Rowan County Senior High School 2010-2011 Page 1

RCHS Rev. 06/2011 Geometry B

Unit 4 Expressing Geometric Properties With Equations Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards

___weeks G.GPE.4 Use Vocabulary: side length, vertex, first quadrant, slope, Note: coordinates to distance, midpoint, parallel, perpendicular, intersecting Same with Unit 5 prove simple geometric theorems Unit Learning targets algebraically. For Students will be able to: example, prove or disprove that a I can recall previous understandings of coordinate figure defined by geometry (including, but not limited to: distance, midpoint four given points in and slope formula, equation of a line, definitions of parallel the coordinate and perpendicular lines, etc.) (K) plane is a rectangle; From Appendix A: This unit has a close connection prove or disprove with the next unit. For example, a curriculum might that the point (1, √3) merge G.GPE.1 and the unit 5 treatment of G.GPE.4 lies on the circle with the standards in this unit. Reasoning with centered at the triangles in this unit is limited to right triangles; origin and containing the point I can represent the vertices of a figure in the coordinate (0, 2). plane using variables. (R) I can connect a property of a figure to the tool needed to verify that property. (R) I can use coordinates and the right tool to prove or disprove a claim about a figure. For example: Use slope to determine if sides are parallel, intersecting, or perpendicular; Use the distance formula to determine if sides are congruent or to decide if a point is inside a circle, outside a circle, or on the circle; Use the midpoint formula or the distance formula to decide if a side has been bisected. (R)

Rowan County Senior High School 2010-2011 Page 1 RCHS Rev. 06/2011 Geometry B

___weeks G.GPE.5 Prove the Vocabulary: slope, parallel, perpendicular, product, line, slope criteria for linear equation, slope-intercept form, point slope form parallel and perpendicular lines Unit Learning targets and use them to Students will be able to: solve geometric problems (e.g., find I can recognize that slopes of parallel lines are equal. (K) the equation of a I can recognize that slopes of perpendicular lines are line parallel or opposite reciprocals (i.e, the slopes of perpendicular lines perpendicular to a have a product of -1). (K) given line that I can find the equation of a line parallel to a given line that passes through a passes through a given point. (K) given point). I can find the equation of a line perpendicular to a given line that passes through a given point. (K) I can prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (R) From Appendix A: Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra 1 involving systems of equations having no solution or infinitely many solutions. Parallel Lines: I can draw a line on a coordinate plane and translate that line to produce its image. (K) I can explain that these lines are parallel since translations preserve angle. (K) I can determine the slope of the original line and its image after translation and show they have the same slope using specific examples and general coordinates (x,y). (R) I can state that parallel lines have the same slope. (K) I can determine if lines are parallel using their slopes. (R)

I can write an equation for a line that is parallel to a given line that passes through a given point. (R) Perpendicular Lines: I can draw a line on a coordinate plane and rotate that line 90° to produce a perpendicular image. (K) I can determine the slope of the original line and its image after a 90° rotation and show they have the opposite reciprocal slopes using specific examples and general coordinates (x,y). (R) I can state that perpendicular lines have the opposite reciprocal slopes. (K) I can determine if lines are perpendicular using their slopes. (R) I can write an equation for a line that is perpendicular to a given line that passes through a given point. (R)

___weeks G.GPE.6 Find the Vocabulary: directed line segment, endpoint, ratio point on a directed line segment Unit Learning targets between two given Students will be able to: points that partitions the I can recall the definition of ratio. (K) segment in a given I can recall previous understandings of coordinate ratio. geometry. (K) I can calculate the point(s) on a directed line segment

whose endpoints are (x1 ,y1) and (x2,y2) that partitions the

line segment into a given ratio, r1 to r2 using the formula

x = r2x1 +r1x2 and y = r2y1 + r1y2

r1 + r2 r1 + r2 (e.g., For the directed line segment whose endpoints are (0,0) and (4,3), the point that partitions the segment into a ratio of 3 to 2 can be found: x = (2 • 0 + 3 • 4) = 12 and y = (2 • 0 + 3 • 3) = 9 (3+2) 5 (3 + 2) 5

So the point is (12/5, 9/5).) (R)

___weeks G.GPE.7 Use Vocabulary: coordinate plane, coordinates, distance coordinates to formula, perimeter, polygon, area, triangle, rectangle compute perimeters of polygons and Unit Learning targets area of triangles and Students will be able to: rectangles, e.g., using the distance Formulate a model of figures in contextual problems to formula.*(*Modeling compute area and/or perimeter. (R) Standard) From Appendix A: G.GPE.7 provides practice with the distance formula and its connection with the Pythagorean theorem.

