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Mathematics - Geometry District Course Course Description Open to: Grades 10, 11, 12 One Year Course Prerequisite: C or better in Algebra 1, or pass Intermediate Algebra 1, Instructor/Counselor approval Content: Students will use an integrated approach to concepts of algebra, geometry, logic and probability with emphasis on geometry. Adopted Materials Instructional Objectives H.S. G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. NO. Performance Objectives 1 Identify vertex, sides, and interior of an angle. Identify the basic parts of a circle (center, radius, and diameter). Identify parallel vs. perpendicular lines. Compare and contrast lines, rays and segments. 2 3 4 Resource Reference Instructional Objectives Standard Reference None Assessment Correlation Standard Reference H.S. G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.4.3.1 NO. Performance Objectives Assessment Correlation 1 Manipulate a given figure to represent the different transformations (rotation, reflection, translation) Represent a translation as a function in coordinate notation. Compare transformations that preserve distance and angle to those that do not. 2 3 Resource Reference Instructional Objectives H.S. G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Standard Reference G.4.3.1 NO. Performance Objectives 1 Given a figure identify the type(s) of symmetry the figure has. If it has line symmetry sketch the figure and the lines of symmetry. If it has rotational symmetry state the angle of rotation. Resource Reference Instructional Objectives Assessment Correlation Standard Reference H.S. G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.4.3.1 NO. Performance Objectives Assessment Correlation 1 Develop a definition for a reflection using perpendicular lines. Develop a definition for a translation using parallel lines. Develop a definition for a rotation using angles and/or circles. 2 3 Resource Reference Instructional Objectives Standard Reference H.S. G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G.4.3.1 NO. Performance Objectives Assessment Correlation 1 Given a geometric figure, draw the new figure under the given transformation. Given a preimage and an image, specify the sequence of transformations that will map the preimage onto the image. 2 Resource Reference Instructional Objectives Standard Reference H.S. G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.4.3.1 G.4.1.3 NO. Performance Objectives Assessment Correlation 1 Determine if two given geometric figures are congruent in terms of rigid motion. Given two geometric figures transformed by rigid motion, determine if the conditions of congruency have been met. 2 Resource Reference Instructional Objectives H.S. G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs Standard Reference G.4.1.3 of sides and corresponding pairs of angles are congruent. NO. Performance Objectives 1 Given two triangles transformed by rigid motion, determine if the conditions of congruency have been met. Resource Reference Instructional Objectives Assessment Correlation Standard Reference H.S. G-CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 9.4.1.1 Good 10.4.1.1 Good NO. Performance Objectives Assessment Correlation 1 Describe why ASA, SAS, and SSS satisfy the congruency conditions for triangles. Resource Reference Instructional Objectives Standard Reference H.S. G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. NO. Performance Objectives 1 Given two parallel lines and a transversal, identify the special angle pairs and justify your answers. Given two parallel lines and a transversal, solve for missing angle measures. Prove the vertical angle theorem, alternate interior angle theorem. Solve for lengths of segments on a perpendicular bisector Prove the perpendicular bisector theorem. 2 3 4 5 Resource Reference Instructional Objectives measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Performance Objectives 1 Verify/prove the triangle sum theorem, isosceles triangle theorem, triangle midsegment theorem (use a variety of Assessment Correlation Standard Reference H.S. G-CO.10. Prove theorems about triangles. Theorems include: NO. G4.1.3 Strong (No proofs included in Idaho grade level standard) Resource Reference G4.1.3 Strong (Proofs not included in Idaho Grade Standards) Assessment Correlation 2 3 methods). Use angle bisectors and perpendicular bisectors to solve for segment lengths and angle measures. Identify the centroid, incenter, orthocenter, and circumcenter of a triangle, Instructional Objectives Standard Reference H.S. G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. NO. Performance Objectives 1 Given the coordinates of the vertices of a quadrilateral, determine the most precise name of the quadrilateral. Justify your answers algebraically. Compare and contrast properties and attributes of the differing parallelograms. Under given conditions identify if a given quadrilateral is a parallelogram. 