Math 3 - Lesson Title: Using the Coordinate Plane for Proofs

Transcription

1 Targeted Content Standard(s): Use coordinates to prove simple geometric theorems algebraically. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). G.GPE.5 Prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems, (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula). Targeted Mathematical Practice(s): 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 an express regularity in repeated reasoning. Supporting Content Standard(s): (optional) Student Friendly Learning Targets I can Find the equation of a line parallel to a given line through a given point. Find the equation of a line perpendicular to a given line through a given point. Use coordinates to show lines are either parallel or perpendicular. Use coordinate geometry, such as the distance formula, to identify and prove properties of geometric figures. Determine perimeter and area of a rectangle, rhombus, and square given its coordinates. Purpose of the Lesson: The overarching emphasis in this unit is for students to use the coordinate plane to verify geometric theorems previously learned in Math 1 or 2. Students will formalize criteria for parallel and perpendicular lines using the coordinate plane in segment 2, and then apply these, along with the distance formula, to special quadrilaterals as they investigate and determine the properties of several midpoint quadrilaterals in segments 3 and 4. Explanation of Rigor: (Fill in those that are appropriate.) Conceptual: Students will prove parallel lines have the same slope and perpendicular lines have slopes which are opposite reciprocals (or whose product is -1). G.GPE. 5 Procedural: Students will use coordinates to compute perimeter and area of quadrilaterals. G.GPE.7 Vocabulary: parallel rectangle rhombus perpendicular square parallelogram slope reciprocal midpoint diagonal Application: Students will use coordinates to classify a quadrilateral by its properties. G.GPE.4

2 quadrilateral coordinates Pre-: Parallel and Perpendicular Lines Pre- (Segment 1) Formative (s): Parallel and Perpendicular Lines Activity (Segment 1) Midpoint Madness (Segments 2 and 3) Summative : G.GPE Summative #1,2,6 Self-:

3 Lesson Procedures: Segment 1 20 minutes Pre-assessing upcoming necessary skills with linear equations. MP#7 Look for and make use of structure. Students should recognize the usefulness of putting an equation in slope-intercept form to identify the slope and y-intercept. Students may have trouble getting the equations into slope intercept form. 1. Give pre-assessment. As students are completing it, observe student responses and look for different methods of determining the lines parallel or perpendicular. 2. Have students share when most are finished. Ask students who solved the problems using a graph, using only the equations, using points, or other methods to share their method with the class. Parallel and Perpendicular Lines Pre- Students who do not recognize from the equation the properties of parallel and perpendicular lines could use a graphing calculator to investigate the graphs of the systems. A picture definition of parallel and perpendicular lines could be provided. Practice (Homework):

4 Segment 2 45 minutes Prove the slope criteria for parallel and perpendicular lines. MP #2 Reason abstractly and quantitatively. Students will be able to use figures and information pertaining to a specific geometric object as an aid in reasoning about that geometric object in general. MP #7 Look for and make use of structure. Students will be able to use the structure of geometric objects to gain insights into, make conjectures about, and create proofs pertaining to these objects. Students may not need assistance determining the effect of a translation or rotation on the coordinates of a point. 1. In small groups, have students work through #1 in the activity. Facilitate learning by encouraging students to record how they know the two lines are parallel with an algebraic method (slope formula). Wrap up #1 as a class, making sure to reiterate that parallel lines can be created by a sequence of two translations. 2. Have students continue working on #2-5 of the activity. Encourage students to use algebraic formulas to calculate the slope, instead of just counting the rise and run on the graph. Observe how the students explain the lines are perpendicular using slope. If different arguments surface, have a whole class discussion about the similarities of their ideas, (e.g., the product of the slopes is -1, the slopes are opposite reciprocals). Parallel and Perpendicular Lines Activity Graph Paper Students may use the corner of a notecard to trace a line on the graph on which the 90⁰ rotation of the original point would lie. Then, mark off the distance to the original point from the origin on the side of the notecard. Find the location of the image by using this marked distance from the origin on the new line. Practice (Homework): #1-7 following Parallel and Perpendicular Lines Activity Students may be interested in investigating any additional transformations that produce parallel lines.

5 Segment 3 45 minutes Use coordinates to prove simple geometric theorems algebraically, specifically focused on definitions of quadrilaterals. Use the slope criteria for parallel and perpendicular lines to solve geometric problems. MP #2 Reason abstractly and quantitatively. Students will be able to use figures and information pertaining to a specific geometric object as an aid in reasoning about that geometric object in general. MP #7 Look for and make use of structure. Students will be able to use the structure of geometric objects to gain insights into, make conjectures about, and create proofs pertaining to these objects. MP #6 Attend to precision. Students will recognize that incorrect initial attempts at definitions, conjectures, and theorems may be corrected through a process of refinement. MP #8 Look for and express regularity in repeated reasoning. Students will recognize a pattern in the classification of the midpoint quadrilaterals and generalize a pattern in the area and perimeter of these quadrilaterals. Students who do not understand the similarities and differences between the types of quadrilaterals may need some remediation. 1. Use whole class instruction to guide students in #1-6. The quadrilateral ABCD is the solution to the homework from Segment 2. Students may need a refresher on the properties of parallelograms, rectangles, rhombi, and squares. Midpoint Madness Graph Paper Students may need to list the properties of quadrilaterals and be guided to see the relationships between all the types. Dynamic geometry software may be helpful to demonstrate these connections. When stating properties of quadrilateral, students may draw a picture to support their reasoning. Practice (Homework): Assign #14 for homework. Sketches are fine, but students could use graph paper to be more precise.

6 2. Allow students to work in small groups for #7-13. Encourage students to state geometric properties using If-Then statements when supporting their answers. Possible observations might include: 1) If a quadrilateral has four congruent sides, then it is a rhombus. 2) If the diagonals of a quadrilateral bisect each other and are perpendicular, then it is a rhombus. 3) If a quadrilateral has 2 pairs of opposite sides congruent, then it is a parallelogram. 4) If a parallelogram has one right angle, then it is a rectangle. 3. When most groups have completed #9, it may be helpful to share different reasoning statements in a whole group. Students may need guidance when placing the notecard on the coordinate plane, as this can be done abstractly, naming the coordinates by the measurements of the notecard, i.e. (0,4)(6,4)(0,0)(6,0). 4. To complete #12, students should be using the distance formula. The pattern n1 1 the area is 24 and for perimeter 2 n n n is odd and 4 13 if n is 2 2 even.

7 Segment 4 30 minutes Use coordinates to prove simple geometric theorems algebraically, specifically focused on definitions of quadrilaterals. Use the slope criteria for parallel and perpendicular lines to solve geometric problems. MP #3 Construct viable arguments and critique the reasoning of others. Students will be able to create and present a proof that the midpoint quadrilateral of a quadrilateral is a rectangle, and be able to critique the proof and deductive reasoning of others. MP #7 Look for and make use of structure. Students will be able to use the structure of geometric objects to gain insights into, make conjectures about, and create proofs pertaining to these objects. Midpoint Madness Graph Paper Students who have difficulty in the abstract case may use numerical values for the coordinates and complete the activity for several different quadrilaterals. Students may have difficulty generalizing and may need to use numerical values at first. 1. Students should work in small groups to devise a proof and support their reasoning using the distance formula and slope criteria for parallel and perpendicular lines. Practice (Homework):

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