Discover the equation of the circle through the Pythagorean Theorem
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- Wendy Lawrence
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1 10 minutes minutes Class: Secondar Math Standard: Performance Quizette - Factoring # 01 Homework Q&A Review right triangle relationships - Pthagorean Theorem Send out calculator document and understand objective Page 1. in TI Nspire doc: Use calculate feature to calculate the hpotenuse of the right triangle using pthagorean theorem. Move triangle and eperiment with various side lengths. Discuss the choice of variables used in the calculator. Wh? Page 1.3 in TI Nspire doc: Reciprocal teaching: Predict What shape will it make now that we have triangles? What if the whole screen were filled with multiple What eplicit teaching strategies need to be emphasized? What will I do if the alread know it? (What additional challenges will I assign?) triangles that adhered to this pattern? Page 1.5 in TI Nspire doc: Geometr trace feature to trace path of point p. Discuss: wh is a circle created? What is the center and radius? * Partner sharing (think, 10 minutes Page 1. Definition of a circle. Discuss: meaning of locus. Radius and hpotenuse relationship. Using, to determine if point is on circle. pair, share) Equation of a circle *eploration through 10 minutes Page 1.7 in TI Nspire doc: eplore changes to radius and center and impact on equati technolog Reciprocal Teaching: Summarize findings 10 minutes Practice writing equations of circles and homework Ke Vocabular: Circle- the locus of points equidistant from a given point in a plane. * reciprocal teaching Pthagorean Theorem- locus - a set or collection of points strategies Hpotenuse - the side oppposite the right angle Center, Radius of a circle Radius of a circle How will ou know that the learned the material? Resources/Materials Needed: TI Nspire Concept Quiz 8.1 Homework Reflect on how the lesson was received b the students: Tooele Count Lesson Plan Template G.GPE.1 - Derive the equation of a circle of given center and radius using the Pthagorean Theorem What do I want m students to learn and be able to do? Learning Objective in Student Friendl Language (Post in class for students to see.) Discover the equation of the circle through the Pthagorean Theorem Tier 1 Instruction - Step b Step Procedure Considerations for Special Populations: *Technolog *Manipulatives What will I do if the don't learn it? (Tier & 3 interventions) * Small group instruction *Reciprocal Teaching *Predict transformations of circles and the impact on the equation of a circle. Eperiment with calculators to verif predictions. TI Nspire Calculators with Navitagor software, TI Nspire Circles document with student handout.
2 Eploring the Equation of a Circle Name Student Activit Class Open the TI-Nspire document Eploring_the_Equation_of_a_Circle.tns. In this lesson, ou will be able to visualize the definition of a circle and the relationship between the radius and the hpotenuse of a right triangle. B manipulating the size and location of different circles, ou will see how the equation of a circle is derived. Move to page Move our cursor to cover point p and hold down the select button until the hand appears to grab the point. Now, drag the point to make an right triangle ou want and click again to drop the point. a. As ou drag point P, a triangle moves along with the point. What changes about the triangle? What stas the same? What are the lengths of the legs of the right triangle ou ve selected? b. Use the Pthagorean Theorem to find the length of the hpotenuse. What do ou notice about this number? c. Select (grab), drag and drop the point P to a different spot. What are the lengths of the legs? What is the length of the hpotenuse? How do ou know? d. As ou drag the point P around, what do all those points have in common? Move to page You have right triangles on our page now. What could we prove about these triangles? 3. Select (grab) and drag the point P. What happens? What relationship do all these triangles share?. As ou drag point P, can ou see what shape is formed? Let s take a closer look Move to page 1.. Carefull read these directions: 1. Select Menu > Trace > Geometr Trace. Select point P 3. Then select (grab) and drag point P to observe the path it traces. 015 Teas Instruments Incorporated 1 education.ti.com
