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Arizona’s Mathematics

Final Draft

ARIZONA DEPARTMENT OF EDUCATION

HIGH ACADEMIC STANDARDS FOR STUDENTS

December 2016

Arizona Mathematics Standards

Introduction

ARIZONA DEPARTMENT OF EDUCATION

HIGH ACADEMIC STANDARDS FOR STUDENTS

December, 2016

The Arizona Mathematics Standards are well articulated across grades K-8 and high school. The Arizona Mathematics Standards are the result of a process designed to identify, review, revise or refine, and create high-quality, rigorous mathematics standards. The Arizona Mathematics Standards are coherent, focus on deep mathematical content knowledge, and address a balance of rigor, which includes conceptual understanding, application, and procedural skills and fluency.

Balanced approach to rigor found in the Arizona Mathematics Standards

What the Arizona Mathematics Standards Are NOT

The standards are not the curriculum. While the Arizona Mathematics Standards may be used as the basis for curriculum, the Arizona Mathematics Standards are not a curriculum. Therefore, identifying the sequence of instruction at each grade – what will be taught and for how long – requires concerted effort and attention at the district and school levels. The standards do not dictate any particular curriculum. Curricular tools, including textbooks, are selected by the district/school and adopted through the local governing board. The Arizona Department of Education defines standards, curriculum, and instruction as: Standards – What a student needs to know, understand, and be able to do by the end of each grade/course. Standards build across grade levels in a progression of increasing understanding and through a range of cognitive-demand levels. Curriculum –The resources used for teaching and learning the standards. Curricula are adopted at the local level by districts and schools. Curriculum refers to the how in teaching and learning the standards. Instruction – The methods used by teachers to teach their students. Instructional techniques are employed by individual teachers in response to the needs of all the students in their classes to help them progress through the curriculum in order to master the standards. Instruction refers to the how in teaching and learning the standards.

The standards are not instructional practices. While the Arizona Mathematics Standards define the knowledge, understanding, and skills that need to be effectively taught and learned for each and every student to be college and workplace ready, the standards are not instructional practices. The educators and subject matter experts who worked on the Mathematics Standards Subcommittee and Workgroups ensured that the Arizona Mathematics Standards are free from embedded pedagogy and instructional practices. The Arizona Mathematics Standards do not define how teachers should teach and must be complimented by well-developed, aligned, and appropriate curriculum materials, as well as effective instructional practices.

The standards do not necessarily address students who are far below or far above the grade level. No set of grade-specific standards can fully reflect the great variety in abilities, needs, learning rates, and achievement levels of students in any given classroom. The Arizona Mathematics Standards do not define the intervention methods or materials necessary to support students who are well below or well above grade level expectations. It is up to the teacher, school, and district to determine the most effective instructional methods and curricular resources and materials to meet all students’ needs.

The following narratives describe the eight Standards for Mathematical Practice. (MP)

