The study of conic sections provides

Transcription

1 Planning the Unit Unit The stud of conic sections provides students with the opportunit to make man connections between algebra and geometr. Students are engaged in creating conic sections based on their definitions. The learn how to identif and appl conics in real-world settings. The make connections between the graphs of a conic, the standard-form equation, and the general second-degree equation of the conic section. Academic Vocabular Blackline masters for use in developing students vocabular skills are located at the back of this Teacher s Edition. Encourage students to eplore the meanings of the academic vocabular words in this unit, using graphic organizers and class discussions to help students understand the ke concepts related to the terms. Encourage students to place their vocabular organizers in their Math notebooks and to revisit these pages to make notes as their understanding of concepts increases. Embedded Assessments The Embedded Assessment for this unit follows Activit.. AP/College Readiness Unit prepares students to use conic sections as models in a variet of problems encountered AP Calculus and promotes epertise in a variet of process skills students need to be successful in AP courses b: Making the connections between algebraic and graphical representations of relations and functions eplicit to students. Modeling a written or graphical description of a phsical situation using implicitl defined relations and models. Using technolog to eplore relationships, make conjectures, and support conclusions. Emphasizing mathematical models in the coordinate plane to prepare students for differential and integral calculus applications of conic sections. Allowing students to communicate their mathematical knowledge verball and in writing. Embedded Assessment Working with Us Identifing equations as the equation of a particular conic Graphing conic sections Writing the equations of conic sections. 0 College Board. All rights reserved. Unit Conic Sections a

2 Planning the Unit Continued Suggested Pacing The following table provides suggestions for pacing either a -minute period or a block schedule class of 0 minutes. Space is left for ou to write our own pacing guidelines based on our eperiences in using the materials. -Minute Period 0-Minute Period Comments on Pacing Unit Overview Activit. Activit. Activit. Activit. Activit. Embedded Assessment Total Unit Practice Practice Problems appear at the end of the unit. Math Standards Review To help accustom students to the formats and tpes of questions the ma encounter on high stakes tests, additional problems are provided at the end of the unit. These problems are constructed for multiple choice, short response, etended response, and gridded responses. 0 College Board. All rights reserved. b SpringBoard Mathematics with Meaning Algebra

3 Conic Sections Unit UNIT OVERVIEW Unit Overview Ask students to read the unit overview and mark the tet to identif ke phrases that indicate what the will learn in this unit. Essential Questions Read the essential questions with students. Encourage them to investigate real-world applications of the conic sections as the stud each tpe of conic. Unit Overview In this unit ou will investigate the curves formed when a plane intersects a cone. You will graph these curves known as the conic sections and ou will identif conic sections b their equations.? Essential Questions How are the algebraic representations of the conic sections similar and how are the different? Academic Vocabular Read through the vocabular list with students. Assess prior knowledge b asking students if the can define an of the terms. Encourage students to eplore these words in depth using a graphic organizer and to add the words to their math notebooks. Embedded Assessment There is one embedded assessment in this unit, with an evaluation rubric. You ma want to review skills needed for the assessment with students prior to the beginning of their work. 0 College Board. All rights reserved. 0 College Board. All rights reserved. Academic Vocabular Add these words and others ou encounter in this unit to our vocabular notebook. conic section ellipse hperbola quadratic relation standard form? How do the conic sections model real world phenomena? EMBEDDED ASSESSMENTS This unit has one embedded assessment, following Activit.. It will give ou the opportunit to demonstrate our abilit to recognize and graph circles, ellipses, parabolas and hperbolas. Embedded Assessment Conic Sections p. 0 Unit Conic Sections

4 UNIT GETTING READY You ma wish to assign some or all of these eercises to gauge students readiness for Unit topics. UNIT Getting Read Write our answers on notebook paper. Show our work.. Model the process for completing the square using the equation + - = 0. Prerequisite Skills Graphing (Items,, ) Writing the equation of lines (Item ) Distance formula (Item ) Simplifing radicals (Item ) Simplifing polnomials (Item ) Completing the square (Item ) Answer Ke a. line with slope of (- ) passing through (, 0) and (, ) b. vertical line, units to the right of the -ais c. horizontal line, units above the -ais. Answers ma var. Sample answer: Pick several points on the graph and substitute their values into each equation to see which ones satisf the equation. Or, pick two or three values of, solve for and see if the ordered pairs are points on the graph.. Describe the geometric representation of each equation. a. + = 0 b. = c. ( - ) = + -. Given the four quadratic equations below, eplain how ou could determine which equation is represented b the graph displaed. = + = - = - - = -. Write the equations of the diagonals of a rectangle that has vertices (-, ), (, ), (-, ), and (, ).. Find the distance between (-, ) and (, ).. Simplif ( - ).. Identif pairs of points that are smmetric about the line of smmetr in the parabola below.. Simplif. a. 00 b. 0 College Board. All rights reserved = 0 + = + + = + ( + ) = 0. =± SpringBoard Mathematics with Meaning TM Algebra. Sample answers: (-, -) and (-, -); (-, ) and (-, ) a. b. 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning Algebra

