Fall 2011 Mathematics SOL Institutes

1. GRADE BAND: 6-8 Fall 2011 Mathematics SOL Institutes Debbie Crawford, Carolyn Holmes, Nöel Klimenko, Amy Lamb,

2. Think about the morning presentation! Which 3 things have hit home for you?

4. Session Outcomes Participants will: analyze tasks for content alignment, process standards, and cognitive depth design learning experiences for students that will increase understanding and long term retention of mathematics.

5. Mindstreaming: How do you choose the problems, tasks or projects that you plan for your students?

6. Directions: With a partner, read each task. Determine if it is a low, medium, or high level task. Sort the tasks accordingly. Discuss your results as a table group and come to a consensus. For each task, write the number of the task in the middle of a Post-It Note. Use the appropriate color and post on your table’s chart. Low-Yellow Medium- Pink High-Blue Task Sort

7. Task #1 This line plot shows the number of letters in the names of 7 students. x x x x x x x 5 6 7 8 9 10 11 Determine the balance point for this set of data and explain how you arrived at this answer. Adapted from EPAT practice items, VDOE, Grade 6, 2010.

8. Task #2 . Determine the value of each expression. 10³ 10² 10¹ 10º Graph each of these values on the same number line. What do you notice? List three true statements about your graph.

9. Task #3 Jordan and Paul were comparing two numbers. Jordan said, “My number is greater than your number.” Paul said, “That may be true, but the absolute value of my number is greater than your number.” Locate Jordan’s and Paul’s number on a number line and explain your reasoning. Compare your answers with other classmates. What do you notice?

10. Task #4 Your job is to design plastic containers for ice cream Sprinkles. Design and sketch containers in the shape of a right triangular prism, a rectangular prism, and a right circular cylinder. Each must fit on a shelf space that is 12 cm tall, 6 cm wide, and 6 cm deep. Sketch each and label the dimensions. Explain which container will hold the most sprinkles for the given shelf space. Which container design would save money by using less plastic? Explain your reasoning.

11. Task #5 Hannah made 54 cupcakes for Erin’s birthday party. She made half of the cupcakes chocolate and half of the cupcakes yellow. She put sprinkles on 1/3 of the chocolate ones. She put one candle on each of the 2/3 cupcakes that did not have sprinkles. How many candles did Erin have to blow out?

12. Task #6 A box shaped like a rectangular prism has a volume of 360 cubic inches. This box has a width of 6 inchesand a length of 10 inches. A. What is the height of the box? B. If you doubled the length, what would be the new volume? Explain how you found each. Adapted from MCAS, Grade 8, 2011

13. Task #7 Identify each number that has an absolute value of 4. 16 4 2 ¼ 0 -2 -4 -16

14. Task #8 Cindy surveyed 60 students about their favorite type of movie. This circle graph represents the results of the survey. Construct a bar graph that could represent the same set of data. Adapted from EPAT practice items, VDOE, Grade 6, 2010.

15. Task #9 What is the value of 2x² + 5(x³ - 4) when x = 4?

16. Task #10 A rectangle as shown has a length of 0.9 centimeters and a length of 0.4 centimeters. A circle is drawn inside that touchesthe rectangle at two points. 0.9 cm 0.4 cm What is the total area of the unshaded region in the rectangle?

17. Content Alignment Process Standards Cognitive Depth What are some key planning resources to use when selecting tasks?

18. Describe the methods you used to solve each task. How did you communicate your reasoning to your table group? Triplet Tasks

19. Content Alignment Based on: • Purpose: student needs • VA SOL • Vertical Alignment Where do you find the content in the tasks in the VA SOL curriculum frameworks? Which related prerequisites did students have previously? How will they use this concept next year?