2 3 4 Resource Reference Instructional Objectives copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Performance Objectives 1 Create a list of steps needed to construct congruent segments, angles, bisect segments and angles, parallel and perpendicular lines. Use a compass and straightedge to construct congruent segments, angles, bisect segments and angles, parallel and perpendicular lines. Use multiple methods to do the above. 2 3 Assessment Correlation Standard Reference H.S. G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; NO. G4.1.1 Good G4.1.3 Strong (Proofs not included in Idaho Grade Standards) Resource Reference Instructional Objectives G4.4.1 Strong (Not included in Idaho Grade Standards) Assessment Correlation Standard Reference H.S. G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G4.4.1 Strong (Not included in Idaho Grade Standards) NO. Assessment Performance Objectives Resource Reference 1 Correlation Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Instructional Objectives Standard Reference H.S. G-SRT.1a. Verify experimentally the properties of dilations given by a center and a scale factor: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. 9.4.1.2; 10.4.1.2 strong G4.3.1 strong NO. Performance Objectives Assessment Correlation 1 Identify the center of dilation and classify it as a reduction or enlargement based on preimage and image. Identify the center of dilation and classify it as a reduction or enlargement based on preimage and image. Compare and contrast the properties of the preimage and image. 2 3 Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.1b.Verify experimentally the properties of dilations given by a center and a scale factor: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 9.4.1.2; 10.4.1.2 strong G4.3.1 strong NO. Performance Objectives Assessment Correlation 1 Identify the ratio of sides of image to preimage and relate it to the scale factor of dilation. Given a preimage and scale factor, determine the lengths of the sides of the image? 2 Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 9.4.1.1; 10.4.1.1 as both apply to similarity and congruency (weak) 9.4.1.2; 10.4.1.2 as both apply to similarity (strong) G4.1.2 Good NO. Performance Objectives Assessment Correlation 1 Determine if two polygons are similar. If so, write a similarity statement and give the similarity ratio. If not justify your answer. Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 9.4.1.1; 10.4.1.1 as both apply to similarity and congruency (weak) 9.4.1.2; 10.4.1.2 as both apply to similarity (strong) G4.1.2 Strong NO. Performance Objectives Assessment Correlation 1 Given limited information about two triangles (two proportional side lengths, one proportional side and two congruent angles, etc.) determine if the triangles are guaranteed to be similar. What is the least amount of information needed to guarantee two triangles are similar? Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G4.1.3 Strong (Proofs not included in Idaho Grade Standards NO. Performance Objectives Assessment Correlation 1 Prove the sidesplitter theorem and its converse. Prove the Pythagorean theorem. 2 Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 9.4.1.1; 10.4.1.1 good G4.1.2 strong NO. Performance Objectives Assessment Correlation 1 Use congruence and similarity criteria to solve for missing information in triangles. Use proportions to identify missing information in similar triangles. 2 Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 9.2.2.1 good G4.1.4 strong NO. Assessment Correlation Performance Objectives Resource Reference 1 2 Using the special right triangles, identify the ratios of sides within the same triangle and how these relate to a similar triangle. Given side lengths of a right triangle, write the sine, cosine, and tangent ratio. Instructional Objectives Standard Reference H.S. G-SRT.7. Explain and use the relationship between the sine and cosine of complementary angles. G4.1.4 weak Does not use relationship between the two. (Not included in Idaho Grade Standards) NO. Performance Objectives Assessment Correlation 1 Compare and contrast the sine and cosine of the two acute angles in multiple right triangles. Compare and contrast the sine and cosine of two complimentary angles. 2 Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 9.4.1.2; 10.4.1.2 strong G4.1.4 strong NO. Performance Objectives Assessment Correlation 1 Solve right triangles given one side length and an acute angle using trigonometric ratios and the Pythagorean theorem. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 2 Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Not in any current 9,10th grade, or Geometry standard. NO. Performance Objectives Assessment Correlation 1 Pre-calc Resource Reference Instructional Objectives Standard Reference H.S. G-SRT.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. Not in any current 9,10th grade, or Geometry standard NO. Assessment Correlation Performance Objectives Resource Reference 1 Pre-cal Instructional Objectives Standard Reference H.S. G-SRT.