© 2013 University Of Pittsburgh Supporting Rigorous Mathematics Teaching and Learning Making Sense of the Expressions and Equations Standards via a Set of Tasks Tennessee Department of Education Middle School Mathematics Grade 6-8 [*Note: Slides for Math CCSS Grades 6-8 are the same except for the tasks and the grade level standards. 6 th & 7 th grade Focus – Making Sense of the Number System Standard via a Set of Tasks.]

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, differentiate between those that require problem-solving, and those that assess for specific mathematical reasoning. 2

© 2013 University Of Pittsburgh Session Goals Participants will: make sense of Expressions and Equations Common Core State Standards (CCSS); determine the cognitive demand of tasks and make connections to the Standards for Mathematical Content and the Standards for Mathematical Practice; and differentiate between assessment items and instructional tasks. 3

© 2013 University Of Pittsburgh Session Goals Participants will: make sense of the Expressions Common Core State Standards (CCSS); determine the cognitive demand of tasks and make connections to the Standards for Mathematical Content and the Standards for Mathematical Practice; and differentiate between assessment items and instructional tasks. 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 (6 th & 7 th )/ Expressions and Equations (8 th ) 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. 5

© 2013 University Of Pittsburgh Analyzing Tasks as a Means of Making Sense of the CCSS Expressions and Equations 6

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 7

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 8

© 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 Mathematical Practice Standards 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. 9

© 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? 10

The CCSS for Mathematical Content − Grade 8 Common Core State Standards, 2010, p. 54, NGA Center/CCSSO Expressions and Equations 8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.C.7 Solve linear equations in one variable. 8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 11

The CCSS for Mathematical Content − Grade 8 Common Core State Standards, 2010, p. 55, NGA Center/CCSSO Expressions and Equations 8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.C.8 Analyze and solve pairs of simultaneous linear equations. 8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. 8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. 12

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 13

© 2013 University Of Pittsburgh A. Same or Different? 14

© 2013 University Of Pittsburgh B. A Friendly Walk Lois, Emile, and Shantay are students at Smith Street School. They all live on Smith Street. Lois is waiting in front of her home. Emile and Shantay plan to meet her there so the three of them can walk to school to watch the school basketball team play. Lois lives x blocks east of the school; Emile lives 2 blocks east of Lois; while Shantay lives 3 times as many blocks east of the school as does Emile. If the total amount of blocks walked by all three students is 23, how many blocks does each student walk? 15

© 2013 University Of Pittsburgh C. Saving Money Sisters Aya and Jun keep track of the amount they have saved using the tables below. Write an equation to describe the amount of money in dollars that each sister has in the bank after any number of months. After how many months will the sisters have the same amount of money in the bank? Explain in words how you know. Aya’s Savings MonthAmount in Bank in Dollars 1$10 2$ $15 4$17.50 Jun’s Savings MonthAmount in Bank in Dollars 2$18 4$23 6$28 8$33 16

© 2013 University Of Pittsburgh D. To Meet or Not to Meet Many algebra books contain the following message to students: When 2 lines are graphed on the same set of axes, one of 3 situations will occur: the lines will meet in exactly one point, or they will be parallel, or they will coincide. a.Predict the conditions under which the 2 lines will meet in one point, will be parallel, or will coincide. Use drawings, equations, and words to write about the thinking behind your predictions. Exploration b. Using graph paper, graph 2 lines that meet in exactly one point. Find the equation of each line. Share your graphs and equations with a partner, then with another pair. What do you notice? Using what you have discovered, revise or add on to your prediction. 17

© 2013 University Of Pittsburgh D. To Meet or Not to Meet cont. 18

© 2013 University Of Pittsburgh E. Calling Plans 1 Long-distance Company A charges a base rate of $5 per month, plus 4 cents per minute that you are on the phone. Long-distance Company B charges a base rate of only $2 per month, but they charge 10 cents per minute used. How much time per month would you have to talk on the phone before subscribing to Company A would save you money? 19

© 2013 University Of Pittsburgh F. Storage Tanks Two large storage tanks, T and W, contain water. T is losing water. W is gaining water. The graph below shows the amount of water in each tank over a period of time; assume that the rates of water loss and water gain continue as shown. When will the two tanks contain the same amount of water? Explain how you found your solution and how your solution relates to the problem. 20

© 2013 University Of Pittsburgh G. To Meet Again! Maya and her brother Devonte both belong to a local fitness club. The club offers several membership plans. Under Maya’s plan, 10 visits cost $150, while 20 visits cost $250. Under Devonte’s plan, 6 visits cost $100, while 9 visits cost $145. Explain to Maya and Devonte how they can decide, without graphing, whether or not the graphs containing the data above from the two plans will meet. 21

© 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? 22

© 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? 23

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. 24

© 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? 25

© 2013 University Of Pittsburgh Bridge to Practice Select a high-level task. Analyze the task and determine the alignment to the Mathematical Content Standards and the Standards for Mathematical Practice. Describe if the task is an instructional task or an assessment task and explain why you decided on the label. If necessary, refer back to slide 33 for assistance. Use the set of examples of tasks to differentiate between the two types of tasks.