20. Which process standards were involved for you as a learner while doing these problems? Communication: Talking and writing about math Multiple Representations Problem-Solving Reasoning Connections

21. Cognitive Demand …students who performed best on a project assessment designed to measure thinking and reasoning processes were more often in classrooms in which tasks were enacted at high levels of cognitive demand (Stein and Lane 1996), that is, classrooms characterized by sustained engagement of students in active inquiry and sense making (Stein, Grover, and Henningsen 1996). For students in these classrooms, having the opportunity to work on challenging mathematical tasks in a supportive classroom environment translated into substantial learning gains. ---Stein & Smith, 2010

22. Characteristics of Rich Mathematical Tasks High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008) Significant content(Heibert et. al, 1997) Require Justification or explanation (Boaler & Staples, in press) Make connections between two or more representations (Lesh, Post & Behr, 1988) Open-ended (Lotan, 2003; Borasi &Fonzi, 2002) Allow entry to students with a range of skills and abilities Multiple ways to show competence (Lotan, 2003)

23. Task Analysis Guide – Lower-level Demands Involve recall or memory of facts, rules, formulae, or definitions Involve exact reproduction of previously seen-material No connection of facts, rules, formulae, or definitions to concepts or underlying understandings. Require limited cognitive demand Focused on producing correct answers rather than developing mathematical understandings Require no explanations or explanations that focus only on describing the procedure used to solve Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

24. Task Analysis Guide – Higher-level Demands • Focus on use of procedures for developing deeper levels of understanding of concepts and ideas • Suggest broad general procedures with connections to conceptual ideas (not narrow algorithms) • Provide multiple representations to develop understanding and connections Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

25. Task Analysis Guide – Higher-level Demands DOING Mathematics Require complex, non-algorithmic thinking and considerable cognitive effort Require exploration and understanding of concepts, processes, or relationships Require accessing and applying prior knowledge and relevant experiences to facilitate connections Require task analysis and identification of limits to solutions Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

26. Factors Associated with Impeding Higher-level Demands Shifting emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient or too much time to wrestle with the mathematical task Letting classroom management problems interfere with engagement in mathematical tasks Providing inappropriate tasks to a given group of students Failing to hold students accountable for high-level products or processes Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

27. Factors Associated with Promoting Higher-level Demands Scaffolding of student thinking and reasoning Providing ways/means by which students can monitor/guide their own progress Modeling high-level performance Requiring justification and explanation through questioning and feedback Selecting tasks that build on students’ prior knowledge and provide multiple access points Providing sufficient time to explore tasks Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

29. Rigor Compare the cognitive demand of the tasks. Where would you place the three tasks on the TAG now?

30. Reflection in Groups How would you modify the two problems to better fit the goals of SOL alignment, deeper cognitive depth and student reasoning and communication (PS)? How can a parallel task be written for students who can already solve these problems?

31. Application Return to the original Task Sort and decide as a table group where each item would fit on the Task Analysis Guide based on the descriptors. Choose one task per table to modify for greater cognitive depth. Share with partner table in Task Exchange.

33. Division Professional Development Share resources from the VDOE 6-8 Fall 2011 Mathematics Institutes How will this training information best get to key stakeholders in your division? How can teachers in your division work with these resources to improve teaching and learning of mathematics through problem solving?

34. LUNCH

36. Building Rafts with Rods Create all possible one-color rafts for your given Cuisenaire Rod Create a table to represent the surface area and volume for rafts 1 through 10, raft 12, raft 25, and raft n Create a graph to represent the surface area and volume  Write a good mathematical explanation using appropriate mathematics vocabulary (explain what and why)

37. Task Analysis Questions would typically fall under alignment, cognitive depth, prerequisite skills and concepts, and predictions about student solution paths and misconceptions.

38. Table Talk Does this task align to our curriculum? Which solutions and strategies do you predict students will use? What misconceptions do you predict students might have with this concept? What level of cognitive demand?

39. Do the Task!

40. Gallery Walk • Post your graph one the wall near your table when you have finished • Walk around and look at the products of the other groups Any similarities? differences? surprises?

41. Task in Action Watch the video Building Rafts with Rods Each table has one Process Standard as the lens for watching the video. Record your observations to the graphic organizer.

42. One Stray • At your table discuss what you observed • One member will be selected to be the strayer. • When the word stray is called, the “strayer” moves to the next table • Participants share information in their new team. • This continues until all groups have completed graphic organizers.

43. Student Work Work as a group to look at the student work. Randomly pull a question out of a bag and use it to guide your conversation

44. Reflection What is the value of examining student work?

45. Next Steps • Resources Available • Timeline & Location • 3-2-1 Action Plan • Reflect on your next 3 action steps related to this training. • Identify 2 key people you will need to speak with. • Think of 1 question for which you will want to find out more information.