1. Laurel County SSA Day 4 Jim Moore moore6346@bellsouth.net Jennifer McDaniel Jennifer.mcdaniel@clay.kyschools.us

2. Agenda 8:00-8:30Sign In, Warm-Up Problems 8:30-8:45 Outline Goals and Expectations 8:45-10:45 Break Out Sessions (Building Algebraic Thinking) 11:00-12:30 Lunch 12:30-1:30 Strategies to Engage Students 1:30-1:40 Break 1:40-2:30 Using Technology to Address the Common Core Standards 2:30-3:00 Wrap-Up, Reflections, and Evaluations

3. MEMORY BOX Summer Review

4. Memory Box How might you use this strategy in your classroom? Can you name five “go to” strategies that you use already in your classroom to review material?

5. SSA Goals - Assist teachers with diagnosing needs specific to learning goals in The Number System, Expressions and Equations and Functions. -Have teachers design instructional experiences that incorporate technology, manipulatives, vocabulary strategies, various representations, communications and connections to engage and motivate students of all learning styles. -Connect the Mathematical Practice Standards to the mathematical content

6. SSA Goals - Teachers understand that strategies will also be useful as interventions for struggling students. - Participants leave each training session with the instructional materials used in training.

7. SSA Goals Some Trainer Stepping Stones to the Stated Goals: - Creating familiarity with KCAS, Career and College Readiness Standards and successful EPAS performance. To this end, trainers will work in partnership with district leadership and MSLN leaders to identify content focus and related assessments as indicators of improvement over the three year project.

8. SSA Goals - give teachers experiences/strategies needed to elicit higher level thinking skills that can be incorporated into their instruction. - give teachers first hand experience with technology tools to use to support instruction (e.g. GeoGebra, Graphing Calculators, TI Nspire Teacher Edition Software, etc.). - give teachers knowledge of and experience with tools to create a classroom climate to ensure that all students are engaged in the learning of mathematics

9. Day One Stumbling Blocks -Prequisite Skills/retention/fear of failure-past experiences; Lack of support from home, motivation, waiting to dropout; Content: generalizations, variables, moving beyond concrete stage, misuse of technology Technology-Inspire CX - Introduction Characteristics of Highly Effective Teaching and Learning (CHETL)  Learning Environment  States of Mind Integer Games and Number Line Activities Persistence in Problem Solving  Connections to Mathematical Practice Standards  Multiple Representations (NAGS)

10. Day 2 Learning Styles & Student Engagement  Task Rotations  Question Museums  Parallel and Open Tasks  Conceptual Skills Practice Problems  Examples: Open Task: You multiply two integers. The result is 50 less than one of them. What might the two integers be?

11. Day 2 Parallel Task: Choice 1: Describe what 10 colored cubes you would put in a bag so that the probability of selecting a red one is high, but not certain. Choice 2: Describe what 10 colored cubes you would put in a bag so that the probability of selecting a red one is 2/5.

12. Day 2 Conceptual Skills Problem: The average of four numbers is negative. Explain your response to each: (a) Can all four numbers be negative? (b) Can all four numbers be positive? (c) Can only two of the four numbers be positive? (d) Can only three of the numbers be negative? (e) Can only one of the four numbers be negative?

13. Day 2 Writing Quality Questions Open Questions, Parallel Tasks and Conceptual Skills Practice Problems

14. Day 3 Modeling with Mathematics Data Analysis  Comparing TI-84 & CX GeoGebra (Free Technology) – Introduction as as dynamic geometry software, all types of graphing and data analysis Grade level grouping for a first effort in writing some open questions, parallel tasks and conceptual skills practice problems.

15. Feedback Technology – requests for TI 84 and TI Nspire, GeoGebra, SmartBoard, Manipulatives, iPads Hands on activities Writing Quality Questions Manipulatives

16. Follow Up Three Full Day On Site Sessions  Sept. 28 th  Oct. 25 th  Nov. 29th Administration Support  TBD Digital Presence Year 2 & Year 3

17. Laurel County SSA Day 4 Middle School Breakout Session Jim Moore moore6346@bellsouth.net Jennifer McDaniel Jennifer.mcdaniel@clay.kyschools.us

18. Today’s Learning Targets I can use the border problem to build algebraic thinking in my classroom. I can determine high and low cognitive tasks. I can identify appropriate mathematical practices addressed during a high level cognitive task. I can develop a stop doing/start doing list to create high cognitive level tasks in my classroom.

19. Connecting Mathematical Ideas Border Problem

20. The Border Problem Without counting, use the information given in the figure above (exterior is 10 x 10 square; interior is an 8 x 8 square; the border is made up of 1x1 squares) to determine the number of squares needed for the border. If possible, find more than one way to describe the number of border squares.

21. What about a 6 in by 6 in grid? What about a 15 in by 15 in grid? What about a 253 in by 253 in grid? What about an n inch by n inch grid? Create a verbal representation Use the verbal representation to introduce the notion of variable If n represents the number of unit squares on one side, give an algebraic expression for the number of unit squares in the border. Develop understanding of function, variables (independent and dependent) and graphing.

