Common Core and 2010 Az Mathematics standards

Common Core and 2010 Az Mathematics standards
An introduction to Common Core and Az Mathematics standards Statewide Instructional Technology Project

What do you already know?
2 The federal government took the lead in developing the Common Core standards for math and language arts. True or False AZ adopted the Common Core standards as our new 2010 Math and Language Arts Standards. True or False Q1 = False Beginning in the spring of 2009, Governors and state commissioners of education from 48 states, 2 territories and the District of Columbia committed to developing a common core of state K-12 English-language arts (ELA) and mathematics standards. The Common Core State Standards Initiative (CCSSI) is a state-led effort coordinated by the National Governors Association (NGA) and the Council of Chief State School Officers (CCSSO). Q2 = True

What do you already know?
3 Full implementation of the AZ 2010Math standards at all levels by True or False is the pilot year for the common core assessment True or False Q1= True Q2= False The assessment begins in , PARC Assessments will e given to grades 3-11.

4/13/2017

Why Common Core State Standards?
Preparation: college-and career-ready Competition: internationally benchmarked Equity: consistent for all Clarity: focused, coherent, and clear Collaboration: across states and districts, pooling resources and expertise Preparation: The standards are college-and career-ready. They will help prepare students with the knowledge and skills they need to succeed in education and training after high school. Competition: The standards are internationally benchmarked. Common standards will help ensure our students are globally competitive. Equity: Expectations are consistent for all. Students must be prepared to compete with not only their American peers in the next state, but with students from around the world. Clarity: The standards are focused, coherent, and clear. Clearer standards help students (and parents and teachers) understand what is expected of them. Collaboration: The standards create a foundation to work collaboratively across states and districts, pooling resources and expertise, to create curricular tools, professional development, common assessments and other materials.

Your Experience What experience/use of the 2010 Mathematics Standards documents have you had working with teachers or incorporated into training? A - Little/no prior experience B - Some experience C - Lots of experience

Outcomes for Today Understand structure and terminology of the AZ 2010 Mathematics Standards Gain awareness of AZ grade level documents Reflect on implications for instructional practice related to implementation of the AZ 2010 Mathematics Standards (cc) and the 2009 Education Technology Standard

Key Ideas K-5 K-7 6-8 High School
Focus on number and operations K-7 Graded ramp up to algebra, through fractions, ratios, and proportional reasoning 6-8 Rich hands-on work with ratios, geometry, statistics, and probability High School Rigorous algebra, geometry, modeling, statistics and probability Emphasis throughout on problem-solving, quantitative reasoning, and modeling This slide demonstrates areas for future professional development.

Strand Concept Performance Objective
Format changes from 2008 Strand Concept Performance Objective 2008 **high school has a fourth level before Domain

Strand Concept Performance Objective
Format changes from 2008 2008 Strand Concept Performance Objective Domain Cluster Standard 2010 **high school has a fourth level before Domain

CC and AZ Mathematics Format
Domains are large groups of related standards. Standards from different domains may sometimes be closely related. Clusters are groups of related standards. Standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Clusters appear inside domains. Standards define what students should know and be able to do Standards are part of a cluster.

Common Core Format K-8 High School
Domain Cluster Standard Conceptual Category Domain Cluster Standard (There are no preK Common Core Standards)

Domain Grades K-8 Format Cluster Standard
The important ideas in the standards are called Domains. Each domain is divided into clusters of mathematical ideas. The clusters are further divided into in related standards, which specify what students should know and be able to do. 14

High School Mathematics Format
Cluster Domain Standard Conceptual Category Conceptual categories are the overarching big ideas (larger than former Strands/Domains) that connect topics across high school. The important ideas in the standards are called Domains. Each domain is divided into clusters of mathematical ideas. The clusters are further divided into in standards, which specify what students should know and be able to do.

This slide shows how K-8 is organized by domains and how High School is organized by conceptual categories and domains. Question – What do you notice about the K-HS table? What information stand out to you at what grade level? Answer through microphone or through chat. What do you notice about the domains around 5th grade? What do you notice about the domains around 8th grade?

