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- 1. Mathematics Concept Map Geometry 1-2 (Regular and Honors) Phoenix Union High School District April 2009 Copyright 2009 Phoenix Union High School District. All rights reserved. No part of this document may be reproduced without the express prior written permission of the Phoenix Union High School District. PHOENIX UNION HIGH SCHOOL DISTRICT NO. 210 4502 North Central Avenue Phoenix, AZ 85012PUHSD 2009 Page 1 of 31Curriculum Division
- 2. Phoenix Union High School District Mathematics Concept Map Table of Contents Section Page # Definition of Concept Map 3 Geometry 1-2 Concept Maps 4 Geometry 1-2 Scope and Sequence 17 Geometry 1-2 Honors Concept Maps 18 Geometry 1-2 Honors Scope and Sequence 31 Acknowledgements The Mathematics Curriculum Guide and Concept Maps were developed through the hard work of the 2008 Curriculum Revision TeamAlgebra Team Geometry Team Algebra 3-4 TeamKathy Long North Oscar Ramirez Cesar Chavez Nicole Fernandez NorthFernando Galvez Maryvale Eddie Fylling Bostrom Diane Welch Metro TechMelanie Dalager Central Lydia Horstman Fairfax Juli Schexnayder Maryvale Joseph Prevost Maryvale Sheryl Filliater Metro Tech Facilitator Mona Toncheff Math Content Specialist CESPUHSD 2009 Page 2 of 31Curriculum Division
- 3. Concept Map Definition Page Topic: Power The organization of Key Vocabulary Standards performance objectives into a Concept Unit common theme that promotes student engagement and focuses student inquiry. Power Standards: Enduring Understanding: The ideas that connect the PO’s to the overarching Key Vocabulary: The Essential Questions: topic. vocabulary that is important for students to know in Examples order to demonstrate an Technology : Resources understanding of a topic. Standards Resources: Lists important lessons, activities, assessments, or links for the instructional .unit. Enduring Understanding: A central and organizing notion Essential Question: The mental Examples: Examples that gives meaning and questions that help students form a illustrate level of how connection to facts. It has conceptual understanding of the students can solve problems lasting value and can transfer to concept or concepts. They point within each concept unit or other inquiries and requires toward key ideas and issues and how concepts can be “uncoverage”. suggest meaningful and provocative assessed. inquiry into content.PUHSD 2009 Page 3 of 31Curriculum Division
- 4. Key Concepts Concept Unit Map Key Vocabulary Grade Level: Geometry 1 collinear/Linear Identify, name, and draw points, lines, rays, segments, coplanar/plane planes and angles. Using Points, Lines, Planes, and Angles skew point Solve problems related to Enduring Understanding: segment complementary, Recognizing the relationship of points, lines, and planes and the geometric figures they ray supplementary, congruent form is the foundation for applying mathematics to the real world line angles, bisectors, angles or segment concepts. angle segment Essential Question(s): perpendicular bisector What is the difference between a postulate and a theorem? Solve problems created by linear pair parallel lines cut by a How can slope be used to determine the relationship between two lines? What is the distance and midpoint between two points? complementary/supplementary transversal. What is the relationship between points, lines and planes? vertical angles transversal Determine congruence of parallel/perpendicular segments. Examples adjacent angles Given: M is the midpoint of AB . Identify a line segment, ray, plane, midpoint Determine the distance and Find AB. line, a pair of skewed lines, postulate midpoint between 2 points in a perpendicular segments, three theorem coordinate system. 4x-6 2x+10 coplanar points, two collinear points. A M B B C Resources Algebra Help D http://www.algebrahelp.com/index.jsp A Textbook resources http://classzone.com H E Segment addition postulate tutorial Cabri Jr. Discovery http://www.hstutorials.net/math/geom activities G F etry/theoPosts/seg_add.htm www.timath.com • Angle bisector • Midpoint of a linePUHSD 2009 Page 4 of 31Curriculum Division