5 0 College Board. All rights reserved. The Conic Sections It s How You Slice It SUGGESTED LEARNING STRATEGIES: Marking the Tet, Visualization, Use Manipulatives, Summarize/Paraphrase/ Retell, Discussion Group, Group Presentation In the rd centur BCE the Greek mathematician Apollonius wrote an eight volume tet, Conic Sections, detailing curves formed b the intersection of a plane and a double cone. Nearl two millennia later Johannes Kepler used one of these intersections to model the path planets follow when orbiting the sun. René Descartes also studied the work of Apollonius, discovering that the coordinate sstem he created, the Cartesian Plane, could be applied to the conic sections and each could be represented b a quadratic relation. Follow the instructions for the figures our teacher has assigned. Figure One Materials: Piece of plain paper Inde card Scissors Instructions:. In the center of a plain piece of paper, place a point and label it C.. Using one corner of an inde card as a right angle cut the inde card to form a right triangle.. Label the verte of the right angle of the triangle Q and the vertices of the acute angles P and P.. Place P on C and mark the point on the paper where P falls.. Repeat step four 0 times keeping P on C and moving P to different locations on the paper.. Join the points formed b P with a smooth curve to form a closed geometric figure.. Using the definitions of the conic sections in the M Notes section, identif the figure ou created, sketch the figure in the space above these instructions, and, near the figure, write its name. The figure is a circle.. a. How would the resulting figure change if P were placed on C and the mark was made where P falls? Answers ma var. Sample answer: You would create a circle with the same radius as the original circle. b. Eplain how the work ou did to create our figure models the definition of the curve ou created. Answers ma var. Sample answer: All the points drawn were the distance P P from the center, C. The segment P P has a constant length. A circle is the set of points equall distant from a fied point. M Notes ACADEMIC VOCABULARY conic sections ACTIVITY. MATH TERMS A circle is the set of all points in a plane that are equidistant from a fied point. ACADEMIC VOCABULARY An ellipse is the set of all points in a plane such that the sum of the distances from each point to two fied points is a constant. ACADEMIC VOCABULARY A hperbola is the set of all points in a plane such that the absolute value of the differences from each point to two fied points is constant. MATH TERMS A parabola is the set of points in a plane that are equidistant from a fied point and a fied line. ACTIVITY. Investigative The Conic Sections Activit Focus Creating conic sections Relating models to the definitions of the conic sections Materials Plain paper Inde cards Scissors String Tape or tacks Patt paper or waed paper Compass Straightedge Chunking the Activit # # TEACHER TO Have students create, TEACHER individuall or in groups, one or more of the conic sections described in Figures One Four. There are several was to assign students to this work. () You ma form groups of four students and assign each student in the group a different figure to create. () You ma have each student create two of the figures. () You ma group students b figure and have them discuss the instructions among themselves as the create the figure. 0 College Board. All rights reserved. Unit Conic Sections Marking the Tet, Visualize, Use Manipulatives, Summarize/ Paraphrase/Retell, Discussion Group, Group Presentation, Debriefing Figures One Four each generate a different conic. The circle and the ellipse are the easiest for students to create. Unit Conic Sections

6 ACTIVITY. Continued ACTIVITY. The Conic Sections It s How You Slice It () Debrief the activit b having students share the figure(s) the created and discuss how their work modeled the definition of the conic section. M Notes SUGGESTED LEARNING STRATEGIES: Marking the Tet, Visualization, Use Manipulatives, Summarize/Paraphrase/ Retell, Discussion Group, Group Presentation Figure Two Materials: Piece of plain paper Piece of string between and inches long Tape or tacks.b. Answers ma var. Sample answer: B putting the ends of the string on the two fied points and pulling it out to a point not on the line, ou are creating two segments of string. As the pencil moves, one segment of string becomes longer as the other becomes shorter. The sum of the segments is alwas the length of the string and is therefore is a constant.. Answers ma var. Sample answer: When a point on the bottom of the paper is placed on the point F there is a point, P, on the fold line that is the same distance from F as it is from the point on the bottom of the piece of paper. When all the points, P, are connected the form a parabola. 0 SpringBoard Mathematics with Meaning TM Algebra Instructions:. Draw a line on the paper.. Place two points on the line and label them F and F.. Using tape or tacks secure one end of the string to F and the other end of the string to F.. Use a pencil to pull the string tight.. With the tip of the pencil on the paper and keeping the string tight, move the pencil until a closed geometric figure is formed.. Using the definitions of the conic sections in the M Notes section on page, identif the figure ou created, sketch the figure in the space above these instructions, and, near the figure, write its name. The figure is an ellipse.. a. What would happen if F and F were closer to each other? Answers ma var. Sample answer: As F and F get closer to each other the ellipse becomes more circular. b. Eplain how the work ou did to create our figure models the definition of the curve ou created. Figure Three Materials: Piece of plain paper, waed paper or patt paper Instructions:. Label the top of one side of the paper A. Then turn the paper over as ou would turn the page of a book and label the top of the other side of the paper B.. Place a point on side A about a third of the wa down the page and in the middle. Label the point F.. On side B, place points along the bottom edge of the page. The points should be evenl spaced out across the bottom of the page.. Fold the paper so that one point on the bottom falls on point F and crease the paper.. Repeat Step for each point on the bottom of side B.. With a pencil trace the smooth curve formed b these folds.. Using the definitions of the conic sections in the M Notes section on page, identif the figure ou created, sketch the figure in the space above these instructions, and, near the figure, write its name. The figure is a parabola.. Eplain how the work ou did to create our figure models the definition of the curve ou created. 0 College Board. All rights reserved. 0 College Board. All rights reserved. 0 SpringBoard Mathematics with Meaning Algebra