This slide shows how K-8 is organized by domains and how High School is organized by conceptual categories and domains. Question – What do you notice about the K-HS table? What information stand out to you at what grade level? Answer through microphone or through chat. What do you notice about the domains around 5th grade? What do you notice about the domains around 8th grade? What do you notice about the K-HS table? What information stands out to you at what grade level?

Flow Between Domains Expressions and Equations
Operations and Algebraic Thinking Number and Operations- Base Ten Algebra The Number System Number and Operations- Fractions K High School

Let’s review the new terminology
K-8 Q 1 - What is the name of Level 1? Level 2 = Cluster Q2 -What is the name of Level 3? Q 3 - What is the name of the over arching level in HS math? Question 1 – multiple choice question Domain Conceptual Category Standard Mathematical Practices Question 2 – multiple choice question Question 3 – multiple choice question

Architecture (Format)
Features of the 2010 Mathematics Standards (Common Core State Standards)

Domain Cluster Grade 4 Grade 4 Overview
Grade 4 Overview Operations and Algebraic Thinking (OA) Use the four operations with whole numbers to solve problems. Gain familiarity with factors and multiples. Generate and analyze patterns. Number and Operations in Base Ten (NBT) Generalize place value understanding for multidigit whole numbers. Use place value understanding and properties of operations to perform multi-digit arithmetic. Number and Operations—Fractions (NF) Extend understanding of fraction equivalence and ordering. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Understand decimal notation for fractions, and compare decimal fractions. Measurement and Data (MD) Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. Represent and interpret data. Geometric measurement: understand concepts of angle and measure angles. Geometry (G) Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Mathematical Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Domain Cluster The architecture of each grade level K-8 is the same. Each grade level document begins with an overview page which details the organization. The grade level is organized by domains. Domains are larger groups of related standards organized around the same topic and may connect across grade levels. Domains illustrate progression of increasing complexity from grade level to grade level. Clusters are smaller groups of related standards and may appear in multiple grade levels. Clusters are most similar to concepts in the 2008 Mathematics Standard.

High School (9-12) Conceptual Category Domain Cluster
The high school standards specify the mathematics that all students should study in order to be college and career ready. Additional mathematics that students should learn in fourth credit courses or advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by (+). All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. Standards with a (+) symbol may also appear in courses intended for all students. The high school standards are listed in conceptual categories including Number and Quantity, Algebra, Functions, Modeling, Geometry, Statistics and Probability, and Contemporary Mathematics. Conceptual categories portray a coherent view of high school mathematics; a student’s work with functions, for example, crosses a number of traditional course boundaries, potentially up through and including calculus. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). High School - Number and Quantity Overview Conceptual Category The Real Number System Extend the properties of exponents to rational exponents Use properties of rational and irrational numbers. Quantities Reason quantitatively and use units to solve problems The Complex Number System Perform arithmetic operations with complex numbers Represent complex numbers and their operations on the complex plane Use complex numbers in polynomial identities and equations Vector and Matrix Quantities Represent and model with vector quantities. Perform operations on vectors. Perform operations on matrices and use matrices in applications. Mathematical Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Domain The architecture of High School differs from K-8 in that the Standards are organized into conceptual categories. Conceptual categories are the overarching big ideas that connect topics across high school. The rest of the nomenclature in high school is identical to K-8, domain, cluster, and standard. Domains are larger groups of related standards organized around the same topic. Standards from different domains may sometimes be closely related. Clusters are smaller groups of related standards. Note that standards from different domains and clusters may sometimes be closely related, because mathematics is a connected subject. Standards define what students should know, understand, and be able to do. Cluster

Grade 4 In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. (1) Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context. (2) Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number. (3) Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving symmetry. Critical Ideas It is important to note that at the beginning of each grade level, the important ideas for that grade level are listed.