- 5. Concept Unit Map Key Vocabulary Key Concepts Grade Level: Geometry 1 parallel lines perpendicular lines Identify parallel lines, Perpendicular and Parallel lines perpendicular lines, skew lines, skew lines parallel planes, and a line parallel planes perpendicular to a plane. line perpendicular to a plane Enduring Understandings: complementary angles Parallel and perpendicular lines provide the foundation for most physical supplementary angles Identify and solve problems structures. using angles formed by transversal perpendicular lines. corresponding angles alternate interior angles Essential Questions: Identify congruent angles and alternate exterior angles What angle relationships are formed by parallel lines cut by a supplementary angles formed same-side interior angles transversal? by parallel lines and a congruent transversal. What angle relationships are formed by perpendicular lines? converse How does slope determine the relationship between two lines? theorem Determine if lines are parallel How can angle relationships be used to determine measures of other postulate or perpendicular based on angles? slope or angle relationships. Examples Resources Apply properties, theorems, Find the value of x. Algebra Help and constructions about http://www.algebrahelp.com/index.jsp parallel lines, perpendicular Find the value of x. Textbook resources lines, and angles to prove theorems. ( 2 x + 25) ° http://classzone.com Geometry lesson on parallel and perp. ( 2 x − 10 ) ° http://www.homeschoolmath.net/teaching/ ( x + 5) ° g/parallel_and_perpendicular.php Cabri Jr. Discovery activities www.timath.com 110° • Vertical and adjacent angles • TransversalsPUHSD 2009 Page 5 of 31Curriculum Division
- 6. Key Concepts Concept Unit Map Key Vocabulary Grade Level: Geometry 1 conjecture Construct a simple formal deductive proof. inductive reasoning Logical Reasoning counterexample Prove a conjecture false by finding patterns a counterexample. deductive reasoning Enduring Understandings: conditional statement Draw valid conclusions from Logical reasoning is essential to synthesize information, put the world in hypothesis conditional statements using law perspective, and apply knowledge. conclusion of detachment and syllogism. law of detachment syllogism List related if…then statements in Essential Questions: inverse logical order. What conjecture can be made based on an observation? converse How can you use a counterexample to prove a conjecture false? contrapositive Identify a valid conjecture using How can a logical conclusion be made using law of detachment and if-then statement inductive reasoning. syllogism? reflexive property How can you use triangle congruence theorems to prove two triangles symmetric property Determine triangle congruency using SSS, SAS, ASA, HL and are congruent? transitive property AAS congruence postulates. How do you write a paragraph proof? proofs writing proofs State the inverse, converse, and paragraph proofs contrapositive of a given if-then Examples two-column proofs statement and determine if it is true Using the Law of Syllogism, which statement follows from the pair of writing a formal proof or false. true statements? valid invalid Critique inductive and deductive arguments concerning geometric If I do not get paid this week, then I cannot pay my bills. ideas and relationships. If I cannot pay my bills, then the bill collectors will call. Resources Algebra Help http://www.algebrahelp.com/index.js Cabri Jr. Discovery activities p www.timath.com What is the next number you expect? 45, 90, 135, 180…….. Textbook resources • Conditional statements http://classzone.com • ASA,SAS,SSS Triangle congruencePUHSD 2009 Page 6 of 31Curriculum Division
- 7. Concept Unit Map Key Vocabulary Key Concepts Grade Level: Geometry 1 triangle equilateral triangle isosceles triangle Classify triangles according to Attributes and Properties of Triangles scalene triangle lengths of sides and angle right triangle measures. obtuse triangle Enduring Understandings: acute triangle Solve problems using triangle Triangles are the support system for most objects that we use on a daily equiangular triangle sum and exterior angles basis. Are you sitting on a triangle now? vertex theorems. length measure corollary Solve problems related to base Essential Questions: interior angles angles, equilateral, and How are triangles classified? exterior angles equiangular triangle theorems. base How do triangle attributes help solve problems related to angles measures and side lengths? legs base angles Use the Pythagorean Theorem How are the Pythagorean Theorem and Distance formula related? hypothesis and Distance formula to find What is the relationship between segments formed by the intersection of Pythagorean theorem unknown lengths. the medians of a triangle? square square root distance Determine and solve problems classify Examples using angle and side length median of a triangle relationships. centroid Decide whether a triangle inequality triangle could have the Classify the triangle with Solve problems using median the most specific name. and centroids of a triangle. given angle measures. Explain your reasoning. 18°, 34°, 128° Resources Solve problems using the triangle 8 9 Algebra Help inequality theorem. http://www.algebrahelp.com/index.js p Find the value of x. Textbook resources 10 http://classzone.com Cabri Jr. Discovery activities www.timath.com 102° Geometry Interactive activities • Triangle Inequalities http://www.shodor.org/interactivate/ activities/ • Ratio of Right Triangles • Distance in the coordinate x° 51° planePUHSD 2009 Page 7 of 31Curriculum Division