7 The Conic Sections It s How You Slice It ACTIVITY. ACTIVITY. Continued SUGGESTED LEARNING STRATEGIES: Marking the Tet, Visualization, Use Manipulatives, Summarize/Paraphrase/ Retell, Discussion Group, Group Presentation M Notes Figure Four Materials: Piece of plain paper Compass and straight edge 0 College Board. All rights reserved. 0 College Board. All rights reserved. Instructions:. Draw a line, l, across the center of a piece of plain paper.. Place two points on the line and label them F and F.. Fold F onto F to find the midpoint of F F and mark the midpoint C.. Pick a length,, that is less than the length of F F and greater than the length of F C or CF.. Place the point of a compass on F and using the compass, mark a point units from F on F F.. Place the point of a compass on F and using the compass, mark a point units from F on F F.. Label the points identified in steps and V and V.. Pick two numbers, a and b, so that a - b =.. Assign a convenient unit of length for a and b. Set the pencil point and the compass point a units apart. Place the point of a compass on F and draw an arc etending above and below line, l.. Move the point of the compass to F and draw an arc of radius a etending above and below line, l.. Set the pencil point and the compass point b units apart. Place the point of a compass on F and draw an arc of radius b etending above and below line, l.. Move the point of the compass to F and draw an arc of radius b etending above and below line, l.. Place a point where the arcs of radius a intersect the arcs of radius b. You should have points.. Repeat steps through with additional values of a and b.. With a pencil connect the points to form two smooth curves.. Using the definitions of the conic sections in the M Notes section on page, identif the figure ou created, sketch the figure in the space above these instructions, and, near the figure, write its name. The figure is a hperbola.. Eplain the work ou did to create our figure models the definition of the curve ou created. Answers ma var. Sample answer: For each point at the arc intersections, the absolute value of the difference between the distance to F and the distance to F is alwas. Unit Conic Sections Unit Conic Sections

8 ACTIVITY. Continued ACTIVITY. The Conic Sections It s How You Slice It First Paragraph and Visual Displa Vocabular Organizer Be sure students understand the parts of the double cone as the will need these terms to describe various conic sections. M Notes SUGGESTED LEARNING STRATEGIES: Visualization, Think/ Pair/Share The four conic sections ou have created are known as non-degenerate conic sections. A point, a line, and a pair of intersecting line are known as degenerate conics. Ais Think/Pair/Share, Visualization Edge Suggested Assignment There are no Check Your Understanding or Practice problems designated for this activit. For homework after the activit is completed, ou ma want to have students investigate a real-life application of the conic(s) the created. circle ellipse parabola Verte Base The figures to the left illustrate a plane intersecting a double cone. Label each conic section as an ellipse, circle, parabola or hperbola.. Describe the wa in which a plane intersects the cone to form each of the conic sections. Answers ma var. Sample answers: Circle: The plane is perpendicular to the ais of the cone and parallel to the base of the cone. Ellipse: The plane intersects onl one cone. It is not perpendicular to the ais, not parallel to the edge or base, and not parallel to the ais of the cone. Parabola: The plane intersects onl one cone and is parallel to the edge of the cone. Hperbola: The plane intersects both cones, but not at the verte and is perpendicular to the bases.. How would a plane intersect the double cone to form a point? The plane would intersect the double cone at the verte of the cones and at no other point. 0 College Board. All rights reserved. 0 College Board. All rights reserved.. How would a plane intersect the double cone to form a line? Plane would be tangent to the edge of the cones. hperbola SpringBoard Mathematics with Meaning TM Algebra SpringBoard Mathematics with Meaning Algebra

9 Ellipses and Circles Round and Round We Go SUGGESTED LEARNING STRATEGIES: Shared Reading, Interactive Word Wall, Vocabular Organizer, Marking the Tet, Look for a pattern, Guess and Check M Notes ACTIVITY. ACTIVITY. Guided Ellipses and Circles Activit Focus Ellipses Circles Materials No additional materials 0 College Board. All rights reserved. Prior to the th centur, astronomers believed the orbit of the planets around the sun was circular. In the earl th centur, Johannes Kepler discovered that the orbital path was elliptical and the sun was not at the center of the orbit, but at one of the two foci. An ellipse is the set of all points in a plane such that the sum of the distances from each point to two fied points, called foci, is a constant. The center of an ellipse is the midpoint of the segment which has the foci as its endpoints. The major (longer) ais of an ellipse contains the foci and the center and has endpoints on the ellipse, the vertices. The minor ais of the ellipse is the line segment perpendicular to the major ais which passes through the center of the ellipse and has endpoints on the ellipse. Minor Ais Vertices. Match the graphs in the table on the following page with the corresponding equations from the list of equations given below b writing the equation in the appropriatel headed column. + = ( - ) + = 0 + = _ ( + ) ( - ) + = ( + ) ( + ) + _ =. For each equation and graph, find the coordinates of the center point, the length of the major ais and the length of the minor ais to complete the chart. Foci Major Ais Chunking the Activit # # # # # # # # TEACHER TO This activit allows TEACHER students to eplore the standard form of an ellipse. Spend some time discussing the vocabular of the ellipse. First Paragraph Shared Reading Second Paragraph Vocabular Organizer, Marking the Tet, Interactive Word Wall Look for a Pattern, Guess and Check, Debriefing To help students complete the chart on the net page, list these equations on the board. This will allow students to see the equations without having to flip back and forth between pages. 0 College Board. All rights reserved. MINI-LESSON: Algebraic Derivation of the Standard Form Have students start with two points (-c, 0) and (c, 0), and epress the sum of the distances from (, ), a point on an ellipse, to these two points using the distance formula. ( + c) + + ( - c) + = a Rewrite as ( - c) + = a - ( + c) + Square both sides and solve for the square root. a ( + c) + = a + c Square again and simplif. a + a c + a c + a = a + a c + c ( a - c ) + a = a ( a - c ) Let b = a - c b + a = a b This can be written as a + b = Unit Conic Sections MINI-LESSON: You ma want to have students algebraicall derive the standard form of the equation for an ellipse. See the mini-lesson. B having students eperience this derivation, students are given the opportunit of seeing wh this is the algebraic form of the ellipse with center at (0, 0). This derivation also shows students that the length of the major ais equals the constant sum to (, ) on the ellipse from the foci (-c, 0) and (c, 0). The also see the relationship between a, b, and c as a = b + c. ( on net page) Unit Conic Sections