Critical Ideas by Domain
High School (9-12) High School - Numbers and Quantity Numbers and Number Systems. During the years from kindergarten to eighth grade, students must repeatedly extend their conception of number. At first, “number” means “counting number”: 1, 2, 3…. Soon after that, 0 is used to represent “none” and the whole numbers are formed by the counting numbers together with zero. The next extension is fractions. At first, fractions are barely numbers and tied strongly to pictorial representations. Yet by the time students understand division of fractions, they have a strong concept of fractions as numbers and have connected them, via their decimal representations, with the base-ten system used to represent the whole numbers. During middle school, fractions are augmented by negative fractions to form the rational numbers. In Grade 8, students extend this system once more, augmenting the rational numbers with the irrational numbers to form the real numbers. In high school, students will be exposed to yet another extension of number, when the real numbers are augmented by the imaginary numbers to form the complex numbers. With each extension of number, the meanings of addition, subtraction, multiplication, and division are extended. In each new number system—integers, rational numbers, real numbers, and complex numbers—the four operations stay the same in two important ways: They have the commutative, associative, and distributive properties and their new meanings are consistent with their previous meanings. Extending the properties of whole-number exponents leads to new and productive notation. For example, properties of whole-number exponents suggest that (51/3)3 should be 5(1/3)·3 = 51 = 5 and that 51/3 should be the cube root of 5. Calculators, spreadsheets, and computer algebra systems can provide ways for students to become better acquainted with these new number systems and their notation. They can be used to generate data for numerical experiments, to help understand the workings of matrix, vector, and complex number algebra, and to experiment with non-integer exponents. Quantities. In real world problems, the answers are usually not numbers but quantities: numbers with units, which involves measurement. In their work in measurement up through Grade 8, students primarily measure commonly used attributes such as length, area, and volume. In high school, students encounter a wider variety of units in modeling, e.g., acceleration, currency conversions, derived quantities such as person-hours and heating degree days, social science rates such as per-capita income, and rates in everyday life such as points scored per game or batting averages. They also encounter novel situations in which they themselves must conceive the attributes of interest. For example, to find a good measure of overall highway safety, they might propose measures such as fatalities per year, fatalities per year per driver, or fatalities per vehicle-mile traveled. Such a conceptual process is sometimes called quantification. Quantification is important for science, as when surface area suddenly “stands out” as an important variable in evaporation. Quantification is also important for companies, which must conceptualize relevant attributes and create or choose suitable measures for them. Critical Ideas by Domain It is important to note that at the beginning of each conceptual category, the important ideas are listed.

Domain Cluster Standard Operations and Algebraic Thinking (OA)
Use the four operations with whole numbers to solve problems. Standards Mathematical Practices Explanations and Examples Students are expected to: 4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5  7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Connections: 4.OA.3; 4.SL.1d; ET04-S1C2-01; ET04-S1C2-02 4.MP.2. Reason abstractly and quantitatively. 4.MP.4. Model with mathematics. A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which quantity is being multiplied and which number tells how many times. 4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (see Table 2) Connections: 4.RI.7; ET04-S1C2-01; ET04-S1C2-02 4.MP.5. Use appropriate tools strategically. 4.MP.7. Look for and make use of structure. Students need many opportunities to solve contextual problems. Table 2 in the glossary includes the following multiplication problem: “A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?” In solving this problem, the student should identify $6 as the quantity that is being multiplied by 3. The student should write the problem using a symbol to represent the unknown. ($6 x 3 = ) Table 2 in the glossary includes the following division problem: A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? In solving this problem, the student should identify $18 as the quantity being divided into shares of $6. Continued on next page Domain Cluster Standard Reinforce the nomenclature with participants specifically the domain and cluster. Point out that standards from different domains and clusters may sometimes be closely related, because mathematics is a connected subject. Standards define what students should know, understand, and be able to do. Standards are most similar to performance objectives in our 2008 Standard.

