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Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome Sample Assessment Connections Critical for Mastery at Grade 10: 10.09.01 Identify possible types of two- or threedimensional figures that would match a set of given conditions. (9A) 10.09.02 Determine if a triangle is possible using side lengths and the triangle inequality. (9A) 10.09.03 Create and critique arguments concerning geometric ideas and relationships such as properties of circles, triangles and quadrilaterals. (9C) Make a chart with properties of a special parallelogram ( four sides congruent, opposite angles congruent, etc) across the top and types of special parallelograms down side. Ask students to check the correct properties or attributes. Use sometimes, always, never. Connect to science: Analyze three-dimensional figures in nature. Consider lines of symmetry, rotational patterns. Make available several sets of three straws of different lengths. Ask students to connect them end to end to form a triangle. Find the minimum and maximum length for a possible third side. Construct a triangle given an adjacent side and opposite side to a given angle. Give students several printed circles. Ask them to draw several chords and their perpendicular bisectors in each circle. Find the center of the circle. Connect to social studies: Research how archeologists reconstruct broken plates found in ruins. Connect the midpoints of the sides of a quadrilateral and conjecture about the shape. 10.09.04 Develop a formal proof for a given geometric situation on the plane. (9C) Prove the properties of an isosceles triangle. Look at pattern generated by the Fibonacci sequence. Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome 10.09.05 Describe and apply properties of a polygon or a circle in a problem-solving situation. (9A) 10.09.06 Classify angle relationships for two or more parallel lines crossed by a transversal. (9A) Sample Assessment Connections Make a sketch showing how to tile a floor using both squares and regular octagons. Connect to science: Note how often regular polygons are found in nature. Cross two parallel lines with a transversal. Mark a pair of alternate interior angles with measures of (3x+20) degrees and (x+50) degrees. Analyze and discuss the appearance of non-parallelism: for example, examine pictures of railroad tracks. Research Rhiemann and Lobochevsky geometry. 10.09.07 Represent, solve, and explain numerical and algebraic relationships using geometric concepts. (9C) 10.09.08 Provide examples or counter-examples to either illustrate or disprove conjectures about geometric characteristics. (9C) 10.09.09 Determine and justify the side length relationships present in 45o-45o-90o triangles and 30o-60o-90o triangles. (9D) Using graph paper, draw two parallel lines and cross with a transversal. Show that the lines are parallel, the slopes are the same, the corresponding angles are congruent. Prove or disprove that an obtuse triangle cannot contain a right angle. Given the altitude of an equilateral triangle, find the length of the side of the equilateral triangle. Repeat for isosceles right triangle. Draw a triangle on a deflated balloon. Blow up the balloon. Discuss triangle angle sum. Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome 10.09.10 Solve problems using 45o-45o-90o and 30o-60o-90o triangles. (9D) 10.09.11 Identify and apply properties of medians, altitudes, angle bisectors, perpendicular bisectors, and mid segments of a triangle. (9A) 10.09.12 Solve problems using triangle congruence and similarity of figures. (9B) 10.09.13 Extend knowledge of plane figure relationships to relationships within and between geometric solids. (9B) 10.09.14 Identify relationships among circles, arcs, chords, tangents, and secants. (9B) Sample Assessment Connections Each side of a regular hexagon measures 10 inches. Find the lengths of the diagonals from one vertex. Repeat for a square. Find the center of gravity in a triangle. Analyze the Sierpinski Triangle. Analyze the nine-point circle. A utility pole is supported by two guy wires of equal length. Reason why or why not the wires are fastened to the ground at the same distance from the base of the pole. Connect to environmental values: Connect to the career of wind prospectors. Determine if a cube can have the same number of cubic inches in volume as square inches in total area. Build a tetrahedral kite; note the space between the tetrahedrons. Two circles of different radii are tangent. Find the length of their common external tangent. Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome 10.09.15 Create and critique arguments concerning geometric ideas and relationships, such as congruence, similarity, the Pythagorean relationship, or formulas for surface areas or volume of simple three-dimensional objects. (9C) 10.09.16 Describe the difference between an inductive argument and a deductive argument. (9C) 10.09.17 Determine distances and angle measures using indirect measurement and properties of right triangles. (9D) 10.09.18 Create and analyze scale models using proportional reasoning. (9B) Sample Assessment Connections Given two similar cones with a sixty degree angle made by the slant height and axis. Discuss and compare the ratio of axes, ratio of surface areas and ratio of volumes. Instruct students to measure the angles of several triangles using a protractor, and then form a conclusion about the sum of angles in a triangle. (inductive) Have students use parallel lines to prove that the sum of the angles of a triangle is 180. Discuss the two methods. Connect to a detective murder mystery. Play the game of Clue. A lumberjack stands 60 feet from the base of a tree and measures the angle of elevation to the top of the tree to be 60 degrees. Is the tree more than 100 feet tall? Create a problem about the line of descent for an airplane and the runways at an airport. Draw a replica of a garden with dimensions 48 ft. by 45 ft. The scale choices are 1 in. = 2 ft., 1 in. = 3 ft., and 1 in. = 4 ft. Decide which choice is most reasonable and explain. Analyze a blueprint of your home. Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome Sample Assessment Connections Significant to Develop at Grade 10: 10.09.19 Solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture using geometric ideas. (9B) Find the length of a golden rectangle given the width. Look at pictures of buildings constructed by the ancient Greeks. Find the golden rectangles in architecture of the past and of today. Using coordinate geometry, prove that the diagonals of a square are perpendicular. Learn about taxicab geometry as it relates to city planning. 10.09.21 Analyze the results of a combination of reflections, rotations, and translations of a figure, and determine alternate motions that could produce the same results. (9A) Draw a polygon in the corner of a piece of patty paper. Fold the paper in half and then in half again. Trace the resulting image. Discuss the rotation and reflection. Determine the minimum length of a full length mirror necessary to see your entire image. 10.09.22 Design a net that will create a given figure when folded. (9A) Construct a net that will form a rectangular based box with given dimensions. How much wrapping paper is needed to wrap a large boxed package. Research the Japanese method of wrapping. . Connect to art and culture: Create origami animals or other shapes. 10.09.20 Represent and analyze the properties of geometric shapes using coordinate geometry. (9A) Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome 10.09.23 Recognize Pythagorean Triples. (9D) Sample Assessment Show that a triangle with sides that are multiples of 9, 40, and 41 is a right triangle. Explain why the sides of a right triangle cannot be 5, 6, 7. 10.09.24 Develop conjectures about geometric situations with and without technology. (9C) Using Geometer’s Sketchpad, list conjectures about the properties of a kite. 10.09.25 Identify the basic trigonometric ratios in terms of lengths of the sides of a right triangle and an acute angle. (9D) Given a 3,4,5 triangle, write the sine, cosine and tangent of each of the acute angles. 10.09.26 Solve for missing side lengths using the trigonometric ratios in right triangles. (9D) 10.09.27 Determine the ratio of lengths of sides of a right triangle with given measures for its acute angles using appropriate technologies. (9D) A guy wire is attached to the top of a 75 foot pole. The wire makes an angle of 63 degrees with the ground. Find the length of the wire. A building casts a 20 foot shadow. The angle of elevation (angle of depression, angle of inclination) of the sun is 40 degrees. Find the height of the building. Connections Connect to history of mathematics and technology: Research the history of triples. Write a calculator program that generates Pythagorean Triples. Research the job of a surveyor. Connect the career to geometry. Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome 10.09.28 Visualize a three-dimensional object from different perspectives and describe their cross sections. (9A) Sample Assessment Connections Find the volume of a frustum given the parent pyramid. Find examples of frustums in pictures in magazines. Find the area of a great circle. Connect to social studies: Relate the great circle with latitude and longitude lines on a globe. Useful to Work On at Grade 10: 10.09.29 Construct a perpendicular bisector of a segment. (9A) 10.09.30 Combine simple construction techniques to construct squares, equilateral triangles, or other simple combinations of equal segments, angles, etc. (9A) 10.09.31 Solve problems using constructions. (9A) 10.09.32 Analyze properties of a shape that enable it to tessellate the plane. (9A) Use Geometer’s Sketchpad, if available to construct the perpendicular bisector. Prove that every point that lies on the perpendicular bisector is equidistant from the endpoints of the segment. Use any convenient length, construct an equilateral triangle. Explain how to construct a 30 degree angle. Use a map. Find the shortest path to a distant road from a marked town using a construction. Analyze a simple translation tessellation by M. C. Escher. Make your own. Connect to maps in geography. Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome 10.09.33 Analyze properties of a combination of shapes that enable them to tessellate the plane. (9B) 10.09.34 Analyze and describe the transformations that lead to successful tessellations of one or more figures. (9B) Sample Assessment Make a tessellation of the block letter F. Find other block letters that will tessellate. Connections Make tessellations in art class. Analyze pictures of tessellations by M. C. Escher. Draw a non-specific quadrilateral. Find the midpoint of one of its sides. Rotate the original figure around the midpoint and describe what happens. 10.09.35 Represent transformations of an object in the plane using sketches, coordinates, and vectors. (9A) Sketch the image of a figure and give the coordinates of the vertices of the image if the figure is reflected over the y=x line. 10.09.36 Examine the congruence or similarity of objects using transformations. (9B) Prove that the diagonals of a parallelogram bisect each other. (Hint: use half-turns) 10.09.37 Justify constructions using geometric properties. (9C) Justify the construction of a segment congruent to a given segment. Connect to technology and art: Create a logo using transformations of your school’s name. Archdiocese of Chicago: Mathematics Curriculum Framework Grade 10: Geometry Focus State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Learning Standard/Outcome 10.09.38 Justify the simple construction methods used to produce angle bisectors, perpendicular lines, and equilateral triangles. (9C) Sample Assessment Justify the construction of an angle bisector. Connections Discuss the importance of accurate measurements in building a bridge. Build a bridge using only constructions, no ruler measurements. Connect to Physics: Enter bridge building contest. 10.09.39 Solve problems using the Law of Sines and Law of Cosines. (9D) Given an oblique triangle with lengths of three sides given, find the measures of the three angles. Find the area of a plot of land in the shape of a non-specific quadrilateral.