I can use coordinates of the vertices of a polygon graphed in the coordinate plane and use the distance formula to compute the perimeter. (S) I can use the coordinates of the vertices of triangles and rectangles graphed in the coordinate plane to compute area. (S)

___weeks G.GPE.2 Derive the Vocabulary: parabola, focus, directrix, distance formula, equation of a factor, perfect square trinomial parabola given a focus and directrix. Unit Learning targets

Students will be able to:

I can define a parabola including the relationship of the focus and the equation of the directrix to the parabolic shape. From Appendix A: The directrix should be parallel to a coordinate axis. (K) I can find the distance from a point on the parabola (x,y) to the directrix. (S) I can find the distance from a point on the parabola (x,y) to the focus using the distance formula (Phytagorean Theorem). (S) I can equate the two distance expressions for a parabola to write its equation. (S) I can identify the focus and directrix of a parabola when given its equation. (R)

Unit 5 Circles Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards

___weeks G.C.1 Prove that all Vocabulary: circle, similar figures, rigid motion, dilation, circles are similar. angle measure, preimage, image, central angle

Unit Learning targets Students will be able to:

I can recognize when figures are similar. (Two figures are similar if one is the image of the other under a

transformation from the plane into itself that multiplies all distances by the same positive scale factor, k. That is to say, one figure is a dilation of the other. ) (K) I can compare the ratio of the circumference of a circle to the diameter of the circle. (R) I can discuss, develop and justify this ratio for several circles. (R) Determine that this ratio is constant for all circles. (R)

I can prove that all circles are similar by showing that for a dilation centered at the center of a circle, the preimage and the image have equal central measure. (R)

___weeks G.C.2 Identify and Vocabulary: central angle, inscribed angle, describe circumscribed angle, diameter, radius, chord, tangent, relationships among circle, intersect, endpoints, right angle, perpendicular inscribed angles, radii, and chords. Include the Unit Learning targets relationship Students will be able to: between central, inscribed, and I can identify inscribed angles, radii, chords, central

circumscribed angles, circumscribed angles, diameter, and tangent. (K) angles; inscribed I can examine the relationship between central, inscribed angles on a and circumscribed angles by applying theorems about diameter are right their measures. (R) angles; the radius of a circle is I can describe the relationship between a central angle perpendicular to the and the arc it intercepts. (R) tangent where the I can describe the relationship between an inscribed angle radius intersects the and the arc it intercepts. (R) circle. I can describe the relationship between a circumscribed angle and the arcs it intercepts. (R) I can recognize that an inscribed angle whose sides intersect the endpoints of the diameter of a circle is a right angle. (K) I can recognize that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (K)

___weeks G.C.3 Construct the Vocabulary: inscribed, circumscribed, angle bisector, inscribed and perpendicular bisector, construction, compass, circumscribed straightedge, intersection, incenter, circle, circumcenter, circles of a triangle, quadrilateral, arc, inscribed angle, ARC Addition Postulate, and prove equation, opposite angles, supplementary properties of angles for a quadrilateral inscribed in a circle. Unit Learning targets Students will be able to: I can define the terms inscribed, circumscribed, angle bisector, and perpendicular bisector (K) I can construct the inscribed circle whose center is the

point of intersection of the angle bisectors (the incenter). (P) I can construct the circumscribed circle whose center is the point of intersection of the perpendicular bisectors of each side of the triangle (the circumcenter). (P) I can apply the Arc Addition Postulate to solve for missing arc measures. (S) I can prove that opposite angles in an inscribed quadrilateral are supplementary. (R)

___weeks G.C. 4 (+) Construct Vocabulary: tangent line, circle, perpendicular, radius, a tangent line from a endpoint, midpoint, line segment, point point outside a given circle to the circle. Unit Learning targets Students will be able to:

I can recall vocabulary: tangent, radius, perpendicular bisector, and midpoint (K) I can identify the center of the circle. (K) I can define and identify a tangent line. (K) I can synthesize theorems that apply to circles and tangents, such as: Tangents drawn from a common external point are congruent. A radius is perpendicular to a tangent at the point of tangency. (R)

I can construct the perpendicular bisector of the line segment between the center C to the outside point P. (P) I can construct arcs on circle C from the midpoint Q, having length of CQ. (P) I can construct a tangent line from a point outside the circle to the circle using construction tools or computer software. (P)

