1. © 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Making Sense of the Number System Standards via a Set of Tasks Tennessee Department of Education Middle School Mathematics Grade 6

2. Rationale Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it, yet not all tasks afford the same levels and opportunities for student thinking. [They] are central to students’ learning, shaping not only their opportunity to learn but also their view of the subject matter. Adding It Up, National Research Council, 2001, p. 335 By analyzing instructional and assessment tasks that are for the same domain of mathematics, teachers will begin to identify the characteristics of high-level tasks and differentiate between those that require problem-solving and those that assess for specific mathematical reasoning. 2

3. © 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: make sense of the Number System Common Core State Standards (CCSS); determine the cognitive demand of tasks and make connections to the Mathematical Content Standards and the Standards for Mathematical Practice; and differentiate between assessment items and instructional tasks. 3

4. © 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: analyze a set of tasks as a means of making sense of the Number System Common Core State Standards (CCSS); determine the Standards for Mathematical Content and the Standards for Mathematical Practice aligned with the tasks; relate the characteristics of high-level tasks to the CCSS for Mathematical Content and Practice; and discuss the difference between assessment items and instructional tasks. 4

5. © 2013 UNIVERSITY OF PITTSBURGH The Data About Students’ Understanding of Rational Numbers 5

6. Linking to Research 6

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8. Linking to Research/Literature Research has shown that children who have difficultly translating a concept from one representation to another are the same children who have difficulty solving problems and understanding computations. Strengthening the ability to move between and among these representations improves the growth of children’s concepts. Lesh, Post, & Behr, 1987. 8

9. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Tasks as a Means of Making Sense of the CCSS The Number System 9

10. TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project 10

11. TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project Setting Goals Selecting Tasks Anticipating Student Responses Orchestrating Productive Discussion Monitoring students as they work Asking assessing and advancing questions Selecting solution paths Sequencing student responses Connecting student responses via Accountable Talk ® discussions Accountable Talk ® is a registered trademark of the University of Pittsburgh 11

12. © 2013 UNIVERSITY OF PITTSBURGH Linking to Research/Literature: The QUASAR Project Low-level tasks High-level tasks 12

13. © 2013 UNIVERSITY OF PITTSBURGH Linking to Research/Literature: The QUASAR Project Low-level tasks – Memorization – Procedures without Connections High-level tasks – Doing Mathematics – Procedures with Connections 13

14. The Mathematical Task Analysis Guide Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press. 14

15. © 2013 UNIVERSITY OF PITTSBURGH The Cognitive Demand of Tasks (Small Group Discussion) Analyze each task. Determine if the task is a high-level task. Identify the characteristics of the task that make it a high-level task. After you have identified the characteristics of the task, then use the Mathematical Task Analysis Guide to determine the type of high-level task. Use the recording sheet in the participant handout to keep track of your ideas. 15

16. © 2013 UNIVERSITY OF PITTSBURGH The Cognitive Demand of Tasks (Whole Group Discussion) What did you notice about the cognitive demand of the tasks? According to the Mathematical Task Analysis Guide, which tasks would be classified as: Doing Mathematics Tasks? Procedures with Connections? Procedures without Connections? 16

17. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Tasks: Aligning with the CCSS (Small Group Discussion) Determine which Content Standards students would have opportunities to make sense of when working on the task. Determine which Standards for Mathematical Practice students would need to make use of when solving the task. Use the recording sheet in the participant handout to keep track of your ideas. 17

18. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Tasks: Aligning with the CCSS (Whole Group Discussion) How do the tasks differ from each other with respect to the content that students will have opportunities to learn? Do some tasks require that students use Standards for Mathematical Practice that other tasks don’t require students to use? 18

19. The CCSS for Mathematical Content: Grade 6 Common Core State Standards, 2010, p. 43, NGA Center/CCSSO The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6.NS.C.6 Understand a rational number as a point on a number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.C.6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. 19

20. The CCSS for Mathematical Content: Grade 6 Common Core State Standards, 2010, p. 43, NGA Center/CCSSO The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.6b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. 6.NS.C.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 6.NSC..7 Understand ordering and absolute value of rational numbers. 6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right. 6.NS.C.7b Write, interpret and explain statements of order for rational numbers in real- world contexts. For example, write -3 o C > -7 o C to express the fact that -3 o C is warmer than -7 o C. 20

21. The CCSS for Mathematical Content: Grade 6 Common Core State Standards, 2010, p. 43, NGA Center/CCSSO The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars. 6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance of less than - 30 dollars represents a debt greater than 30 dollars. 6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 21