Domain Conceptual Category Cluster Standard
Number and Quantity: The Real Number System – (N-RN) Extend the properties of exponents to rational exponents. Standards Label Mathematical Practices Explanations and Examples Students are expected to: HS.N-RN.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.  HS.MP.2. Reason abstractly and quantitatively. HS.N-RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. HS.MP.7. Look for and make use of structure. Examples: Rewrite using fractional exponents: Rewrite in at least three alternate forms. Solution: Rewrite using only fractional exponents. Rewrite in simplest form. Key to Labels: 9-10 = Standards for Grades 9 and  = Algebra 2 Standards + = 4th Credit Standards  = Standards connected to Mathematical Modeling Domain Conceptual Category Cluster High School - Reinforce the nomenclature with participants specifically the conceptual category, domain, and cluster. Point out that standards from different domains and clusters may sometimes be closely related, because mathematics is a connected subject. Standards define what students should know, understand, and be able to do. Standards are most similar to performance objectives in our 2008 Standard. Standard

Explanations & Examples Standard
Operations and Algebraic Thinking (OA) Use the four operations with whole numbers to solve problems. Standards Mathematical Practices Explanations and Examples Students are expected to: 4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5  7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Connections: 4.OA.3; 4.SL.1d; ET04-S1C2-01; ET04-S1C2-02 4.MP.2. Reason abstractly and quantitatively. 4.MP.4. Model with mathematics. A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which quantity is being multiplied and which number tells how many times. 4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (see Table 2) Connections: 4.RI.7; ET04-S1C2-01; ET04-S1C2-02 4.MP.5. Use appropriate tools strategically. 4.MP.7. Look for and make use of structure. Students need many opportunities to solve contextual problems. Table 2 in the glossary includes the following multiplication problem: “A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?” In solving this problem, the student should identify $6 as the quantity that is being multiplied by 3. The student should write the problem using a symbol to represent the unknown. ($6 x 3 = ) Table 2 in the glossary includes the following division problem: A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? In solving this problem, the student should identify $18 as the quantity being divided into shares of $6. Continued on next page Explanations & Examples Standard The format of the grade level documents is modeled after the 2008 Standard. The far left column lists the standards and the far right column contains the explanations and examples. The explanations and examples is the living part of the document and will be revisited, reviewed, and updated in September of each year. This slide also highlights the Mathematical Practices and connections. S5C2 performance objectives have been replaced with the Mathematical Practices in the middle column. Please share how the committee indicated technology connections when appropriate and included examples in the Explanations and Examples column. Mathematical Practices Connection to other Subjects

Mathematical Practices
Number and Quantity: The Real Number System – (N-RN) Extend the properties of exponents to rational exponents. Standards Label Mathematical Practices Explanations and Examples Students are expected to: HS.N-RN.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.  HS.MP.2. Reason abstractly and quantitatively. HS.N-RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. HS.MP.7. Look for and make use of structure. Examples: Rewrite using fractional exponents: Rewrite in at least three alternate forms. Solution: Rewrite using only fractional exponents. Rewrite in simplest form. Key to Labels: 9-10 = Standards for Grades 9 and  = Algebra 2 Standards + = 4th Credit Standards  = Standards connected to Mathematical Modeling Mathematical Practices Labels Explanations & Examples Standard The format of the grade level documents is modeled after the 2008 Standard. The far left column lists the standards and the far right column contains the explanations and examples. The explanations and examples is the living part of the document and will be revisited, reviewed, and updated in September of each year. This slide also highlights the Mathematical Practices and connections. S5C2 performance objectives have been replaced with the Mathematical Practices in the middle column. Please share how the committee indicated technology connections when appropriate. Review the meaning of the label column and what each label represents.

Domain: Measurement and Data
Coding for K-8 Domain: Measurement and Data Grade 4 4.MD.1 The new coding for the standards is distinctly different. Please note that the cluster is not included in the coding because the standards are numbered sequentially within a domain even though there may be several clusters contained within a domain. Standard 1

Domain: Seeing Structure in Expressions Conceptual Category: Algebra
Coding for High School Domain: Seeing Structure in Expressions High School HS.A-SSE.2 High School’s coding is the same as K-8 with the addition of the conceptual category. Standard 2 Conceptual Category: Algebra

Connections within the Standards
What else is needed to help make more technology integration connections to the 2010 AZ Mathematics Standards? Use microphone or chat to answer the questions Some specific examples might be given in the standards document of how to integrate technology Smart board