Unit 5 Circles Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards Vocabulary: similarity, rigid motion, dilation, angle ___weeks G.C. 5 Derive using measure, length, proportional, arc, constant of similarity the fact proportionality, radian, angle, area, circle, sector, formula, that the length of intercepted arc the arc intercepted by an angle is Unit Learning targets proportional to the Students will be able to: radius, and define I can recall how to find the area and circumference of a the radian measure circle.(K) of the angle as the I can explain that 1° = Π/180 radians (K) constant of I can recall (from G.C.1) that all circles are similar. (K) proportionality; I can determine the constant of proportionality (scale derive the formula factor). (K) for the area of a I can justify the radii of any two circles (r1 and r2) and the sector. arc lengths (s1 and s2) determined by congruent central angles are proportional, such that r1 /s1 = r2/s2 (R) I can verify that the constant of a proportion is the same as the radian measure, Θ, of the given central angle. Conclude s = r Θ (R) From Appendix A: Emphasize the similarity of all circles.

Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. I can define similarity as rigid motions with dilations, which preserves angle measures and make lengths proportional (K) I can use similarity to calculate the length of an arc. (S) I can define the radian measure of an angle as the ratio of an arc length to its radius and calculate a radian measure when given an arc length and its radius. (R) I can convert degrees to radians using the constant of proportionality (2π x angle measure / 360° ). (K) I can calculate the area of a circle.(S) I can define a sector of a circle. (K) I can calculate the area of a sector using the ratio of the intercepted arc measure and 360° multiplied by the area of the circle. (R)

Unit 5 Expressing Geometric Properties with Equations Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards

___weeks G.GPE.1 Derive the Vocabulary: distance formula, Pythagorean Theorem, equation of a circle difference, coordinates, radius, circle, hypotenuse, of given center and equation, center, complete the square, quadratic equation, radius using the conic equation, standard form, general form Pythagorean Theorem; complete Unit Learning targets the square to find Students will be able to: the center and radius of a circle I can define a circle. (K) given by an I can use Pythagorean Theorem. (K) equation. I can complete the square of a quadratic equation. (K) I can derive equation of a circle using the Pythagorean Theorem – given coordinates of the center and length of the radius. (R) I can determine the center and radius by completing the square. (R) From Appendix A: Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. I can identify the center and radius of a circle given its equation. (K) I can draw a right triangle with a horizontal leg, a vertical leg, and the radius of a circle as its hypotenuse. (K) I can use the distance formula (Pythagorean Theorem), the coordinates of a circle’s center, and the circle’s radius to write the equation of the circle. (R) I can convert an equation of a circle in general (quadratic)

form to standard form by completing the square. (S) I can identify the center and radius of a circle given its equation. (R)

Unit 5 Expressing Geometric Properties with Equations Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards Vocabulary: side length, vertex, first quadrant, slope, ___weeks G.GPE.4 Use distance, midpoint, parallel, perpendicular, intersecting coordinates to prove simple Unit Learning targets geometric theorems Students will be able to: algebraically. For example, prove or I can recall previous understandings of coordinate disprove that a figure geometry (including, but not limited to: distance, midpoint defined by four given and slope formula, equation of a line, definitions of parallel points in the and perpendicular lines, etc.) (K) coordinate plane is a I can use coordinates to prove simple geometric theorems rectangle; prove or algebraically. (R) disprove that the For example, prove or disprove that a figure defined by point (1, √3) lies on four given points in the coordinate plane is a rectangle; the circle centered at prove or disprove that the point (1, √3) lies on the circle the origin and centered at the origin and containing the point (0, 2). containing the point From Appendix A: Include simple proofs involving circles. (0,2). I can represent the vertices of a figure in the coordinate plane using variables. (R) I can connect a property of a figure to the tool needed to verify that property. (R) I can use coordinates and the right tool to prove or

disprove a claim about a figure. For example: Use slope to determine if sides are parallel, intersecting, or perpendicular; Use the distance formula to determine if sides are congruent or to decide if a point is inside a circle, outside a circle, or on the circle; Use the midpoint formula or the distance formula to decide if a side has been bisected. (R) Unit 5 Modeling with Geometry Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards

___weeks G.MG.1 Use Vocabulary: circumference, area, perimeter, volume geometric shapes, their measures, and their properties to Unit Learning targets describe objects Students will be able to: (e.g., modeling a tree trunk or a I can use measures and properties of geometric shapes to human torso as a describe real world objects. (K) cylinder).*(*Modelin Given a real world object, I can classify the object as a g Standard) known geometric shape - use this to solve problems in context. (R) From Appendix A: Focus on situations in which the analysis of circles is required.