10 ACTIVITY. Continued ACTIVITY. Ellipses and Circles Round and Round We Go For some students, ou ma want to use specific numbers for a, b, and c. For eample, if a = and c =, we have ( + ) + + ( - ) + = Following the steps in the Mini-Lesson this becomes + =. Look for a Pattern, Guess and Check, Debriefing These items are designed to have students use their knowledge of transformations of functions to match the graphs. For students that are still having trouble identifing transformations, it is possible for them to use guess and check to determine which graph matches the functions. Students should be allowed to struggle with and eplore this item with little support. Graph Equation Coordinates of Center of Ellipse Length of Major Ais Length of Minor Ais Graph M Notes ( + ) ( - ) + = SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Guess and Check 0 + = ( - ) (-, ) (0, 0) (, 0) 0 + = 0 College Board. All rights reserved. Equation Coordinates of Center of Ellipse Length of major Ais Length of Minor Ais 0 College Board. All rights reserved. + = ( + ) ( + ) + = (0, 0) (-, -) SpringBoard Mathematics with Meaning TM Algebra SpringBoard Mathematics with Meaning Algebra

11 0 College Board. All rights reserved. Ellipses and Circles Round and Round We Go SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Note taking, Vocabular Organizer, Interactive Word Wall, Think/Pair/Share, Create Representations. How are the denominators of the equations related to the major and minor aes of an ellipse? Answers ma var. Sample answers: The larger number is equal to the square of the length of the major ais. The square root of the smaller number is the length of the minor ais.. How are numerators of the equations related to the center of the ellipse? Answers ma var. Sample answer: The coordinates of the center are the opposite of the number being added to and in the numerators.. How can ou determine the orientation of the major ais from the form of the equation of the ellipse? Answers ma var. Sample answer: If the larger denominator is in the term the major ais is horizontal and parallel to the -ais. If the larger denominator is in the -term the major ais is vertical and parallel to the -ais.. If a > b, the standard form of an ellipse is ( - h) + ( - k) =. a. Where is the center of the ellipse located? (h, k) b. How is the major ais oriented in the coordinate plane? parallel to -ais c. How long is the major ais and what are the coordinates of the endpoints? Length is a. Coordinates of the endpoints are (h ± a, k) d. How long is the minor ais? b. If a > b, the standard form of an ellipse is ( - h) + ( - k) =. a. Where is the center of the ellipse located? (h, k) b. What direction is the major ais? parallel to -ais c. How long is the major ais and what are the coordinates of the endpoints? Length is a. Coordinates of the endpoints are (h, k ± a) d. How long is the minor ais? b. Using what ou found in Items and, find the following ( - ) information for the ellipse ( + ) + =. a. the coordinates of the center (, -) b. the length and coordinates of the endpoints of the major ais Length is. End points are (, -) and (-, -). c. the length and coordinates of the endpoints of the minor ais Length is. Endpoints are (, -) and (, 0). d. In the M Notes section, graph the ellipse and label the center and endpoints of the aes. a b b a M Notes ACTIVITY. ACADEMIC VOCABULARY standard form of the equation of an ellipse (, 0) (, ) (, ) (, ) 0 (, 0) ACTIVITY. Continued Look for a Pattern, Quickwrite, Think/Pair/ Share This item, along with the net two, helps students verbalize the understanding the gleaned from eploring the table. Look for a Pattern, Quickwrite, Think/Pair/Share Students should make some connection to the numerators of the fractions ( - h) + ( - k) = and state a b the center is located at (h, k). Since the have not seen the standard form of an ellipse, their verbalization of this ma not be precise at this point. Items and will solidif the language. Look for a Pattern, Quickwrite, Think/Pair/Share Students should sa something along the lines of If the larger denominator is in the term the major ais is horizontal. If the larger denominator is in the -term the major ais is vertical. Look for a Pattern, Note Taking, Vocabular Organizer, Interactive Word Wall, Group Presentation, Debriefing Students should formalize the standard form and how it relates to the graph of an ellipse. If students need more scaffolding, do the Mini-Lesson. 0 College Board. All rights reserved. MINI-LESSON: Scaffolding for Standard Form of an Ellipse. a + = is an ellipse centered at the origin. If a > b, eplain b how to find the lengths of the major and minor aes and tell the coordinates of the endpoints of the major ais.. ( - h) b + ( - k ) a = is not centered at the origin. If a > b, find the lengths of the major and minor aes and find the coordinates of the endpoints of the major ais. (For Mini-Lesson answers, see net page.) Unit Conic Sections Create Representations This item acts as another check to verif student understanding of the relationship between the equation and the graph of an ellipse. If students need more practice, use these equations. a. b. + ( + ) = ( + ) ( - ) + = Unit Conic Sections

12 ACTIVITY. Continued MINI-LESSON (answers). Since a > b, the length of the major ais is equal to twice the square root of a, or a. The minor ais is equal to twice the square root of b, or b. The endpoints of the major ais are on the -ais at (-a, 0), and (a, 0).. Since a > b, the length of the major ais is a and the length of the minor ais if b. The major ais is parallel to the -ais and has endpoints (h, k ± a). Suggested Assignment CHECK YOUR UNDERSTANDING p. 0, # UNIT PRACTICE p., # ACTIVITY. Ellipse Ellipses and Circles Round and Round We Go M Notes Center Length and orientation of major ais Length and Orientation of Minor Ais Equation of Ellipse SUGGESTED LEARNING STRATEGIES: Create Representations, Work Backward/Think/Part/Share. Complete the table below using the information given. (0, ) (-, -) units horizontal units vertical units vertical units horizontal + ( - ) = ( + ) ( + ) + = Create Representations, Work Backward, Think/ Pair/Share This item is designed to build the students capacit to translate between graphical, analtic, and numeric representations of an ellipse. Students should once again be allowed to struggle with the item. Graph Center (0, 0) (-, ) 0 College Board. All rights reserved. Length and orientation of major ais Length and Orientation of Minor Ais Equation of Ellipse units vertical units horizontal 0 College Board. All rights reserved. units horizontal units vertical + = ( + ) ( - ) + = SpringBoard Mathematics with Meaning TM Algebra SpringBoard Mathematics with Meaning Algebra