The Standards for Mathematical Practices
Describe characteristics and traits that mathematics educators at all levels should seek to develop in their students Rest on important “processes and proficiencies” with long standing importance in mathematics education. Explicitly link to some of the 4 C’s

Operations and Algebraic Thinking (OA)
Use the four operations with whole numbers to solve problems. Standards Mathematical Practices Explanations and Examples Students are expected to: 4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5  7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Connections: 4.OA.3; 4.SL.1d; ET04-S1C2-01; ET04-S1C2-02 4.MP.2. Reason abstractly and quantitatively. 4.MP.4. Model with mathematics. A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which quantity is being multiplied and which number tells how many times. 4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (see Table 2) Connections: 4.RI.7; ET04-S1C2-01; ET04-S1C2-02 4.MP.5. Use appropriate tools strategically. 4.MP.7. Look for and make use of structure. Students need many opportunities to solve contextual problems. Table 2 in the glossary includes the following multiplication problem: “A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?” In solving this problem, the student should identify $6 as the quantity that is being multiplied by 3. The student should write the problem using a symbol to represent the unknown. ($6 x 3 = ) Table 2 in the glossary includes the following division problem: A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? In solving this problem, the student should identify $18 as the quantity being divided into shares of $6. Continued on next page Reinforce the nomenclature with participants specifically the domain and cluster. Point out that standards from different domains and clusters may sometimes be closely related, because mathematics is a connected subject. Standards define what students should know, understand, and be able to do. Standards are most similar to performance objectives in our 2008 Standard.

Standards for Mathematical Practice
Make sense of problems and persevere in solving them Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with Mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.

Standards for Mathematical Practice
Explanations and Examples Students are expected to: Mathematical Practices are listed throughout the grade level document in the 2nd column to reflect the need to connect the mathematical practices to mathematical content in instruction. 4.MP.1. Make sense of problems and persevere in solving them. In fourth grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. 4.MP.2. Reason abstractly and quantitatively. Fourth graders should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions and decimals. Students write simple expressions then record calculations with numbers and represent or round numbers using place value concepts. 4.MP.3. Construct viable arguments and critique the reasoning of others. In fourth grade, students may construct arguments using concrete referents, such as objects, pictures, and drawings. They explain their thinking and make connections between models and equations. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking. 4.MP.4. Model with mathematics. Students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Fourth graders should evaluate their results in the context of the situation and reflect on whether the results make sense. 4.MP.5. Use appropriate tools strategically. Fourth graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper or a number line to represent and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative size of units within a system and express measurements given in larger units in terms of smaller units. (5 minutes) The grade level documents also contain explanations and examples of the 8 Standards for Mathematical Practice that are specific for each grade level. Direct participants to read through the practices.

The Mathematical Practices

The Importance of Mathematical Practices

Explicit Intersection with P21–4 C’s
Critical Thinking and Problem Solving MP 1. Make sense of problems and persevere in solving them MP 2. Reason abstractly and quantitatively MP 4. Model with Mathematics MP 7. Look for and make use of structure Communication & Information Literacy MP 3. Construct viable arguments and critique the reasoning of others MP 6. Attend to precision MP 8. Look for and express regularity in repeated reasoning ICT Literacy MP 5. Use appropriate tools strategically. What would an integration of technology and a specific Mathematical Practice look like in the classroom? Use chat, microphone or drawing board.

Intersections with P21 – 4 C’s
Connections to Collaboration and Creativity and Innovation might not be “explicit” in the standards but can still be integrated throughout. What Mathematical Practices do you think lend themselves to integrating… Collaboration Creativity and Innovation What might this integration look like in a mathematics classroom? Report out through chat or microphone or drawing board?

