# Classroom Connections Examining the Intersection of the Standards for Mathematical Content and the Standards for Mathematical Practice

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1 Classroom Connections Examining the Intersection of the Standards for Mathematical Content and the Standards for Mathematical Practice Title: Considering Coordinate Geometry Common Core State Standards Addressed in the Student Work Task: G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. Evidence of Standards for Mathematical Practice in the Student Work: 2: Reason abstractly and quantitatively. 3: Construct viable arguments and critique the reasoning of others. 5: Use appropriate tools strategically. 6: Attend to precision. 7: Look for and make use of structure. 8: Look for and express regularity in repeated reasoning. Task Components: Part I: Mathematical Background (Page 2) Today s Content Part II: Math Metacognition (Page 3) Part III: Unpacking the Rigor of the Mathematical Task (Pages 4 5) Part IV: Looking at Student Work (Page 6 7) Triangle ABC Task (HS Geometry) Protocol for LASW Part V: Vertical Content Alignment (Page 8) Charting Coherence through Mathematical Progressions Writing a Grade Level Problem or Task Part VI: Wrap up (Page 9) Handouts Included: Math Metacognition: Page 10 Protocol for LASW: Page 11 Mathematical Task Triangle ABC: Page 12 Student Work Samples: Pages Student Work Analysis Grid: Page 18 Unpacking the Rigor: Page 19 Materials Needed: Grid paper or graph boards Ruler or straight-edge Classroom Connections Considering Coordinate Geometry Page 1

2 Part I: Mathematical Background Approximate Time: 10 minutes Grouping Structure: Whole Group A. Today s Content: a. The mathematics during this session focuses on coordinate geometry, and the student work task itself includes several different elements and skills. For example, students must prove whether or not a figure is a right triangle, determine a midpoint, and find an equation of a circle. Coordinate geometry provides a nice opportunity to blend students prior knowledge of algebra with new concepts and skills learned in geometry. b. What do we need to know about: i. Locating ordered pairs ii. Properties of shapes, in particular of triangles iii. Distance iv. Slope v. Midpoint vi. Pythagorean Theorem vii. Circles and equations of circles before we can solve real-life problems using the coordinate plane accurately and efficiently? c. Chart ideas to refer to during the Protocol for LASW. Classroom Connections Considering Coordinate Geometry Page 2

4 Part III: Unpacking the Rigor of the Mathematical Task Approximate Time: 30 minutes Grouping: Whole Group A. Comparing Different Versions of the Mathematical Task: Let s compare the rigor of two related problems to the Triangle ABC task. The level of rigor is based on which of the Standards for Mathematical Practice we could expect to see when examining the student solutions. Pass out the Unpacking the Rigor handout (see Page 19). See completed chart on the next page for more details of what this would look like. B. In addition to the Mathematical Practices, consider discussing the following with your group as you compare the variations above: a. Cognitive demand b. Task accessibility to a variety of learners c. Real-life applications and math connections d. Assessment of student learning C. If time allows, you can use a Venn Diagram to compare and contrast the elements of each version of the task. Classroom Connections Considering Coordinate Geometry Page 4

6 Classroom Connections Considering Coordinate Geometry Page 6

8 about what the student knows and does not understand. c. The content of Question 3 could serve as an extension, however the question format is more basic than the first two. In student work samples collected, we saw that many students showed little justification for their proofs yet were still able to identify the correct equation of the circle. What implications might this have? What might you do differently? E. Questions for Evidence-based, Whole Group Discussion: a. Does the student work exhibit proficiency of the Standards for Mathematical Content? b. Consider the Standards for Mathematical Practice that are embedded in the task design. Which of these Practices do you see exhibited in the student work? c. What is the evidence in the student work that the student is moving towards the intentions of the task design? (i.e., understanding and demonstrating mastery of the content as well as engaging in math practices) d. How far removed from the intent of the task is the student s thinking? Which pieces of understanding are present? Which are not? Is there evidence that they are close? Is there a misconception present? Part V: Vertical Content Alignment Approximate Time: 25 Minutes Grouping: Partners or Small Groups Classroom Connections Considering Coordinate Geometry Page 8

