1. CAAT Follow Up Meeting #1 Clay County High School Clay County Middle School

2. MEMORY BOX Summer Review

3. 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?

4. Our Norms… Negative experiences with PD Cell phones ringing, texting, distractions Not starting on time No real Agenda Others? Positive experiences with PD Agenda Norms Clear Targets Opportunities to share Others?

5. Goals for CAAT Follow Up Sessions #1 Concept Goal – To help teachers conceptually understand slope/rate and be able to identify the misconceptions students have with slope/rate. #2 Conceptual Goal – To help teachers develop an understanding of how students learn mathematics, in order to better engage student learning. Hi-cognitive/hi-engaging lessons Student misconceptions Formative/Summative assessments

6. Today’s Learning Targets I can share and appreciate the station activity experiences of each group member. 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.

7. Station Activities Think/Pair/Share High vs. Low Cognitive Tasks Student Engagement Pick two station activities that fit unto your units of study between now and Christmas. Try them in that unit and put them in your curriculum map.

8. Connecting Mathematical Ideas Border Problem

9. 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

10. 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.

11. 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.

12. 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.

13. 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.

14. 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

15. 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.

16. 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

17. 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

18. Border Problem Lesson 3 Students conjectured that the following expressions were equivalent to the original. The class was challenged to verify their conjectures. b 2 - (b - 2) 2 = ? b 2 - b - 2 2 ? b 2 - b 2 - 2 ? b 2 - 2 - b 2 ? (b 2 - 2) 2

19. 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

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

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

22. 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?

23. 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.

24. 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

25. Solution Strategies: Martha’s Carpeting Task

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

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

28. Solution Strategies: The Fencing Task

29. The Fencing Task Diagrams on Grid Paper

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

31. The Fencing Task Graph of Length and Area

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

33. 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

34. 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

35. 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

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

37. 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

38. Task Analysis Use Cognitive level Handout and printed tasks

39. To Do List To STOP DoingTo START Doing

40. How did we do? I can share and appreciate the station activity experiences of each group member. 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.

41. Assignment for next meeting Pick one strategy and implement it in your classroom before next meeting. Be able to justify why you picked this strategy and be prepared to share. Read Chapters 1 & 9 from: Total Participation Techniques