1. ACT IX Cohort Facilitator: Sohael Abidi October 3 rd, 2008 Day # 1

2. Act Cohorts - Key Focuses  Student & Teacher roles during lesson  Facilitating Discussion: effective questioning - Engaging; Refocusing; Clarifying  The Backward design of a lesson  Assessment ‘For’ Learning; Assessment ‘As’ PD  Embedding the Process Standards in lessons  Mental Math Strategy Development  Student & Teacher reflection on learning  The Three Part Lesson Model Refer to ‘Today’s Mathematics Classroom Grades 7 – 9’ handout

3. ‘ACT’ Through Enduring Questions  What is the math behind this activity?  What are you assuming students need to know to do this activity?  What process standards do you see in this lesson?  How can this activity be accessible to all students?  How will you know when your students have learned the math?  What will you do with your assessment information?  When would you do this activity?

4. ‘Three-Part’ Format for Problem-Based Lessons BEFORE DURING AFTER ▪ Get students mentally prepared for task ▪ Be sure the task is understood ▪ Clearly establish expectations ▪ Let go; listen carefully ▪ provide hints but not solutions ▪ Observe and assess ▪ Students discover the mathematics ▪ Engaging, redirecting, clarifying questions ▪ Allow enough time ▪ Share solutions and strategies ▪ Accept solutions without judgment ▪ Students justify and evaluate results and methods

5. Process Standards How Students Acquire and Learn Math Knowledge  Problem Solving – (build new math knowledge; using variety of strategies; monitoring and reflect on processes)  Reasoning & Proof – (making math conjectures; develop and evaluate arguments/proofs; select and utilize various forms of reasoning and proof)  Communication – (organize and consolidate thinking; clear and coherent; evaluate thinking strategies of others; use math language to express mathematical ideas)  Connections – (recognize and utilize connections among math ideas; understand the interconnectedness of ideas; apply mathematics to external contexts)

6. Process Standards How Students Acquire and Learn Math Knowledge  Representations – (create and use to organize, record, and communicate ideas; use representations to model and interpret math phenomena; apply and select mathematical representations to solve problems)

7. Representations Pictures Oral Language Written SymbolsManipulatives Real-World Situations Elementary and Middle School Mathematics: Teaching Developmentally by John A. Van de Walle

8. Activity: The Human Number Line  Obtain a number card from the facilitator  Form a number line, from least to greatest  Their should be NO TALKING!  Once in line, you may discuss and explain your strategy that helped you make your decision.

9. Human Number Line: Answer Least 0.01, 1/9, 0.2, ¼, 27%, 2/7, 1/3, 2/5, 0.45, ½, 53%, 4/7, 2/3, 5/7, 4/5, 0.82, 5/6, 9/10 3/3 Greatest

10. Debriefing Discussion:  Share our strategies  Difficulties?  What was one thing that you learned from the people beside you?  How do we make this accessible to all learners?  Reflect on the ‘Enduring Questions’ for each classroom activity…

11. ‘ACT’ Through Enduring Questions Human Number Line  What is the math behind this activity?  What are you assuming students need to know to do this activity?  What process standards do you see in this lesson?  How can this activity be accessible to all students?  How will you know when your students have learned the math?  What will you do with your assessment information?  When would you do this activity? We will revisit these ‘core questions’ throughout the cohort

12. Introducing a Mental Math Strategy Comparing & Ordering Real Numbers  Warm-up – Pattern Blocks & Fraction Factory  The Before: - If this is the whole, find ½. Find 1/6, ¼, 2/3, 9/12 - If this is the whole, find 2/6, ½, 2,3

13. Warm-up Continued:  If this is 2/3 of the whole, find the whole.  Display each of the following fractions using pattern blocks or Fraction Factory: 1/3, 5/6, 5/12, 2/4

14. Ordering Real Numbers (During)  Indicate whether each of the following fractions are closer to either 0 or 1. (jot your answer down) Ready??

15. Ordering Real Numbers

17. 0.78

18. Ordering Real Numbers 0.3

19. Ordering Real Numbers

20. 0.52

21. Ordering Real Numbers

22. The “After”  What were some strategies used to make your decisions of ‘closer to 0 or 1?’  Share difficulties & discoveries  Could we have used a ‘reference point’ along the number line? (benchmark)  How would this have helped?  Let’s model this using Fraction Factory or Pattern Blocks… (using ½ as our benchmark)  How could we differentiate this activity? (Section #2 – Teacher Resource)

23. Lesson #1: “How Close is Close?” (10 by 10 Grids / Faction Factory)  With a partner, list 4 fractions between 1/9 & 8/9.  Share your responses with your table members. Discuss methods as a group

24. Lesson #1: “How Close is Close?” (10 by 10 Grids / Faction Factory)  With a partner, list 4 fractions between ½ & 9/10.  Share your responses with your table members. Discuss methods as a group

25. Lesson: “How Close is Close?” (10 by 10 Grids / Faction Factory)  Activity: Select two fractions that you believe are really close.  The Task: Find 10 fractions that are between the two that you chose.  Use any method you want, but you must be able to explain the method to your partner.  Partner sharing session.