I can represent real-world objects as geometric figures. (R) I can estimate measure (circumference, area, perimeter, volume) of real-world objects using comparable geometric shapes or three-dimensional figures. (R) I can apply the properties of geometric figures to comparable real-world objects (e.g., The spokes of a wheel of a bicycle are equal lengths because they

represent the radii of a circle.). (R)

Unit 6 Conditional Probability and the Rules of Probability Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards Vocabulary: event, sample space, subset union, ___weeks S.CP.1 Describe intersection, complement events as subsets of a sample space (the Unit Learning targets set of outcomes) Students will be able to: using characteristics (or categories) of the I can define event and sample space. (K) outcomes, or as I can establish events as subsets of a sample space. (S) unions, intersections, I can define union, intersection, and complement. (K) or complements of I can establish events as subsets of a sample space (the other events (“or”, set of outcomes) using characteristics of the outcomes, “and”, “not”). based on the union, intersection, or complement of other events (‘or’, “and”, “not’). (S)

___weeks S. CP 2 Understand Vocabulary: independent events, probability, product, that two events A and event B are independent if the probability of A Unit Learning targets and B occurring Students will be able to: together is the product of their I can define and identify independent events. (K) probabilities, and use I can explain and provide an example to illustrate that for

this characterization two dependent events, the probability of the events to determine if they occurring together is the product of the probability of each are independent. event. (R) Statistics and I can calculate the probability of an event. I can predict if two events are independent, explain my reasoning, and check my statement by calculating P(A and B) and P(A) x P(B). (R) Unit 6 Conditional Probability and the Rules of Probability Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards

___weeks S.CP 3 Understand Vocabulary: probability, dependent events, conditional the conditional probability, independent events probability of A given B as P(A and B)/P(B), and interpret independence of A Unit Learning targets and B as saying that Students will be able to: the conditional probability of A I can define dependent events and conditional probability. given B is the same (K) as the probability of A, and the I can explain that conditional probability is the probability conditional of an event occurring given the occurrence of some other probability of B event and give examples that illustrate conditional given A is the same probability. (R) as the probability of B. I can explain that for two events A and B, the probability of Statistics and event A occurring given the occurrence of event B is Probability is a P(A│B) = P(A and B) and give examples to show how Modeling P(B) to use the formula (R) Conceptual

Category. I can explain that A and B are independent events if the occurrence of A does not impact the probability of B occurring and vice versa (i.e., A and B are independent events if P(B│A) = P(B) and P(A│B) = P(A)). (R)

I can determine if two events are independent and justify my conclusion. (R) Unit 6 Conditional Probability and the Rules of Probability Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards Use the two-way table as a ___weeks S.CP. 4 Construct Vocabulary: two-way frequency table, display, data, sample space to decide if events and interpret two- variable, category, random sample, probability, event, are independent and to way frequency independent events, formula, conditional probability approximate conditional tables of data when probabilities. two categories are From Appendix A: Build on work associated with with two-way tables from each object being Unit Learning targets Algebra 1 Unit 3 (S.ID.5) to classified. Use the Students will be able to: develop understanding of two-way table as a conditional probability and sample space to I can determine when a two-way frequency table is an independence. decide if events are appropriate display for a set of data. (R) independent and to I can collect data from a random sample. (S) Interpret two-way frequency approximate I can construct a two-way frequency table for the data tables of data when two conditional using the appropriate categories for each variable. (S) categories are associated with probabilities. For I can decide if events are independent of each other by each object being classified. example, collect data comparing P(B│A) and P(B) or P(A│B) and P(A). (R) (For example, collect data from from a random sample I can calculate the conditional probability of A given B a random sample of students in of students in your using the formula P(A│B) = P(A and B) . (S) your school on their favorite school on their favorite P(B) subject among math, science, subject among math, I can pose a question for which a two-way frequency is and English. Estimate the science, and English. Estimate the probability appropriate, use statistical technique to sample the probability that a randomly that a randomly population, and design an appropriate product to selected student from your

selected student from summarize the process and report the results. (P) school will favor science given your school will favor that the student is in 10th grade. science given that the Do the same for other subjects student is in 10th grade. and compare the results.) Do the same for other subjects and compare the results.

Unit 6 Conditional Probability and the Rules of Probability Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards

___weeks S.CP.5 Recognize Vocabulary: conditional probability, dependent events, and explain the independence, independent events concepts of conditional probability and independence in everyday language and everyday situations. For example, compare Unit Learning targets the chance of Students will be able to: having lung cancer if you are a smoker I can illustrate the concept of conditional probability using with the chance of everyday examples of dependent events. (R) being a smoker if you have lung I can illustrate the concept of independence using cancer. Statistics everyday examples of independent events. and Probability is a Modeling Conceptual Category.

