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Regional Integrated Geometry Curriculum UNIT: Informal & Formal Proofs TOPIC: Logic Time frame for unit: Time frame for topic: 10 days Prior Knowledge A.RP.2 Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture A.RP.11 Use a Venn diagram to support a logical argument Content Strand G.G.24 Determine the negation of a statement and establish its truth value G.G.25 Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true G.G.26 Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalence G.G.27 Write a proof (ex. indirect proof) arguing from a given hypothesis to a given conclusion Concepts Logic a. translate written expressions to symbolic representation (p, q, V, →, etc) b. negation c. conjunction d. disjunction e. conditional, hypothesis, conclusion, hidden conditional f. biconditional g. truth values h. converse (change order) i. inverse (insert negation) j. contrapositive (change order and negate) k. logically equivalent Logic Proofs a. b. c. d. e. contrapositive detachment syllogism/chain rule (Modus Tollens) deMorgan’s disjunctive inference Essential Questions How will I use logical reasoning to help me make decisions in my life? How could logic be used to determine if I’ve been tricked by someone? Or by an advertisement? Or a commercial? Process/skills G.PS.4 Construct various types of reasoning, arguments, justifications and methods of proof for problems G.RP.3 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion G.RP.7 Construct a proof using a variety of methods (e.g., deductive, analytic, transformational) G.RP.8 Devise ways to verify results or use counterexamples to refute incorrect statements G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid G.CM.12 Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing G.CN.6 Recognize and apply mathematics to situations in the outside world G.R.7 Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person’s argument have a logical foundation) Vocabulary antecedent argument biconditional chain rule (syllogism) compound statement conclusion conditional statement conjecture conjunction consequence contradiction contrapositive converse deductive reasoning deMorgan’s detachment disjunctive inference hypothesis inverse justify logic logical argument logical equivalence Modus Tollens negation reason statement tautology truth value valid/invalid argument Suggested assessments Pre-assessment Regents/State exams Projects Participation Q-A Responses Oral responses Conversations Resources On-spot checks of class work Homework Observations of students’ work/behaviors Students’ explanations Draw a picture Tests and Quizzes T/F Multiple Choice Written Response Ticket out the door See Blackboard for specific Assessments* Regional Integrated Geometry Curriculum UNIT: Informal/ Formal Proofs TOPIC: Triangles - congruent Time frame for unit: Time frame for topic: 20 days Prior Knowledge 5.G.6 Classify triangles by properties of their angles and sides 5.G.9 Identify pairs of congruent triangles 5.G.10 Identify corresponding parts of congruent triangles 7.G.5 Identify the right angle, hypotenuse, and legs of a right triangle 8.G.1 Identify pairs of vertical angles as congruent 8.G.2 Identify pairs of supplementary and complementary angles 8.G.4 Determine angle pair relationship with two parallel lines cut by a transversal Content Strands G.G.27 Write a proof arguing from a given hypothesis to a given conclusion G.G.28 Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles G.G.29 Identify corresponding parts of congruent triangles Concepts Identify corresponding parts (sides, angles, appropriate naming) Identify included side, included angle Congruent () – define & recognize Establish the format of a two-column proof SSS SAS ASA AAS HL – right triangle CPCTC—Corresponding Parts of Congruent Triangles are Congruent Essential Questions How do you analyze the given information to determine the strategy(s) needed to complete a proof? How much information do you need to know to determine whether two triangles are congruent? How can you show the logical progression of a proof? Why would it be important to know that two triangles (polygons) are congruent? Process/skills G.PS.4 Construct various types of reasoning, arguments, justifications and methods of proof for problems G.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions G.RP.3 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion G.RP.7 Construct a proof using a variety of methods (e.g., deductive, analytic, transformational) G.CM.1 Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid G.