Thomas Jefferson High School for Science and Technology Program of Studies TJ Math 1

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1 Course Description: This course is designed for students who have successfully completed the standards for Honors Algebra I. Students will study geometric topics in depth, with a focus on building critical thinking and reasoning skills. Topics of study include inductive and deductive reasoning, understanding logic statements, writing proofs, parallel and perpendicular lines and their properties, congruent triangles and similarity, properties of triangles, quadrilaterals, and circles. The process standard focus will be reasoning. This set of standards includes emphasis on two- and three-dimensional reasoning skills, coordinate and transformational geometry, and the use of geometric models to solve problems. A variety of applications and some general problem-solving techniques, including algebraic skills, shall be used to implement these standards. Appropriate technology tools will be used to assist in teaching and learning. Any technology that will enhance student learning shall be used.

2 Resources and Activities Unit 1: Tools of ½ weeks Standard 4: PERFORM A VARIETY OF GEOMETRIC CONSTRUCTIONS The student will construct and justify the constructions of a) a line segment congruent to a given line segment; f) an angle congruent to a given angle Construction techniques are used to solve real-world problems in engineering, architectural design, and building construction. Construction techniques include using a straightedge and compass, paper folding, and dynamic geometry software. Note: Constructions are performed throughout the course as they are necessary and appropriate. The same essential understandings apply. Therefore, Standard 4 will implicitly appear throughout this document. Benchmark 4.a Construct a Line Segment Congruent to a Given Line Segment The student will construct and justify the constructions of a line segment congruent to a given line segment. Indicator 4.a.1 The student will construct a line segment congruent to a given line segment. Indicator 4.f.1 The student will construct an angle congruent to a given angle. TJHSST Expanded POS The student will understand and use the basic undefined terms and defined terms of geometry. The student will use the Segment Addition Postulate, Angle Addition Postulate, Linear Pair Postulate and Ruler Postulate The student will use angle postulates to find the measures of angles. The student will classify angles as acute, right, obtuse, or straight. The student will define and use angle pairs such as vertical angles, linear pairs, complementary and supplementary angles. Vocabulary: point, line, plane, collinear points, coplanar points, line segment, endpoint, ray, initial point, opposite rays, intersect, intersection, postulate, theorem, congruent segments, angle, sides and vertex of angle, congruent angles, measure of an angle, interior and exterior of an angle, acute-rightobtuse-straight angles, adjacent angles, midpoint, bisect, segment bisector, angle bisector, vertical angles, linear pair, complementary and supplementary angles

3 Unit 2: Reasoning and Proof ½ weeks Standard 1: CONSTRUCT AND JUDGE THE VALIDITY OF A LOGICAL ARGUMENT The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying the converse, inverse, and contrapositive of a conditional statement; b) translating a short verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; and d) using deductive reasoning. Essential Understandings Inductive reasoning, deductive reasoning, and proof are critical in establishing general claims. Deductive reasoning is the method that uses logic to draw conclusions based on definitions, postulates, and theorems. Inductive reasoning is the method of drawing conclusions from a limited set of observations. Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, and theorems. Logical arguments consist of a set of premises or hypotheses and a conclusion. Euclidean geometry is an axiomatic system based on undefined terms (point, line and plane), postulates, and theorems. When a conditional and its converse are true, the statements can be written as a biconditional, i.e., iff or if and only if. Logical arguments that are valid may not be true. Truth and validity are not synonymous. The Law of Detachment (also known as Modus Ponens) says that if p q is true, and p is true, then q must be true. Resources and Activities (Optional) Truth in Advertising Project Benchmark 1.a Construct and Judge the Validity of a Logical Argument The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include identifying the converse, inverse, and contrapositive of a conditional statement. Indicator 1.a.1 The student will identify the converse, inverse, and contrapositive of conditional statements. Indicator 1.a.2 Given and conditional statement, the student will identify the hypothesis and conclusion of the statement. Indicator 1.a.3 The student will disprove conjectures by counterexample. Indicator 1.a.4 The student will justify conjectures involving geometric relationships. Indicator 1.a.5 The student will write a plan for a geometric proofs. Indicator 1.a.6 The student will write geometric proof with more than four steps.

