Overview of Math Standards

Transcription

1 Geometry Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse audiences, Quality Producers who create intellectual, artistic and practical products which reflect high standards Complex Thinkers who identify, access, integrate, and use available resources Collaborative Workers who use effective leadership and group skills to develop positive relationships within diverse settings. Community Contributors who use time, energies and talents to improve the welfare of others Self-Directed Learners who create a positive vision for their future, set priorities and assume responsibility for their actions. Click here for more. Overview of Math Teams of teachers and administrators comprised the pk-12+ Vertical Alignment Team to draft the maps below. The full set of Kansas College and Career () for Math, adopted in 2010, can be found here. To reach these standards, teachers use Holt curriculum, resources, assessments and supplemented instructional interventions. of Mathematical Practice 1: Make sense of problems and persevere in solving them 2: Reason abstractly and quantitatively 3: Construct viable arguments and critique the reasoning of others 4: Model with mathematics 5: Use appropriate tools strategically 6: Attend to precision 7: Look for and make use of structure 8: Look for and express regularity in repeated reasoning. Click here for more. Additionally, educators strive to provide math instruction centered on: 1: Focus - Teachers significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards. 2: Coherence - Principals and teachers carefully connect the learning within and across grades so that students can build new understanding onto foundations. 3: Fluency - Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions. 4: Deep Understanding - Students deeply understand and can operate easily within a math concept before moving on. They learn more than the trick to get the answer right. They learn the math. 5: Application - Students are expected to use math concepts and choose the appropriate strategy for application even when they are not prompted. 6: Dual Intensity - Students are practicing and understanding. There is more than a balance between these two things in the classroom both are occurring with intensity. Click here for more. 1

2 Notes: Manhattan-Ogden USD 383 Math Year at a Glance Geometry Vocabulary terms are listed only in the unit they are first introduced. 1. Basics of Geometry 1.1 Points, Lines and Planes 1.2 Measuring and Constructing Angles 1.3 Using Midpoint and Distance Formulas 1.4 Perimeter and Area in the Coordinate Plane??? 1.5 Measuring and Constructing Angles 1.6 Describing Pairs of Angles G-CO.A.1 G-CO.D.12 G-GPE.B.7 G-MG.A.1 Undefined terms Point Line Plane Collinear points Coplanar points Defined terms Line segment Endpoints Ray Opposite ray Intersection Postulate Axiom Coordinate Distance Construction Congruent segments Between Midpoint Segment bisector Angle Vertex Sides of an angle Interior of an angle Exterior of an angle Measure of an angle Acute angle Right angle How can you solve real life problems involving lines and planes? How can you measure and construct a line segment How can you find the midpoint and length of a line segment in a coordinate plane? How can you find the perimeter and area of a polygon in a coordinate plane? How can you measure and classify an angle? How can you describe angle pair relationships and use these descriptions to find angle measures? name points, lines, planes, segments, and rays. find segment lengths using the Ruler Postulate, the Segment Addition Postulate, midpoints, segment bisectors, and Distance Formula. classify polygons and angles. find perimeters and areas of polygons in the coordinate plane. Make sure students have a good grasp on this chapter. 1

3 2. Reasoning and Proofs 2.1 Conditional Statements? 2.2 Inductive and Deductive Reasoning?? 2.3 Postulates and Diagrams?? 2.4 Algebraic Reasoning 2.5 Proving Statements about Segments and Angles 2.6 Proving Geometric Relationships 2 G-CO.C.9, 10, 11 G-SRT.B.4 Obtuse angle Straight angle Congruent angles Angel bisector Complementary angles Supplementary angles Adjacent angles Linear pair Vertical angles Conditional statement If-then form Hypothesis Conclusion Negation Converse Inverse Contrapositive Equivalent statements Perpendicular lines Biconditional statement Truth value Truth table Conjecture Inductive reasoning Counterexample Deductive reasoning Line perpendicular to a plane Proof Two-column proof Theorem When is a conditional statement true or false? How can you use reasoning to solve problems? In a diagram, what can be assumed and what needs to be labeled? How can algebraic properties help you solve an equation? How can you prove a mathematical statement? How can you use a flowchart to prove a mathematical statement? construct congruent segments and angles, and bisect segments and angles. write conditional and biconditional statements. use inductive and deductive reasoning. use properties of equality to justify the steps in solving equations and to find segment lengths and angle measures. write twocolumn proofs, flowchart proofs, and focus on the basics