13 Ellipses and Circles Round and Round We Go ACTIVITY. ACTIVITY. Continued SUGGESTED LEARNING STRATEGIES: Create Representations, Work Backward, Vocabular Organizer, Interactive Word Wall, Note Taking, Quickwrite. Use the information below. Write the equation and then graph the ellipse described. a. length of vertical major ais: length of minor ais: center: (, ) b. endpoints of major ais: (, ) and (-, ) endpoints of minor ais: (-, 0) and (-, ) ( - ) ( - ) + = ( + ) ( - ) + = M Notes CONNECT TO AP In calculus, ou will have to quickl recognize a particular conic section from its equation and produce its sketch. 0 Create Representations, Work Backward, Discussion Group This item allows for formative assessment on student understanding. Walk around the room and question students to help them see solutions and gain clarit. TEACHER TO Eccentricit ma be TEACHER interpreted as a measure of how much this shape deviates from a circle. Under standard assumptions, eccentricit ( c a) is defined for all circular, elliptical, parabolic and hperbolic orbits and ma take the following values. circular orbits: ( c a) = 0 elliptic orbits: 0 < ( c a) < 0 College Board. All rights reserved. 0 College Board. All rights reserved. The foci of an ellipse are located on the major ais c units from the center. The values a, b, and c are related b the equation c = a - b. The eccentricit of a conic section is a c. The eccentricit of a conic section or an orbit s eccentricit indicates the roundness or flatness of the shape.. Give the coordinates of the foci of each ellipse. a. + = (-, 0), (, 0) ( + ) b. + = (-, ), (-, - ) Unit Conic Sections parabolic trajectories: ( c a) = hperbolic trajectories: ( c a) > Paragraph Vocabular Organizer, Marking the Tet, Note Taking, Interactive Word Wall Connect to AP When students stud calculus, the will have to quickl recognize a particular conic section from its equation and produce a sketch. Once that is done, the can perform a variet of calculations including finding the area of the conic section, the equation of a line tangent to the curve at a point, or the volume of a solid formed b rotating a portion of the curve about a horizontal or vertical line. Unit Conic Sections

14 ACTIVITY. Continued ACTIVITY. Ellipses and Circles Round and Round We Go b Create Representations c Quickwrite, Think/Pair/ Share The eccentricit ( c a) for the ellipse satisfies 0 < ( c a) < and gives a comparative measure of the "flatness" of the ellipse. Students can be asked to discuss what it means about the shape of the ellipse when the eccentricit is close to 0 and when it is close to. d Create Representations, Quickwrite, Think/Pair/Share, Debriefing While a circle is technicall not an ellipse, this item allows students to work with what the know about an ellipse and eccentricit to develop the equation of a circle. TEACHER TO The eccentricit of an TEACHER ellipse, a c, is a number between 0 and. Knowing that ever ellipse must obe the relationship c = a - b can help ou understand eccentricit geometricall. c = a - b c a = a - b a c a = - b a c a = - b a This epresses eccentricit in terms of a and b. If a and b are close in value, an ellipse is close in shape to a circle. Then a c = - b a will be close to zero (since b will be a close to one). If b is relativel small compared to a, an ellipse will have a long, thin shape. Then a c = - b a will be close to one (since b a will be close to zero). For eample, in Item a eccentricit = 0. and in Item b eccentricit = = 0.. M Notes SpringBoard Mathematics with Meaning TM Algebra. Graph each ellipse. Determine the eccentricit. a. SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Think/Pair/Share 0 + = eccentricit 0. b. + = eccentricit = 0.. What does the eccentricit tell ou about the graph of an ellipse? Answers ma var. Sample answer: A narrow ellipse has an eccentricit close to. A wide ellipse has an eccentricit close to Consider the equation for an ellipse in which a = b. a. Give a verbal, visual, and smbolic representation of the conic. Answers ma var. Sample answer: It is a circle; ; ( - h) + ( - k) = a b. What is the eccentricit of the ellipse? 0 c. What does the eccentricit tell ou about the major and minor aes of the ellipse? The are the same length. 0 d. What does the eccentricit tell ou about foci of the ellipse? The are in the center. 0 College Board. All rights reserved. 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning Algebra

15 Ellipses and Circles Round and Round We Go SUGGESTED LEARNING STRATEGIES: Vocabular Organizer, Interactive Word Wall, Marking the Tet, Note taking, Activating Prior Knowledge, Think/Pair/Share, Create Representations A circle is the set of all points in a plane that are equidistant from a fied point the center. The standard form of the equation of a circle is ( - h) + ( - k) = r where the center is (h, k) and the radius is r. ( + ). Write ( - ) + = in the standard form of a circle. Identif the center and radius and then graph the circle.. Graph each circle and label the center and radius. ( + ) + ( - ) = ; center: (-, ); radius: a. ( - ) + ( - ) = b. ( - ) + ( + ) = M Notes ACTIVITY. ACADEMIC VOCABULARY standard form of the equation of a circle ACTIVITY. Continued Paragraph Vocabular Organizer, Interactive Word Wall, Marking the Tet, Note Taking e Activate Prior Knowledge, Create Representations Students are given the opportunit to make a connection between the equation of an ellipse in standard form and that of a circle. f Create Representations, Think/Pair/Share Have students share their graphs on white boards. The can then do an self or peer editing on their paper. g Create Representations, Think/Pair/Share, Debriefing 0 College Board. All rights reserved. 0 College Board. All rights reserved. C (, ) r = C (, ) r =. Write the equation of each circle. a. center (-, ), radius ( + ) + ( - ) = b. center (, ) and passing through (, ) ( - ) + ( - ) = Unit Conic Sections Unit Conic Sections

