1. Presented by Taryn DiSorbo and Kim Vesper

2.  Developing students’ capacity to engage in the mathematical practices specified in the Common Core State Standards will ONLY be accomplished by engaging students in solving challenging mathematical tasks, providing students with tools to support their thinking and reasoning, and orchestrating opportunities for students to talk about mathematics and make their thinking public. It is the combination of these three dimensions of classrooms, working in unison, that develop students habits of mind and promote understanding of mathematics.

3.  The tasks or activities in which students engage should provide opportunities for them to “figure things out for themselves.” (NCTM, 2009, pg. 11), and to justify and communicate the outcome of their investigation;

4.  Tools (i.e. language, materials, and symbols) should be available to provide external support for learning. (Hiebert, et al, 1997); and  Productive classroom talk should make students’ thinking and reasoning public so that it can be refined and/or extended. (Chapin, O’Conner, & Anderson, 2009).

5.  Significant Content (i.e. they have the potential to leave behind important residue)(Hiebert et al, 1997).  High Cognitive Demand (Stein et. al, 1996; Boaler & Staples 2008)  Multiple ways to enter the task and show competence (Lotan, 2003)  Require justification or explanation (Boaler & Staples 2008)  Make connections between different representations (Lesh, Post, & Behr, 1988)  Provide a context for sense making (Van De Walle, Karp, & Bay-Williams, 2013)  Provide opportunities to look for patterns, make conjectures, and form generalizations. (Stylianides, 2008; 2010)

6. PracticeTask Features ToolsTalk 1 Make sense of problems and persevere in solving them High Cognitive Demand Multiple Entry Points Requires Explanation 2 Reason Abstractly and quantitatively Contextual 3 Construct Viable arguments and critique the reasoning of others Requires justification or proof 4 Model with mathematicsContextual 5 Use appropriate tools strategically 6 Attend to precision 7 Look for and make use of structure Opportunity to look for patterns and make conjectures 8 Look for and express regularity in repeated reasoning Opportunity to make generalizations Teacher Actions

7. PracticeTask Features ToolsTalk 1 Make sense of problems and persevere in solving them High Cognitive Demand Multiple Entry Points Requires Explanation Make resources available that will support entry and engagement 2 Reason Abstractly and quantitatively ContextualEncourage use of different representational forms 3 Construct Viable arguments and critique the reasoning of others Requires justification or proof 4 Model with mathematicsContextual 5 Use appropriate tools strategically Can be solved using different tools Make tools available that will support entry and engagement 6 Attend to precision 7 Look for and make use of structure Opportunity to look for patterns and make conjectures 8 Look for and express regularity in repeated reasoning Opportunity to make generalizations Teacher Actions

8.  Students must talk, with one another as well as in response to the teacher. When the teacher talks most, the flow of ideas and knowledge is primarily from teacher to student. When students make public conjectures and reason with others about mathematics, ides, and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community.  NCTM, 1991, p. 34

9. PracticeTask Features ToolsTalk 1 Make sense of problems and persevere in solving them High Cognitive Demand Multiple Entry Points Requires Explanation Make resources available that will support entry and engagement Ask students to explain their thinking Ask questions that assess and advance student understanding 2 Reason Abstractly and quantitatively Contextual 3 Construct Viable arguments and critique the reasoning of others Requires justification or proof 4 Model with mathematicsContextual 5 Use appropriate tools strategically Can be solved using different tools 6 Attend to precision 7 Look for and make use of structure Opportunity to look for patterns and make conjectures 8 Look for and express regularity in repeated reasoning Opportunity to make generalizations Teacher Actions

10. PracticeTask Features ToolsTalk 1 Make sense of problems and persevere in solving them High Cognitive Demand Multiple Entry Points Requires Explanation Make resources available that will support entry and engagement Ask students to explain their thinking Ask questions that assess and advance student understanding 2 Reason Abstractly and quantitatively ContextualEncourage use of different representational forms Prompt students to make connections between symbols and what they represent in context 3 Construct Viable arguments and critique the reasoning of others Requires justification or proof Ask students to argue for their point of view and evaluate and make sense of the reasoning of their peers 4 Model with mathematicsContextualAsk students to justify their models 5 Use appropriate tools strategically Can be solved using different tools Make tools available that will support entry and engagement 6 Attend to precisionEncourage students to be clear and use appropriate terminology 7 Look for and make use of structure Opportunity to look for patterns and make conjectures Encourage students to look for patterns 8 Look for and express regularity in repeated reasoning Opportunity to make generalizations Prompt students to consider the connections between the current task and prior tasks Teacher Actions

