MADISON ACADEMY GEOMETRY PACING GUIDE

Transcription

1 MADISON ACADEMY GEOMETRY PACING GUIDE Standards (ACT included) ALCOS#1 Know the 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. ACT #G501 Use several angle properties to find an unknown angle measure. ACT #G50 Recognize that real-world measurements are typically imprecise and that an appropriate level of precision is related to the measuring device and procedure. ACT #G505 Compute the perimeter of simple composite geometric figures with unknown side lengths. ACT #G506 Compute the area of triangles and rectangles when one or more additional steps are required. 1st 9 Week Resources *Honors only objectives Glencoe: Chapter 1, Sections 1,2,3,,5 Identify and model points, lines, planes, intersecting lines and planes. Measure and calculate line segments. Find distance and midpoint. Measure and classify angles. Idendify and use congruent angles and angle bisectors. Identify and use angle pairs and perpendicular lines. Approximate pacing number of days 5 ACT #G507 Compute the area and circumference of circles after identifying necessary information. ACT #G511 Find the midpoint of a line segment. ACT #G605 Use the distance formula. ALCOS #1 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy ACT #G70 Analyze and draw conclusions based on a set of conditions. ALCOS #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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Glencoe: Chapter 2, Sections 1,3,,5 Make conjectures based on inductive reasoning. Conditional statements and deductive reasoning. Identify and use postulates. Glencoe: Chapter 2 Sections 6,7,8 Use algebra and properties of equality to write two-column geoemtric proofs. Write proofs involving segment addition and congruence. Write proofs involving supplementary and complementary angles. Write proofs involving congruent and right angles. *Proofs Notebook

2 ALCOS #1 Know the 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. ALCOS #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; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. ALCOS #31 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). ACT #G510 Determine the slope of a line from points or a graph. ACT #G606 Use properties of parallel and perpendicular lines to determine an equation of a line or coordinates of a point. Glencoe: Chapter 3 Sections 1,2,5 Identify relationship between two lines or two planes. Name angle pairs formed by parallel lines and transversals. Use theorems to determine relationships between angle pairs. Use algebra to find angle measurements. Prove that two lines are parallel. *Proofs Notebook Glencoe: Chapter 3 Sections 3, Find slopes of lines. Use slopes to identify parallel and perpindicular lines. Write and equation of a line based on a graph. Solve problems by writing equations. 3

3 MADISON ACADEMY GEOMETRY PACING GUIDE Standards ALCOS # 7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. ALCOS #10 Prove theorems about triangles. Theorems include 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, and the medians of a triangle meet at a point. ALCOS #12 Make formal geoemtric constructions with a variety of tools and methods such as compass and straight edge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; coying an angle; bisecting a segment; bisecting an angle; constructing perpindicular 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. 2nd 9 Weeks Resources *Honors only objectives Glencoe: Chapter, Sections 1,2,3 Identify and classify triangles by angle and sides. Apply Triangle-Sum Theorem and Exterior Angle Theorem. Use corresponding parts of congruent polygons. Prove triangles congruent using the definition of congruence. *Constructions Notebook Approximate pacing number of days ALCOS #18 Use congruence and similarity criteria for triangles to solve problems andn to prove relationships in geometric figures. ALCOS #10 Prove theorems about triangles. Theorems include 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, and the medians of a triangle meet at a point. ALCOS #18 Use congruence and similarity criteria for triangles to solve problems andn to prove relationships in geometric figures. ACT #G503 Use symmetry of isosceles triangles to find unknown side lengths or angle measures. ALCOS #10 Prove theorems about triangles. Theorems include 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, and the medians of a triangle meet at a point. ALCOS #1 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy Glencoe: Chapter, Sections,5,6 Use the SSS and SAS Postulates to test for triangle congruence. Use the ASA and AAS Posutlates to test for triangle congruence. Use properties of isosceles and equilateral triangles. Glencoe: Chapter 5, Sections 1,2,3,5 Identify and use perpendicular bisectors and angle bisectors in triangles. Identify and use medians and altitudes in triangles. Recognize and apply properties of inequalities to the measures of the angles of a triangle. Recognize and apply properties of inequalities to the relationships between the angles and the sides of a triangle.