16 ACTIVITY. Continued Suggested Assignment CHECK YOUR UNDERSTANDING p. 0, # UNIT PRACTICE p., # a. ( - ) + ( + ) = b. ( - ) + = ( + ) c. ( + ) + =. a. and b. See below right.. + =. ( + ) + ( - ) =. (h, k ± a). Answers ma var. Possible answers: Circles are a limit for ellipses. The are like ellipses whose aes are congruent; or, circles are like ellipses whose eccentricit is zero. Answers ma be compared graphicall as well. ACTIVITY. Ellipses and Circles Round and Round We Go M Notes CHECK YOUR UNDERSTANDING Write our answers on notebook paper or grid paper. Show our work.. Write the equation of each graph. a. b. c. c. SUGGESTED LEARNING STRATEGIES: Create Representations C (, ). Graph each equation. Label the center and endpoints of the major and minor aes. a. + = b. ( + ) + ( + ) =. Write the equation of an ellipse that has the endpoints of the major ais at (, 0) and (-, 0) and endpoints of the minor ais at (0, ) and (0, -).. Write the equation of a circle that has center (-, ) and a diameter of length.. If a > b, what are the endpoints of the major ais of the ellipse ( - h) + ( - k) =? b. MATHEMATICAL REFLECTION ( - ) + ( + ) = a How are circles and ellipses related? 0 College Board. All rights reserved. 0 SpringBoard Mathematics with Meaning TM Algebra a. b. 0 0 College Board. All rights reserved. 0 SpringBoard Mathematics with Meaning Algebra

17 Hperbolas What s the Difference? SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Vocabular Organizer, Quickwrite, Close Reading, Graphic Organizer Recall the definitions of ellipse and hperbola: An ellipse is the set of all points in a plane such that the sum of their distances to two fied points is a constant. A hperbola is the set of all points in a plane such that the absolute value of the difference of their distances to two fied points, the foci, is a constant. The ellipse + = 0 and the hperbola - = 0 are graphed on the right. 0 0 M Notes ACTIVITY. ACTIVITY. Guided Hperbolas Activit Focus Hperbolas Materials No additional materials Chunking the Activit # # # # # # E. TT A # 0 College Board. All rights reserved.. Tell the coordinates of the center and the endpoints of the major and minor aes of the ellipse. Center: (0, 0); Endpoints of major ais: (, 0), (, 0); Endpoints of minor ais: (0, ), (0, ). a. Using dashed line segments draw an auiliar rectangle with vertices (, ), (, -), (-, ), and (-, -). Also using dashed lines, draw two diagonal lines that pass through the center and vertices of the rectangle and etend to the edges of the grid. b. What relationships do the rectangle and lines have to the ellipse and hperbola? Answers ma var. Sample answer: The lines go through the center of the ellipse and the branches of the hperbola approach the lines. The ellipse is enclosed within the rectangle. The ellipse and the hperbola have their vertices on the rectangle. c. Wh are dashed lines used when sketching the rectangle and diagonals of the rectangle? Answers ma var. Sample answer: Since the lines are onl used to help draw the relation and are not part of it, the should not be drawn as solid lines. The transverse ais of a hperbola has endpoints on the hperbola. The center of a hperbola is the midpoint of the transverse ais. The foci are on the line containing the transverse ais. The conjugate ais of the hperbola is the line segment perpendicular to the transverse ais passing through the center of the hperbola. The hperbola has asmptotes, lines which the branches of the hperbola approach. The asmptotes contain the center of the hperbola and pass through the vertices of the auiliar rectangle. MATH TERMS transverse ais conjugate ais asmptote focus c Endpoint of conjugate ais b (h, k) focus c First Paragraph Activating Prior Knowledge, Vocabular Organizer, Interactive Word Wall Activating Prior Knowledge This item connects students back to what the have learned about ellipses and connects to hperbolas. Quickwrite Students need to recognize that the tools that are used to help graph relations are not part of the relation itself. Second Paragraph Close Reading, Graphic Organizer, Vocabular Organizer, Interactive Word Wall It ma be helpful to use a graphic organizer to arrange the vocabular. Students should draw a picture of a hperbola, and place terms where the belong. This will help to make connections to the concepts the are pulling from the reading and make the ideas more concrete. 0 College Board. All rights reserved. (h, k) a b verte verte a Endpoint of conjugate ais Unit Conic Sections Unit Conic Sections

18 ACTIVITY. Continued Think/Pair/Share, Self/Peer Revision This table is similar to those students completed in Activit.. It gives students a chance to eplore multiple quadratic relations that represent hperbolas. The can use their understanding of quadratic relations or guess and check to determine the information. Students should once again be allowed to struggle with little assistance from ou. Have groups share answers that ou saw as ou walked around the room and allow time for students to correct an errors the ma have had before continuing to the net two questions. ACTIVITY. Hperbolas What s the Difference? M Notes Graph SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Self/Peer Revision, Guess and Check. Complete the table below. The first row has been done for ou using the hperbola - = as an eample. Equation - = - = Length, Endpoints, and Orientation of Transverse Ais units (, 0), (-, 0) horizontal units (, 0), (-, 0) horizontal Length, Endpoints, and Orientation of Conjugate Ais units (0, ), (0, -) vertical units (0, ), (0, -) vertical Equations of Asmptotes = = ± = ± = units (0, ), (0, -) vertical units (, 0), (-, 0) horizontal = ± 0 College Board. All rights reserved. 0 0 College Board. All rights reserved. - = units (0, ), (0, -) vertical units (, 0), (-, 0) horizontal = ± 0 SpringBoard Mathematics with Meaning TM Algebra SpringBoard Mathematics with Meaning Algebra