11.  No matter where you live-Pennsylvania, California, Virginia, or Canada-students need opportunities to engage in the habits of practice embodied in the Standards for Mathematical Practices in CCSSM  The features of the tasks that you select for students to work on set the parameters for opportunities they have to engage in these practices-high-level tasks are necessary but not sufficient conditions.  The way in which you support students-the questions you ask and the tools you provide-help good tasks live up to their potential.

12.  According to Webster, RIGOR is strict precision or exactness.  According to mathematicians RIGOR is having theorems that follow from axioms by means of systematic reasoning.  According to most people RIGOR means “too difficult” and “only limited access is possible.”  Also they believe RIGOR is an excuse to avoid high quality math teaching and learning

13.  Rigor is teaching and learning that is active, deep, and engaging. ACTIVE DEEP ENGAGING

14.  Active learning involves conversation and hands-on/minds-on activities. For example, questioning and discovery learning goes on. ACTIVE DEEP ENGAGING

15.  Deep learning is focused, attention given to details and explanations, via problem solving or projects. Students concentrate on the intricacies of a skill, concept, or activity. ACTIVE DEEP ENGAGING

16.  When learning is engaging, students make a real connection with the content. There is a feeling that, although learning may be challenging, it is satisfying. ACTIVE DEEP ENGAGING

17.  Rigor is a process—not a problem ACTIVE DEEP ENGAGING

18. USING THE NCTM PRACTICE STANDARDS AND THE CCSSM HOW DO WE GET STUDENTS ACTIVE AND ENGAGED, THINKING DEEPLY ABOUT MATH?

19. AB C DE F

20. 7. Is this a regular or irregular polygon? Write a descriptive paragraph to support your answer. Include diagram. 8. Explain a method you would use to find the perimeter of the polygon. 9. Using a ruler, determine the perimeter to the nearest centimeter. 10. Describe a method to describe the area. Label your steps in sequential order. Use pictures to describe your steps if you want. AB C DE F

21. 11. Formulate an expression that represents the area of the polygon. 12. Implement your method to solve for the area. 13. If the lengths of the sides were doubled predict how that would affect the perimeter of the figure. 14. If the lengths of the sides were doubled predict how that would affect the area of the figure. 15. If the measures of some angles increased, how would the lengths of the sides change? Justify your answer. AB C DE F

22. 16. Measure each angle and find the sum of the angle measures. Compare the sum of the angle measures to the sum of the angle measures in a triangle, a quadrilateral, and a pentagon. What pattern do you notice? 17. If the polygon were the base of a 3 dimensional figure, what type of figure could it be? Explain your answer. 18. If the polygon is the base of a hexagonal prism, what would its sides look like? AB C DE F

23. 19. How many faces, vertices, and edges would the hexagonal prism have? 20. Explain how you could determine the volume of the hexagonal prism. Compare your method to a classmate’s. How are the two methods alike? How are the two methods different? 21. How many lines of symmetry can you draw in the polygon? 22. Name a line segment that shows a line of symmetry. 23. Use mathematical notation to identify parallel sides. AB C DE F

24. 24. Draw the polygon in Quadrant I of a coordinate plane. 25. Identify the coordinate pairs of each vertex of the polygon. 26. If you translated the polygon 2 units to the right and 3 units down, what would the new coordinate pairs be for each vertex? 27. If you rotate the polygon 90 o, in which quadrant would it be located? 28. Draw a 90 o rotation. AB C DE F

25. 29. Reflect the original polygon in Quadrant I over the x-axis. Identify the coordinate pairs of the image polygon. 30. What type of transformation would have occurred if the image of the original polygon in Quadrant I were in Quadrant 3? Illustrate your answer. 31. If the original polygon in Quadrant I were dilated by a scale factor of ½, what would the coordinate pairs of the new polygon be? 32. Draw a similar figure and write a proportion that shows the scale factor. AB C DE F

26. Overwhelming evidence suggests that we have greatly underestimated human ability by holding expectations that are too low for too many children, and by holding differential expectations where such differentiation is not necessary.