4 ALCOS # 39 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Glencoe: Chapter 6, Sections 1,2,3,,5,6 ALCOS #11 Prove theorems about parallelograms. Theormes 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. ALCOS #30 Use coordinates to prove simple geometric theorems algebraically. ALCOS #1 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy Find and use the sum of the measures of the interior and exterior angles of a polygon. Recognize and apply the properties of the sides, angles and diagonals of parallelograms. Recognize the conditions that ensure a quadrilateral is a parallelogram. Prove that a set of points form a parallelogram in the coordinate plane. Recognize and apply the properties of rectangles. Determine whether parallelograms are rectangles. Recognize and apply properties of rhombi and squares. Determine whether quadrilaterals are rectangles, rhombi, or squares. Recognize and apply the properties of trapezoids, including the medians of trapezoids. Recognize and apply the properties of kites. 6

5 MADISON ACADEMY GEOMETRY PACING GUIDE Standards ALCOS #15 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; expain 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. ALCOS #16 Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. ALCOS #17 Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; ane the Pythagorean Theorem proved using triangle similarity. ALCOS #18 Use congruence and similarity criteria for triangles to solve problems andn to prove relationships in geometric figures. 3rd 9 Weeks Resources *Honors only objectives Glencoe: Chapter 7, Sections 1,2,3 Write ratios and solve proportions. Use proportions to identify similar polygons. Solve problems using the properties of similar polygons. Identify similar triangles using the AA Similarity Postulate and the SSS and SAS Similarity Theorems. Use similar triangles to solve problems. Approximate pacing number of days ALCOS #1 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy ACT #505 Compute the perimeter of simple composite geoemtric figures with unknown side lengths. ACT #703 Use scale factors to determine the magnitude of a size change. Glencoe: Chapter 7, Sections,5,7 Use proportional parts within triangles and parallel lines. Recognize and use proportional relationships of corresponding segments of similar triangles. Use the Triangle Angle Bisector Theorem. Interpret scale models. Use scale factors to solve problems.

6 ALCOS #17 Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; ane the Pythagorean Theorem proved using triangle similarity. ALCOS #18 Use congruence and similarity criteria for triangles to solve problems andn to prove relationships in geometric figures. ALCOS #19 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. ALCOS #20 Explain and use the ralationship between the sine and cosine of coplenmentary angles. ALCOS #21 Use trigonomeric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ALCOS #1 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy Glencoe: Chapter 8, Sections 1,2,3, Find the geometric mean between two numbers. Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse. Use the Pythagorean Theorem and its converse. Use the properties of triangles and triangles. Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle measures in right triangles. 5 ACT #G508 Given the length of two sides of a right triangle, find the third when the lengths are Pythagorean triples. ACT #G509 Express the sine, cosine, and tangent of an angle in a right triangle as a ratio of given side lengths. ACT #G602 Use the Pythagorean theorem. ACT #G603 Apply the properties of , , similar, and congruent triangles. ACT #G60 Apply basic trigonometric ratios to solve right-triangle problems. ALCOS #21 Use trigonomeric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Glencoe: Chapter 8, Sections 5,6,7 ALCOS #22 Prove the Law of Sines and Cosines and use them to solve problems. ALCOS #32 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Solve problems involving angles of elevation and depression. Use angles of elevation and depression to find the distance between two objects. Use the Law of Sines and the Law of Cosines to solve triangles. Find the magnitudes and directions of vectors. Add and subtract vectors.

7 ALCOS #2 Represent transformations in the plane using, e.g. transparenies 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 horzontal stretch). ALCOS #3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry onto itself. ALCOS # Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. ALCOS #5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geoemtry software. Specify a sequence of transformations that will carry a given figure onto another. ALCOS #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. ALCOS #1 Verify experimentally the properties of dilations given by a center and a scale factor. ACT #502 Count the numbers of lines of symmetry of a geometric figure. ACT #512 Find the coordinates of a point rotated 180 degrees around a given center point. ACT #607 Find the coordinates of a point reflected across a vertical or horizontal line or across y=x. ACT #608 Find the coordinates of a point rotated 90 degrees about the origin. Glencoe: Chapter 9, Sections 1,2,3,,5,6 Draw reflections, translations, rotations, and dilations in the coordinate plane. Draw glide reflections and other compositions of isometries in the coordinate plane. Draw compositions of reflections in parallel and intersecting lines. Identify line and rotational symmetries in twodimensional figures. Identify plane and axis symmetries in three-dimensional figures. 6