4 Benchmark 1.b Translate a Short Verbal Argument into Symbolic Form The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include translating a short verbal argument into symbolic form. Indicator 1.b.1The student will translate verbal arguments into symbolic form. Indicator 1.b.2 The student will recognize and use the symbols of formal logic. Indicator 1.b.3 The student will translate from symbolic argument to verbal argument. Benchmark 1.c Use Venn Diagrams to Represent Set Relationships The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include using Venn diagrams to represent set relationships. Indicator 1.c.1 The student will use Venn diagrams to represent set relationships. Indicator 1.c.2 The student will interpret Venn diagrams. Indicator 1.c.3 The student will solve problems with three sets using Venn diagrams. Benchmark 1.d Use Deductive Reasoning The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include using deductive reasoning. Indicator 1.d.1The student will determine the validity of a logical argument. Indicator 1.d.2 The student will use deductive reasoning to determine the validity of a logical argument. The student will use the following laws of deductive reasoning: Law of Syllogism, Modus Ponens (Law of Detachment), Modus Tollens, and the Law of Contrapositive. Indicator 1.d.3 The student will justify each step in solving a linear equation. Indicator 1.d.4 The student will evaluate the truth value of simple and compound statements as well as biconditionals using truth tables. The student will prove and apply theorems about angles including use of the Vertical Angle Theorem, Congruent Supplements Theorem, and Congruent Complements Theorem. TJHSST Expanded POS The students will use inductive reasoning when making conclusions based on observed patterns. The student will evaluate the truth value of conditional and biconditional statements as well as conjunctions, disjunctions and negations using truth tables Vocabulary: conjecture, inductive reasoning, deductive reasoning, counterexample, conditional statement, biconditional statement, hypothesis, conclusion, converse, negation, inverse, contrapositive, equivalent statements, logical argument, Law of Syllogism, Modus Ponens, Modus Tollens, Law of Contrapositive, truth table, Venn diagram, conjunction, disjunction, negation

5 Unit 3: Parallel and Perpendicular Lines weeks Standard 2: USE ANGLE RELATIONSHIPS TO DETERMINE IF TWO LINES ARE PARALLEL The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. Parallel lines intersected by a transversal form angles with specific relationships. Some angle relationships may be used when proving two lines intersected by a transversal are parallel. The Parallel Postulate differentiates Euclidean from non-euclidean geometries such as spherical geometry and hyperbolic geometry. Resources and Activities (Optional) Benchmark 2.a Use Angle Relationships to Determine if Two Lines Are Parallel The student will use the relationships between angles formed by two lines cut by a transversal to determine whether two lines are parallel. Indicator 2.a.1 The student will identify and classify the types of angles formed by two lines and a transversal and will solve problems using the angle relationships. Indicator 2.a.2 The student will use algebraic methods to solve problems using angle relationships. Benchmark 2.b Verify Parallelism Using Algebraic, Coordinate, and Deductive Methods The student will use the relationships between angles formed by two lines cut by a transversal to verify the parallelism, using algebraic and coordinate methods as well as deductive proofs. Indicator 2.b.1 The student will verify whether two lines are parallel using deductive proofs.

6 Benchmark 4.b Construct and Justify the Perpendicular Bisector of a Line Segment The student will construct and justify the constructions of the perpendicular bisector of a line segment. Indicator 4.b.1 The student will construct and justify the perpendicular bisector of a line segment. Benchmark 4.c Construct a Perpendicular to a Given Line from a Point not on the Line The student will construct and justify the constructions of a perpendicular to a given line from a point not on the line. Indicator 4.c.1 The student will construct a perpendicular to a given line from a point not on the line. Benchmark 4.d Construct a Perpendicular to a Given Line at a Point On the Given Line The student will construct and justify the constructions of a perpendicular to a given line at a given point on the line Indicator 4.d.1 The student will construct a perpendicular to a given line at a point on the line. Benchmark 4.a Construct a Line Segment Congruent to a Given Line Segment The student will construct and justify the constructions of a line segment congruent to a given line segment Indicator 4.a.1 The student will construct a line parallel to a given line. Vocabulary: parallel lines, skew lines, parallel planes, transversal, corresponding angles, alternate exterior angles, alternate interior angles, consecutive interior angles, flow proof, perpendicular, perpendicular bisector of a segment, triangle angle-sum theorem, parallel postulate, triangle exterior angle theorem, perpendicular postulate

7 Unit 4: Congruent Triangles ½ weeks Standard 6: PROVE TWO TRIANGLES ARE CONGRUENT USING A VARIETY OF METHODS The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Congruence has real-world applications in a variety of areas, including art, architecture, and the sciences. Congruence does not depend on the position of the triangle. Concepts of logic can demonstrate congruence or similarity. Congruent figures are also similar, but similar figures are not necessarily congruent. Benchmark 6.a Prove Triangle Congruence Using Algebraic/Coordinate/Deductive Methods The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Resources and Activities (Optional) Indicator 6.a.1 The student will use definitions/postulates/theorems to prove triangles congruent. Indicator 6.a.4 The student will map corresponding parts of congruent figures onto each other. Indicator 6.a.5 The student will use properties of congruent triangles to plane and write proofs using SSS, SAS, ASA, AAS and HL. Indicator The student will apply the theorems and corollaries about isosceles triangles.