19 Hperbolas What s the Difference? ACTIVITY. ACTIVITY. Continued SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Notetaking, Vocabular Organizer, Create Representations. How do the equations of the asmptotes relate to the equation of the hperbola? Answer ma var. Sample answer: The equations of the asmptotes are found b setting the quadratic terms equal to each other and solving for.. How can the direction in which the branches of the hperbola open be determined b the equation? Answer ma var. Sample answer: The branches open towards the ais that is in the positive term of the hperbola. The standard form of a hperbola is ( - h) - ( - k) =, when a b the transverse ais is horizontal. The standard form of a hperbola ( - k ) is _ - ( - h ) = when the transverse ais is vertical. The a b endpoints of the transverse ais are the vertices of the branches, and are located a units from the center of the hperbola that is located at the point (h, k). The equations of the asmptotes are found b setting the quadratic terms equal to each other and solving for. M Notes ACADEMIC VOCABULARY Standard form of the equation of a hperbola Look for a Pattern, Quickwrite Students look for patterns to determine the relationship between the asmptotes and the equation of the hperbola. Having the opportunit to verbalize the process will connect students to a different learning modalit and enhance the learning process. Look for a Pattern, Quickwrite, Debriefing This item allows students to make a conjecture about how the hperbola opens. Having students come up with the concept will help them retain the idea. 0 College Board. All rights reserved. Suggested Assignment CHECK YOUR UNDERSTANDING p., # UNIT PRACTICE p., # Boed Tet Note Taking, Vocabular Organizer EXAMPLE Note Taking, Create Representations Walk students through the eample. Have students verbalize how each of the important tools (central rectangle, asmptotes) for graphing hperbolas is found and used. 0 College Board. All rights reserved. EXAMPLE Sketch the hperbola ( - ) - =. Tell the coordinates of the center and the vertices, and give the equations of the asmptotes The positive term is ( - ), so the transverse ais is horizontal. Since a is, then a = and the transverse ais is units long. The center is (, 0). The vertices on the transverse ais are units from the center: (-, 0) and (, 0). ( - ) Setting = and solving for gives the equations of the asmptotes. = ( - ) = ± ( - ) 0 0 Unit Conic Sections Unit Conic Sections

20 ACTIVITY. Continued EXAMPLE Note Taking, Create Representations Walk students through the eample. Have students compare this eample to the previous one and again discuss the important tools used for graphing. TRY THESE A Create Representations Use these questions as an opportunit to formativel assess students. Verif that the are opening the branches in the correct direction and the are able to find the equations of the asmptotes. Use questioning techniques to bring the students to understanding. Identif a Subtask This item gives students an opportunit to work with a hperbola that is in standard form and centered at the origin with less scaffolding than in the beginning of the activit. ACTIVITY. Hperbolas What s the Difference? M Notes 0 a. b. c. (0, 0) 0 (0, ) (0, ) (, 0) 0 (, ) 0 (, ) SpringBoard Mathematics with Meaning TM Algebra SUGGESTED LEARNING STRATEGIES: Notetaking, Create Representations, Identif a Subtask EXAMPLE Sketch the hperbola ( + ) - ( + ) =. Tell the coordinates of the center and the vertices, and give the equations of the asmptotes. The positive term is ( + ), so the transverse ais is vertical. Since a is, then a = and the transverse ais is units long. The center is (-, -). The vertices on the transverse ais are units from the center: (-, ) and (-, -). ( + ) Setting = ( + ) and solving for gives the equations of the asmptotes. ( + ) = ( + ) ( + ) = ± ( + ) = - ± ( + ) TRY THESE A Write our answers on notebook or grid paper. Show our work. Sketch each hperbola. Tell the coordinates of the center, label the vertices and give the equations of the asmptotes. a. 0 - = b. - = c. - ( + ) = center: (0, 0); center: (0, 0); center: (0, -); equations of equations of the equations of the asmptotes: asmptotes: = ± = ± asmptotes: = - ± d. ( + ) d. - ( - ) = center: (-, ); equations of the asmptotes: = ± ( + ). - a = is a hperbola centered at the origin. Find each item. b a. the direction of the transverse ais horizontal b. the length and endpoints of the transverse ais length: a; endpoints: (a, 0) and (-a, 0) c. the length of the conjugate ais length: b d. the equation of the asmptotes = ± b a (, ) 0 0 (, ) 0 College Board. All rights reserved. 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning Algebra

21 Hperbolas What s the Difference? SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Self/Peer Revision, Create Representations. Complete the table below using the information given. Hperbola 0 0 Center (0, 0) Length and Orientation of Transverse Ais horizontal Length and Orientation of Conjugate Ais vertical M Notes Equation of Hperbola - = ACTIVITY. ACTIVITY. Continued Think/Pair/Share, Self/ Peer Revision, Create Representations This item is designed to build the students capacit to translate between graphical, analtic, and numeric representations of a hperbola. Students should once again be allowed to struggle with the question. Make sure the are aware that the need to label aes. Students should share answers and an student that needs to make corrections should do so before moving on. 0 (-, 0) vertical horizontal - ( + ) = 0 0 College Board. All rights reserved. 0 College Board. All rights reserved. 0 0 (0, 0) units vertical units horizontal - = 0 (-, ) units horizontal units vertical ( + ) ( - ) - = 0 Unit Conic Sections Unit Conic Sections