- 8. Concept Unit Map Key Vocabulary Grade Level: Geometry 1 ratio Key Concepts proportion Similarity Solve applied problems cross product property using the attributes of similar polygons similar triangles. scale factor Enduring Understandings: midsegment of a triangle Identify similar Properties of similar shapes can be used to solve problems and create similarity polygons. and critique inductive and deductive arguments Angle-Angle Similarity (AA) Side-Side-Side Similarity Show that two triangles (SSS) are similar using AA, Essential Questions: Side-Angle-Side Similarity SSS, and SAS Theorems How do I apply my knowledge of ratios and proportions to similiarity? (SAS) What makes polygon similiar? triangle proportionality Can I show two triangles are similar using AA, SSS, or SAS? theorem Can I use the Triangle Proportionality Theorem? midsegment theorem Use the triangle proportionality and midsegment theorem. Examples Resources Solve problems using Textbook resources ratio and proportions. 5 7 http://classzone.com Solve the proportion = similarity lesson x−2 x What value of x will make the http://www.gogeometry.com/geomet two triangles similar? ry/similarity_ratio_proportion_index Cabri Jr. Discovery 12 14 28 .html A). 14 B). 7 C). D). x activities 7 12 34 www.timath.com 51 • Constructing 40 similar triangles Write the ratio of the values of 4 42 nickels to the values of 10 dimes? 1 2 4 10 A). B). C). D). 5 5 10 4PUHSD 2009 Page 8 of 31Curriculum Division
- 9. Concept Unit Map Grade Level: Geometry 1 Key Concepts Quadrilaterals Identify and classify polygons. Key Vocabulary polygon side of a polygon Find the angle measures of Enduring Understandings: vertex of a polygon quadrilaterals. The building blocks of society are based on Polygons, especially four diagonal of a polygon sided objects (Quadrilaterals). parallelogram Identify properties of parallelograms. rhombus Essential Questions: rectangle Prove that a given quadrilateral Can I identify and classify polygons? square is a parallelogram. Can I find measures of quadrilaterals? trapezoid Can I apply properties of parallelograms to solve problems? bases, legs, and base angles of a Solve problems using Can I prove that a quadrilateral is a parallelogram? trapezoid properties of special isosceles trapezoid quadrilaterals including midsegment of a trapezoid rhombuses, rectangles, and squares. Solve problems using Examples Resources Classify the figure in as Textbook resources properties of trapezoids. Find the value of x and y. many way as possible http://classzone.com 14 x Cabri Jr. Discovery activities www.timath.com • y • Properties of 9 parallelograms • Angles in a quadrilateral Trapezoid JKLM JKLM is a rhombus. Find the value of x. x= y= 50° K 5x° J K (4y + 3)° 103° J L MPUHSD 2009 (11x)° Page 9 of 31 M 89°Curriculum Division L
- 10. Concept Unit Map Key Concepts Key Vocabulary Grade Level: Geometry 2 outcome Determine the theoretical theoretical probability probability of events, estimate Probability probabilities using experiments, experimental probability and compare the results. event counting principle Use concepts and formulas of Enduring Understandings: factorial notation area to calculate geometric Understanding theoretical probability makes you better informed when arrangements probabilities. making future decisions. combinations Determine the number of permutations possible outcomes of an event. Essential Questions: Apply appropriate means of How do you determine the number of possible outcomes of an event? computing the number of possible arrangements of items What is the difference between theoretical and experimental probability? using permutations where order How can you determine the number of arrangements that can be made matters, and combinations where given an event? Resources order does not matter. Algebra Help How can you determine the number of combinations in an event? http://www.algebrahelp.com/index.js How can you use theoretical probability to make predictions? p Apply the addition and multiplication principles of Textbook resources counting and represent these http://classzone.com principles algebraically using Examples The radii of the concentric Combinations and permutations calculator factorial notation. circles on a dart board are 2 in, http://www.mathsisfun.com/combinat Which of the following 4 in, 6 in, and 8 in respectively. orics/combinations-permutations- expressions could be used to find calculator.html the number of four-letter arrangements that can be made • Use simulators to from the letters in the word generate experimental ALGEBRA? probability data 7! • Compute the number of A) 7! B) 7 P4 C) 4 P7 D) What is the probability of getting 4! possible arrangement at least 20 points on the first with permutations and throw? combinations 5 3 1 9 • Computational use A) B) C) D) 16 4 2 16PUHSD 2009 Page 10 of 31Curriculum Division
