coordinate Find the coordinates of the midpoint of a segment having the given endpoints. Big Ideas Geometry from one end of a line

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1 G.(2) Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one- and two-dimensional coordinate systems to verify geometric conjectures. The student is expected to: G.2(A) determine the coordinates of a point that is a given fractional distance less than one How can we find the midpoint of a line segment on a number line or in a coordinate plane? Find the coordinates of the midpoint of a segment having the given endpoints. Coordinate Distance Formula Midpoint Formula Compare the methods of counting lines on the number line or coordinate plane and using the from one end of a line midpoint or distance segment to the other A. (-1,8) formula to calculate the in one- and twodimensional coordinate C. (6,1) B. (-10,-8) distances. systems, including D. (-5,-4) finding the midpoint. Answer: C SAT: Find the coordinates of the other endpoint given one endpoint of (1, -3) and a midpoint of (6, 1) A. (-1, 8) B. (-5, -4) C. (11, 5) D. (10, 8) Answer: C Page 1

2 G.2(B) derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines. Compare the Distance Formula to the Pythagorean Theorem Given two points on the coordinate grid, (1, 5) and (2,-2), find the distance between the points in reduced radical form. Answer:5 2 Graph the points D (3, 4) E (0, 4) and F (-7, 4). Connect the points with a segment. Calculate mdddd and. EEEE Use the Segment Addition Postulate (Postulate 1.2) to find the length of DDDD. Verify the length of DF using the Distance Formula. Coordinate Distance Formula Midpoint Formula Demonstrate how the Pythagorean Theorem can be used to calculate distances and how the Distance formula is related to the Theorem The student may substitute the x- and y-values incorrectly when using the formulas. The student may divide a value by 2 instead of taking the square root when using the distance formula. The student may add the x-value to the y-value, instead of computing the sum of the x-values and computing the sum of the y-values before dividing by 2 in the midpoint formula. The student may incorrectly write the ratio of the slope of a line as the ratio of horizontal change divided by vertical change Page 2

3 G.(4) Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to: G.4(A) distinguish between undefined terms, definitions, postulates, conjectures, and theorems. Develop verbal descriptions to define Geometric Terms throughout the curriculum Use number line and coordinate plane to represent points, lines, rays, line segments and geometric figures. If two lines intersect, then their intersection is exactly one point. Is this a definition, a postulate or an undefined term? Answer: Postulate Conjecture Definition Postulate Theorem Undefined Term Develop verbal descriptions to define geometric terms throughout the curriculum Use manipulatives and technology to draw conclusions and discover relationships about geometric shapes and their G.4(B) identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse. Make, interpret and/or understand statements such as if p, then q as applied to attributes of geometric drawings, figures, etc. Develop conjectures in the form of a conditional statement Use counter-examples to prove why statements are false Use inductive or deductive reasoning to prove statements true. Given a conditional statement: If you are a guitar player, then you are a musician. Write the converse, inverse and contrapositive of the statement and state whether each statement is true or false. Can the statement and converse be a biconditional? If not, give a counterexample. Conclusion Conditional Statement Contra-Positive Converse Hypothesis Inverse Negation Truth Value properties Write conditional statements, converse, inverse and contrapositive Use discussions and brainstorming to determine the validity of each statement and provide a counter-example, if false Page 3

4 G.4(C) verify that a conjecture is false using a counterexample. Draw conclusions from number or picture patterns, specific examples or events. Draw conclusions by using inductive reasoning to form conjectures Determine if the following conjecture is true or false: The value of x^2 is always greater than the value of x. If false, give a counter-example. Conjecture Counter-example Use facts, definitions, postulates, theorems and properties to prove statements true or false. The student may think all conjectures are false. The student may confuse counterexamples with examples. The student may not correctly interpret phrases such as, at least one and exactly one. G.(5) Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to: G.5(A) investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. Draw conclusions from number or picture patterns, specific examples or events. Draw conclusions by using inductive reasoning to form conjectures How are angles 1 and 8 related? A. Same side interior B. Alternate exterior C. Alternate interior D. Corresponding Answer: B Alternate Exterior Alternate Interior Corresponding Diagonal Parallel Lines Perpendicular Lines Same-side Interior Segment Skew Lines Transversal Teacher may wish to use patty paper for students to draw/construct parallel lines and investigate angle pairs The student may make a conjecture based on limited investigation of patterns. The student may randomly state a conjecture without investigating and recognizing patterns. The student may not know how to use a construction to make a conjecture. The student may not be able to perform constructions correctly. The student may not state a conjecture using precise geometric vocabulary Patty Paper Geometry by Michael Serra Page 4

5 G.5(B) construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge. Use a compass and a straight edge to construct: Congruent segments Congruent angles An angle bisector What construction is shown in the accompanying diagram? A. The bisector of angle PJR. B. The midpoint of line PQ C. The Perpendicular bisector of line segment PQ. D. A perpendicular line to PQ through point J. Answer: C Compass Construction Drawing Sketch Straight Edge Focus on constructing geometric figures with only a straight edge and a compass. Ensure students can construct congruent segments. Use two column notes that have students write the steps needed to construct on one side while performing the task of construction in the other. 1.2, f.com/tocs/construction stoc.html Page 5

6 G.(6) Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to: G.6(A) verify theorems about angles formed by Verify theorems about angles formed by Adjacent Alternate Exterior Explore different proof methods to verify the 1.5 the intersection of lines The intersection of theorems concerning 1.6 and line segments, including vertical lines including vertical angles Alternate Interior segments, angles and their relationships angles, and angles The intersection of formed by parallel lines line segments, Based on the illustration, what is the Corresponding cut by a transversal and prove equidistance including vertical angles. Diagonal between the endpoints Parallel lines cut by Linear Pairs of a segment and points a transversal Parallel Lines on its perpendicular Apply these Perpendicular bisector and apply these relationships to solve Lines relationships to solve problems. Same-side Interior problems. How are the different relationship between angle 1 and Segment angle pairs related? angle 3? Skew Lines A. They are congruent angles. Supplementary B. They are supplementary. Transversal C. They form a right angle. How do their measures compare? How and why do we know this? D. They are complementary. Answer: A The student may not use logical reasoning correctly to work through proofs. The student may not apply justification to support statements n a twocolumn proof Page 6

7 G.11 Two-dimensional and three-dimensional figures. The student uses the process skills int eh application of formulas to determine measures of two-and threedimensional figures. The student is expected to: G.(11B) determine the area of composite twodimensional figures comprised of a combination of Determine the area of composite twodimensional figures comprised of a combination of Detemine the area of quadrilateral ABCD. Area Circle Composite twodimensional Kite This is focus on area on a coordinate plane. Connect the use of the distance formula to Big Ideas Math Geometry 1.4 triangles, Triangles parallelograms, Parallelograms Parallelogram Regular polygon determine measurements of figures to determine area of trapezoids, kites, regular Trapezoids Sector composite figures. polygons, or sectors of Kites Trapezoid circles to solve problems Regular polygons Triangle using appropriate units Sectors of circles of measure. More depth to this student expectation in the 5 th Six Weeks. Answer: 26 square units Student may not be able to decompose a composite figure into triangles, trapezoids, parallelograms, etc. Students may not recognize that they need to use the distance formula to obtain measurements Page 7