SFUSD Mathematics Core Curriculum Development Project

Transcription

1 1 SFUSD Mathematics Core Curriculum Development Project Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own understanding

2 Geometry Unit G.5: Coordinate Geometry 2 Number of Days Lesson Reproducibles Number of Copies Materials 1 Entry Task Parallel and Perpendicular Lines 1 per student Graph paper (optional) 2 Lesson Series 1 Proving Lines are Parallel or Perpendicular Top Gun 1 per student 1 per student Graph paper 1 Apprentice Task Finding Equations of Parallel and Perpendicular 1 per pair Lines cards (1 set per group) Computer with projector for slides 3 Lesson Series 2 Distance Formula Exploration (2 pages) Midpoint High (2 pages) It s a Parallelogram! 1 per pair 1 per pair 1 per pair 1 Expert Task Proving with Distance and Slope 1 per pair Graph paper 3 Lesson Series 3 Midpoint Quadrilateral of a Square Midpoint Quadrilaterals (3 pages) 1 per pair 1 per pair 1 Milestone Task Putting It All Together 1 per student Graph paper

3 3 Unit Overview Big Idea Coordinate geometry can be used to prove geometric theorems about polygons and lines and to find the distance between two points in order to solve problems. Unit Objectives Students will be able to use coordinates to prove simple geometric theorems algebraically. Students will be able to prove the relationship between parallel and perpendicular lines. Students will be able to find a point on a given line segment between two points that partitions the segment into a specific ratio. Given coordinates, students will be able to find the perimeter and area of polygons using the distance formula. Unit Description Students will review characteristics of parallel and perpendicular lines, develop midpoint and distance formulas, and apply these skills to proving geometric theorems. Expressing Geometric Properties with Equations CCSS-M Content Standards 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.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

4 Progression of Mathematical Ideas Prior Supporting Mathematics Current Essential Mathematics Future Mathematics 4 slope of a line writing equation of a line graphing in the coordinate plane understanding and applying the Pythagorean Theorem understanding the meaning of parallel and perpendicular segments in a plane definitions of different quadrilaterals, triangles, circle finding area and perimeter of polygons. Use coordinate geometry to formalize the relationship between parallel and perpendicular lines. Develop the distance formula through an understanding of the Pythagorean Theorem, and use them in conjunction with the midpoint formula to prove characteristics of polygons. Students will use coordinate geometry to develop equations of a circle, ellipse, and other conic sections.

5 Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit? 5 Entry Task Apprentice Task Expert Task Lesson Series 1 Lesson Series 2 Lesson Series 3 Milestone Task Total days: 12 1 day 2 days 1 day 3 days 1 day 3 days 1 day

6 6 Entry Task Finding Equations of Parallel & Perpendicular Lines: Assessment Task Apprentice Task Finding Equations of Parallel & Perpendicular Lines: Matching Task and Lines & Rectangles Expert Task Proving Distance and Slope Milestone Task Putting It All Together CCSS-M Standards G.GPE.4, G.GPE.5 G.GPE.4, G.GPE.5 G.GPE.4, G.GPE.6 G.GPE.4, G.GPE.7 Brief Description of Task Students will determine which group of four linear equations will create a rectangle using slope. Students will pair parallel and perpendicular lines, given linear equations. Students will prove a square given four equations. They will prove that the connected midpoints of a given quadrilateral form a parallelogram. Given three points, students will complete the rectangle and prove a rectangle (solo). Source FAL, 2013, Shell Center FAL, 2013, Shell Center Teacher Created Teacher Created Lesson Series 1 Lesson Series 2 Lesson Series 3 CCSS-M Standards G.GPE.5 G.GPE.6, G.GPE.7 G.GPE.4, G.GPE.5 Brief Description of Lessons Students will decide whether given lines fit the definitions of parallel and perpendicular, and create equations for lines that are parallel or perpendicular to a given line through a given point. Students will learn the midpoint and distance formulas, and prove if a quadrilateral is also a parallelogram Students prove geometric theorems using coordinate geometry. Sources SFUSD teacher created SFUSD teacher created SFUSD teacher created