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts G.R.8 Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent) Vocabulary Alternate interior angles Altitude Analytical proof Angle Angle addition & subtraction Angle bisector Assumption statement Bisector Complementary angles Corresponding parts Deductive reasoning Equidistant Given Hypotenuse Included side & angle Indirect proof Inductive reasoning Interior angles Leg Midpoint Paragraph Proof Parallel Lines Perpendicular Perpendicular bisector Proof Proof by Contradiction Reflexive Statement/Reason Two column proofs Substitution Supplementary angles Transversal Suggested assessments Pre-assessment Regents/State exams Projects Participation Q-A Responses Oral responses Conversations On-spot checks of class work Homework Observations of students’ work/behaviors Students’ explanations Resources Review & extend from the previous units addition & subtraction properties (angles and sides) altitude (GG21) base angles of an isosceles triangle (GG31) bisector & angle bisector (GG17) median (GG21) midpoint (GG18) // lines, transversals, and angles (GG35) perpendicular lines & right angles (GG18) reflexive, symmetric, and transitive properties substitution/transitive property two angles forming a linear pair vertical angles Draw a picture Tests and Quizzes T/F Multiple Choice Written Response Ticket out the door See Blackboard for specific Assessments* Regional Integrated Geometry Curriculum UNIT: Informal/ Formal Proofs TOPIC: Triangles - similar Time frame for unit: Time frame for topic: 15 days Prior Knowledge 5.G.2 Identify pairs of similar triangles 5.G.3 Identify the ratio of corresponding sides of similar triangles 6.G.1 Calculate the length of corresponding sides of similar triangles, using proportional reasoning 7.G.5 Identify the right angle, hypotenuse, and legs of a right triangle 7.G.6 Explore the relationship between the lengths of the three sides of a right triangle to develop the Pythagorean Theorem 7.G.8 Use the Pythagorean Theorem to determine the unknown length of a side of a right triangle 7.G.9 Determine whether a given triangle is a right triangle by applying the Pythagorean Theorem and using a calculator A.G.1 Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only). A.A.25 Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable. A.A.26 Solve algebraic proportions in one variable which result in linear or quadratic equations Content Strands G.G.42 Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle. G.G.44 Establish similarity of triangles, using the following theorems: AA, SAS, and SSS G.G.45 Investigate, justify, and apply theorems about similar triangles G.G.46 Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle G.G.47 Investigate, justify, and apply theorems about mean proportionality: --the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse --the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg G.G.48 Investigate, justify, & apply the Pythagorean theorem & its converse Review & extend from the previous units Various theorems on working with angles and segments Concepts Similar triangles - dilation AA (~) Proofs SSS (~) Proofs SAS (~) Proofs Pythagorean theorem & its converse Mean proportional in right triangles (geometric mean) CPSTP—Corresponding Parts of Similar Triangles are in Proportion Essential Questions How are similar triangles different from congruent triangles? How are similar triangles the same as congruent triangles? How can similar triangles be used in a career/real-life? Why is the Pythagorean theorem useful in building a house? Process/skills G.PS.4 Construct various types of reasoning, arguments, justifications and methods of proof for problems G.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions G.RP.2 Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies G.RP.5 Present correct mathematical arguments in a variety of forms G.CM.12 Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing G.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts G.R.4 Select appropriate representations to solve problem situations G.R.8 Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent) Vocabulary Alternate interior angles Altitude Analytical proof Angle Angle addition & subtraction Angle bisector Area Bisector Complementary angles Corresponding parts Cross-products Deductive reasoning Equidistant External angle Hypotenuse Included side & angle Indirect measurement Inductive reasoning Interior angles Leg Mean proportional Means Midpoint Mid-segment of a triangle Paragraph Proof Parallel Lines Perimeter Perpendicular Perpendicular bisector Proof Proof by Contradiction Proportion Proportional segments Pythagorean theorem Ratio Reflexive Remote angle Scale factor Similar Statement/Reason – 2column proofs Substitution Extremes Geometric mean Given Suggested assessments Pre-assessment Regents/State exams Projects Participation Q-A Responses Oral responses Conversations Resources Fractals On-spot checks of class work Homework Observations of students’ work/behaviors Students’ explanations Draw a picture Tests and Quizzes T/F Multiple Choice Written Response Ticket out the door See Blackboard for specific Assessments* Regional Integrated Geometry Curriculum UNIT: Informal/ Formal Proofs TOPIC: Triangles – properties Time frame for unit: Time frame for topic: 7 days Prior Knowledge 5.G.6 Classify triangles by properties of their angles and sides 5.G.7 Know that the sum of the interior angles of a triangle is 180 degrees 5.G.8 Find a missing angle when given two angles of a triangle Content Strands G.G.30 Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle G.G.31 Investigate, justify, and apply the isosceles triangle theorem and its converse G.G.32 Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem G.G.33 Investigate, justify, and apply the triangle inequality theorem G.G.34 Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle Concepts Angle sum in a triangle (measuring angles: protractor, G.S. software?) *review classification using acute, obtuse, right, scalene, isosceles, equilateral Isosceles triangles & properties *investigate, apply, *proof, indirect proof (assumption statement, proof by contradiction) Exterior angle theorem *investigate, apply, proof, indirect proof Triangle inequality theorem *investigate, apply, proof, indirect proof Essential Questions How are triangles used in the construction industry? Why do all interior angles of a triangle add up to 180 degrees? Why is the longest side of a triangle opposite the largest angle? Why does the sum of any two sides of a triangle have to be greater than the measure of the third side? Process/skills G.PS.5 Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic) G.RP.8 Devise ways to verify results or use counterexamples to refute incorrect statements G.RP.9 Apply inductive reasoning in making and supporting mathematical conjectures G.CM.3 Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form G.CM.12 Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing G.CN.3 Model situations mathematically, using representations to draw conclusions and formulate new situations G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts G.R.8 Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent) Vocabulary Acute Angle Angle bisector Alternate interior angles Altitude Base angles Bisector Complementary angles Exterior angle Suggested assessments Pre-assessment Regents/State exams Projects Participation Q-A Responses Oral responses Conversations Resources Hypotenuse Indirect proof Inductive reasoning Interior angles Isosceles Leg Median Midpoint Obtuse On-spot checks of class work Homework Observations of students’ work/behaviors Students’ explanations Parallel Lines Paragraph Proof Perpendicular Perpendicular bisector Proof Proof by Contradiction Scalene Statement/Reason 2column proofs Substitution Draw a picture Tests and Quizzes T/F Multiple Choice Written Response Ticket out the door See Blackboard for specific Assessments* Regional Integrated Geometry Curriculum UNIT: Formal & Informal Proofs Time frame for unit: TOPIC: Centroid: Concurrent Lines of Triangles Time frame for topic: 1 Day Prior Knowledge 8.G.0 Construct the following, using a straight edge and compass: Segment congruent to a segment Angle congruent to an angle Perpendicular bisector Angle bisector G.G.21 Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles Content Strand G.G.43 Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1 Concepts Proportions Parallel lines Median Centroid Essential Questions Why is the centroid dividing the median into a 2:1 proportion? Process/skills G.PS.2 Observe and explain patterns to formulate generalizations and conjectures G.RP.2 Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid G.CN.6 Recognize and apply mathematics to situations in the outside world G.R.4 Select appropriate representations to solve problem situations Vocabulary Altitude Centroid Construct Median Perpendicular bisector Point of Concurrency Suggested assessments Pre-assessment Regents/State exams Projects Participation Q-A Responses Oral responses Conversations Resources On-spot checks of class work Homework Observations of students’ work/behaviors Students’ explanations Draw a picture Tests and Quizzes T/F Multiple Choice Written Response Ticket out the door See Blackboard for specific Assessments* Regional Integrated Geometry Curriculum UNIT: Formal & Informal Proofs TOPIC: Proportional Parts of Triangles Midsegments Time frame for unit: Time frame for topic: 1 day Prior Knowledge 6.G.1 Calculate the length of corresponding sides of similar triangles, using proportional reasoning 8.G.0 Construct the following, using a straight edge and compass: a. Segment congruent to a segment b. Angle congruent to an angle c. Perpendicular bisector d. Angle bisector Content Strand G.G.42 Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle Concepts Proportions Properties of the line segment joining the midpoints of 2 sides of the triangle Similar Triangles Essential Questions How are parallel lines and triangles used in real life? Why are the midsegments proportional to the parallel side? Process/skills G.PS.2 Observe and explain patterns to formulate generalizations and conjectures G.RP.2 Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid G.CN.6 Recognize and apply mathematics to situations in the outside world G.R.4 Select appropriate representations to solve problem situations Vocabulary Geometric relationship Median Midpoint Mid-Segment Parallel Proportions Similar triangles Suggested assessments Pre-assessment Regents/State exams Projects Participation Q-A Responses Oral responses Conversations Resources On-spot checks of class work Homework Observations of students’ work/behaviors Students’ explanations Draw a picture Tests and Quizzes T/F Multiple Choice Written Response Ticket out the door See Blackboard for specific Assessments* Regional Integrated Geometry Curriculum UNIT: Formal and Informal Proofs TOPIC: Polygons Time frame for unit: Time frame for topic: 2 Days Prior Knowledge 7.G.7 Find a missing angle when given angles of a quadrilateral 8.G.2 Identify pairs of supplementary and complementary angles 8.G.3 Calculate the missing angle in a supplementary