Geometry: Concept Categories Concept Category 1 (CC1): Transformations & Basic Definitions

Transcription

1 Concept Category 1 (CC1): Transformations & Basic Definitions Concept Category 2 (CC2): Triangles (Similarity and Properties) Concept Category 3 (CC3): Triangle Trigonometry Concept Category 4 (CC4): Triangle Congruence Concept Category 5 (CC5): Proof & Quadrilaterals Concept Category 6 (CC6): Polygons, Circles, Solids, & Constructions Concept Category 7 (CC7): Conditional Probability Concept Category 8 (CC8): Solids & Conics Concept Category 1 (CC1): Transformations & Basic Definitions Identify rotated, reflected, and translated figures, with the option of using tracing paper, as in problems 1-64, 1-108, and CL (a) and (b). Rotate, reflect, and translate figures on a grid, with the option of using tracing paper, as in problems 1-85, 1-97, 1-109, 1-116, and CL (c). Find the slope or equation of a perpendicular line as in problems 1-88 (e), 1-77, and 1-105, 2-42, 2-58, 2-69, 2-105, and CL (d) Identify angle relationships, including Triangle Angle Sum Theorem, and solve problems using those relationships as in problems 2-34, 2-55, 2-56, 2-66, 2-67, 2-76, 2-107, 2-116, and CL In many of these problems there will be more than one correct way to find missing angles and arrive at a solution, thus students ability to justify the steps they take is critical. Using the Triangle Inequality to determine if three given side lengths could form a triangle, or to find the possible range of lengths of a third side given the lengths of the two other sides as in problem Using information given in a problem to choose an appropriate tool from the Triangle Toolkit to find a missing side or angle. Further, students can be expected to find missing sides or angles and then to apply that information to finding area or perimeter of a shape. Concept Category 2 (CC2): Triangles (Similarity and Properties) Finding missing angles in shapes and parallel line diagrams. Students are to write equations to find missing values or to solve for variables based on angle relationships. Explore dilations using rubber bands, as in problems 3-5, 3-18, 3-46(a), and CL Identify corresponding parts on similar figures and use common ratios to find missing side lengths on figures, as in problems 3-58, 3-65, 3-80, 3-113, and CL Determine if two shapes are similar. Support conjectures with justification, including demonstrating equal ratios between corresponding sides and finding missing angles, as in problems 3-54, 3-55, 3-69, 3-81, 3-90, CL 3-115, and CL Create if-then conditional statements, and use that logic in forming flowcharts of situations, as in problems 2-17(f), 2-26(c), 3-10, 3-23, 3-33, 3-44, 3-53, 3-68, 3-92, and CL Formally determining if two shapes are similar using the SSS ~, AA ~, or SAS ~ conditions, as in problem Concept Category 3 (CC3): Triangle Trigonometry Create a diagram based on information in a word problem, and identify a right triangle within that diagram, as in problems 4-43, 4-50, 4-83, 4-113, and CL In right triangles where one leg and an angle are given, use a calculator to find missing sides using the tangent ratio, as in the problems in the above bullet, and as in problems 4-39, 4-63, 4-74, CL 4-122, and CL Find the missing sides or angles of right triangles using sine, cosine, tangent, as in problems 5-17, 5-18, 5-44, 5-46, 5-100, 5-137, CL 5-139(a-d), and CL 5-143(a), and their inverse functions, as in problems 5-30, 5-77, 5-103, and CL 5-139(e-g). Problems may be presented as triangle diagrams in different orientations, or word problems with a diagram provided or instructions to draw one.