- 11. Concept Unit Map Key Vocabulary Key Concepts Grade Level: Geometry 2 measures of center mean, median, mode, range Draw inferences about data sets Data Analysis from lists, tables, matrices, and collect data plots. data analysis frequency Organize collected data into an Enduring Understandings: tables appropriate graphical Data analysis is used to make important decisions in all facets of our matrices representation. lives, including education and corporate America. lists make predictions and Display data, make predictions observations and observations. Essential Questions: make inferences Can I draw inferences about data sets from lists, tables, matrices and interval Make inferences by comparing plots? data sets. sample Can I organize collected data into an appropriate graphical population Determine the most appropriate representation? misrepresentation and distortions measure of center for a given Can I make inferences by comparing data sets? of data display situation. Can I identify misrepresentations and distortions in displays of data? network graphs Identify misrepresentations and adjacency matrices distortions in displays of data. Examples Determine the shortest route for the Vertex, edges recycling trucks that start and end at Solve network problems using A? Resources graphs and matrices Danesha is choosing the number Textbook resources of events to have this year for her http://classzone.com community service club. 4 Comparing data activity Members of the club e-mail http://www.northcanton.sparcc.org/~tec • Enter data sets into lists Danesha with the number of 2 hnology/excel/files/comparing_data.ht to create graphical 3 ml representations, then events they prefer. If Danesha Path puzzles use 1-VAR Stats under wants to choose the most popular 6 http://www.mathmaniacs.org/lessons/1 number of events that were e- 2 4 LISTS to compute the 2-euler/PencilPuzzles.html measures of central mailed to her, which measure of tendency centreer should she use to make 7 • Computational use 5 her choice? 11 A) Mean B)Median C)Mode D) RangePUHSD 2009 Page 11 of 31Curriculum Division
- 12. Concept Unit Map Grade Level: Geometry 2 Key Vocabulary Key Concepts transformation Know that after any of Transformations translation image transformation (rotation, reflection, and translation), the shape still has pre-image the same size, area, angles and reflection side lengths. Enduring Understandings: line of symmetry A clear understanding of transformations allows students to visualize the dilation Know that if one shape can become movement of objectives in three dimensional space and on a plane. glide reduction another using roation, reflection, enlargement and translation, then the two shapes rotation are called congruent. Essential Questions: center of rotation What does the word transformation means? angle of rotation Understand that during What is the relationship of Geometry transformations and symmetry? rotational symmetry transformation called a dilation, Can you use more than one transformation to describe the relationship? (enlargement or reduction), the tessellation shape becomes bigger or smaller. isometry plane figure Dilation does not result in scale factor congruent shapes, the shapes will be similar. Same shapre, yet different sizes are similar shapes. TRANSFORMATIONS Resources Discover the line of reflection, the Cabri Jr. Translation center of rotation, and the center of Java Applet dilation. Sketchpad Examples TI Calculator Textbook Resources Cabri Jr. Discovery activities http://classzone.com www.timath.com Reflection Dilation • Students can use the transformation tool on Cabri Jr. to explore translation, reflections, rotation or dilations Rotation 45°PUHSD 2009 Page 12 of 31Curriculum Division