11 Math Metacognition Three of the vertices of figure ABCD have been drawn on the coordinate plane below. A(-3, 4) B(2, 4) C(4, 1) D(, ) 1) Decide on a location for vertex D. 2) What type of figure is ABCD? How do you know? 3) Come up with 3 questions you could ask about Figure ABCD. 4) Decide on a new location for vertex D so that a different geometric figure is formed. What type of figure is ABCD now? How do you know? 5) Can you ask the same questions as in Question 3 above? If so, why? If not, why not? 6) Come up with at least 1 question you could now ask about your new Figure ABCD. Classroom Connections Considering Coordinate Geometry Page 11

12 Protocol for Looking at Student Work Read the task and discuss what it is assessing. Solve the problem individually Share your thinking with a partner Discuss the mathematics of the task as a whole group Look at how students solved the same task Identify evidence of the Standards of Mathematical Practice exhibited in the student work Discuss evidence of the Standards of Mathematical Practice exhibited in the student work as a whole group Based on the Mathematics Learning Community (MLC) Protocol for LASW, 2011 Commonwealth of Massachusetts [Department of Elementary and Secondary Education Classroom Connections Considering Coordinate Geometry Page 12

13 Mathematical Task Triangle ABC Solve this problem in the space provided below. Show and explain all of your work. On the coordinate grid, sketch the triangle with these three vertices: A(-2, -5), B(2,3), and C(4, -3) Use your sketch to answer the following questions: (1) Is this a right triangle? Explain mathematically how you know the answer to this question. (2) Find the coordinates of the midpoint of the longest side of this triangle and prove that it is equidistant from each of the triangle s vertices. (3) Write the equation for a circle that will go through all three vertices of the triangle and verify that the coordinates of the vertices satisfy this equation. Classroom Connections Considering Coordinate Geometry Page 13

14 Problem: Triangle ABC Student Work Analysis Grade Level: HS Geometry Student A Classroom Connections Considering Coordinate Geometry Page 14

15 Problem: Triangle ABC Student Work Analysis Grade Level: HS Geometry Student B Classroom Connections Considering Coordinate Geometry Page 15

16 Problem: Triangle ABC Student Work Analysis Grade Level: HS Geometry Student C Classroom Connections Considering Coordinate Geometry Page 16

17 Problem: Triangle ABC Student Work Analysis Grade Level: HS Geometry Student D Classroom Connections Considering Coordinate Geometry Page 17

18 Problem: Triangle ABC Student Work Analysis Grade Level: HS Geometry Student E Classroom Connections Considering Coordinate Geometry Page 18

19 Student Work Analysis for: Triangle ABC Student MP 2: Reason abstract. MP 3: Critique Reason. MP 5: Use tools MP 6: Precision MP 7: Structure MP 8: Repeated Reason. What comes next in instruction for this student? A B C D E Classroom Connections Considering Coordinate Geometry Page 19

20 Unpacking the Rigor Comparing Different Versions of the Triangle ABC Mathematical Task Task Level of Rigor On the coordinate grid, sketch the triangle with these three vertices: A(-2, -5), B(2,3), and C(4, -3) Using the Pythagorean Theorem, explain how ABC is a right triangle. On the coordinate grid, sketch the triangle with these three vertices: A(-2, -5), B(2,3), and C(4, -3) 4. Using the Pythagorean Theorem, explain how ABC is a right triangle. 5. Using the midpoint formula, identify the midpoint of AB. 6. Using the distance formula, show that this midpoint is equidistant from points A, B, and C. Solve this problem in the space provided below. Show and explain all of your work. On the coordinate grid, sketch the triangle with these three vertices: A(-2, -5), B(2,3), and C(4, -3) Use your sketch to answer the following questions: (1) Is this a right triangle? Explain mathematically how you know the answer to this question. (2) Find the coordinates of the midpoint of the longest side of this triangle and prove that it is equidistant from each of the triangle s vertices. (3) Write the equation for a circle that will go through all three vertices of the triangle and verify that the coordinates of the vertices satisfy this equation. Classroom Connections Considering Coordinate Geometry Page 20

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