22. The CCSS for Mathematical Practice 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 22

23. © 2013 UNIVERSITY OF PITTSBURGH A. Walking Task Mary Jane and her brother Paul each go on a walk starting from the same location. Mary Jane walks north 3 miles. Paul walks south 1.5 miles. 1.Use a number line to represent their starting and ending points. 2.Determine their distance from each other at the end of their walks using 2 different methods. 23

24. © 2013 UNIVERSITY OF PITTSBURGH B. Saving Money Task 5 friends are keeping track of their finances. Some of them have saved money. Others have borrowed money from their parents. 1.Plot points on the number line below representing the amount each friend has saved or owes. Explain why you located the points where you did. 2.Explain how you can use the points you placed on the number line to determine which friend has saved the most money and which has the greatest debt. 3.Brandon claims that the points representing 2 people have the same absolute value. Is he correct? Why or why not? NameMoney Saved or Owed AbbeySaved $12.10 BrandonSaved $5.50 ClaireOwes $12.10 DanteSaved $14 ElizabethOwes $8.25 24

25. © 2013 UNIVERSITY OF PITTSBURGH C. Location, Location, Location Joe made a map on the coordinate plane showing the location of several places in relationship to his home. Each tick mark represents one block. 1.Describe the location of the school in relationship to Joe’s home. 2.How are the positions of the museum and park related? Explain your reasoning. 3.Joe’s grandmother’s house is located at the coordinates (-7, 5). Plot a point to represent his grandmother’s house. Explain how you determined this location. 25

26. © 2013 UNIVERSITY OF PITTSBURGH D. Absolute Value Points A and B are the same distance from 0 on the number line. What can you determine about the absolute values of A and B? Explain your reasoning. 26

27. © 2013 UNIVERSITY OF PITTSBURGH E. Weight at Birth The average weight of a newborn infant is 7.5 pounds. At Riverside Hospital, every infant is assigned a number indicating their weight relative to the average newborn weight. 1.What does it mean if an infant is assigned a positive value? A negative value? 0? 2.On Monday, 4 babies were born at Riverside Hospital. Clarence weighed 6.9 pounds. Travis weighed 7.8 pounds. Brea weighed 8.9 pounds. Ginger weighed 6.1 pounds. Do any of the babies have weights equally distant from the average birth weight? Explain your reasoning. 3.On Tuesday twin boys, Larry and Harry, were born. Larry was assigned the value -0.7 and Harry was assigned the value -0.4. Write an inequality comparing the values assigned to each twin. Which baby’s weight is farther from the average birth weight? Explain your reasoning. 27

28. © 2013 UNIVERSITY OF PITTSBURGH F. Isosceles Xavier drew a triangle with vertices at (3, 2), (3, -5), and (-5, 2). Teacher: How would you classify this triangle by its angle measures and side lengths? Xavier: This is a right triangle because the horizontal and vertical sides meet at a right angle. It is isosceles because the hypotenuse is the longest side and the other 2 sides have the same length. 1.Explain how you know that Xavier’s triangle is not isosceles. 2.How could Xavier change the location of one or more vertices to make the triangle isosceles? 28

29. © 2013 UNIVERSITY OF PITTSBURGH Reflecting and Making Connections Are all of the CCSS for Mathematical Content in this cluster addressed by one or more of these tasks? Are all of the CCSS for Mathematical Practice addressed by one or more of these tasks? What is the connection between the cognitive demand of the written task and the alignment of the task to the Standards for Mathematical Content and Practice? 29

30. © 2013 UNIVERSITY OF PITTSBURGH Differentiating Between Instructional Tasks and Assessment Tasks Are some tasks more likely to be assessment tasks than instructional tasks? If so, which and why are you calling them assessment tasks? 30

31. Instructional Tasks Versus Assessment Tasks Instructional TasksAssessment Tasks Assist learners to learn the CCSS for Mathematical Content and the CCSS for Mathematical Practice. Assesses fairly the CCSS for Mathematical Content and the CCSS for Mathematical Practice of the taught curriculum. Assist learners to accomplish, often with others, an activity, project, or to solve a mathematics task. Assess individually completed work on a mathematics task. Assist learners to “do” the subject matter under study, usually with others, in ways authentic to the discipline of mathematics. Assess individual performance of content within the scope of studied mathematics content. Include different levels of scaffolding depending on learners’ needs. The scaffolding does NOT take away thinking from the students. The students are still required to problem-solve and reason mathematically. Include tasks that assess both developing understanding and mastery of concepts and skills. Include high-level mathematics prompts. (The tasks have many of the characteristics listed on the Mathematical Task Analysis Guide.) Include open-ended mathematics prompts as well as prompts that connect to procedures with meaning. 31

32. © 2013 UNIVERSITY OF PITTSBURGH Reflection So, what is the point? What have you learned about assessment tasks and instructional tasks that you will use to select tasks to use in your classroom next school year? 32