- 13. Key Concepts Concept Unit Map Key Vocabulary Grade Level: Geometry 2 radical Understand that the term “solving the triangle” means that if we start radicand with a right triangle and know any Right Triangle Trigonometry 45°-45°-90° triangle two sides, we can find or solve for 30°-60°-90° triangle the unknown side. trigonometric ratios Enduring Understandings: Demonstrate understanding of the following equation types to solve leg opposite an angle Investigate the fundamental concepts problems involving right triangles and use them in real life situations. leg adjacent to an angle behind trigonometry: three basic trig tangent functions and how to determine • Pythagorean Theorem which trig function to use. sine • Definition of Sine cosine • Definition of Cosine special right triangle • Definition of Tangent Know that recognizing special right • Sum of Angles triangles in geometry can help you to problem solve. A special right Resources triangle is a right triangle whose sides Essential Questions: Cabri Jr. are in a particular ratio. Could we use Pythagorean Theorem with triangles that are not right Java Applet ( 30 , 60 , 90 ) & ( 45 , 45 , 90 ) triangles? Sketchpad How do you determine if a triangle is a right triangle? Computer Use SOHCAHTOA to memorize the TI Calculator three main trigonometric functions. Examples Pythagorean Theorem Power Point opposite If we know the two legs of B Textbook Resources SOH → sin x = hypotenuse a right triangle, we can http://classzone.com a 2 + b2 = c 2 c2 adjacent solve for the hypotenuse CAH → cos x = a2 hypotenuse using the formula: opposite C A TOA → tan x = b2 adjacent where a and b are the lengths of the two legs of Cabri Jr. Discovery activities the triangle, and Relative to www.timath.com h is the hypotenuse angle A, this • Investigating special is how the triangles sides of a right • Use the calculator to find sine, cosine, or tangents triangle would be labeledPUHSD 2009 Page 13 of 31Curriculum Division
- 14. Concept Unit Map Key Concepts Grade Level: Geometry 2 Key Vocabulary chord Identify and name parts of a Circles secant circle tangent minor arc Find the circumference and area Enduring Understandings: major arc of circles Circular objects are abundant in our Universe formed by nature, arc length specifically by the force of gravity. inscribed angle Use properties of tangents intercepted arc diameter Use properties of arcs and Essential Questions: radius central angles; identify and How are segments and lines related to circles? name semi circle How can properties of tangents, arcs, chords and inscribed angles be circumscribed Solve problems by applying the used to solve problems? point of tangency relationship between radii, What is the relationship between radii, diameters, chords and tangents? circumference diameters, chords, and tangents. area inscribed polygon Determine measures of central Examples locus and inscribed angles and their intercepted arcs. Use the diagram below to The diagram below show special find the intercepted arc or segments and lines of a circle. Resources inscribed angle Textbook resources http://classzone.com Cabri Jr. Discovery activities A F www.timath.com A J • Relationship between 146 circumference and 37° diameter 100° I • Measure angles and G C D related arc C D 70° H B E B EPUHSD 2009 Page 14 of 31Curriculum Division
- 15. Concept Unit Map Key Concepts Grade Level: Geometry 2 Identify how many vertices, sides, Two-Dimensional Figures Key Vocabulary equilateral polygon edges a polygon has. equiangular polygon regular polygon Compare and contrast polygons Enduring Understandings: concave/convex two-dimensional figures. Analyze characteristics and properties of two-geometric shapes and apothem develop mathematical arguments about the relationships of geometric area Identify and classify polygons perimeter figures in the real world. circumference using manipulatives and create two-dimensional figures. circle center of the circle Essential Questions: radius Identify and describe polygons What attributes define a polygon? diameter (concave, convex, regular, What is the relationship between the angles and the number of sides in a sector area pentagon, hexagon, and n-gonal, polygon? central angle interior angle circles and sector areas). How do you calculate the measure of the interior and exterior angles of a exterior angle polygon? height of trapezoid Identify and describe properties of How do you measure area and perimeter of polygons? midsegment of trapezoid a circle, kite, trapezoid, Polygon Names parallelogram, rectangle, square, triangle and rhombus. Examples quadrilateral Find the area of a circle with a radius parallelogram of 10 millimeters. rhombus Use congruent relationships of Find the area of the polygon. Round to the nearest tenth. square two-dimensional figures to A. 31.4 mm2 C. 314.2 mm2 rectangle 2 determine unknown values, such B. 314.1 mm D. 3,142 mm2 trapezoid as angles, side lengths, perimeter kite or circumference and areas. pentagon hexagon Apply the interior and exterior A. 42 ft2 C. 30 ft2 Find the area of the parallelogram. heptagon angle sum of convex polygons to B. 60 ft2 D. 36 ft2 octagon solve problems. decagon n-gon A. 16 m2 C. 36 m2 Cabri Jr. Discovery activities B. 81 m2 D. 13 m2 Resources www.timath.com Cabri Jr. • Volume Sketchpad • Area of the circle TI Calculator Power PointPUHSD 2009 Page 15 of 31 Textbook ResourcesCurriculum Division http://classzone.com