___weeks S.CP.6 Find the Vocabulary: probability, event, dependent event, conditional conditional probability, intersection, set probability of A given B as the fraction of B’s Unit Learning targets outcomes that also Students will be able to: belong to A and interpret the answer I can find the conditional probability of A given B as the in terms of the fraction of B’s outcomes that also belong to A. model. Statistics and I can interpret the answer in terms of the model. Probability is a I can calculate the probability of the intersection of two Modeling events. (S) Conceptual I can calculate the conditional probability of A given B Category. using the model P(A│B) = P(A and B) . (S) P(B) I can interpret probability based on the context of the given problem. (R)

___weeks S.CP.7 Apply the Vocabulary: probability, event, intersection, union, Additional Rule, P(A Addition Rule or B) = P(A) + P(B) – P(A and B) and Unit Learning targets interpret the answer Students will be able to: in terms of the model. I can apply the Addition rule to determine the probability of Statistics and the union of two events using the formula P(A or B) = P(A) Probability is a + P(B) – P(A and B). (S) Modeling I can interpret the probability of unions and intersections Conceptual based on the context of the given problem. (R) Category.

___weeks S.CP.8 (+) Apply the Vocabulary: probability, event, conditional probability, general General Multiplication Rule, intersection Multiplication Rule in a uniform probability model, P(A and B) = Unit Learning targets P(A)P(B|A) = Students will be able to: P(B)P(A|B), and

interpret the answer I can use the multiplication rule with correct notation. in terms of the I can apply the general Multiplication Rule to calculate the model. probability of the intersection of two events using the Statistics and formula P(A and B) = P(A)P(B|A) = P(B)P(A|B). Probability is a I can Interpret conditional probability based on the context Modeling of the given problem. Conceptual Category. Unit 6 Conditional Probability and the Rules of Probability Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards

___weeks S.CP.9 (+) Use Vocabulary: fundamental counting principle, outcomes, permutations and sample space, factorial, permutation, combination, combinations to compound event, probability compute probabilities of Unit Learning targets compound events Students will be able to: and solve problems. I can apply the fundamental counting principle to find the Statistics and total number possible outcomes in a sample space. (S) Probability is a Modeling I can define factorial, permutation, combination and Conceptual compound event. (K) Category. I can distinguish between situations that require permutations and those that require combinations. (R)

I can apply the permutation formula to determine the number of outcomes in an event nPr =n!/(n-r)! (S)

I can apply the combination formula to determine the number of outcomes in an event.

nCr =n!/(n-r)!r! (S)

I can compute probabilities of compound events. (S) I can solve problems involving permutations and combinatations. (S) I can write and solve original problems involving compound events, permutations, and/or combinations (P) Unit 6 Using Probability to Make Decisions Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards Vocabulary: probability, event, theoretical probability, S.MD.6 (+) Use experimental probability, ___weeks probabilities to make fair decisions (e.g. drawing by lots, using a random number generator.) Unit Learning targets This unit sets the Students will be able to: stage for work in Algebra II, where the I can compute Theoretical and Experimental Probabilities. ideas of statistical (K) inference are introduced. I can use probabilities to make fair decisions (e.g. drawing Evaluating the risks by lots, using a random number generator.) (R) associated with conclusions drawn from sample data (i.e. I can analyze decisions and strategies using probability incomplete concepts (e.g., product testing, medical testing, pulling a information) requires hockey goalie at the end of a game.) an understanding of probability concepts. From Appendix A: This unit sets the stage for work in Algebra II, where the ideas of statistical inference are introduced. Evaluating the risks associated with

conclusions drawn from sample data (i.e. incomplete information) requires an understanding of probability concepts.

Unit 6 Using Probability to Make Decisions Length of KY Common Core Vocabulary/Learning Targets Activities/Assessments/ Unit Academic Resources Standards Vocabulary: probability, event, product testing, medical S.MD.7 (+) Analyze testing ___weeks decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game.) Unit Learning targets Statistics and Students will be able to: Probability is a Modeling I can recall prior understandings of probability. Conceptual Category. I can analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game.)

From Appendix A: This unit sets the stage for work in Algebra II, where the ideas of statistical inference are introduced. Evaluating the risks associated with conclusions drawn from sample data (i.e. incomplete information) requires an understanding of probability

concepts.

Rowan County Senior High School 2010-2011 Page 24