1. CCSS Mathematics Grades 6-8 Curriculum Review Week 2012

2. Today’s Learning Targets I can understand that CCSS math standards require a different type of teaching, which also requires a different type of assessing. I can understand that this different way of teaching and assessing has an impact on the work that we will be doing this week, as we create our Baseline and Benchmark Assessments.

3. teachers & students: Focus- Teach less, go deeper Coherence- Build on previous learning and connect concepts Rigor- Balance of conceptual understanding, procedural skill and fluency, and application of skills in problem solving situations Math Shifts

4. When planning, ask “What task can I give that will build student understanding?” rather than “How can I explain clearly so they will understand?” Grayson Wheatley, NCCTM, 2002

5. Definitions Understanding mathematics  Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like?  One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y).  Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. http://www.corestandards.org/

6. Understanding the Math The “Why” behind the math- students need to know why and be able to explain why! Develop the concept then teach the procedure that goes along with that. Concrete  Representational  Abstract

7. Timeline for Common Core Mathematics Implementation YearStandards To Be TaughtStandards To Be Assessed 2011 – 20122003 NCSCOS 2012 – 2013CCSSCCSS (NC) 2013 – 2014CCSSCCSS (NC) 2014 – 2015CCSSCCSS (SBAC) Common Core State Standards Adopted June, 2010

8. Let’s look at a familiar problem…

9. Which of the following represents ? a. b. c. d.

10. Same problem- a new twist…

11. For numbers 1a – 1d, state whether or not each figure has of its whole shaded. 1a. 1b. 1c. 1d. ο Yes ο No

12. For numbers 1a – 1d, state whether or not each figure has of its whole shaded? 1a. 1b. 1c. 1d. ο Yes ο No

13. This item is worth 0 – 2 points depending on the responses. What series of the yes and no responses would give a student: “Turn and Talk” 2 points? 1 point? 0 points?

14. 2 points: YNYN 1 point: YNNN, YYNN, YYYN 0 point: YYYY, YNNY, NNNN, NNYY, NYYN, NYNN NYYY, NYNY, NNYN, NNNY, YYNY, YNYY For numbers 1a - 1d, state whether or not each figure has of its whole shaded.

15. The kind and level of thinking required of students to successfully engage with and solve a task. Teaching for Understanding Through Cognitive Demand

16. Cognitive Demand Spectrum MEMORIZATION PROCEDURES WITHOUT CONNECTIONS (TO UNDERSTANDING, MEANING, OR CONCEPTS) PROCEDURES WITH CONNECTIONS (TO UNDERSTANDING, MEANING, OR CONCEPTS) DOING MATHEMATICS Tasks that require memorized procedures in routine ways Tasks that require engagement with concepts, and stimulate students to make connections to meaning, representation, and other mathematical ideas

17. Procedures Without Connections Convert the fraction to a decimal and a percent.

18. Memorization What are the decimal and percent equivalents for the fractions: and

19. Using a 10x10 grid, identify the decimal and percent equivalents of Procedures With Connections

20. Level 1: Recalling and Recognizing Student is able to recall routine facts or knowledge and can recognize shape, symbols, attributes and other qualities. Level 2: Using Procedures Student uses or applies procedures and techniques to arrive at solutions or answers. Depth of Knowledge

21. Level 3: Explaining and Concluding Student reasons and derives conclusions. Student explains reasoning and processes. Student communicates procedures and findings. Level 4: Making Connections, Extending and Justifying: Student makes connections between different concepts and strands of mathematics. Extends and builds on knowledge to a situation to arrive at a conclusion. Student uses reason and logic to prove and justify conclusions.

22. Balance Defined Problem Solving Computational & Procedural Skills DOING MATH Conceptual Understanding “WHERE” THE MATHEMATICS WORK “HOW” THE MATHEMATICS WORK “WHY” THE MATHEMATICS WORK

23. Let’s do some Mathematics

24. Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine each of the following: a.the fractional part of the area shaded. b. the decimal part of the area shaded. c. the percent of the area that is shaded. Doing Mathematics

25. Different Objects, Same Ratios Trucks and Boxes On which cards is the ratio of trucks to boxes the same? Given one card, students are to select a card on which the ratio of the two types of objects is the same. This task moves students to numeric approach rather than a visual one and introduces the notion of ratios as rates. A unit rate is depicted on a card that shows exactly one of either of the two types of objects. For example the card with three boxes and one truck provides one unit rate. A unit rate for the other ratio is not shown. What would it be?

