1. Don’t Be So Symbol Minded Problem Solving, Reasoning, and Sense Making in the Core Standards Environment Jim Rubillo JRubillo@verizon.net

2. An Important Notice The opinions expressed in this presentation are solely those of the presenter. They are offered for your consideration and reflection. They are aligned with the speaker’s lifelong motto: “Seldom right, but never in doubt.”

6. What Do You See? A List of Specific Content Skills or A Plan with a Purpose?

7. Primary Purpose of the Endeavor! Ensuring College and Workforce Readiness for ALL Students

8. While You Are Teaching Them Math, Teach Them … (Judy Spitz, CIO, Verizon) How to see the big picture. How to see the forest for the trees. How to be good at story telling. Understand the difference between leadership and power. “Leadership is the ability to get people to move in a consistent direction when you have no power over them.” Think non-linearly, but execute in a linear fashion.

11. These Standards endeavor to follow such a design, not only by stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value or the laws of arithmetic to structure those ideas.

12. 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 and express regularity in repeated reasoning. Standards for Mathematical Practice

13. SHOW ME!

14. What is PROBLEM SOLVING? Engaging in an activity for which the method of solution is not known in advance.

15. A Problem for You! Prove that NO number in the following “Fibonacci-Like” sequence is divisible by 5: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682,...

16. A Fundamental Example I have quarters, dimes, and nickels in my pocket. If I take three coins out of my pocket, how much money could I have in my hand? 15 cents 75 cents 20 cents 25 cents 30 cents 35 cents 40 cents 45 cents 50 cents 55 cents 60 cents 65 cents 70 cents

17. Solve the following using only the numerals used in our base ten number system: 0,1,2,3,4,5,6,7,8,9 In the equation: The same letter stands for the same digit and different letters stand for different digits. There are no repeated digits. There are ten letters and ten digits. What is the value of the product ? Solution: F, O, U, and R can not equal zero, why? The value of T must equal zero, why? The values of E, I, G, H, F, O, U, R, and W can scramble the values 1,2,3,4,5,6,7,8,and 9.

18. Basic Skills are important, but so is the understanding when it is useful! YES, we want a 4 th grader to know how to correctly complete:

19. But, can you think of four unique situations where knowing how to compute would be useful to a 4th grader?

20. Situation A: There are 4 students and 26 cookies. If the cookies are equally divided among the students, how many cookies should each student receive? 26  4 = ?

21. Situation B: You have 26 quarters. You go to the bank and ask the teller to swap your quarters for $1 bills. How many one dollar bills will you receive? 26  4 = ?

22. Situation C: There are 26 students going on a class field trip. We are driving to the site in cars. We can place a maximum of four students in each car. What is the minimum number of cars required for the field trip? 26  4 = ?

23. Situation D: There are 26 students in a classroom. The teacher wants to arrange the students’ into four rows? How many students are in each row? 26  4 = ?

24. Is the “Standard” Algorithm Just a Set of Rote Procedures? 1 24 x13 72 24__ 312

25. Why Does It Work? 20 + 4 10 + 3 x 10 + 3126040 + 200 312

26. An Alternate Algorithm?

27. The Link to Area 24 x 13 12 60 60 40 40 + 200 + 200 312 3 10 20 4

28. The Far Too Typical Experience! 1.Here is an equation: y = 3x + 4 2.Make a table of x and y values using whole number values of x and then find the y values, 3.Plot the points on a Cartesian coordinate system. 4.Connect the points with a line. Opinion: In a student’s first experience, the equation should come last, not first.

29. Situation 3:The Mirror Problem Parts Corner Edge Center A company makes “bordered” square mirrors. Each mirror is constructed of 1 foot by 1 foot square mirror “tiles.” The mirror is constructed from the “stock” parts. How many “tiles” of each of the following stock tiles are needed to construct a “bordered” mirror of the given dimensions?