8 MADISON ACADEMY GEOMETRY PACING GUIDE Standards ALCOS #1 Know the 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. ALCOS #2 Prove that all circles are similar. ALCOS#25 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. ALCOS#28 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. ALCOS #1 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy ACT #G609 Recognize special characteristics of parabolas and circles. (e.g. the vertex of a parabola and the center or radius of a circle) ACT #G701 Use relationships among angles, arcs, and distances in a circle. th 9 Weeks Resources *Honors only objectives Glencoe: Chapter 10, Sections 1,2,3 Identify and use parts of circles. Solve problems involving the circumference of a circle. Identify central angles, major arcs, minor arcs, and semicircles, and find their measures. Find arc lengths. Recognize and use relationships between arcs and chords. Recognize and use relationships between arcs, chords, and diameters. Approximate pacing number of days

9 ALCOS #12 Make formal geoemtric constructions with a variety of tools and methods such as compass and straight edge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; coying an angle; bisecting a segment; bisecting an angle; constructing perpindicular 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. ALCOS #25 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. ALCOS #26 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. ALCOS #27 Construct a tangent line from a point outside a given circle to the circle. ALCOS #29 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 equaiton. ALCOS #32 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Glencoe: Chapter 10, Sections,5,6,7,8 Find measures of inscribed angles. Find measures of angles of inscribed polygons. Use properties of tangents. Solve problems involving circumscribed polygons. Find measures of angles formed by lines intersecting on or inside a circle. Find measures of angles formed by lines intersecting outisde the circle. Find measures of segments that intersect in the interior of a circle. Find measures of segments that intersect in the exterior of a circle. *Constructions Notebook 5 ACT #G609 Recognize special characteristics of parabolas and circles. (e.g. the vertex of a parabola and the center or radius of a circle) ACT #G701 Use relationships among angles, arcs, and distances in a circle. ACT #G705 Solve multistep geometry problems that involve integrating concepts, planning, and/or visualization. ALCOS #28 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. ALCOS #33 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ALCOS #3 Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. ALCOS #35 Give an informal argument for the formulas for the circumference of a circle; area of a circle; and Glencoe: Chapter 11, Sections 1,2,3 Find perimeters and areas of parallelograms. Find perimeters and areas of triangles. Find areas of trapezoids, rhombi, and kites. Find areas of circles and sectors of circles.

10 ALCOS #35 Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Glencoe: Chapter 11, Sections,5 ALCOS #37 Determine the relationship between surface areas of similar figures and volumes of similar figures. ALCOS #39 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ALCOS #1 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy Find areas of regular polygons and composite figures. Find areas of similar figures by using scale factors. Find scale factors or missing measures given the areas of similar figures. ACT #G505 Compute the perimeter of simple composite geometric figures with unknown side lengths. 3 ACT #G506 Compute the area of triangles and rectangles when one or more additional simple steps are required. ACT #G507 Compute the area and circumference of circles after identifying necessary information. ACT #G702 Compute the area of composite geometric figures when planning and/or visualization is required. ALCOS #35 Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. ALCOS #36 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ALCOS #38 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. ALCOS #39 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Glencoe: Chapter 12, Sections 1,2,3,6 Draw isometric views of three dimensional figures. Investigate cross sections of three-dimensional figures. Find lateral areas and surface areas of prisms and cylinders. Find lateral areas and surface areas of pyramids and cones. Find surface areas and volumes of spheres. ALCOS #1 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy ACT #G601 Use relationships involving area, perimeter, and volume of geometric figures to compute another measure. (e.g. surface area for a cube of a given volume and simple probability) Glencoe: Chapter 12, Sections,5 Find volumes of prisms, cylinders, pyramids, and cones. 3 ACT #G602 Use the Pythagorean theorem.