26. Solve this problem without using any numeric algorithms such as cross-products. This could include pictures or counters, but there is no prescribed method. Two camps of scouts are having pizza parties. The Bear Camp ordered enough so that every 3 campers will have 2 pizzas. The leader of the Raccoons ordered enough so that there would be 3 pizzas for every 5 campers. Did the Bear or the Raccoon campers have more pizza to eat?

27. A bird flew 20 miles in 100 minutes at a constant speed. At that speed: (a) How long would it take the bird to fly 6 miles? Explain. (b) How far would the bird fly in 15 minutes? Explain. (c) How fast is the bird flying in miles per hour? (d) What is the bird’s pace in minutes per mile? (e) Explain the meaning of the answers for parts c and d? How would using the table help?

28. Assessments Assessment Specifications Summary is in your CRW folders and on our ISS Common Core wiki to guide you as you create your Baseline and Benchmarks.

29. Thank You!! Let’s put our brains to work, creating quality baselines and benchmarks that reflect the teaching and understanding of the CCSS!!

30. Given: x 2 – 9 Which is true? I.x = 3 II.x = -3 III.x 2 = 9 a.I onlyb. I and II only c.I and III onlyd. I, II, and III e.none Given: (a – b)(a 2 – b 2 ) = 0, which is true? I.a = b II.a = -b III.a 2 = b 2 a.I onlyb. I and II only c.I and III onlyd. III only http://mathnotations.blogspot.com/2007/07/understanding-algebra-conceptually.html

31. A high level task… Has no predictable pathway Requires student to access and use relevant knowledge Requires student to explain Has the potential to engage student in higher level thinking

32. Soup and Beans

33. Consider these two Measurement Problems Carpeting Task Martha is re-carpeting her bedroom, which is 15 feet wide. How many square feet of carpet will she need to purchase, and how much baseboard will she need to run around the edge of the carpet? Explain your thinking. The Fencing Task Ms Brown’s class will raise rabbits for their science fair. They have 24 feet of fencing with which to build a rectangular pen. If Ms. Brown’s students want their rabbits to have as much room as possible, how long should each of the sides of the pen be? Explain your thinking. For each problem, what kind of thinking is required? How are they alike? How are they different?

34. Carpeting Task vs. Fencing Task Alike Both require Area and perimeter calculations Both require students to “explain your thinking” Both are word problems, set in a “real world” context Not Alike Fencing requires a systematic approach Fencing leads to generalization and justification The “thinking in Fencing is complex – requires more than applying a memorized formula

35. Technology and Testing Content of the North Carolina assessments is aligned to the CCSS-M; however, the technology will not be as sophisticated as in assessments created by the Smarter Balanced Assessment Consortium (SBAC).

36. www.smarterbalanced.org

37. www.ncdpi.wikispaces.net

38. DPI Contact Information DPI Contact Information Kitty Rutherford Elementary Mathematics Consultant 919-807-3934 kitty.rutherford@dpi.nc.gov Amy Scrinzi Elementary Mathematics Consultant 919-807-3839 amy.scrinzi@dpi.nc.gov Robin Barbour Middle Grades Mathematics Consultant 919-807-3841 robin.barbour@dpi.nc.gov Johannah Maynor Secondary Mathematics Consultant 919-807-3842 johannah.maynor@dpi.nc.gov Barbara Bissell K – 12 Mathematics Section Chief 919-807-3838 barbara.bissell@dpi.nc.gov Susan Hart Program Assistant 919-807-3846 susan.hart@dpi.nc.gov

39. Let’s do some MORE math! a) Pick any integer greater than 10: __________ b) Square the number you picked: __________ c) Find the product of the integer that is one more than your original number and the integer that is one less than your original number: ___________ d) Compare your result in part a to your result in part b. What do you notice? e) Repeat steps a – c three or four times, using a different positive integer as your starting number each time. f) Create an algebraic equation to represent what you see happening, but for ANY starting number. g) Turn to someone sitting near you and explain as clearly as you can what your equation represents.

40. Place Value Traditional worksheet vs Common Core worksheet What do you notice?

41. Mathematical Language Growing Staircase Building staircase Blocks Pattern Total blocks for nth step Total blocks for a staircase with n number of steps Base of a staircase Top step of staircase Step vs. stage Generalize Conjecture Explicit or general/closed formula or function Recursive formula or function Justify/prove

42. Growing Staircases