4 3. Parallel and Perpendicular Lines 3.1 Pairs of Lines and Angles 3.2 Parallel Lines and Transversals 3.3 Proofs with Parallel Lines 3.4 Proofs with Perpendicular Lines 3.5 Equations of Parallel and Perpendicular Lines G-CO.A.1 G-CO.C.9, D.12 G-GPE.B.5, 6 Flowchart proof Paragraph proof Parallel lines Skew lines Parallel lines Transversal Corresponding angles Alternate interior angles Alternate exterior angles Consecutive interior angles Distance from a point to a line Perpendicular bisector Directed line segment What does it mean when two lines are parallel, intersecting, coincident, or skew? When two parallel lines are cut by a transversal which of the resulting pairs of angles are congruent? For which of the theorems involving parallel lines and transversals is the converse true? What conjectures can you make about perpendicular lines? How can you write an equation of a line that is parallel or perpendicular to a given line and passes through a given point? paragraph proofs. identify planes, pairs of angles formed by transversals, parallel lines, and perpendicular lines. use properties and theorems of parallel lines. prove theorems about parallel lines and about perpendicular lines. write equations of parallel lines and about perpendicular lines. find the distance from a point to a line. 3

5 4. Transformations 4.1 Translations 4.2 Reflections 4.3 Rotations 4.4 Congruence and Transformations 4.5 Dilations 4.6 Similarity and Transformations 4 G-CO.A.2, 3, 4, 5 G-CO.B.6 G-MG.A.3 G-SRT.A.1a, 1b, 2 Vector Initial point Terminal point Horizontal component Vertical component Component form Transformation Image Preimage Translation Rigid motion Composition of transformations Reflection Line of reflection Glide reflection Line symmetry Line of symmetry Rotation Center of rotation Angles of rotation Rotational symmetry Center of symmetry Congruent figures Congruence transformation Dilation Center of dilation Scale factor Enlargement Reduction Similarity transformation How can you translate a figure in a coordinate plane? How can you reflect a figure in a coordinate plane? How can you rotate a figure in a coordinate plane? What conjectures can you make about a figure reflected in two lines? What does it mean to dilate a figure? When a figure is translated, reflected, rotated, or dilated in the plane, is the image always similar to the original figure? perform translations, reflections, rotations, dilations, and compositions of transformations. solve real-life problems involving transformations. identify lines of symmetry and rotational symmetry. describe and perform congruence transformations and similarity transformations.

6 Similar figures 5. Congruent 5.1 Angles of 5.2 Congruent Polygons 5.3 Proving Triangle Congruence by SAS 5.4 Equilateral and Isosceles 5.5 Proving Triangle Congruence by SSS 5.6 Proving Triangle Congruence by ASA and AAS 5.7 Using Congruent 5.8 Coordinate Proofs??? G-CO.C.10 G-MG.A.1, 3 G-CO.B.7, 8 G-CO.D.13 G-SRT.B.5 G-GPE.B.4 Interior angles Exterior angles Corollary to a theorem Corresponding parts Legs Vertex angle Base Base angles Hypotenuse Coordinate proof How are the angle measures of a triangle related? Given two congruent triangles, how can you use rigid motions to map one triangle to the other triangle? What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? What conjectures can you make about the side lengths and angles measures of an isosceles triangle? What can you conclude about two triangles when you know the corresponding sides are congruent? identify and use corresponding parts. use theorems about the angles of a triangle. Use SAS, SSS, HL, ASA, and AAS to prove two triangles congruent. prove constructions. write coordinate proofs. 5

7 What information is sufficient to determine whether two triangles are congruent? How can you use congruent triangles to make an indirect measurement? How can you use a coordinate plane to write a proof? What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle? 6. Relationships Within 6.1 Perpendicular and Angle Bisectors 6.2 Bisectors of 6.3 Medians and Altitudes of 6.4 Triangle and Midsegment Theorem 6.5 Indirect Proof?? And Inequalities in One Triangle 6.6 Inequalities in Two G-CO.C.9, 10 G-MG.A.1, 3 G-CO.D.12 G-C.A.3 Equidistant Concurrent Point of concurrency Circumcenter Incenter Median of a triangle Centroid Altitude of a triangle Orthocenter Midsegment of a triangle Indirect proof What conjectures can you make about the perpendicular bisectors and the angle bisectors of a triangle? What conjectures can you make about the medians and altitudes of a triangle? How are the midsegments of a triangle related to the sides of the triangle? How are the sides related to the angles of a triangle? How understand and us angle bisectors and perpendicular bisectors to find measures. find and use the circumference, incenter, centroid, and orthocenter of a triangle. use the Triangle Midsegment Theorem and the Triangle 6