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Not in any current 9,10th grade, or Geometry standard NO. Performance Objectives 1 Pre-cal Resource Reference Assessment Correlation Resource Reference Standard Reference None Assessment Correlation Instructional Objectives H.S. G-C.1. Prove that all circles are similar. NO. Performance Objectives 1 Using diagrams or a geometric software package, show that all circles are similar and justify your results. Instructional Objectives Standard Reference H.S. 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. G2.2.2 Strong (Not included in Idaho Grade Standard) NO. Performance Objectives Assessment Correlation 1 Solve for the measures of inscribed and central angles and their corresponding intercepted arcs. Solve for the lengths of segments and measures of angles formed by radii, chords, secants and tangents of circles. 2 Resource Reference Instructional Objectives Standard Reference H.S. G-C.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G4.4.1 strong (No grade level standard) NO. Performance Objectives Assessment Correlation 1 Construct an inscribed and circumscribed circle of a triangle. Prove opposite angles are supplementary in an inscribed quadrilateral. 2 Instructional Objectives Resource Reference Standard Reference H.S. G-C.4.(+) Construct a tangent line from a point outside a given circle G4.4.1 strong to the circle. No grade content standard NO. Performance Objectives 1 Pre-cal Resource Reference Instructional Objectives H.S. 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. NO. Performance Objectives 1 Given circles of differing radii, and specified central angle, determine the relationship between arc length and radius (identify radian measures). Derive the formula for the area of a sector. 2 Resource Reference Instructional Objectives H.S. 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. NO. Performance Objectives 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. 2 Resource Reference Instructional Objectives H.S. G-GPE.2. Derive the equation of a parabola given a focus and directrix. NO. Performance Objectives 1 Given the equation of a parabola, identify the coordinates of the vertex, directrix, and focus and graph the parabola. Write the equation of a parabola given any two of the following: vertex, focus, or directrix. 2 Resource Reference Instructional Objectives Assessment Correlation Standard Reference None Assessment Correlation Standard Reference None Assessment Correlation Standard Reference None Assessment Correlation Standard Reference H.S. G-GPE.3. (+) Derive the equations of ellipses and hyperbolas given Not in any the foci, using the fact that the sum or difference of distances from the foci Geometry is constant. or Grade standard Advanced Course NO. Performance Objectives Resource Assessment Reference 1 Pre-cal Instructional Objectives Standard Reference H.S. G-GPE.4. 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). NO. Performance Objectives 1 Given three points in a coordinate plane, find a fourth point to make the figure a parallelogram and prove your results. Prove or disprove that a given point lies on a circle given the center and a point on the circle. 2 Correlation Resource Reference Instructional Objectives G4.2.1 weak G3.1.1 weak New standard for circles Assessment Correlation Standard Reference H.S. G-GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). 9.4.4.3 Good 10.4.4.3 Good G3.1.1 Strong G4.1.3 Strong NO. Performance Objectives Assessment Correlation 1 Construct parallel or perpendicular lines and calculate the slopes to compare relationships. Given the equations of two lines, determine if they are parallel, perpendicular or neither. Find the equation of a line parallel or perpendicular to a given line that passes through a given point. 2 3 Resource Reference Instructional Objectives Standard Reference H.S. G-GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 9.2.2.1 weak 10.2.2.1 weak Does not correlate to existing ID content standards NO. Performance Objectives Assessment Correlation 1 Given the endpoints of a segment, find a point on the segment that divides it into a given ratio. Given one endpoint of a segment and a point on the segment with a given ratio, find the other endpoint. Find the midpoint of a segment. 2 3 Resource Reference Instructional Objectives Standard Reference H.S. G-GPE.7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G4.2.1 Strong Modeling is new No Grade Level Standard NO. Performance Objectives Assessment Correlation 1 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. Resource Reference Instructional Objectives H.S. G-GMD.1. 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. NO. Performance Objectives 1 Establish the relationship using manipulatives that circumference divided by diameter is equal to pi. Solve for C. Area of a circle argument, more wedges give flatter top and bottom. 2 3 Resource Reference Standard Reference None Assessment Correlation Using a prism and a pyramid with congruent bases and height, determine the relationship between the volumes by filling the pyramid and pouring it into the prism. Instructional Objectives Standard Reference H.S. G-GMD.2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. 