- 16. Concept Unit Map Key Concepts Grade Level: Geometry 2 Identify how many vertices, sides, Three-Dimensional Figures Key Vocabulary edges, and/or faces of a three- polyhedron dimensional figure. base Enduring Understandings: face Compare and contrast three- Analyze characteristics and properties of three-dimensional geometric edge dimensional figures. shapes and develop mathematical arguments about geometric surface area relationships in the real world. lateral area Use manipulatives to create three-dimensional figures slant height height Essential Questions: prism Apply surface area and volume formulas for prisms, pyramids, How do you determine which formula to use with a given solid? pyramid cylinders, cones and spheres. How does changing a dimension affect the volume? cylinder How do I find lateral areas of prisms and cylinders? cone How do I find lateral areas of cones? sphere Draw three-dimensional figures with appropriate labels and make How do I identify and use three-dimensional figures and use hemisphere a three-dimensional model from nets to draw them? net a net. Draw, describe, and analyze solid Examples geometry figures. Find the surface area Which solid has a net Find the volume of of the cylinder below like the one shown? the rectangular prism Resources Examples Cabri Jr. to the nearest tenth. below. Java Applet Cabri Jr. Discovery activities Sketchpad www.timath.com Compute • Volume TI Calculator • Area of the circle Power Point A. Triangular prism Textbook Resources B. Triangular A. 25 cm3 www.classzone.com E. 1256.6 cm 2 pyramid B. 200 cm3 F. 5026.5 cm 2 C. Rectangular C. 392 cm3 G. 1658.8 cm 2 prism D. 480 cm3 H. 2513.3 cm 2 D. Square pyramidPUHSD 2009 Page 16 of 31Curriculum Division
- 17. Geometry 1-2 Scope and Sequence Quarter 1 Quarter 2 Quarter 3 Quarter 4Points, Lines & Planes Attributes & Properties of Triangles Probability & Statistics 2-3 dimensional figures • Lines • Triangle Sum = 180 degrees • Measures of center • Changing dimensions Distance/Midpoint formula • Types (obtuse, acute, scalene, right, • Data collection with/without technology • Area of circle and sector Segment addition isosceles, equilateral, equiangular) • Data analysis • Nets Segment congruence • Angle relation to side length • Theoretical probability • Volume & Surface Area Bisectors • Triangle inequality • Geometric probability Cylinders Collinear • Distance formula • Permutation Cones • Angles • Pythagorean theorem • Combinations Sphere Angle addition • Triangle congruence • Network Graphs Prisms Angle congruence SSS, SAS, ASA, AAS, HL Pyramids Angle bisectors Label triangles by • Angle Measures corresponding parts Transformations (single and multiple) Circles Vertical Construct a congruent triangle • Translations, Rotations, Reflections, • Segments and angles formed Linear pair Dilations (scale factor) by circles Complementary and Similarity • Arcs and chords supplementary • Triangle Similarity • Secants and tangents Obtuse, right or acute SSS, SAS, AA, Right Triangles & Trigonometry • Central, inscribed, andPerpendicular & Parallel Lines Label triangles by • Special Triangles circumscribed • Lines and angles corresponding parts 45-45-90 and 30-60-90 • Degrees of arcs • Proof and perpendicular lines • Scale factor similar Polygons • Right Triangles • Identify center and radius • Parallel lines and transversals Proportions Pythagorean theorem • Proving lines are parallel Corresponding parts Pythagorean inequality • Using properties of parallel lines Quadrilaterals • Trigonometry • Parallel & Perpendicular Lines in the • Hierarchy of Quadrilaterals Set up basic ratios coordinate plane • Attributes of quad. Solve using special triangles Logical Reasoning (30/60/45) • Simple formal proof • Counterexample • If...then • Inductive • Deductive • Conjecture • Inverse, converse and contrapositivePUHSD 2009 Page 17 of 31Curriculum Division
