Subject Geometry Honors Grade 10 Unit # 4 Pacing 8 10 weeks Unit Name Circles, Area and Volume, and Probability Overview Unit 4 explores properties of circles. Students will be familiar with all properties and demonstrate understanding by solving problems and involving various types of angles and lines within circles. Students will transition into 3-D and calculate volume and apply formulas to real world situations. Students will then apply area formulas of all figures to calculate geometric probability. Probability will be explored in depth and will include calculations involving permutations and combinations. Standard # Standard, SC, or AC SLO # Student Learning Objectives Depth of Knowledge G.C.1 Prove that all circles are similar. 1 Investigate that all circles are similar by showing the size of the circle is related to the radius r. (You can change the size of the circle by dilating the radius and a dilation = similarity transformation.) Level 3 2 Students will use proportions to determine if a common scale factor exists (making two circles similar.) or 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. 3 4 Differentiate between central, inscribed, and circumscribed angles. Investigate the properties of an angle inscribed in a semi circle.
5 Investigate the relationship between a line tangent to a circle, and a radius of that circle drawn to the point of tangency. 6 Investigate the relationships between two lines that intersect inside a circle and outside a circle. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 7 8 9 Construct a circle that is inscribed in a triangle. Construct a circle that circumscribes a triangle. Prove that opposite angles of an inscribed quadrilateral are supplementary. G.C.4 (+) Construct tangent and secant lines from a point outside a given circle 10 Construct a tangent line from a point outside the circle. Level 3 G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. 11 12 Define radian measure. Convert from degree to radians and vice versa. Investigate how arc measurement relates to central angles/inscribed angles and apply to solve problems. 13 Derive formula for area of a sector. Use formula to solve problems. G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 14 15 Derive the equation of a circle that has a center at the origin, using Pythagorean Theorem. Derive the equation of a circle that has a center at (h, k) using the Pythagorean Theorem.
16 Given the equation of a circle, find the center and radius. (Will need to complete the square REVIEW.) G.GPE.2 Derive the equation of a parabola given a focus and directrix. 17 18 Define focus and directrix. Find the equation of a parabola, given the focus and directrix. G.GPE.3 (+) G.GPE.4 G.GMD.1 Derive the equation of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. 19 20 Derive the equation of an ellipse given its foci. Derive the equation of a hyperbola given its foci. 21 Prove that a circle with a given point lies on a circle with a given radius. 22 23 Identify circumference as a special kind of perimeter. Understand the ratio of the circumference to the perimeter is pi. Argue that slicing a circle in 8 pieces and lining them up will resemble a parallelogram. Compare the base of the parallelogram to half the circumference of a circle, and the height to the radius. A=bh A=pi*r2 Level 4
24 Compare the Leaning Tower of Pisa to an Upright Tower of Pisa. Explain why the towers would have the same volume using horizontal cross sections. G.GMD.2 (+) Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures. 25 Apply the formula for volume of a sphere and other solid figures using Cavalieri s Principle. G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 26 27 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply formulas to oblique figures. G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 28 29 Investigate the results of rotating various 2D shapes. Investigate the results of creating various 2D cross sections of 3D figures. Level 2 G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). 30 Use geometric shapes, their measures, and their properties to describe objects. (Example: a pair of dice are cubes, scrabble tiles are rectangular prisms, calculate the amount of area left uncovered by a circular rug in rectangular room.) Level 3 G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). 31 32 Define density ratio Extend the ideas of area and volume to real world situations such as population density. Level 3
G.MG.3 S.CP.1 S.CP.2 S.CP.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent Understand the conditional probability of A given B as P(A andb)/p(b), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 33 Create a geometric approximation of a real world object using 3D figures. 34 Apply the knowledge of surface area and volume to solve problems relating to the geometric model created. SC 35 Describe events as unions, intersections, or complements of other events. SC 36 Determine if two events A and B are independent SC 37 38 Understand the conditional probability of A given B Interpret the independence of A given B S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to SC 39 40 Construct and interpret two-way frequency tables. Use two-way frequency tables to decide if events are independent.
S.CP.5 S.CP.6 approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. 41 Use two-way frequency tables to approximate conditional probability. SC 42 Explain concepts of conditional probability as it applies in everyday situations. SC 43 Find the conditional probability of A given B. S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. SC 44 Apply the addition rule interpret the answer in terms of the model S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. SC 45 Apply the general multiplication rule and interpret the answer in terms of the model
S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. SC 46 Use permutations and combinations to compute probabilities of compound events and solve problems. S.MD.6 (+) S.MD.7 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). 47 Use combinations to compute probabilities of compound events. SC 48 Evaluate outcomes of decisions involving probability. SC 49 Analyze outcomes of decisions and strategies involving probability. Mathematical Practice # Selected Opportunities for Connections to Mathematical Practices MP.1 MP.3 MP.4 MP.5 Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Model with mathematics Use appropriate tools strategically.
Big Ideas If you triple the radius of a circle, the resulting circle will be similar and nine times larger. You can derive the equation of a circle using Pythagorean theorem given the center and point on the circle. Geometric probability uses the area of different figures to calculate the probability an event will occur. Cross sections are useful to help visualize the make-up of a three-dimensional figure and help derive formulas for volume. Essential Questions Why does tripling the radius of a circle not triple the area of the resulting circle? How can you derive the equation of a circle using Pythagorean Theorem? How do you determine the geometric probability of a variety of events occurring? Why are various cross sections used to help visualize different three-dimensional figures? Assessments Tests, Quizzes, and homework aligned to the curriculum, including: Quizzes Equations of circles/arc length/sector area, properties within circles, Volume formulas, Volume real world applications, probability/geometric probability. Tests Circles, Volume, Probability Key Vocabulary Volume Geometric Probability Conditional Probability Focus Directrix Suggested Resources (list specific chapters and or page numbers from existing text that correspond to the SLOs and Standards) Chapter 11 Sections: 11-1, 11-2, 11-3, 11-4, 11-5 Prentice Hall Chapter 10 Sections: 10-2 (cross sections), 10-5, 10-6, 10-8 Geometry Textbook Chapter 7 Sections: 7-6, 7-7, 7-8 Additional Resources www.learnzillion.com -- provides video lesson demonstration of many core standards.
Learning Experiences (last area to be completed) Instructional Focus Student Learning Objectives Assessments What can I do to make the work maximally engaging and effective? What content should we cover? What content needs to be uncovered? List SLOs that are addressed via instructional focus All SLOs should be addressed; if listed in unit then they should be taught How will you assess these learning events? What types of assessments will you use to check for understanding? When should the basics come first? When should they be on a need to know basis? When should I teach, when should I coach, and when should I facilitate student discovery? How do I know who and where the learners are? In order to truly meet the standard, what should they be able to do independently (transfer)? What should I be doing to make them more independent and able to transfer? What events will help students practice & get feedback in transfer using the learning in realistic ways? What mathematical practices will my students engage in to make connections to the content?