22 ACTIVITY. Continued Create Representations, Quickwrite This item is similar to Item, but here students are given different tpes of information and have to both write the equation and create the graph. ACTIVITY. Hperbolas What s the Difference? M Notes SUGGESTED LEARNING STRATEGIES: Create Representations. Write the equation and graph the hperbola described. a. center (-, ), transverse ais units, vertical conjugate ais units 0 0 ( + ) - ( - ) = b. asmptotes = ±, vertices (, 0), (-, 0) 0 0 College Board. All rights reserved. 0 College Board. All rights reserved. - = 0 SpringBoard Mathematics with Meaning TM Algebra SpringBoard Mathematics with Meaning Algebra

23 Hperbolas What s the Difference? SUGGESTED LEARNING STRATEGIES: Create Representations The foci of a hperbola are located on the transverse ais c units from the center. The values a, b, and c are related b the equation c = a + b.. Graph each hperbola and label the foci with their coordinates. a. - = M Notes ACTIVITY. ACTIVITY. Continued Create Representations This item allows for formative assessment. Students must complete similar actions to those done in Questions and, but with less scaffolding. Suggested Assignment CHECK YOUR UNDERSTANDING p., # UNIT PRACTICE p., # (0.0, 0) (.0, 0) ( from page ) a. (, ) b. vertical c. = ± ( - ) b. ( + ) - = d. 0 College Board. All rights reserved. (0,.) (, ) (0,. ) (, ) 0 College Board. All rights reserved. Unit Conic Sections. Answers ma var. Sample answer: The branches of the hperbola approach, but never reach the asmptotes. The indicate the limiting edge of the hperbola when it gets graphed. The also contain the center of the hperbola. a. (0, 0) b. vertical, units c. =± d. - = a. (-, 0) b. horizontal, units c. =± ( + ) d. ( + ) - = Unit Conic Sections

24 ACTIVITY. Continued ACTIVITY. Hperbolas What s the Difference? a. (0, 0) b. horizontal c. =± d. (, 0) 0 (, 0) a. (0, 0) b. vertical c. =± d. (0, ) 0 (0, ) CHECK YOUR UNDERSTANDING Write our answers on notebook or grid paper. Show our work. For each hperbola in Questions : a. Give the coordinates of the center. b. Tell the direction of the transverse ais. c. Write the equations of the asmptotes. d. Sketch the hperbola and label the endpoints of the transverse ais.. - =. - = ( + ). - ( + ) = ( - ). - ( + ) = ( - ). 0 - ( - ) = For each hperbola in Questions : a. Give the coordinates of the center. b. Tell the direction and length of the transverse ais. c. Write the equations of the asmptotes. d. Write the equation of the hperbola MATHEMATICAL How do the asmptotes of REFLECTION a hperbola help ou graph the hperbola? 0 College Board. All rights reserved. a. (-, -) b. horizontal c. =- ± ( + ) d. SpringBoard Mathematics with Meaning TM Algebra -_SB_A_-_SE.indd 0 (, ) (, ) a. (, -) b. horizontal c. =- ± ( - ) d. : See page. (, ) (, ) // :0:0 P 0 College Board. All rights reserved. SpringBoard Mathematics with Meaning Algebra

25 Parabolas A Parabola on the Roof SUGGESTED LEARNING STRATEGIES: Shared Reading, Questioning the Tet, Marking the Tet, Vocabular Organizer, Create Representations In previous units ou learned about quadratic functions. The graph of a quadratic function is a parabola, one of the conic sections ou have studied in this unit. In this activit, ou will learn more about geometric properties of parabolas, their applications in real world settings, and how to recognize and graph them. Man people have a parabola on the roof of their homes. The satellite television dishes used to detect television signals are parabolic reflectors. The reason these dishes are shaped like a parabola is due to the following geometric propert of a parabola. When an line parallel to the ais of a parabola hits its surface, the line is reflected through the focus. M Notes ACTIVITY. ACTIVITY. Guided Parabolas Activit Focus Properties of parabolas Graphing parabolas given the equation in standard form Standard form of a parabola Materials Graphing calculator (optional) Chunking the Activit # # # E. TT A # # 0 College Board. All rights reserved. In a satellite dish, the device collects satellite signals over the surface area of the dish. The overall signal is amplified when the individual signals are all reflected to the focus point, where the actual antenna is located at 0. A parabola is the set of points in a plane that are equidistant from a fied point and a fied line. The fied point is called the focus and the fied line is called the directri.. Graph =.. 0. Differentiating Instruction Students ma need a review of sketching parabolas using transformations prior to beginning this activit. The terms focus and directri are introduced on the first page of this unit. Refer students back to the paper folding the did in Activit. and have them label the focus point and the directri. First and Second Paragraphs Shared Reading, Questioning the Tet Third and Fourth Paragraphs Marking the Tet 0. Form the inverse relation b echanging and and use our knowledge of the properties of inverses to graph this relation on the graph in Item. The inverse relation is =. Fifth Paragraph Interactive Word Wall, Vocabular Organizer 0 College Board. All rights reserved. Unit Conic Sections TEACHER TO There are man applications of parabolic surfaces including TEACHER solar collectors and parabolic mirrors. You ma want students to investigate other real-world uses of the parabolic shape. Create Representations Make sure students are using ke points such as (-, ), (-, ), (0, 0), (, ), and (, ) when the make their graphs. Create Representations, Debriefing For groups that struggle with this question, ask them if the recall from a previous unit how to form the inverse of a relation. The can also graphicall form the inverse b echanging the - and -coordinates of the points on the original function. Unit Conic Sections