- 18. Concept Unit Map Key Concepts Grade Level: Geometry 1 Honors Key Vocabulary: collinear/linear Using Points, Lines, Planes, and Angles coplanar/plane Identify points, lines, rays, line segments, angles, and planes in a two- skew and three-dimensional figure and point notation. segment Enduring Understanding: ray Solve problems related to Recognizing the relationship of points, lines, and planes and the geometric figures they line form is the foundation for applying mathematics to the real world angle complementary, supplementary, congruent angles, bisectors, angles or segment segment concepts. perpendicular bisector Essential Question(s): linear pair What is the difference between a postulate and a theorem? How do you use slope to determine the relationship between two lines? complementary/supplementary Solve problems created by parallel lines cut by a transversal. What is the distance and midpoint between two points? vertical angles How can you find the solutions to a system of equations? transversal parallel/perpendicular Identify lines, segments, points, Examples adjacent angles angles, and arcs of circles. Identify a line segment, ray, plane, midpoint Given: M is the midpoint of AB . postulate line, a pair of skewed lines, Determine the distance and midpoint Find AB. theorem perpendicular segments, three between 2 points in a coordinate coplanar points, two collinear points. angle bisectors system. 4x-6 2x+10 congruent segments A M B B C Construct and apply concepts of angle bisectors, perpendicular bisectors and D Resources A congruent segments, in order to solve Algebra Help related problems. H E http://www.algebrahelp.com/index.js p Textbook resources G F http://classzone.com Cabri Jr. Discovery activities www.timath.com • Angle bisector • Midpoint of a linePUHSD 2009 Page 18 of 31Curriculum Division
- 19. Concept Unit Map Key Concepts Grade Level: Geometry 1 Honors Identify parallel lines, Perpendicular and Parallel lines Key Vocabulary perpendicular lines, skew lines, parallel lines parallel planes, and a line perpendicular lines perpendicular to a plane. skew lines Enduring Understandings: parallel planes Parallel and perpendicular lines provide the foundation for most physical line perpendicular to a plane Identify and solve problems using angles formed by structures. complementary angles perpendicular lines. supplementary angles transversal Identify congruent angles and Essential Questions: corresponding angles supplementary angles formed by What angle relationships are formed by parallel lines cut by a alternate interior angles parallel lines and a transversal. transversal? alternate exterior angles What angle relationships are formed by perpendicular lines? same-side interior angles Determine if lines are parallel How does slope determine the relationship between two lines? congruent or perpendicular based on slope How can angle relationships be used to determine measures of other converse or angle relationships. angles? theorem postulate perpendicular bisectors Apply properties, theorems, and constructions about Examples parallel lines, perpendicular Find the value of x. lines, and angles to prove theorems. Find the value of x. ( 2 x + 25) ° Resources Algebra Help Cabri Jr. Discovery ( 2 x − 10 ) ° http://www.algebrahelp.com/index.jsp activities www.timath.com ( x + 5) ° Textbook resources http://classzone.com • Vertical and Geometry lesson on parallel and perp. http://www.homeschoolmath.net/teaching/ adjacent angles 110° g/parallel_and_perpendicular.php • TransversalsPUHSD 2009 Page 19 of 31Curriculum Division
- 20. Key Concepts Concept Unit Map Key Vocabulary Grade Level: Geometry 1 Honors conjecture Use inductive reasoning to find patterns and use them to make inductive reasoning Logical Reasoning conjectures. counter-example patterns Prove a conjecture false by deductive reasoning finding a counterexample. Enduring Understandings: conditional statement Logical reasoning is essential to synthesize information, put the world in hypothesis Draw valid conclusions from perspective, and apply knowledge. conclusion conditional statements using Law of Detachment Law of Detachment and syllogism Syllogism. Essential Questions: inverse What conjecture can be made based on an observation? converse Determine triangle congruency using SSS, SAS, ASA, and How can you use a counter-example to prove a conjecture false? contrapositive AAS congruence postulates. How can a logical conclusion be made using Law of Detachment and if-then statement Syllogism? reflexive property How can you use triangle congruence theorems to prove two triangles symmetric property State the inverse, converse, and contrapositive of a given if-then are congruent? transitive property statement and determine if it is How do you write a paragraph proof? proofs true or false. writing proofs paragraph proofs Examples two-column proofs Critique inductive and deductive arguments concerning geometric ideas and Using the Law of Syllogism, which statement follows from the pair of relationships. true statements? If I do not get paid this week, then I cannot pay my bills. Resources Construct a formal deductive Algebra Help proof. http://www.algebrahelp.com/index.js If I cannot pay my bills, then the bill collectors will call. p Textbook resources http://classzone.com Cabri Jr. Discovery activities What is the next number you expect? 45, 90, 135, 180…….. www.timath.com • Conditional statements • ASA,SAS,SSS Triangle congruencePUHSD 2009 Page 20 of 31Curriculum Division