Mathematical Practices look like in the classroom?
Great resource – Inside Mathematics Link to videos that demonstrate what each of the Mathematical Practices look like in a classroom setting. Click here to see a sample Two color counters Click here for online version

Mathematical Practices and the 4 C’s in action
Looking at this video, which Mathematical Practices and which of the 4 C’s might pertain to the students learning experience. Girls Angle =COy6qLKG2K0CFaQbQgod104aoQ For sure.. Creativity and Innovation – design of the proof Critical thinking & Problem solving – How to design the proof Communication – product communicates the proof Collaboration – names on the website state who created what part of the product

Sophistication of knowledge
Students use the same basic skills but in a more sophisticated way. Same rules, higher level game…. It is critical to consider the complexity of the content. Even though some of the learning progressions across grade level contain similar content, the levels of complexity increase. The cognitive demand is determined by looking at what is expected of a student both in content and performance., so it is critical to consider both the content (nouns) and skill (verbs).

Cognitive Demand (Expectations for Student Performance)
Skill/Concept Recall Strategic Thinking Extended Use Extend Memorize Perform Procedures Demonstrate Understanding Conjecture, Generalize Prove Solve non- routine/ make connections Acquire Share the following information about taxonomies: There are many different taxonomies for determining the complexities of skills, such as Bloom’s Taxonomy. The top taxonomy is called the Dimensions of Knowing and Inquiry. With only 3 categories there is a lot of overlap between the categories. The next taxonomy is the Depth of Knowledge. This particular taxonomy gives one ranking for assessment items or a standard. The ranking is determined as the highest level present. The last taxonomy shows more of a hierarchical continuum, knowing that there is some overlap between dimensions of knowing and thinking. We are going to apply this continuum when we unwrap our new standards. This particular taxonomy is used in the Surveys of Enacted Curriculum that was developed by the Council of Chief State School Officers (CCSSO). The Surveys of Enacted Curriculum (SEC) are a practical, reliable set of data collection tools being used with teachers of math, English language arts, science, and social studies to collect and report consistent data on current instructional practices and content being taught in classrooms. Think of an example of what you would observe in a classroom when students are demonstrating each of the 5 levels.

Cognitive Demand Sorting Activity
Memorize Facts, Definitions, Formulas E Solve Non-Routine Problems, Make Connections B Perform Procedures C Demonstrate Understanding of Mathematical Ideas D Conjecture, Analyze, Generalize, Prove Pass out posters. Position 5 participants at each poster. Have a discussion about what the different levels mean. What types of student performance could be sorted into these categories? (5 minutes for discussion) After this discussion, distribute cards with indicators. 46 46

Pass out posters. Position 5 participants at each poster
Pass out posters. Position 5 participants at each poster. Have a discussion about what the different levels mean. What types of student performance could be sorted into these categories? (5 minutes for discussion) After this discussion, distribute cards with indicators. 47 47

SEC Math Expectations for Students
Ask groups to compare their sorting to the summary indicator sheet and notice the similarities and differences. Let them know they will be using the sheets in an upcoming sorting activity with a cluster of standards and throughout the Unwrapping Process. Be sure to wrap up with the benefits of engaging in this type of analysis particularly with the increase of content complexity and cognitive demand in our new standards. (5 minutes)

How our students will be assessed will change in Fall 2014.

In the Meantime . . . Moving Forward with AIMS
AIMS will continue to assess the Mathematics Standard through Spring 2014 Blueprint will remain the same Proficiency levels will remain the same Multiple choice (MC) questions will continue Writing extended response will continue using holistic rubric Field testing of new MC items will continue in 2011 and 2012 The 2010 Mathematics Standards Crosswalk will assist in determining where classroom instruction will need to address both standards Field Testing for PARCC will be in &

ADE Support for Implementation
Crosswalks between the Mathematics Standards and the Mathematics Standard Summary of Changes Documents Professional Development Crosswalks from the 2008 Standard to the new 2010 Standards are available on the ADE website. The Summaries of Changes documents highlight the changes at each grade level. Both the crosswalk and the changes documents should be reviewed side-by-side to gain a comprehensive picture of the changes. There will be periodic updates to the third column on the grade level documents (once/year in September).