8 are any two sides of a triangle related to the third side? Inequality Theorem. 7. Quadrilaterals and Other Polygons 7.1 Angles of Polygons 7.2 Properties of Parallelograms 7.3 Proving that a Quadrilateral is a parallelogram 7.4 Properties of Special Parallelograms 7.5 Properties of Trapezoids and kites 8. Similarity 8.1 Similar Polygons 8.2 Proving Triangle Similarity by AA 7 G-CO.C.11 G-SRT.B.5 G-MG.A.1, 3 G-SRT.A.2, 3 G-MG.A.1, 3 G-SRT.B.4, 5 G-GPE.B.5, 6 Diagonal Equilateral polygon Equiangular polygon Regular polygon Parallelogram Rhombus Rectangle Square Trapezoid Bases of a trapezoid Base angles of a trapezoid Legs of a trapezoid Isosceles trapezoid Midsegment of a trapezoid kite Corresponding parts of similar polygons Corresponding lengths of similar polygons If the two sides of one triangle are congruent to two sides of another triangle, what can you say about the third sides of the triangles? What is the sum of the measures of the interior angles of a polygon? What are the properties of parallelograms? How can you prove that a quadrilateral is a parallelogram? What are the properties of the diagonals of rectangles, rhombuses, and squares? What are some properties of trapezoids and kites? How are similar polygons related? What can you conclude about two triangles when you know write indirect proofs. find and use the interior and exterior angle measures of polygons. use properties of parallelograms and special parallelograms. prove that a quadrilateral is a parallelogram. identify and use properties of trapezoids and kites. use AA, SSS, and SAS Similarity Theorems to

9 8.3 Proving Triangle Similarity by SSS and SAS 8.4 Proportionality Theorems 9. Right and Trigonometry 9.1 The Pythagorean Theorem 8 G-SRT.B.4, 5 G-SRT.C.6, 7, 8 G-SRT.D.9, 10, 11 G-MG.A.1, 3 Perimeters of similar polygons theorem Areas of similar polygons theorem Angle-Angles (AA) similarity theorem Side-Side-Side (SSS) similarity theorem Side-Angle-Side (SAS) similarity theorem Triangle proportionality theorem Converse of the Triangle proportionality theorem Three Parallel Lines Theorem Triangle angle bisector theorem Pythagorean triple Geometric mean Trigonometric ratio Tangent Angle of elevation Sine that two pairs of corresponding angles are congruent? What are two ways to use corresponding sides of two triangles to determine that the triangles are similar? What proportionally relationships exist in a triangle intersected by an angle bisector or by a line parallel to one of the sides? How can you prove the Pythagorean Theorem? What is the relationship among the side lengths of 45 - prove triangles are similar. decide whether polygons are similar. use similarity criteria to solve problems about lengths, perimeters, and areas. prove the slope criteria using similar triangles. use the Proportionality Theorem and other similar proportionality theorems. use the Pythagorean Theorem and the Converse of the Pythagorean Theorem.

10 9.2 Special Right 9.3 Similar Right 9.4 The Tangent Ratio 9.5 The Sine and Cosine Ratios 9.6 Solving Right 9.7 Laws of Sines and Cosines?? Cosine Angle of depression Inverse tangent Inverse sine Inverse cosine Solve a right triangle Law of Sines Law of Cosines triangles? triangles? How are altitudes and geometric means of right triangles related? How is a right triangle used to find the tangent of an acute angle? Is there a unique right triangle that must be used? How is a right triangle used to find the sine and cosine of an acute angle? Is there a unique right triangle that must be used? When you know the lengths of the sides of a right triangle, how can you find the measures of the two acute angles? use geometric means. find side lengths and solve real-life problems involving special right triangles. find the tangent side, sine, and cosine ratios and use them to solve real-life problems. use the Law of Sines and Law of Cosines to solve triangles. What are the Law of Sines and the Law of Cosines? 10. Circles 10.1 Lines and Segments That Intersect Circles 9 G-CO.A.1 G-C.A.1, 2, 3, 4 G-MG.A.1, 3 G-CO.D.13 G-GPE.A.1 Circle Center Radius Chord Diameter Secant What are the definitions of the lines and segments that intersect a circle? How are circular arcs measures? identify chords, diameters, radii, secants, and tangents of circles.