- 21. Concept Unit Map Key Concepts Key Vocabulary Grade Level: Geometry 1 Honors triangle Classify triangles according to equilateral triangle, isosceles lengths of sides and angle Attributes and Properties of Triangles triangle, scalene triangle, right measures. triangle, obtuse triangle, acute triangle, equiangular triangle Solve problems using triangle vertex sum and exterior angles Enduring Understandings: length theorems. Triangles are the support system for most objects that we use on a daily measure basis. Are you sitting on a triangle now? corollary Solve problems related to base interior angles angles, equilateral, and exterior angles equiangular triangle theorems. base Essential Questions: legs Use the Pythagorean Theorem How are triangles classified? base angles and Distance formula to find How do triangle attributes help solve problems related to angle hypotenuse unknown lengths. measures and side lengths? Pythagorean Theorem How are the Pythagorean Theorem and Distance formula related? square Determine and solve problems What is the relationship between segments formed by the intersection of square root using angle and side length distance the medians of a triangle? classify relationships. Can I show two triangles are congruent using AAS, ASA, SSS, SAS, or median of a triangle Solve problems using median and HL? centroid centroids of a triangle. triangle inequality coordinate proof Examples Decide whether a triangle could corresponding Solve problems using the triangle have the given angle measures. midsegment inequality theorem. Explain your reasoning. Angle-Angle-Side Theorem Find the value of x. 18°, 34°, 128° (AAS) Identify midsegments and the Angle-Side-Angle Postulate points of concurrency of a 102° (ASA) triangle. Side-Side-Side Postulate (SSS) Side-Angle-Side Postulate (SAS) x° 51° Classify the triangle with Show that two triangles are Hypotenuse-Leg Theorem (HL) the most specific name. congruent using AAS and HL Theorems, and ASA, SSS, and SAS postulates. Resources Algebra Help Cabri Jr. Discovery activities 8 9 www.timath.com http://www.algebrahelp.com/index.js p • Triangle Inequalities Textbook resources • Ratio of Right Triangles http://classzone.com • Distance in the coordinate 10 Geometry Interactive activities plane http://www.shodor.org/interactivate/PUHSD 2009 Page 21 of 31 activities/Curriculum Division
- 22. Concept Unit Map Key Concepts Grade Level: Geometry 1 Honors Key Vocabulary Identify and classify polygons. Quadrilaterals polygon side of a polygon vertex of a polygon Find the angle measures of Enduring Understandings: diagonal of a polygon quadrilaterals. The building blocks of society are based on polygons, especially four parallelogram sided objects called quadrilaterals. rhombus Identify properties of rectangle parallelograms. square Essential Questions: trapezoid What information on quadrilaterals are used to identify and classify polygons? Given some measures of quadrilaterals, how do I find missing measures of kite Prove that a given quadrilateral bases, legs, and base angles of a is a parallelogram. quadrilaterals? Can I apply properties of parallelograms, trapezoids and special quadrilaterals trapezoid to problem-solve? isosceles trapezoid Solve problems using Can I show that a quadrilateral is a Parallelogram? midsegment of a trapezoid properties of trapezoids. How do I justify the classification of a quadrilateral? convex concave Solve problems using Examples equilateral Classify the figure in as properties of special equiangular quadrilaterals including Find the value of x and y. many way as possible rhombuses, rectangles, and squares. 14 x Resources Justify the classification of a • y Textbook resources 9 http://classzone.com quadrilateral using coordinate geometry. Cabri Jr. Discovery activities Trapezoid JKLM JKLM is a rhombus. www.timath.com x= Find the value of x. • Properties of y= K parallelograms 50° • Angles in a quadrilateral J K 5x° (4y + 3)° 103° J L MPUHSD 2009 Page 22 of 31Curriculum Division 89° (11x)° M L

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