3 Eploring the Equation of a Circle Name Student Activit Class Move to page Drag point P. What shape is created? 5. Wh is this shape created?. Where is the center and what is the radius? 7. Using the relationship we found for the triangles on page 1.3, what relationship do all these points P share? Move to page 1.. Read the definition of a circle given on this page. Discuss in our group what the word locus might mean? What parts of a circle have to be defined to draw it? Read page 1.7, then move on to page The equation of the circle and the coordinates of point P are given. a. What is the relationship between the hpotenuse of the right triangle and the radius of the circle? b. What is the relationship between the legs of the right triangle and point P? c. When given an right triangle and the lengths of its legs, what formula is used to find the length of its hpotenuse? Wh is that helpful in this situation? d. Since point P lies on the circle, what must be true about its coordinates? Pick a point and verif. 9. Change the radius of circle O b dragging point Q along the -ais. a. When the radius of the circle changes, what changes in the equation? What stas the same? b. Wh does the constant in the equation change? 015 Teas Instruments Incorporated education.ti.com
4 Eploring the Equation of a Circle Student Activit Name Class 10. Move the center of the circle awa from the origin b dragging point O. a. How are the coordinates of the center of the circle related to the equation? b. Where have we seen this happen in equations before? c. What formula is used to find the length of radius OP? d. Wh is this formula similar to the equation of the circle? 11. Suppose a circle has the equation ( 1) + ( + ) = 5. a. What is the radius of the circle? b. What are the coordinates of the center of the circle? c. How can ou determine whether the point (1, 9) lies on the circle? You re now read for our Concept Quiz 015 Teas Instruments Incorporated 3 education.ti.com
5 Name: Read,'Set,'Go' Circles(and(Other(Conics( 8.1( 5 'Read' Topic:Specialproductsandfactors 013http://flic.kr/p/5mEpJm Factor'the'following'as'the'difference'of''squares'or'as'a'perfect'square'trinomial.''Do'not' factor'if'the'are'neither.' Set''' ' Topic:Writingtheequationsofcircles.' Write'the'equation'of'each'circle'centered'at'the'origin.' "013"MATHEMATICS"VISION"PROJECT" "MV P" In"partnership"with"the"Utah"State"Office"of"Education""" LicensedundertheCreativeCommonsAttributionNonCommercialShareAlike3.0Unportedlicense"
6 Circles(and(Other(Conics( 8.1( Read,'Set,'Go'' Circles(and(Other(Conics( 8.( ' Read' Topic:Makingperfectsquaretrinomials. Fill'in'the'number'that'completes'the'square.''Then'write'the'trinomial'in'factored'form.' www.flickr.com/photos/panomind/ "013"MATHEMATICS"VISION"PROJECT" "MV P" In"partnership"with"the"Utah"State"Office"of"Education""" LicensedundertheCreativeCommonsAttributionNonCommercialShareAlike3.0Unportedlicense"
7 10 Set' ' Circles(and(Other(Conics( 8.( Topic:Writingequationsofcircleswithcenter(h,k)andradiusr.' Write'the'equation'of'the'circle.' Write'the'equation'of'the'circle'with'the'given'center'and'radius.''Then'write'it'in'epanded' form.' 1. Center:(5,),Radius:13 15.Center:(,10),Radius:9 1. Center:(0,8),Radius:15 17.Center:(19,13),Radius:1 18. Center:(1,),Radius:10 19.Center:(3,),Radius:8 M Yp0n1db ikoutttal WSBocfstQwtarreek glal^cg.[ L GAGlYlk Xr_ivgYhStOso srfersmeorbvme]du. Identif the center and radius of each. Then sketch the graph. 1) ( - 3) + ( + 1) = 8 ) + ( + 3) =
8 M Yp0n1db ikoutttal WSBocfstQwtarreek glal^cg.[ L GAGlYlk Xr_ivgYhStOso srfersmeorbvme]du. Identif the center and radius of each. Then sketch the graph. 3) ( - ) + ( - 3) = Use the information provided to write the equation of each circle. ) 7 7) ) e l0w1r JKBuitba psco`fwtlwmairteu RLTLCe.g D kaklylg urzicgihsttsw ureenskearlv[egdu.c O GMjaSdSeg gwhijt_hm sixntffimnniitdep qgeno_m[elteryd. -1- Worksheet b Kuta Software LLC
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