correlation: 10-6 Circles and Arcs (Think About a Plan)
www.pearsonsuccessnet.com
· Online Lesson Plan (aligned with Glencoe Geometry Book-Lesson 10.2):
http://cllenz.wmwikis.net/file/view/Angles+and+Arcs+Lesson+Plan.pdf
Note: Warm-Up Activity #1 can be used as an investigation.
· Visual Aids for Core Content
Math Open Reference – use the link(s) below:
http://www.mathopenref.com/arc.html - An arc of a circle
http://www.mathopenref.com/circlecentral.html - Central angle of a circle
http://www.mathopenref.com/arclength.html - Length of an arc in a circle
Sample Formative Assessment Tasks
Skill-based task
Note: This task is the same as the skill-based task in the Geometry Curriculum Guide Unit 8, Standard G.C.5. If repeated in Geometry, Foundations of Geometry students should show increased ability.
Find the arc measure and arc length of each darkened arc.
Leave your answer in terms of π.
1. 2. 3. / Problem Task
Note: This task is very similar to the problem task in the Geometry Curriculum Guide Unit 8, Standard G.C.5. If repeated in Geometry, Foundations of Geometry students should show increased ability when working the problem.
An analog clock hanging on a classroom wall shows that the time is 3:00 pm in the afternoon. Answer the following:
1. What is the measure of arc formed by the hands of the analog clock hanging on a classroom wall?
2. Is the arc a major or minor arc? How do you know? Sketch a wall clock with the time 3pm to support your answer
3. Lastly, what is the arc length if the radius of the clock is 4 inches?
CORE CONTENT
Cluster Title: Understand and apply theorems about circles.Standard: 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.
Concepts and Skills to Master
· Tangent Lines
· Chord and Arc Measures
· Central and Inscribed Angles
· Angle Measures and Segment Lengths
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Prior knowledge of a circle and its parts, Triangle Angle Sum Theorem, Pythagorean Theorem, Perimeter of Polygons, Congruence
Academic Vocabulary
tangent to a circle, point of tangency, inscribed circles, chord, arc, semicircle, inscribed angles, circumscribed polygons, secant
Suggested Instructional Strategies
· Have students use a circle puzzle as a fun way to test mathematical and conceptual knowledge of core content; i.e. determining angle measurements using central angles, inscribed angles, arcs, angles between tangents and chords and their relationships (see first two resources at right)
· Students sometimes get confused identifying central and inscribed angles and, therefore, use the wrong formula to compute angle measures. Perhaps making a connection that a central angle has its vertex in the center of the circle will help students distinguish between the two.
· Students may benefit from tracing intercepted arcs from central and/or inscribed angles with colored pencils or highlighters.
· Paper folding activities offer students a good way to develop key concepts related to tangents, central angles, chords, and arcs. / Resources
· Textbook Correlation: Lessons 12.1 – 12.4
· Circle Puzzle http://www.fayette.k12.il.us/isbe/mathematics/stageJ/math9BJ.pdf
· Circle Puzzle – Book: Patty Paper Geometry by Michael Serra p.123 #2 and p.124 #3.
· Concept Byte Exploration Activity: p.770 - Paper Folding With Circles
· Group Work: Chord Properties Using Patty Paper
On the CMS secondary wiki resources for Unit 5– you will have access to download this file. (File name: Chord Patty Paper) http://secondarymath.cmswiki.wikispaces.net/Geometry
· More Paper Folding Activities
Patty Paper Geometry Book by Michael Serra
Use pages 107 - 114 or use this link for an online copy of the pages: http://www.pflugervilleisd.net/curriculum/math/documents/Circles_properties.pdf
Sample Formative Assessment Tasks - Note: Tasks are the same in Geometry Curriculum Guide Unit 8, Standard G.C.2. If repeated in Geometry, Foundations of Geometry students should show increased ability when working the problems.
Skill-based task
Refer to C above for Exercises 1–3. Segment is tangent to C.
1. If DE = 4 and CE = 8, what is the radius?
2. If DE = 8 and EF = 4, what is the radius?
3. If mÐC = 42°, what is mÐE? / Problem Task
Reasoning Challenge
Is the statement true or false? If it is true, give a convincing argument. If it is false, give a counterexample.
1. If two angles inscribed in a circle are congruent, then they intercept the same arc.
2. If an inscribed angle is a right angle, then it is inscribed in a semicircle.
3. A circle can always be circumscribed about a quadrilateral whose opposite angles are supplementary.
(See Teacher Edition – Chapter 12 p.786 #35-37 for answers)
CORE CONTENT
Cluster Title: Translate between the geometric description and the equation for a conic sectionStandard: G.GPE.1 Derive the equation of a circle given a center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Concepts and Skills to Master
· Prove the standard equation of the circle using the distance formula
· Write the equation of a circle using a problem or graph that gives a circle’s center and radius and a circle’s center and point on the circle
· Find the center and radius of a circle when given a graph on the coordinate plane or the standard equation of a circle
· Graph a circle on the coordinate plane given its equation
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Distance Formula
· Pythagorean Theorem
· Sketching/plotting graphs on the coordinate plane (x-y axis).
Academic Vocabulary
standard form of an equation of a circle, center of a circle on the coordinate plane (h, k), radius (r)
Suggested Instructional Strategies
· Review the definition of a circle as a set of points whose distance from a fixed point (center) is constant (radius)
· Have students use the distance formula to derive the equation of a circle given a center and a radius. Begin with the case where the center is the origin. (Optional: Teachers may extend this suggestion by first using the Pythagorean theorem to derive the distance formula.)
· Emphasize that writing the equation for a circle in standard form makes it easier to identify the center (h, k). Remind students to use the opposites of h and k from the equation.
· Remind students to take the square root of the value r2 in order to find the radius.
· Investigate practical applications of circles. / Resources
· Textbook Correlation: Lesson 12.5
· How do I find the equation of a circle?
EQUATION FOR A CIRCLE – YouTube video – 6min20secs
http://www.youtube.com/watch?v=HjN9TTRrQiA
(EQUATION OF A CIRCLE – brightstorm video via YouTube – 2min34sec)
http://www.youtube.com/watch?v=Sl0VeTcL-s4
· Equations of Circles Interactive Applet
http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php
· Online Teacher Resource Center
www.pearsonsuccessnet.com - Geometry
Dynamic Activity 12-5: Circles in the Coordinate Plane
12-5 Activities, Games, and Puzzles (classroom game)
Sample Formative Assessment Tasks
Skill-based task
1. Write the standard equation of the circle with center (0, 0) and radius of 3. Also sketch the graph.
2. Write an equation of a circle with diameter if A (3, 0) and B (7, 6).
Find the center and radius of each circle?
3. (x + 4)2 + (y – 1)2 = 16 4. (x – 8)2 + y2 = 9 / Problem-based task
1. Reasoning - Describe the graph of x2 + y2 = r2 when r = 0?
2. Think About a Plan – Find the circumference and area of a
circle whose equation is: (x – 9)2 + (y – 3)2 = 64.
Leave your answer in terms of pi.
· What essential information do you need?
· What formulas will you use?
Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.