2 Students can be asked to find the area and/or perimeter of a shape where they will need to use their knowledge of trigonometry to find missing lengths, as in problems 5-11, 5-33, 5-52, 5-129, and Find the slope angle or the slope of the line segment on a coordinate grid using the tangent ratio, as in problems 5-42, 5-102, 5-113, and CL First semester ends here, CC3 will pick up Second Semester with the rest of CC3 & continue on. Recognize and apply the side ratios in triangles and triangles, as in problems 5-52, 5-53(d), 5-64(b) and (d), 5-91, 5-127, 5-135, and CL Problems may present triangles independently or joined as part of a larger shape. Recognize the smallest Pythagorean Triples and use them as shortcuts to find missing sides in other triangles, as in problems 5-64(a) and (c), 5-129, parts of 5-138, and CL (a) and (b). Understand that it is possible to find missing sides on non-right triangles, as in problems 5-90, 5-111, 5-126, 5-134, and CL It is acceptable for students to create right triangles within a diagram in order to find missing parts. Law of Sines and Law of Cosines. However, students can find missing sides on non-right triangles by making right triangles within a diagram. From Chapter 5, the Laws of Sines and Cosines, as in problems 6-15(c), 6-47, 6-55, 6-74(d), and CL Additionally, know the vocabulary, Learning Log Entries, Toolkit Entries, and Math Notes boxes at the end of each chapter in the Closure section. See Pages 70-71, , , , , Concept Category 4 (CC4): Triangle Congruence Finding missing angles in shapes and parallel line diagrams. Students are to write equations to find missing values or to solve for variables based on angle relationships. From Chapter 3, use flowcharts to prove similarity, as in problems 4-7, 4-41, 4-70, 4-72, 4-118, and CL Identify pairs of triangles as similar or congruent, as in problems 6-23, 6-35, 6-47, 6-58, 6-83, and CL In the case of triangles that are congruent, justifications that use a similarity conjecture (AA ~, SSS ~, SAS ~) with added information that the ratio of similarity is 1 or that use a triangle congruence conjecture (SSS, ASA, AAS, HL and SAS ) are acceptable. Use flowcharts to organize arguments about triangle similarity and congruence, as well as to evaluate the logic of other arguments, as in problems 6-8, 6-23, 6-35, , 6-73, 6-83, CL 6-101, and CL Write the converse of a conditional statement, as in problems 6-48, 6-64, 6-86, and CL Students can be expected to use the converse of parallel line conjectures to determine if two lines are parallel, based on given angle information. See problem CL for an example of this kind of problem. Deciding if pairs of triangles are similar or congruent. Students can be expected to use parallel line theorems and other angle and side relationships in order to determine if they have enough information to draw a conclusion. Concept Category 5 (CC5): Proof & Quadrilaterals Construct a proof structured as a flowchart using given information, reasons for each step, and an organized progression of information, as in problems 7-46, 7-69, 7-76, 7-86, 7-100, 7-108, 7-123, 7-134(a), and CL Work with a simple two-column proof, as in problems 7-113, 7-124, 7-132, and CL Identify different shapes based on a description in words ( must be, could be ) or by their central angle (hinged mirror), as in problems 7-9, 7-43, 7-55, 7-137, 7-142, 7-144, CL 7-147, and CL Analyze a shape on a coordinate grid, and use information about the length and slope of sides and measures of angles, to name the shape and justify conclusions, as in problems 7-35, 7-79, 7-131, 7-135, 7-146, CL 7-150, and CL

3 Find the midpoint of segment on a coordinate grid, as in problems 7-20, 7-29, 7-45, 7-119, 7-140, and CL Concept Category 6 (CC6): Polygons, Circles, Solids, & Constructions Move flexibly around the Regular Polygon Angle Web to find interior angles, exterior angles, or the number of sides of the regular polygon, given any one of the other pieces of information, as in problems 8-44, 8-53, 8-95, 8-107, 8-119, and CL 8-138, and contained in the area problems of the third bullet below. See the Math Notes box in Lesson Find missing angles or solve for variables using the sums of interior or exterior angles of non-regular polygons, as in problems 8-60, 8-61(c), 8-95, 8-97(a), 8-107(a), and CL Find the area of a regular polygon given a side length or radius, as in problems 8-73, 8 84, 8-107(b), 8-105, 8-126, and CL Students develop an algorithm to do this by dividing the polygon into congruent isosceles triangles from a center point. Find the area and circumference of circles. Problems may show whole circles alone, or as part of a larger diagram, as in problems 8-116, 8-123, 8-128, CL 8-130, and CL Find the area of sectors of circles when given a central angle (using proportional reasoning strategies). Understand and use the relationships between central and inscribed angles and their intercepted arcs to calculate arc and angle measures, as in problems 10-33, 10-68(d), 10 78(b), 10-87, , and CL (b). You should be able to find the measures of both arcs and inscribed and central angles, using information about any one of the three as a starting place. Combine your new knowledge about arcs and angles with their understanding of circumference and circle area in order to calculate arc length and sector area, as in problems and 10-87(c). Understand and use the facts that an angle inscribed in a semicircle (the angle of a diameter) measures 90, as in 10-43(a) and 10-68(a), and that a line tangent to a circle is perpendicular to the radius of the circle at the point of tangency, as in problems 10-54, 10 59, and CL (a). Understand chord relationships and use those relationships to find the length of a chord. Find the length of a segment formed by intersecting chords, as in problems 10-60, 10-78(c), (b), and CL or to find the length of a chord given a radius and the measure of an arc or angle, as in problem (a). Apply the relationship between tangents, secants, and chords, as in problems and , 12-11, 12-52, 12-93, and CL Use basic construction techniques to create shapes with specific relationships, as in problems 9-79, 9-80, 9-98, 9-104, CL 9-110, and CL 9-113(a). Specifically, students should be able to use construction techniques to copy angles and line segments, and to create a perpendicular bisector of a segment and an angle bisector. Problems on an assessment might ask students to describe steps to complete simple constructions, as in problem Students should not memorize specific constructions; rather, they should be able to justify what they create based on the construction techniques that they apply. Concept Category 7 (CC7): Probability & Constructions Use counts, especially in two-way tables, to determine probabilities, conditional probabilities, and association, as in problems 10-85, , , , and CL Use relative frequencies (probabilities) in two-way tables to determine other probabilities, conditional probabilities, and association, as in problems , , , , and CL Use the alternative definition for independence derived from the Multiplication Rule to determine independence, and vice versa, as in problems (d), , (c), and CL (b). Count the number of arrangements using a decision chart, as in problems , , (d), CL , and CL Count the number of arrangements for situations which can be put into one of these categories:

4 Permutations (order matters with no repetition), as in problems , , , and (a). Anagrams (arrangements in which there are duplicates of some of the elements), as in problems , , and Combinations (order does not matter with no repetition), as in problems , (c), and Counting the number of outcomes in complex cases that require combining smaller counts of permutations and/or combinations, as in problem and in the classwork problems of Lesson Counting the number of outcomes of n items, choose r when order does not matter with repetition allowed as in some of the classwork problems of Lesson Add and multiply combinations in subsets of problems, as in problems 11-73, 11-74, 11-87(e), 11-98, and Concept Category 8 (CC8): Area, Volume, & Conics Represent three-dimensional solids using a side view, a mat plan, and/or a net, as in problems 9-7, 9-21, 9-49, 9-56(a), 9-91, and CL Students should also be able to use these representations to calculate the surface area or volume of the solid. Find the surface area and volume of three-dimensional shapes, as in problems 9-26, 9-34, 9-38, 9-40, 9-46(a), 9-57, 9-69, 9-83, 9-95, 9-103, and CL Problems can include calculating surface area and volume of three-dimensional solids composed of cubes, or surface area and volume of prisms and cylinders. Solids can be presented as labeled pictures, or be described in words. Find density, as in problems 9-40, 9-57(b), 9-69(d), 9-93(c), and CL Understand how the area and perimeter of similar figures are related, and use those relationships to find areas of enlarged and reduced shapes, as in problems 8-71(c), 8-83, 8-106(b), 8-123(c), and 8-125(b). Understand and apply the relationship between a linear scale factor and the ratio of areas or ratio of volumes of similar shapes, as in problems 9-8, 9-20(b), 9-45, 9-46(b), 9-56, 9-91(c), CL and CL 9-113(c). Students should be able to find a ratio of areas or of volumes given a linear scale factor. Students should be able to use the ratios to find side lengths, surface areas, or volumes of similar solids given measurements of one shape in a pair of similar shapes. Given a ratio of areas or volumes, work backwards to find a linear scale factor, as in problems 9-8(c), 9-86, and CL 9-112(b). Find area as part of a three-dimensional situation, as in problems 9-34, 9-95(b), 9-103(b), CL 9-114(a), and CL Working with solids requires students to revisit many two-dimensional shapes in a new context. Problems about volume or surface area may require students to find the area of a regular polygon or a trapezoid, using trigonometry, as a base of a prism. Students could be asked to find the volume of a slice of a cylinder, where the base is a sector of a circle. Find the volume of a pyramid or cone, as in problems 11-25, 11-40, 11-52, 11-58(b), 11-75, 11-86, , and CL Students should recognize that a pyramid s volume is one-third of the volume of a prism with the same base and height, a cone is one-third of a cylinder, and a sphere is twothirds of a cylinder. Students need to be able to distinguish between the height and the slant height of a pyramid or cone, and to choose the appropriate measurement to find volume. Find the total surface area of a pyramid, as in problems 11-40, 11-76(c), , and CL Apply the r :r 2 :r 3 ratios of similarity, as in problems 11-12, 11-43, 11-86(b), 11-90, , and CL (d). Even though the ratios are review (Math Notes box in Lesson 9.1.5). Master Checkpoint 11: Volumes and Surface Areas of Prisms and Cylinders, as in problems 11-10, 11-17, 11-47, 11-58(a), , , CL , and CL Surface areas of cones or spheres and volumes of spheres, as in problems 11-58(b), 11-72, and 11-97, 12-21, 12-40, 12-54, CL , and CL

5 Graph a circle on a coordinate grid, as in problems 12-6, 12-27, 12-31, 12-43, 12-51, 12-64, 12-74, 12-87, and CL Problems can start with an equation and ask students to make a graph, identify the length of a radius or diameter, or find circumference and area. Students can be asked to analyze a graph of a circle in order to write an equation. Complete the square to change the equation of a circle from general form to graphing form, as in problems 12-24, 12-51(d), , and CL Graph a parabola on focus-directrix paper, as in problems 12-66, 12-96, and CL Although students should be comfortable graphing parabolas on a coordinate grid based on their experiences in algebra, doing so on focus-directrix paper pushes them to apply the geometric definition of a parabola. Finding the equation of a parabola from a geometric description, as in classwork problem Additionally, know the vocabulary, Learning Log Entries, Toolkit Entries, and Math Notes boxes at the end of each chapter in the Closure section. See Pages , , , , ,