8 Benchmark 4.e Construct the Bisector of a Given Angle The student will construct and justify the constructions of the bisector of a given angle. Indicator 4.e.1 The student will construct the bisector of a given angle. Standard 7: PROVE TWO TRIANGLES ARE SIMILAR USING A VARIETY OF METHODS The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Similarity has real-world applications in a variety of areas, including art, architecture, and the sciences. Similarity does not depend on the position of the triangle. Congruent figures are also similar, but similar figures are not necessarily congruent. Benchmark 7.a Prove Triangle Similarity Using Algebraic/Coordinate/Deductive Methods The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Indicator 7.a.4 The student will solve practical problems using congruence. TJHSST Expanded POS The student will use a compass and straightedge to construct a copy of a triangle using the SSS, SAS, and ASA Postulates. Vocabulary: triangle, vertex, adjacent sides of a triangle, legs of a right triangle, hypotenuse, lets of right triangle, base, interior angles, exterior angles, corollary, congruent, corresponding angles of congruent triangles, corresponding sides of congruent triangles, base angles of an isosceles triangle, legs of an isosceles triangle, vertex angle of an isosceles triangle, third angles theorem, isosceles triangle theorem, SSS, SAS, AAS, HL, isosceles triangle theorem, converse of isosceles triangle theorem, overlapping triangles

9 Unit 5: Relationships within Triangles ½ weeks Standard 5: APPLY THE TRIANGLE INEQUALITY PROPERTIES TO SOLVE PRACTICAL PROBLEMS The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a) order the sides by length, given the angle measures; b) order the angles by degree measure, given the side lengths; c) determine whether a triangle exists; and d) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. The longest side of a triangle is opposite the largest angle of the triangle and the shortest side is opposite the smallest angle. In a triangle, the length of two sides and the included angle determine the length of the side opposite the angle. In order for a triangle to exist, the length of each side must be within a range that is determined by the lengths of the other two sides. Resources and Activities (Optional) Hinge Theorem Activity Benchmark 5.a Order Sides of a Triangle by Length, Given Angle Measures The student, given information concerning the lengths of sides and/or measures of angles in triangles, will order the sides by length, given the angle measures Indicator 5.a.1 The student will, given angle measures, order the sides of a triangle by lengths. Benchmark 5.b Order Angles of a Triangle By Degree Measure Given Side Lengths The student, given information concerning the lengths of sides and/or measures of angles in triangles, will order the angles by degree measure, given the side lengths Indicator 5.b.1 The student will, given side lengths, order angles of a triangle by their measures. Benchmark 5.c Determine Whether a Triangle Exists The student, given information concerning the lengths of sides and/or measures of angles in triangles, will determine whether a triangle exists. Indicator 5.c.1 The student will, given lengths of three segments, determine if a triangle could be formed.

10 Benchmark 5.d Determine Range in which Length of the Triangle's Third Side Must Lie The student, given information concerning the lengths of sides and/or measures of angles in triangles, will determine the range in which the length of the third side must lie. Indicator 5.d.1 The student will, given two side lengths, determine the range of the third side length. Benchmark 5.e Solve Real-World Problems Given Information About Triangles The student, given information concerning the lengths of sides and/or measures of angles in triangles, will solve real-world problems. Indicator 5.e.1 The student will solve real-world problems given information about triangles. Benchmark 5.f Investigate Special Lines/Segments/Points Related to Triangles The student will investigate special lines, segments, and points related to triangles. Indicator 5.f.1 The student will investigate various attributes of a triangle. Indicator 5.f.2 The student will investigate and construct the points of concurrency in a triangle. Indicator 5.f.3 The student will determine the midsegments of a triangle, and use related properties. Indicator The student will use the Hinge Theorem and its converse to compare side lengths and angle measure in two triangles. Benchmark 4.h Construct polygons inscribed in a circle Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Indicator 4.h.1 Construct polygons inscribed in a circle. Indicator 4.i.1 Construct the inscribed and circumscribed circles for a triangle. TJHSST Expanded POS The student will write indirect proofs. Vocabulary: perpendicular bisector, equidistant from two points, distance from a point to a line, equidistant from to lines, perpendicular bisector of a triangle, concurrent lines, point of concurrency, circumcenter of a triangle, angle bisector of a triangle, median of a triangle, altitude of a triangle, midsegment of triangle, incenter of a triangle, centroid of a triangle, orthocenter of a triangle, indirect proof.

11 Unit 6: Polygons and Quadrilaterals weeks Standard 9: USE PROPERTIES OF QUADRILATERALS TO SOLVE PROBLEMS The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. The terms characteristics and properties can be used interchangeably to describe quadrilaterals. The term characteristics is used in elementary and middle school mathematics. Quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals. Characteristics of quadrilaterals can be used to identify the quadrilateral and to find the measures of sides and angles. Benchmark 9.a Use Properties of Quadrilaterals to Solve Real-World Problems The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. Resources and Activities (Optional) Polygon Sort Indicator 9.a.1 The student will solve problems using quadrilateral properties. Indicator 9.a.2 The student will prove that quadrilaterals have specific properties. Indicator 9.a.3 The student will prove the characteristics of quadrilaterals using a variety of methods. Indicator 9.a.4 The student will prove properties of angles for a quadrilateral inscribed in a circle. Indicator 9.a.5 The student will use properties to find measures of sides/diagonals/angles.