22. Border Problem Video Part One (Use printed transcripts to follow the dialogue) As you watch the video, concentrate specifically on the activity, the teacher, the students, and the learning environment.

23. The Teacher’s Strategy The teacher used the experience of the 10 by 10 border problem to built algebraic understanding. She asked the students to think about a smaller square, 6 by 6, and asked the students to determine a set of equations of the 6 by 6 that matched the ways the students thought about the 10 by 10 square. They had to write new equations in the same manner that Sharmane, Colin and the others had in the first problem. Next the teacher asked the students to color a picture of the border problem, to match each equation and also write the process to find each total in a paragraph. Now she felt the students were ready to use algebraic notation to generalize each equivalent equation.

24. Video Discussion Why without talking? Why without writing? Why without counting one by one? Why not give them each a grid to facilitate their thinking? Why did the teacher act as the recorder for the arithmetic expressions? Boaler, J. & Humphreys, C. (2005). Building on student ideas: The border problem, part I. Connecting mathematical ideas: Middle school video cases to support teaching and learning (pp.13-39). New Hampshire: Heineman Publications.

25. The Border Problem Sharmane: 410 - 4 = 36 Colin: 10+9+9+8 = 36 Joseph: 10+10+8+8 = 36 Melissa: 1010 - 88 = 36 Tania:49 = 36 Zachery:48 + 4 = 36

26. Border Problem Video Part two (Use printed transcript to follow the dialogue) As you watch the video, concentrate specifically on the activity, the teacher, the students, and the learning environment.

27. Student Equations Generalizing For Any Size Square 10+10+8+8=36 Let x be the number of unit square along the side of the square. x + x + m + m = total x + x + (x- 2 ) + (x- 2 ) = total

28. Introducing Algebraic Notation Moving from the specific to the general case. Developing an understanding of variable and its uses. Tying abstract ideas to concrete situations. Fostering meaning to notation. Developing the concept of equivalent expressions. Encouraging efficiency and brevity in notation

29. The Border Problem allowed for most (if not all) students to develop an algebraic expression, which would calculate the square units in the border of a square frame. What I found is that many of the students did not naturally use a variable in their expression. In the future, I would require students to work with several different size square borders; then have them present their expressions while I compiled a list of correct ones. We would then look for similarities and as a Part II, I would have the expectation that generalizations be made, and that a variable represent the same “part” of different sized frames. Teacher Reflections

30. Comparing Two Mathematical Tasks zMartha’s Carpeting Task zThe Fencing Task

31. Comparing Two Mathematical Tasks How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?

32. Martha’s Carpeting Task Martha was re-carpeting her bedroom, which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase?

33. The Fencing Task Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen to keep the rabbits.  If Ms. Brown’s students want their rabbits to have as much room as possible, how long would each of the sides of the pen be?  How long would each of the sides of the pen be if they had only 16 feet of fencing?  How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.

34. Comparing Two Mathematical Tasks Think privately about how you would go about solving each task (solve them if you have time) Talk with you neighbor about how you did or could solve the task  Martha’s Carpeting  The Fencing Task

35. Solution Strategies: Martha’s Carpeting Task

36. Martha’s Carpeting Task Using the Area Formula A = l x w A = 15 x 10 A = 150 square feet

37. Martha’s Carpeting Task Drawing a Picture 10 15

38. Solution Strategies: The Fencing Task

39. The Fencing Task Diagrams on Grid Paper

40. The Fencing Task Using a Table LengthWidthPerimeterArea 1112411 2102420 392427 482432 572435 662436 752435

41. The Fencing Task Graph of Length and Area

42. Comparing Two Mathematical Tasks How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?

43. Similarities and Differences Similarities Both are “area” problems Both require prior knowledge of area Differences The amount of thinking and reasoning required The number of ways the problem can be solved Way in which the area formula is used The need to generalize The range of ways to enter the problem

44. Similarities and Differences Similarities Both are “area” problems Both require prior knowledge of area Differences The amount of thinking and reasoning required The number of ways the problem can be solved Way in which the area formula is used The need to generalize The range of ways to enter the problem

45. A Critical Starting Point for Instruction Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking. Stein, Smith, Henningsen, & Silver, 2000

46. The level and kind of thinking in which students engage determines what they will learn. Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

47. There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics. Lappan & Briars, 1995

48. Task Analysis Use Cognitive level Handout and printed tasks

49. To Do List To STOP DoingTo START Doing

50. Laurel County SSA Day 4 High School Breakout Session Jim Moore moore6346@bellsouth.net Jennifer McDaniel Jennifer.mcdaniel@clay.kyschools.us

51. The House that Jack Built Building Algebraic Thinking

52. Lunch 11:00-12:30

53. Ideas for engagement Turnover Cards Card Sorts Math Strings

54. Technology Human Wave/Pass the ball Regression activity for graphing calculator How does the CX compare to the TI-84?

55. Solving Equations using the graphing calculator. Standard (A-REI.1) Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

56. New Ideas: The hidden potential in QR Codes

57. Shared Resources Content Network Updates NROC website