30. The Mirror Problem

31. Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft 3 ft x 3 ft 4 ft x 4 ft 5 ft x 5 ft 6 ft x 6 ft 7 ft by 7 ft 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft

32. The Mirror Problem Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft 4 ft x 4 ft 5 ft x 5 ft 6 ft x 6 ft 7 ft by 7 ft 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft

33. The Mirror Problem Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft441 4 ft x 4 ft 5 ft x 5 ft 6 ft x 6 ft 7 ft by 7 ft 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft

34. The Mirror Problem Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft441 4 ft x 4 ft484 5 ft x 5 ft 6 ft x 6 ft 7 ft by 7 ft 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft

35. The Mirror Problem Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft441 4 ft x 4 ft484 5 ft x 5 ft4129 6 ft x 6 ft 7 ft by 7 ft 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft

36. The Mirror Problem Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft441 4 ft x 4 ft484 5 ft x 5 ft4129 6 ft x 6 ft416 7 ft by 7 ft 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft

37. The Mirror Problem Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft441 4 ft x 4 ft484 5 ft x 5 ft4129 6 ft x 6 ft416 7 ft by 7 ft42025 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft

38. The Mirror Problem Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft441 4 ft x 4 ft484 5 ft x 5 ft4129 6 ft x 6 ft416 7 ft by 7 ft42025 8 ft by 8 ft42436 9 ft by 9 ft 10 ft x 10 ft

39. The Mirror Problem Mirror Size Number of 2 borders tiles Number of 1 border tiles Number of No border tiles 2 ft x 2 ft400 3 ft x 3 ft441 4 ft x 4 ft484 5 ft x 5 ft4129 6 ft x 6 ft416 7 ft by 7 ft42025 8 ft by 8 ft42436 9 ft by 9 ft42849 10 ft x 10 ft

40. The Mirror Problem Mirror Size Number of “Tiles” (2 borders) Number of “Tiles” (1 border) Number of “Tiles” (No borders) Total Number of “Tiles” 2 ft x 2 ft4004 3 ft x 3 ft4419 4 ft x 4 ft48416 5 ft x 5 ft412925 6 ft x 6 ft416 36 7 ft by 7 ft4202549 8 ft by 8 ft4243664 9 ft by 9 ft4284981 10 ft x 10 ft43264100

41. The Mirror Problem sidetiles 20 31 44 59 616 725 836 949 1064 1181 1 2 3 4 5 6 7 8 9 10 11

42. The Mirror Problem 1 2 3 4 5 6 7 8 9 10 11

43. The Mirror Problem Mirror Size Number of “Tiles” (2 borders) Number of “Tiles” (1 border) Number of “Tiles” (No borders) Total Number of “Tiles” 2 ft x 2 ft4004 3 ft x 3 ft4419 4 ft x 4 ft48416 5 ft x 5 ft412925 6 ft x 6 ft416 36 7 ft by 7 ft4202549 8 ft by 8 ft4243664 9 ft by 9 ft4284981 8 ft by 8 ft43264100 ::::: n ft by n ft 44(n-2)(n-2) 2 n2n2

44. 7 6 5 4 3 2 17 03 012345678 PENCILSPENCILS ERASERS Total Cost Table: Example 1

45. 7 6 5 4 28 3 2 24 1 0 012345678 PENCILSPENCILS ERASERS Total Cost Table Example 2

46. The Graph Tells a Story! Interpretation is More Important than Drawing You are at the movie and you buy a cup of popcorn. The graph shows the level of the popcorn in your cup. What happened?

47. The Graph Tells a Story! Interpretation is More Important than Drawing

48. What are Today’s Big Ideas? Problem Solving and Reasoning should be a vital part of every mathematics lesson. “Why?” and “How do you know that?” should be the most frequently asked teacher questions. There are at least five ways of representing a mathematical concept. Each way can enhance and extend our understanding.

49. Key to Implementing Any Set of Standards!

50. 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 and express regularity in repeated reasoning. ENCORE! The Standards for Mathematical Practice

51. Keys to Implementing the Standards! Integrate the Standards of Mathematical Practice Collaboration/Teamwork among Teachers Articulation through the Grades Assessment to Improve Teaching and Learning

52. 8Q + 2Q = ? email: JRubillo@verizon.net

53. The Mirror Problem Mirror Size Number of “Tiles” (2 borders) Number of “Tiles” (1 border) Number of “Tiles” (No borders) Total Number of “Tiles” 2 ft x 2 ft 3 ft x 3 ft 4 ft x 4 ft 5 ft x 5 ft 6 ft x 6 ft 7 ft by 7 ft 8 ft by 8 ft 9 ft by 9 ft 10 ft x 10 ft