12 Standard 10: SOLVE REAL-WORLD PROBLEMS INVOLVING ANGLES OF POLYGONS The student will solve real-world problems involving angles of polygons. Two intersecting lines form angles with specific relationships. An exterior angle is formed by extending a side of a polygon. The exterior angle and the corresponding interior angle form a linear pair. The sum of the measures of the interior angles of a convex polygon may be found by dividing the interior of the polygon into nonoverlapping triangles. Benchmark 10.a Use Polygon Angle Measures to Solve Real-World Problems The student will solve real-world problems involving angles of polygons. Indicator 10.a.1 The student will solve problems involving the measures of interior and exterior angles. Indicator 10.a.3 The student will find the sum of the measures of the angles of a convex polygon. Indicator 10.a.4 The student will find the measure of each interior and exterior angle of a regular polygon. Indicator 10.a.5 The student will find the number of sides of a regular polygon given angle measures. Indicator 10.a.6 The student will identify, name, and classify polygons. Vocabulary: polygon, side, vertex, convex, nonconvex, concave, equilateral, equiangular, regular, diagonal, parallelogram, rhombus, rectangle, square, trapezoid, bases of trapezoid, base angels of a trapezoid, legs of a trapezoid, isosceles trapezoid, midsegment of a trapezoid, kite, interior and exterior angles of a polygon, pentagon, hexagon, heptagon, octagon, nonagon, dodecagon, n-gon.

13 Unit 7: Similarity weeks Standard 7: PROVE TWO TRIANGLES ARE SIMILAR USING A VARIETY OF METHODS The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Similarity has real-world applications in a variety of areas, including art, architecture, and the sciences. Similarity does not depend on the position of the triangle. Congruent figures are also similar, but similar figures are not necessarily congruent. Benchmark 7.a Prove Triangle Similarity Using Algebraic/Coordinate/Deductive Methods The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Indicator 7.a.1 The student will use definitions, postulates and theorems to prove triangles similar. Indicator 7.a.4 The student will solve practical problems using similarity. Indicator 7.a.5 The student will solve practical problems using properties of proportions. Indicator 7.a.6 The students will map corresponding parts of similar figures onto each other. TJHSST Expanded POS Indicator The student will state and apply the properties of similar polygons. Indicator The student will determine the geometric mean between two numbers. Indicator The student will apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Resources and Activities (Optional) For All Practical Purposes: On Size and Shape video Vocabulary: ratio, extended ratio, proportion, cross products property, properties of proportions, extremes, means, geometric mean, similar polygons, scale factor, side-splitter theorem, triangle-angle bisector theorem

14 Unit 8: Circles 10.4, weeks Standard 11: INVESTIGATE AND SOLVE PROBLEMS INVOLVING CIRCLES The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and Many relationships exist between and among angles, arcs, secants, chords, and tangents of a circle. All circles are similar. A chord is part of a secant. Real-world applications may be drawn from architecture, art, and construction Benchmark 11.a Investigate, Verify, and Apply Properties of Circles The student will use angles, arcs, chords, tangents, and secants to investigate, verify, and apply properties of circles. Indicator 11.a.1 The student will find angle and arc measures associated with central and inscribed angles. Indicator 11.a.2 The student will find and arc measures. Indicator 11.a.3 The student will find segment lengths. Indicator 11.a.4 The student will verify properties of circles using deductive reasoning. Indicator 11.a.5 The student will describe ways circles intersect. Indicator 11.a.6 The student will define and state properties of tangents, secants and chords. Indicator 11.a.7 The student will identify and apply appropriately the parts of a circle. Benchmark 11.b Solve Real-World Problems Involving Properties of Circles The student will use angles, arcs, chords, tangents, and secants to solve real-world problems involving properties of circles. Indicator 11.b.1 The student will solve real-world problems associated with circles. Resources and Activities (Optional) Benchmark 4.j Construct a tangent line from a point outside a circle to the circle Construct a tangent line from a point outside a given circle to the circle. Indicator 4.j.1 Construct a tangent line from a point outside a circle to the circle. Vocabulary: circle, center, radius, radii, diameter, congruent circles, chord, secant, tangent, tangent circles, concentric circles, common tangents, interior of a circle, exterior of a circle, point of tangency, central angle, minor arc, major arc, semicircle, congruent arcs, inscribed angle, intercepted arc, inscribed polygon, circumscribed polygon, tangent segment, secant segment, external segment,