1. Breathitt/Jackson Independent IMPACT Training Day 3 (first half) Ruth Casey ruthcasey@aol.com Jennifer McDaniel Jennifer.mcdaniel@clay.kyschools.us

2. Agenda 3:45-4:00 Sign In, Memory Box Activity, Evidence of Habits of Mind (Instruction) 4:00-4:45 Building Algebraic Thinking: The Border Problem (Mathematics, Problem Solving) 4:45-5:15 Building Algebraic Thinking: Multiple Representations (Mathematics, Problem Solving, Assessment) 5:15-5:30 Break 5:30-6:00 Mathematical Modeling: Linear or KNOT? (Mathematics, Problem Solving, Technology) 6:00-6:45 Engaging Students: Card sorts, Turnover Cards, Always, Sometimes, Never (Instruction, curriculum)

3. Today’s Learning Targets I can use problems such as: the border problem and manipulatives such as: algebra tiles to strengthen my pedagogy skills to build algebraic thinking in my classroom. I can recognize the need for multiple representations of solutions in my classroom. I can utilize simple strategies to enhance engagement in my classroom.

4. MEMORY BOX IMPACT Review

5. 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?

6. IMPACT Goals I (Instruction)-Learn about effective strategies and activities for you classroom M (Mathematics)- Strengthen and expand your subject area expertise A (Assessment)- Experience and discuss effective uses of formative and summative assessment C (Content)-Identify and align resources designed to satisfy content requirements T (Technology)- Gain hands-on experience using technology as a tool for teaching and learning mathematics.

7. Needs Assessment Instruction- projects, hands-on activities, researched based strategies, sharing ideas about specific activities, more diverse strategies, address different learning styles, different approaches to teaching. Mathematics-Core Content updates, research based practices, address needs of special needs students. Problem Solving-FALS, resources (books, websites, problems, empower students) Assessment- Open Response and K-Prep materials, ways to manage paper work for mastery learning, website resources, data. Curriculum-Pacing Maps, revise units, how to integrate common core standards into real world application, ideas to close gaps. Technology-Effective and appropriate use, equity in access, enhance activities, computer lab ideas.

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. 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.

13. 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.

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. 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

19. Standards for Mathematical Practices 1.Make sense of problems and persevere in solving them. (Organize and consolidate mathematical thinking through communications.) 2.Reason abstractly and quantitatively. (Communicate mathematical thinking coherently and clearly.) 3.Construct viable arguments and critique the reasoning of others. (Analyze and evaluate the mathematical thinking of others.) 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. (Including the use of the language of mathematics to express ideas precisely.) 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning.

20. Habits of Mind Word Splash Persisting means…

21. Y-Chart Persistence What does it look like? What does it sound like? What does it feel like?

22. Picture of Persistence http://www.youtube.com/watch?v=vP2P8mhif- g

23. Question #1 Which of the Mathematical Practices are currently embedded in your instruction and where are they embedded?

24. Question #2 Where do you see opportunities to embed the Mathematical Practices in your instruction? Take a few minutes to brainstorm individually, then at your table.

25. Using Pattern Tiles to build Algebraic Thinking

26. What do we mean by Algebraic Thinking?

27. Algebraic Thinking… “The set of understandings that are needed to interpret the world by translating information or events in the language of mathematics in order to explain or predict phenomena”

28. Developing Algebraic Thinking often requires… Using or setting up mathematical models Gathering and recording data Organizing data and looking for patterns Describing and extending patterns Generalizing findings, often into a rule Using findings, including rules, to make predictions

29. To develop true understanding… Students must work with problem situations that arise throughout the stands of mathematics and in various contexts that are familiar or make sense to them.

30. As students solve engaging, meaningful problems, teachers must focus on… Analyzing change, especially rates of change Understanding functions, especially linear functions Understanding and using variables in different ways Interpreting, creating, and moving fluently between multiple representations for data sets

31. Pedagogical strategies to develop Algebraic Thinking Exposing students to a variety of patterns over time Using relevant/real-world patterns early and often Working with students to recognize and describe, extend, and generalize each pattern whenever possible Starting with patterns created using concrete objects and encouraging the use of drawings, words, and symbols when ready Beginning with simple patterns, but progressing rapidly to more complicated ones Spiraling use of vocabulary such as stage, constant, variable, iterative rule, and explicit rule Encouraging multiple interpretations of each pattern Incorporating the use of a t-chart Validating correct iterative rules while encouraging searches for appropriate explicit rules Asking students to use their rule(s) to predict what each pattern will look like or what its value will be for several stages, larger stage, any stage Asking students to find the appropriate stage number for a particular stage of a pattern.

32. Using Pattern Tiles to build Algebraic Thinking Compare the number of tables to the number of people that can be seated using different pattern tiles. Begin with one “table” and record the number of people that can be seated around the “table” Connect two tables and record the number of people that can be seated. Keep adding tables and recording the number of people that can be seated until you discover a pattern. Draw pictures, describe the pattern in words, and then write the function rule.

34. Multiple Representations Key understandings are built upon working with concrete materials that form the foundations for working with pictorial, tabular, graphic, and eventually, symbolic representations. Students need many varied experiences in meaningful contexts with each of these representations before they can truly understand the symbolic expressions and rules of formal algebra.

41. Top Hat Organizer Pattern Tile: Similarities Differences

43. How Do I Know that I Know?

44. Multiple Representations http://www.cpm.org/pdfs/information/conference /AC_Con_Mult_Rep.pdfhttp://www.cpm.org/pdfs/information/conference /AC_Con_Mult_Rep.pdf

45. Mathematical Modeling & Technology Activity: Linear or KNOT?

46. In this activity you will explore the relationship between the number of knots in a rope and the length of the rope. Question: What do you think will happen to the rope as you tie knots in it? Think, Pair, Ink, Share Linear or “Knot”?

47. Linear or “Knot” Directions Choose a piece of rope. Measure the rope using the meter stick and record your data. Tie a knot in the rope, then re-measure it. Record your data. Repeat this process until you have tied a total of 7 knots in the rope. As you measure after tying each knot, be sure to record your data. Questions A.Does there appear to be a relationship between the two variables? Explain your reasoning using NAGS rule. B.Can you predict the length of the rope if it has 10 knots? Explain your reasoning using mathematical language. C.Using appropriate math tools, be prepared to show how you can justify your answer in part B.

48. Learning Targets? Bell Ringer? Grouping? Vocabulary? Technology? Sharing? Similarities/Differences? Next Steps? Activity vs. Unit

49. Check this out http://www.acps.k12.va.us/curriculum/design/sa mple-algebra-course.pdfhttp://www.acps.k12.va.us/curriculum/design/sa mple-algebra-course.pdf

50. Ideas for engagement Turnover Cards Card Sorts Always Sometimes Never Math Strings Integer Card Games

51. Shared Resources Content Network Updates NROC website www.mathshell.org

52. Next Steps Formative Assessment Lessons www.mathshell.org