7 Entry Task Finding Equations of Parallel and Perpendicular Lines - Assessment Task 7 What will students do? Mathematics Objectives and Standards Math Objectives: Check to see what students know about graphing and slopes of parallel and perpendicular lines in relation to a rectangle. Check if students know the characteristics of a rectangle. CCSS-M Standards Addressed: G.GPE.4, G.GPE.5 Notes: Only give the Assessment Task portion of the Formative Assessment Lesson (FAL) Finding Equations of Parallel and Perpendicular Lines. The rest of this FAL will be used in the Apprentice Task. Framing Student Experience Launch: Let students know the math objective and do the best they can. It is important to ask questions and record what they know. During: This task is to be completed in groups or pairs with each student doing their own write-up; it is only a formative assessment. Circulate to hear conversations to gauge students knowledge. Closure/Extension: Discuss common questions and/or student assumptions and consider putting them on chart paper to revisit over the course of the unit. Potential Misconceptions: What type of slopes define perpendicular and parallel lines. Relying on the look of the angle to determine if it s a right angle

8 8 Finding Equations of Parallel and Perpendicular Lines - Assessment Task Focus Standards for Mathematical Practice: Make sense of problems and persevere in solving them Construct viable arguments and critique the reasoning of others Look for and make use of structure Structures for Student Learning: Academic Language Support: Vocabulary: parallel; perpendicular; slope How will students do this? Sentence frames: These lines form a rectangle:. I know this because Differentiation Strategies: Participation Structures (group, partners, individual, other): Small groups

9 Lesson Series #1 9 Lesson Series Overview: Students will decide whether given lines fit the definitions of parallel and perpendicular, and create equations for lines that are parallel or perpendicular to a given line through a given point. CCSS-M Standards Addressed: G.GPE.5 Time: 2 days Lesson Overview Day 1 Description of Lesson: Students use the Converse of the Pythagorean Theorem to prove that two lines are perpendicular. Notes: Be sure to debrief the activity by generalizing that if lines are perpendicular, their slopes are opposite reciprocals. Include the special case of slopes of horizontal and vertical lines. Lesson Overview Day 2 Description of Lesson: Students create linear equations given two points or a slope and a point, including equations for lines parallel or perpendicular to a given line through a given point. Resources Proving Lines are Parallel or Perpendicular activity Suggested HW: Discovering Geometry, Using Your Algebra Skills (UYAS) #2, pp , selected problems Top Gun activity Resources Suggested HW: Discovering Geometry, UYAS #3, pp , #1-9

10 Apprentice Task Parallel and Perpendicular Lines: Matching Task and Lines and Rectangles 10 What will students do? Mathematics Objectives and Standards Math Objectives: Students will determine which sets of linear equations are parallel, and which are perpendicular. CCSS-M Standards Addressed: G.GPE.4, G.GPE.5 Notes: Students do the last two parts of the FAL from the Entry Task: Matching Task and Lines and Rectangles. The teaching notes for this FAL are included in the Entry Task resources. Students need to provide both oral (to each other and to you) and written justification for each match. Slide P-1 provides an opportunity to summarize student learning and check for understanding. Stress that there are many possible solutions. Framing Student Experience Launch: The Do Now should be a review of the characteristics of a rectangle (opposite sides parallel, vertices are right angles). During: Students will work in pairs or groups of three or four. An emphasis is justification; make sure students are writing and can also explain to you and to their group. Closure/Extension: Suggested HW: Discovering Geometry, UYAS #3, #10-13 Potential Misconceptions: Manipulating equations to check slope.

11 11 Parallel and Perpendicular Lines: Matching Task and Lines and Rectangles Focus Standards for Mathematical Practice: Make sense of problems and persevere in solving them Construct viable arguments and critique the reasoning of others Look for and make use of structure How will students do this? Structures for Student Learning: Academic Language Support: Vocabulary: x-intercept; y-intercept Sentence frames: I know these lines are because Differentiation Strategies: Reduce # of cards if needed Participation Structures (group, partners, individual, other): Pairs or small groups

12 Lesson Series #2 12 Lesson Series Overview: Students will learn the midpoint and distance formulas, and prove whether a given quadrilateral is a parallelogram, rectangle, rhombus or square. CCSS-M Standards Addressed: G.GPE.6, G.GPE.7 Time: 3 days Lesson Overview Day 1 Description of Lesson: Students will learn the distance formula through use of Pythagorean Theorem. Pigeon Race: Distance Formula Resources Notes: Students should be made to see that the distance formula is derived from the Pythagorean Theorem (and, therefore, never needs to be memorized). Lesson Overview Day 2 Description of Lesson: Students will derive the midpoint formula and extend it to finding points a given ratio away from an endpoint. Lesson Overview Day 3 Description of Lesson: Students will prove the quadrilateral is a parallelogram, and move on to prove that it is a rhombus (but not a square!). Notes: If needed, provide a quick review of the definitions, attributes, and relationships between different types of quadrilaterals Suggested HW: Discovering Geometry, Lesson 9.5 (pp ) #1-6 Resources Midpoint High Suggested HW: DG 4, UYAS #1, pp , selected problems Resources Is It a Parallelogram? Suggested HW: Discovering Geometry, Lesson 9.5, #7-10