10.2.1.1 strong Advanced Course NO. Performance Objectives Assessment Correlation 1 Pre-cal Resource Reference Instructional Objectives Standard Reference H.S. G-GMD.3. Use volume formulas for cylinders, pyramids, cones, and Weak comparison spheres to solve problems. NO. Performance Objectives 1 Identify base, lateral sides, slant height, altitude, and lateral edge of three dimensional figures. Calculate the volume of cylinders, pyramids, cones, and spheres. 2 with 9.2.1.1 due to no surface area of 3-D figures. 10.2.1.1 strong Resource Reference Instructional Objectives Assessment Correlation Standard Reference H.S. 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. 10.4.1.2 weak NO. Performance Objectives Assessment Correlation 1 Identify the shapes of two-dimensional cross-sections of three-dimensional objects. Visualize the relationship between twodimensional and three-dimensional objects. 2 Resource Reference Instructional Objectives H.S. 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). NO. Performance Objectives 1 Use the properties and measures of geometric shapes to approximate real world objects. Resource Reference Instructional Objectives H.S. 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). NO. Performance Objectives 1 Given the dimensions of a water bed and the weight of the water it took to fill it, what is the weight of a cubic foot of water? Resource Reference Instructional Objectives H.S. G-MG.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) NO. Performance Objectives 1 Given one geometric solid, design a different Resource Reference Standard Reference New Standards Assessment Correlation Standard Reference New Standard Assessment Correlation Standard Reference New Standard Assessment Correlation geometric solid that will hold the same amount of substance (i.e. cone to prism). Instructional Objectives Standard Reference H.S. S-CP.1. 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”). NO. Performance Objectives 1 2 Identify the sample space of given events. Identify any intersections in the given subsets from the sample space. Given a sample space and two subsets identify the compliment elements 3 Resource Reference Instructional Objectives Assessment Correlation Standard Reference H.S. S-CP.2. 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. New standard for Geometry NO. Performance Objectives Assessment Correlation 1 Understand if two events are independent by looking at all possible outcomes. Given two independent events, A and B, then P(A and B) = P(A)P(B). 2 Resource Reference Instructional Objectives Standard Reference H.S. S-CP.3. Understand the conditional probability of A given B as P(A and B)/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. New standard for Geometry NO. Performance Objectives Assessment Correlation 1 Identify the probability of two events happening given the first either occurs or does not occur. Resource Reference Instructional Objectives Standard Reference H.S. 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 approximate conditional probabilities. For example, collect New standard for Geometry 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. NO. Performance Objectives 1 Construct a two way table for categorical data. Find probabilities based off the data in the two way table of specified events occurring. 2 Resource Reference Instructional Objectives Assessment Correlation Standard Reference H.S. S-CP.5. 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. New standard for Geometry NO. Performance Objectives Assessment Correlation 1 What is the probability of drawing a heart from a standard deck of cards on a second draw, given that a heart was drawn on the first draw and not replaced? Are these events independent or dependent? At Johnson Middle School, the probability that a student takes computer science and French is 0.062. The probability that a student takes computer science is 0.43. What is the probability that a student takes French given that the student is taking computer science? 2 Resource Reference Instructional Objectives Standard Reference H.S. S-CP.6. Find the conditional probability of A given B as the fraction New standard for of B’s outcomes that also belong to A, and interpret the answer in terms of Geometry the model. NO. Performance Objectives 1 Determine if A and B are dependent or independent. Identify the common elements of A and B. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A. 2 3 Resource Reference Instructional Objectives Assessment Correlation Standard Reference H.S. 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. New standard for Geometry NO. Performance Objectives Assessment Correlation 1 Determine if A and B are dependent or independent. Identify the P(A or B) as P(A) and P(B). 2 Resource Reference 3 4 If A and B are dependent find P(A and B). P(A or B) = P(A) + P(B) – P(A and B) if dependent otherwise P(A or B) = P(A) + P(B) if independent.