1. Day 2 Summer 2011 Algebraic Reasoning Institute 1

2. Mathematics Content Standards Algebraic Reasoning Institute 2

3. Key Advances Algebraic Reasoning Institute 3 Focus and coherence Focus on key topics at each grade level. Coherent progressions across grade levels. Balance of concepts and skills Content standards require both conceptual understanding and procedural fluency. Mathematical practices Foster reasoning and sense-making in mathematics. College and career readiness Level is ambitious but achievable.

4. Toward Greater Focus and Coherence Algebraic Reasoning Institute 4 The composite standards (of Hong Kong and Singapore) have a number of features that can inform an international benchmarking process for the development of K-6 mathematics standards in the U.S. First, the composite standards concentrate the early learning of mathematics on the number, measurement, and geometry strands with less emphasis on data analysis and little exposure to algebra. Ginsburg, Leinwand, and Decker, 2009

5. Algebraic Reasoning Institute 5 Key Changes Focuses in early grades on number (arithmetic and operations) to build a solid foundation in math Evens out pace across the grades Focuses on using math and solving complex problems in high school math - the real world emphasis Stresses problem-solving and communication http://www.corestandards.org/about-the-standards/key-points-in-mathematics

6. Toward Greater Focus and Coherence Algebraic Reasoning Institute 6 Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole number, operations and relations) and (2) geometry, spatial relations and measurement, with more mathematics learning time devoted to number than to other topics. Mathematical process goals should be integrated in these content areas. Mathematics & Learning in Early Childhood, National Research Council, 2009

7. Design and Organization Algebraic Reasoning Institute 7 Standards for Mathematical Content K-8 standards presented by grade level Organized into domains that progress over several grades Grade introductions give 2–4 focal points at each grade level High school standards presented by conceptual theme (Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability)

8. Algebraic Reasoning Institute 8 Design and Organization: K-8 Focal points at each grade level

9. Grade Level Overviews: K-8 Mathematics Standards Algebraic Reasoning Institute 9 Each grade includes an overview of cross-cutting themes and critical areas of study

10. Grade Level Overview: K-8 Mathematics Standards Algebraic Reasoning Institute 10 Clusters: illustrate progression of increasing complexity from grade to grade Standards: define what students should know and be able to do at each grade level Domains: overarching ideas that connect topics across the grades

11. Common Core-Clusters Algebraic Reasoning Institute 11 May appear in multiple grade levels with increasing developmental standards as the grade levels progress Indicate WHAT students should know and be able to do at each grade level in their study of mathematics Reflect both mathematical understandings and skills, which are equally important

12. Overview of the K-8 Standards Algebraic Reasoning Institute 12  The K- 8 standards: The K-5 standards provide students with a solid foundation in whole numbers, addition, subtraction, multiplication, division, fractions and decimals The 6-8 standards describe robust learning in geometry, algebra, and probability and statistics Modeled after the focus of standards from high-performing nations, the standards for grades 7 and 8 include significant algebra and geometry content Students who have completed 7 th grade and mastered the content and skills will be prepared for algebra, in 8 th grade or after

13. Standards for Mathematical Content Algebraic Reasoning Institute 13

14. Algebraic Reasoning Institute 14 Number and Operations, Grade 1 Number and Operations in Base Ten Extend the counting sequence. Understand place value. Use place value understanding and properties of operations to add and subtract. Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Understand and apply properties of operations and the relationship between addition and subtraction. Add and subtract within 20. Work with addition and subtraction equations.

15. Algebraic Reasoning Institute 15 Fractions, Grades 3–6 3. Develop an understanding of fractions as numbers. 4. Extend understanding of fraction equivalence and ordering. 4. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4. Understand decimal notation for fractions, and compare decimal fractions. 5. Use equivalent fractions as a strategy to add and subtract fractions. 5. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 6. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

16. Algebraic Reasoning Institute 16 Statistics and Probability, Grade 6 Develop understanding of statistical variability  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

17. Algebraic Reasoning Institute 17 Algebra, Grade 8 Graded ramp up to Algebra in Grade 8  Properties of operations, similarity, ratio and proportional relationships, rational number system. Focus on linear equations and functions in Grade 8  Expressions and Equations  Work with radicals and integer exponents.  Understand the connections between proportional relationships, lines, and linear equations.  Analyze and solve linear equations and pairs of simultaneous linear equations.  Functions  Define, evaluate, and compare functions.  Use functions to model relationships between quantities.