27.  Say “Hi” to everyone at your table and shake hands.  How many handshakes were exchanged at your table?  If we exchange handshakes for the entire room, how many handshakes would there be?

28. We could actually have the 25 people perform the activity of shaking each other’s hands and keep track!

29. Number of Students, n Number of handshakes, H n 21 33 46 510 615 721 828 936 1045...... H 4 =H 3 +3=3+3=6 H 6 =H 5 +5=10+5=15 H n =H n-1 +(n+1)

31.  A Geometric/spatial representation of the problem  4+3+2+1=10

32. Triangular numbers can be Represented by dots arranged In a triangle Triangular number 1234… Number of dots 13610…

33. Create instructional strategies that will address: 1. common misconceptions 2. errors 3. differentiation of instruction 4. student engagement 5. reflection opportunities 6. mathematical communication 7. vocubalury 8. multiple representations of mathematical concepts. PRO EQUITY MODEL

34. 1. Identity Property of Addition (0+n=n+0=n) 2. Zero Property of Multiplication 3. Identity Property of Multiplication (1xn=nx1=n) 4. Golden Rule of Equations 5. The Distributive Property of Multiplication over Addition 6. Commutative Property of Addition and Multiplication 7. Associative Property of Addition and Multiplication. 8. Isolate the Unknown

35.  Solve: x+4=16 x+4=16 -4 -4 x+0=12 x=12

36. STATEMENTS 1. Solve: x+4=16 2. x+4-4=16-4 3. x+0=12 4. x=12 REASONS 1. Given 2. Golden Rule of Eq 3. Simplify (CLT) 4. Identity property of addition

37. STATEMENTS 1. Solve: 3x=15 2. 3x÷3=15÷3 3. 1x=5 4. x=5 REASONS 1. Given 2. Golden Rule of Eq 3. Simplify (divide) 4. Identity property of multiplication

38. STATEMENTS 1. Solve: 3x+4=16 2. 3x+4-4=16-4 3. 3x+0=12 4. 3x=12 5. 3x÷3=12÷3 6. 1x=4 7. x=4 REASONS 1. Given 2. Golden Rule of Eq 3. Simplify (CLT) 4. Identity property of addition 5. Golden Rule of Eq 6. Simplify (divide) 7. Identity property of Multiplication

39. STATEMENTS 1. Solve: 8(x-2)=3x+4 2. 8x-16=3x+4 3. 8x+16-16=3x+4+16 4. 8x+0=3x+20 5. 8x=3x+20 6. 8x-3x=3x+20-3x 7. 5x=20+0 8. 5x=20 9. 5x÷5=20÷5 10. 1x=4 11. X=4 REASONS 1. Given 2. Simplify (distribute) 3. Golden Rule of Eq 4. Simplify (CLT) 5. Identity Property of addition 6. Golden Rule of Eq 7. Simplify (CLT) 8. Identity Property of addition 9. Golden Rule of Eq 10. Simplify (divide) 11. Identity Property of Multiplication

40. Lee V. Stiff, North Carolina State University and EDSTAR Analytics, Inc. http://www.nctm.org/uploadedFiles/Prof essional_Development/Institutes/High_Sc hool_Institute/2013/Workshop_Materials/ NCTM_HS_Institute_Presentation_DC_Augu st2013_PDFVersion.pdf Peg Smith (2013, August). Tasks, Tools, and Talk: A Framework for Enacting Mathematical Practices. http://www.nctm.org/uploadedFiles/Prof essional_Development/Institutes/High_Sc hool_Institute/2013/Workshop_Materials/ Smith-Tasks,%20Tools,%20Talk.pdf Taryn DiSorbo tdisorbo@cheshire.k12.ct.us Kim Vesper kvesper@cheshire.k12.ct.us