1. Connie Laughlin Hank Kepner Rosann Hollinger Dennis Cary Kevin McLeod Mary Mooney 1

2. We are learning to… recognize and apply connections across conceptual categories in the CCSSM. 2

3. We will know we are successful when we can… explain a specific example of connections among conceptual categories. 3

4. 4

5. Drilling teams from oil companies search around the world for new sites to place oil wells. Increasingly, oil reserves are being discovered in offshore waters. The Gulf Oil Company has drilled two high-capacity wells in the Gulf of Mexico about 5 km and 9 km from shore. MODELING AND STRATEGIC CMT USE Optimal Refinery Location

6. The company wants to build a refinery to pipe oil from the two wells to a single new refinery on shore. Assume the 28 km of shoreline is nearly straight. What are important considerations in locating the refinery? What is your best estimate for the location of the refinery?

7. Checkpoint 7

8. Table Talk How did your table use the technology? How did you find an answer to the problem? 8

9. Let’s agree to a location 9

13. The Logical Argument Let D' be the reflection of D across line AB, and let P be any point on segment AB. Then triangles PBD and PBD' are congruent (SAS), and so the distances PD and PD' are equal. Therefore CP + PD = CP + PD'. Therefore finding the smallest total distance CP + PD is the same as finding the smallest total distance CP + PD'. 13

14. The Logical Argument But the distance CP + PD' is least when it is the straight line distance between C and D'; i.e. when P is equal to R, the intersection of segments AB and CD'. In this case, angles ARC and D'RB are congruent (vertical angles), and so triangles RAC and RBD' are similar (AAA). Since triangles RBD and RBD' are congruent (from the second step above), triangles RAC and RBD are similar. 14

15. Model with Mathematics Students: Apply the math they know to solve problems in everyday life, society, and the workplace Use geometry to solve a design problem or Use a function to describe how one quantity of interest depends on another Are comfortable making assumptions and approximations to simplify a complicated situation Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas

16. High School Mathematics A challenge in high school mathematics: too many separately memorized techniques. For example, The Distance Formula sin 2 (t) + cos 2 (t) = 1 Both are manifestations of the Pythagorean Theorem. Help build an understanding that helps students reconstruct these formulas rather than memorize them. From Bill McCallum, lead writer, CCSS-M.

17. High School Conceptual Categories Algebra Geometry Modeling Number and Quantity Statistics and Probability Functions 17

19. We are learning to recognize and apply connections across conceptual categories in the CCSSM. We will be successful when we can explain a specific example among conceptual categories. 19

20. Making Connections  What have I learned in this session? 20