or complementary pair 8.G.4 Determine angle pair relationships when given two parallel lines cut by a transversal 8.G.5 Calculate the missing angle measurements when given two parallel lines cut by a transversal 8.G.6 Calculate the missing angle measurements when given two intersecting lines and an angle Content Strands G.G.36 Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons G.G.37 Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons Concepts Interior/Exterior Angles of Polygons a. Exterior angle of a triangle = sum of remote interior angles b. Sum of the measures of the interior angles c. Sum of the measures of the exterior angles d. Measure of an interior angle of a regular polygon e. Measure of an exterior angle of a regular polygon Polygons a. Classify polygons by sides b. Classify polygons by properties Essential Questions Where does the formula for the sum of the interior angles come from? How does the sum of the interior/exterior angles relate to the number of sides? How can we determine the measure of an interior angle of a regular polygon given the number of sides? What is the correlation between the number of sides to the sum of the interior angle measures? How many diagonals can be drawn in a polygon? Process/skills G.PS.2 Observe and explain patterns to formulate generalizations and conjectures G.RP.3 Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion G.CM.4 Explain relationships among different representations of a problem G.CN.1 Understand and make connections among multiple representations of the same mathematical idea G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts Vocabulary Adjacent angles Decagon Diagonal Dodecagon Exterior angles Heptagon Suggested assessments Pre-assessment Regents/State exams Projects Participation Q-A Responses Oral responses Conversations Hexagon Interior angles Linear pair Nonagon Octagon Pentagon On-spot checks of class work Homework Observations of students’ work/behaviors Students’ explanations Polygon Quadrilateral Regular polygon Supplementary angle Triangle Undecagon Vertices Draw a picture Tests and Quizzes T/F Multiple Choice Written Response Ticket out the door See Blackboard for specific Assessments* Resources Review and Extend Solve geometry problems algebraically Regional Integrated Geometry Curriculum UNIT: Formal & Informal Proofs TOPIC: Quadrilaterals Time frame for unit: Time frame for topic: 15 days Prior Knowledge 5.G.4 Classify quadrilaterals by properties of their angles and sides 8.G.2 Identify pairs of supplementary and complementary angles 8.G.3 Calculate the missing angle in a supplementary or complementary pair 8.G.4 Determine angle pair relationships when given two parallel lines cut by a transversal 8.G.5 Calculate the missing angle measurements when given two parallel lines cut by a transversal 8.G.6 Calculate the missing angle measurements when given two intersecting lines and an angle Content Strands G.G.38 Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals G.G.39 Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals G.G.40 Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals G.G.41 Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids Concepts Properties: a. Parallelogram b. Rectangle c. Rhombus d. Square e. Trapezoid f. Isosceles Trapezoid Given a specific quadrilateral (parallelogram, rhombus, square, rectangle, trapezoid), prove a specified property. Given specific properties, prove a specific quadrilateral (parallelogram, rhombus, square, rectangle, trapezoid). Applications of properties of quadrilaterals a. solving linear equations b. solving quadratic equations Essential Questions Why is it important to compare and contrast different types of quadrilaterals? Why is a square a rectangle but a rectangle not a square? What are some essential steps/information that are needed to start a proof of a specific quadrilateral? Process/skills G.PS.3 Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations) G.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving G.RP.4 Provide correct mathematical arguments in response to other students’ conjectures, reasoning, and arguments G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid G.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts G.R.7 Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person’s argument have a logical foundation) Vocabulary Adjacent angles Direct proof Parallelogram Adjacent sides Equal Perpendicular Alternate interior angles Formal Proof Perpendicular bisector Altitude Given Quadrilateral Angle bisector Height Reason Base angles of a Indirect proof Rectangle trapezoid Isosceles trapezoid Rhombus Complementary angles Kite? Segment bisector Congruent Legs of a trapezoid Square Congruent triangles Consecutive angles Corresponding angles CPCTC Diagonal Suggested assessments Pre-assessment Regents/State exams Projects Participation Q-A Responses Oral responses Conversations Resources Linear pair Median Midpoint Opposite angles Parallel On-spot checks of class work Homework Observations of students’ work/behaviors Students’ explanations Statement Supplementary angles Transversal Trapezoid Vertical angles Draw a picture Tests and Quizzes T/F Multiple Choice Written Response Ticket out the door See Blackboard for specific Assessments*