13 Expert Task Proving with Distance and Slope What will students do? 13 Mathematics Objectives and Standards Math Objectives: Assess if students can recognize when two lines are parallel or perpendicular when just given the equation of the lines. Assess if students can recognize when two lines are parallel or perpendicular when just given two points on a line. Assess if students can find the midpoint of segment when given the coordinates of the endpoints. Students will be able to make conjectures after an inductive process. Assess if students can organize and justify their ideas clearly. CCSS-M Standards Addressed: G.GPE.4, G.GPE.7 Potential Misconceptions: Framing Student Experience Launch: Start of with a warm-up of figuring out if two lines are parallel, perpendicular, or neither, with just the equations of the line. During: Students can perform this in pairs or as a group assessment. If done in groups, use the participation structure of checkpoints. As each group approaches a checkpoint, you will stop by to check for understanding from different students. If students cannot explain, go back in a few minutes while students continue to discuss. Groups cannot move on until you clear the group of checkpoint. Closure/Extension: Ask a group or two to share their work and talk to about the solutions as a class. The bigger question of Will the Midpoint Quadrilateral always be a parallelogram? can be asked among the whole class. This can be a way to check conjectures inductively, and it can also be a way to start a conversation around a formal proof (which they start to do in Lesson Series 3).

14 14 Focus Standards for Mathematical Practice: Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Structures for Student Learning: Proving with Distance and Slope How will students do this? Academic Language Support: I know that this quadrilateral is also a because This quadrilateral is a in addition to a parallelogram because Differentiation Strategies: Shorten task or provide student choice: Choose parts 1 and 2, or part 3 Part 3 #5 is an extension for advanced learners Participation Structures (group, partners, individual, other): Pairs or groups

15 Lesson Series #3 15 Lesson Series Overview: Students prove geometric theorems using coordinate geometry. CCSS-M Standards Addressed: G.GPE.4, G.GPE.5 Time: 3 days Lesson Overview Day 1 Description of Lesson: Students prove that connecting the midpoints of the segments of a square will yield another square. Notes: Have a few groups present their solution. Look for different strategies: coordinate geometry, isosceles right triangles, etc. Midpoint Quadrilateral of a Square Resources Suggested HW: Discovering Geometry UYAS 13, pp ; selected problems over 2-3 nights Lesson Overview Days 2-3 : Description of Lesson: Student prove that the midline of a triangle is parallel to and half the length of the third side, and that connected midpoints of any quadrilateral create a parallelogram IMP Year 3 Orchard Hideout, Midpoint Quadrilaterals Suggested HW: See Day 1 above Notes: Stress that for Part I, students should start with a labeled sketch in the coordinate plane.

16 Milestone Task Putting It All Together What will students do? 16 Mathematics Objectives and Standards Math Objectives: Assess students knowledge of when line segments are parallel or perpendicular when given the coordinates of the endpoints. Assess students knowledge of creating a rectangle with only three points having to find the fourth point. CCSS-M Standards Addressed: G.GPE 4, 5, 6, 7 Potential Misconceptions Framing Student Experience Launch: Remind students that this milestone task is an individual task, and a chance for them to do their best work and show their understanding. Consider sharing your scoring rubric ahead of time. During: Circulate in order to check on progress. Provide sentence frames or academic language support as needed for EL students and students with learning differences. Closure/Extension: Some students may be ready for challenge task afterwards, such as a proof that the coordinates of the centroid of a triangle are the average of the coordinates of the triangles three vertices. (See DG 4, Special Proofs of Special Conjectures, p 741)

17 17 Focus Standards for Mathematical Practice: Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Attend to precision Structures for Student Learning: Academic Language Support: Vocabulary: rectangle, quadrilateral, midpoint, endpoint Putting It All Together How will students do this? Sentence frames: I know that this quadrilateral is a because. Differentiation Strategies: Part II can be optional. Proofs can be given orally if a student is not able to demonstrate understanding through writing. Participation Structures (group, partners, individual, other): Individual