26. Discussion  Did your group make any discoveries?  Share strategies of our group  Can you make a conjecture regarding ordering fractions ?  In what way has this activity changed your understanding of fractions?

27. Lesson #1 - Debrief  Refer back to the ‘Enduring Questions’ (slide 11)  What were the 3-parts of this previous lesson (before, during, & after)?  How Close is Close? – Lesson Plan Handout – discuss (designing a lesson backwards) - Big ideas, process standards, assessment  Student Understanding - Handout

28. Reinforcing a Mental Math Strategy  What are some MM strategies we have discussed so far?  Activity: - State whether the following real numbers are closer to ½ or 1. Ready??

29. Reinforcing a Mental Math Strategy

36.  Okay, now…  Place these values from least to greatest on a number line. (Fraction Factory available) 5/6, 2/3, 0.63, 12/24, 0.75, 6/8, 10/16

37. Discussion/Debrief  What were some strategies used for determining closer to ½ or 1?  What were some strategies used for the number line placement activity?  What might we see that could indicate the level of student thinking or understanding?  What types of questions would lend to promoting student thought and exploration?

38. LUNCH

39. Back to The ‘Enduring Questions’ Closer to ½ or 1?  Did any answers change from the introductory activity? (i.e.: closer to 0 or 1?)  What is the math behind this activity?  What are you assuming students need to know to do this activity?  What process standards do you see in this lesson?  How can this activity be accessible to all students?  How will you know when your students have learned the math?  What will you do with your assessment information?  When would you do this activity?

40. Mental Math Focus  Handout select pgs. Of MM Booklet  Please read select pgs. from grade 9 MM Booklet.  “Introducing, reinforcing, & assessing” MM  In groups, share ways in which you reinforce and assess MM.  Assign one recorder for each group.  Share strategies with the entire group.

41. Things to remember…  The idea of the “3 second response” expectation  Variety of forms of assessment: observations; oral responses; explanations of strategies  Assessment “For” vs. “Of” learning  Importance of visuals when helping students understand a MM strategies: Alge-tiles; fraction factory; pattern blocks etc.

42. Getting Back 49 units 2 36 units 2 45 units 2 ? ? Where/When would this activity be useful? Student Thinking?

43. Lesson #2 “One integer is double another. The sum of their squares is 45. What are the integers?” Debrief Strategies

44. “The Town Hall” (See Handout) A town council set up their town map in such way that the Town Hall was at the center (0, 0). This was then overlaid by a four-quadrant grid so that all locations were determined using positive and negative coordinates. The Hospital is located at (-5, -4), and the community swimming pool is located at (1, 4). One grid unit represents 1 km of actual distance. (a) How far are the hospital and the pool from city hall? (b) How far is the hospital from the pool? (Discuss your solution and procedure with a partner)

45. Discussion/Debrief  Strategies? (volunteers?)  Areas of struggles?  What about the Absolute Value and Principal Square root?  Restrictions on Distance?

46. Lesson #2 - Debrief  Possible ‘Follow-Up’ ideas that will Reinforce student learning?  See Portfolio Assignment  Refer to the ‘Enduring Questions.” What outcome(s) did this activity cover?  What are the three parts in the ‘3-Part Lesson Model?’ – What were they in the previous lesson?

47. Mental Math – Reinforcement of a Strategy Ordering Real Numbers  Let’s recall some strategies used to place numbers in order on a number line.  Activity: Place the following numbers on #line using our Mental Math strategies. Do this activity alone, first. Then discuss with a partner.

48. 22/7, 6/7, 7/8, -3/4, 2/3, 0.62, ∏, -0.58

49. Discussion  What was the focus of conversations?  Difficulties of placing certain numbers?  Share solutions and strategies (volunteers?)  Brainstorm: what types of assessment would be appropriate for this MM section? (Question levels; Question Types; pg.94 T.R) Portfolio Idea: Have students write about the strategies that are helpful to them, and when they apply.

50. Manipulative Activity – Fraction Factory Refer to Handouts  5 – Minutes to Play!  Demo and practice – work on sheets  Addition, Subtraction, Multiplication, Division

51. Q & A  Participant questions  Homework  Sub-claim Forms