11 10.2 Finding Arc Measures 10.3 Using Chords 10.4 Inscribed Angles and Polygons 10.5 Angle Relationships in Circles 10.6 Segment Relationships in circles 10.7 Circles in the Coordinate plane?? 11. Circumference, Area, and Volume 10 G-GPE.B.4 G-GMD.A.1,2, 3 G-GMD.B.4 G-C.B.5 Tangent Point of tangency Tangent circles Concentric circles Common tangent Central angle Minor arc Major arc Semicircle Measure of a minor arc Measure of a major arc Adjacent arcs Congruent circles Congruent arcs Similar arcs Inscribed angle Intercepted arc Subtend Inscribed polygon Circumscribed circle Circumscribed angle Segments of a chord Tangent segment Secant segment External segment Standard equation of a circle Circumference Arc length Radian Population density What are two ways to determine when a chord is a diameter of a circle? How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other? When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed? What relationships exist among the segments formed by two intersecting chords or among segments of two secants that intersect outside a circle? What is the equation of a circle with center (h, k) and radius r in the coordinate plane? How can you find the length of a circular arc? find angle and arc measures. use inscribed angles and polygons and circumscribed angles. use properties of chords, tangents, and secants to solve problems. write and graph equations of cirlces. measure angles in radians.

12 11.1 Circumference and Arc Length 11.2 Areas of Circles and Sectors 11.3 Areas of Polygons 11.4 Three- Dimensional Figures 11.5 Volumes of Prisms and Cylinders 11.6 Volumes of Pyramids 11.7 Surface Areas and Volumes of Cones 11.8 Surface areas and Volumes of Spheres 12. Probability 12.1 Sample Space and Probability 12.2 Independent and Dependent Events 11 G-CO.A.1 G-MG.A.1, 2, 3 G-C.B.5 S-CP.A.1, 2, 3, 4, 5 S-CP.B. 6, 7, 8, 9 Sector of a circle Center of a regular polygon Radius of a regular polygon Apothem of a regular polygon Central angle of a regular polygon Polyhedron Face Edge Vertex Cross section Solid of revolution Axis of revolution Volume Cavalieri s Principle Density Similar solids Lateral surface of a cone Chord of a sphere Great circle Probability experiment Outcome Event Sample space Probability of an event Theoretical probability How can you find the area of a sector of a circle? How can you find the area of a regular polygon? What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? How can you find the volume of a prism or cylinder that is not a right prism or right cylinder? How can you find the volume of a pyramid? How can you find the surface area and the volume of a cone? How can you find the surface area and the volume of a sphere? How can you list the possible outcomes in the sample space of an experiment? How can you determine whether two evens are independent or dependent? find arc lengths and areas of sectors of circles. find areas of rhombuses, kites, and regular polygons. find and use volumes of prisms, cylinders, pyramids, cones, and spheres. describe crosssections and solids of revolution. find the experimental and theoretical probability of an event.

13 12.3 Two-way tables and Probability 12.4 Probability of Disjoint and Overlapping Events 12.5 Permutations and Combinations 12.6 Binomial Distributions Geometric probability Experimental probability Independent events Dependent events Conditional probability Two-way table Joint frequency marginal frequency joint relative frequency marginal relative frequency conditional relative frequency compound event overlapping events disjoint events mutually exclusive events permutation n factorial combination random variable probability distribution binomial distribution binomial experiment How can you construct and interpret a two-ways table? How can you find probabilities of disjoint and overlapping events? How can a tree diagram help you visualize the number of ways in which two or more events can occur? How can you determine the frequency of each outcome of an event? find probabilities of independent and dependent events. use conditional relative frequencies to find conditional probabilities. use formulas for the number of permutations and the number of combinations. construct and interpret probability distributions and binomial distributions. 12