18. High School Conceptual Categories Algebraic Reasoning Institute 18  High school standards are organized around five conceptual categories, which are the big ideas that connect mathematics across high school  Number and Quantity,  Algebra,  Functions,  Geometry, and  Statistics and Probability

19. Algebraic Reasoning Institute 19 High School Level Conceptual Theme Overview (High School- Algebra page 62 in CCSS) Focus: Instructional Time Overview by domain (1 page) Provides in depth explanation for the Conceptual Theme

20. Format of High School Mathematics Standards Algebraic Reasoning Institute 20 Each conceptual category includes an overview of the content found within it

21. Format of High School Mathematics Standards Algebraic Reasoning Institute 21  Modeling:  Modeling standards are distributed under the five major headings and are indicated with a (  ) symbol.  College and Career Readiness Threshold:  Standards indicated as (+) are beyond the college and career readiness level but are necessary for advanced mathematics courses, such as calculus, discrete mathematics, and advanced statistics. Standards with a (+) may still be found in courses expected for all students

22. Algebraic Reasoning Institute 22 Geometry, High School Middle school foundations Hands-on experience with transformations. Low tech (transparencies) or high tech (dynamic geometry software). High school rigor and applications Properties of rotations, reflections, translations, and dilations are assumed, proofs start from there. Connections with algebra and modeling

23. Overview of High School Standards The high school mathematics standards: Call on students to practice applying mathematical ways of thinking to real world issues and challenges Require students to develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly are called to do Emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions Identify the mathematics that all students should study in order to be college and career ready. 23 Algebraic Reasoning Institute

24. Understanding Mathematics Algebraic Reasoning Institute 24 These standards define what students should understand and be able to do in their study of mathematics. This means teachers need to assess whether the student has understood it.

25. Time to start exploring… Algebraic Reasoning Institute 25 Frayer Model WHAT IS THEREWHAT IS NOT THERE INSTRUCTIONAL IMPLICATIONS CONNECTIONS TO THE MATH PRACTICES Common Core Critical Areas

26. Time to start exploring… Algebraic Reasoning Institute 26 Some things to consider as you immerse yourself in the CCSS-M You will NOT have time to complete your exploration today. Decide in advance if you are going to explore from a 10, 000 ft view or the ground level Consider dividing and conquering

27. Foci for exploration Algebraic Reasoning Institute 27 What implications do the Common Core Mathematics Standards pose for your individual classrooms? Your building as a whole? Your district as a unit? Which of these resources do you find useful? How will you use them?

28. Algebraic Reasoning Institute 28

29. The promise of standards Algebraic Reasoning Institute 29 These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep.

30. Algebraic Reasoning Institute 30 At the beginning At one minuteAt two minutes Growing Dots Task Describe the pattern. Assuming the sequence continues in the same way, how many dots are there at 3 minutes? 100 minutes? t minutes?

31. Growing Dots Task Algebraic Reasoning Institute 31 Individually complete the task. Then work with a partner to compare your work. (Look for as many ways to solve the task as possible.)

32. Mathematical Practices Algebraic Reasoning Institute 32 Which mathematical practices are needed to complete the task? Indicate the primary 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.

33. The Growing Dots Task Algebraic Reasoning Institute 33 What mathematics is inherent in this task? How did we structure this activity to give you an opportunity to engage in the math practices ? How would this look in your classroom? How might you structure your classroom to allow for engagement in the mathematical practices?

34. Acknowledgement This material is based on work supported by the SW PA MSP 2010 funds administered through the USDOE under Grant No. Project #: RA-075-10-0603. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the granting agency. Math & Science Collaborative at the Allegheny Intermediate Unit