CROSSWALK: Grade 4 - Operations and Algebraic Thinking – (OA)
CLUSTER 2010 STANDARD ITEM DESCRIPTION 2008 PO Item Description Use the four operations with whole numbers to solve problems. 4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5  7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. * 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (see glossary Table 2) 4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. M04-S1C2-04 Use multiple strategies to divide whole numbers. M04-S1C3-02 Make estimates appropriate to a given situation or computation with whole numbers and fractions.(Includes estimating) M04-S3C3-01 Use a symbol to represent an unknown quantity in a simple algebraic expression involving all operations. M04-S3C3-02 Create and solve one-step equations that can be solved using addition, subtraction, multiplication, and division of whole numbers. (Includes one-step equations only) This is an example crosswalk from grade 4. The crosswalk is a planning tool for teachers as they incorporate the new standards into existing lesson(s) and unit plans. Point out the alignment of the Mathematical Practices to S5C2 towards the end of the document. Identify in the crosswalk where removed performance objectives are listed along with moved or redistributed. Explain how new standards are labeled with an asterisk.

SUMMARY OF CHANGES: GRADE 4
Removed Moved to a Different Grade Level Moved from another Grade Level New Standards M04-S1C1-05 (2008) Use simple ratios to describe problems in context. M04-S1C1-03 (2008) Express fractions as fair sharing, parts of a whole, parts of a set, and locations on a real number line. MOVED TO 3.NF.2a (2010) & 3.NF.2b (2010) M03-S1C1-02 (2008) MOVED TO 4.NBT.2 (2010) Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 4.OA.1 (2010) Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5  7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. M04-S2C1-01 (2008) Collect, record, organize, and display data using double bar graphs, single line graphs, or circle graphs. M04-S2C2-01 (2008) Describe elements of theoretical probability by listing or drawing all possible outcomes of a given event and predicting the outcome using word and number benchmarks. MOVED TO 7.SP.5 (2010) M03-S2C3-01 (2008) MOVED TO AZ.4.OA.3.1a (2010) Solve a variety of problems based on the multiplication principle of counting. Represent a variety of counting problems using arrays, charts, and systematic lists, e.g., tree diagram. 4.OA.2 (2010) Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (see glossary Table 2) M04-S2C1-03 (2008) Use median, mode, and range to describe the distribution of a given data set. M04-S3C1-01 (2008) Recognize, describe, create, extend, and find missing terms in a numerical sequence involving whole numbers using all four basic operations. MOVED TO 5.OA.3 (2010) M03-S2C3-02 (2008) MOVED TO AZ.4.OA.3.1 (2010) Solve a variety of problems based on the multiplication principle of counting. 4.NBT.1 (2010) Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700  70 = 10 by applying concepts of place value and division. This is the Summary of Changes for Grade 4. The first two left columns list the 2008 performance objective language. This content is no longer in the grade level. The far two right columns list the new standards language and what is new to the grade level. Content that has been moved from another grade level will only show up in the third column. If the content is moved from one grade level to more than one grade level, it will be labeled as redistributed. The crosswalk will designate the other grade levels included in the distribution. Sometimes content from one grade level is aligned to several grade levels including the initial grade level. For instance, a performance objective from grade 4 is aligned to new standards in grade 3, 4, and 5, so it is not marked as moved. There is a NOTE included in the Moved from another Grade Level to identify the content that may be more rigorous.

PARCC Assessment The PARCC Assessment System will be designed to measure the knowledge, skills and understandings essential to achieving college and career readiness. In mathematics, this includes conceptual understanding procedural skill and fluency application and problem solving, To measure the full range of the standards, the assessments will include tasks that require students to connect mathematical content and mathematical practices. First PARCC support product - Frameworks

Resources for Further Exploration
Common Core Tools Click at topics under Categories Illustrative Mathematics Project Professional development Progressions Key Points In Mathematics Achieve the Core

The Promise of Standards
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. How can the integration of technology assist teachers in the implement the 2010 Mathematics Standards? Chat, drawing board, microphone

Next Steps What can you do to continue to increase your awareness and knowledge of the 2010 Arizona Mathematics Standards (cc)? How can you assist other teachers in getting to know the standards through technology?

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Question? Sharing is Caring…Find a great idea or lesson to share…let us know. The articulated standards document is updated every September based on the input and feedback we have from teachers. Contact ADE Mary Knuck Suzi Mast We are not the experts…we are part of the PLC

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