1. Make sense of problems and persevere in solving them. Mathematically proficient students explain to themselves the meaning of a problem, look for entry points to begin work on the problem, and plan and choose a solution pathway. While engaging in productive struggle to solve a problem, they continually ask themselves, “Does this make sense?" to monitor and evaluate their progress and change course if necessary. Once they have a solution, they look back at the problem to determine if the solution is reasonable and accurate. Mathematically proficient students check their solutions to problems using different methods, approaches, or representations. They also compare and understand different representations of problems and different solution pathways, both their own and those of others. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. Students can contextualize and decontextualize problems involving quantitative relationships. They contextualize quantities, operations, and expressions by describing a corresponding situation. They decontextualize a situation by representing it symbolically. As they manipulate the symbols, they can pause as needed to access the meaning of the numbers, the units, and the operations that the symbols represent. Mathematically proficient students know and flexibly use different properties of operations, numbers, and geometric objects and when appropriate they interpret their solution in terms of the context. 3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students construct mathematical arguments (explain the reasoning underlying a strategy, solution, or conjecture) using concrete, pictorial, or symbolic referents. Arguments may also rely on definitions, assumptions, previously established results, properties, or structures. Mathematically proficient students make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. Mathematically proficient students present their arguments in the form of representations, actions on those representations, and explanations in words (oral or written). Students critique others by affirming, questioning the reasoning of others. They can listen to or read the reasoning of others, decide whether it makes sense, ask questions to clarify or improve the reasoning, and validate or build on it. Mathematically proficient students can communicate their arguments, compare them to others, and reconsider their own arguments in response to the critiques of others. 4. Model with mathematics. Mathematically proficient students apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. When given a problem in a contextual situation, they identify the mathematical elements of a situation and create a mathematical model that represents those mathematical elements and the relationships among them. Mathematically proficient students use their model to analyze the relationships and draw conclusions. They interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5. Use appropriate tools strategically. Mathematically proficient students consider available tools when solving a mathematical problem. They choose tools that are relevant and useful to the problem at hand. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful; recognizing both the insight to be gained and their limitations. Students deepen their understanding of mathematical concepts when using tools to visualize, explore, compare, communicate, make and test predictions, and understand the thinking of others. 6. Attend to precision. Mathematically proficient students clearly communicate to others using appropriate mathematical terminology, and craft explanations to convey their reasoning. When making mathematical arguments about a solution, strategy, or conjecture, they describe mathematical relationships and connect their words clearly to their representations. Mathematically proficient students understand meanings of symbols used in mathematics, calculate accurately and efficiently, label quantities appropriately, and record their work clearly and concisely. 7. Look for and make use of structure. Mathematically proficient students use structure and patterns to assist in making connections among mathematical ideas or concepts when making sense of mathematics. Students recognize and apply general mathematical rules to complex situations. They are able to compose and decompose mathematical ideas and notations into familiar relationships. Mathematically proficient students manage their own progress, stepping back for an overview and shifting perspective when needed. 8. Look for and express regularity in repeated reasoning. Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency.

The eight Standards for Mathematical Practice describe ways in which students are expected to engage within the mathematical content standards. The Arizona Standards for Mathematical Practice reflect the interaction of skills necessary for success in math coursework as well as the ability to apply math knowledge and processes within real-world contexts. The Standards for Mathematical Practice highlight the applied nature of math within the workforce and clarify the expectations held for the use of mathematics in and outside of the classroom. The Standards for Mathematical Practice complement the math content standards so that students increasingly engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years.

Reading the Mathematical Content Standards in Kindergarten – Grade 8

In kindergarten through grade 8, the Arizona Mathematics Standards are organized by grade level and then by domains (clusters of standards that address “big ideas” and support connections of topics across the grades), clusters (groups of related standards inside domains), and the standards (what students should know, understand, and be able to do). The standards do not dictate curriculum or pedagogy. The numbering of standards within a grade level does not imply an instructional sequence (3.NBT.A. 2 is not required to be taught before 3.NBT.A. 3 ), nor does the numerical coding imply vertical alignment from one grade to the next.

There are eleven domains within the K-8 Standards. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades. Critical areas are included before each grade to support the implementation of the mathematical content standards.

1. CC: Counting and Cardinality
2. OA: Operations and Algebraic Thinking
3. NBT: Number and Operations in Base Ten
4. MD: Measurement and Data
5. NF: Number and Operations – Fractions
6. G: Geometry
7. RP: Ratios and Proportional Relationships
8. NS: The Number System
9. EE: Expressions and Equations
10. F: Functions
11. SP: Statistics and Probability

Domains are intended to convey coherent groupings of content. All domains are bold and centered. Clusters are groups of related standards. Cluster headings are bolded. Standards define what students should know, understand and be able to do. Standards are numbered.

Number and Operations – Fractions (NF) Note: Grade 3 expectations are limited to fractions with denominators: 2,3,4,6,8. 3.NF.A Understand fractions as numbers.

3.NF.A.1 Understand a unit fraction (1/ b ) as the quantity formed by one part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts 1/ b.

Domain

Cluster Standard

The code for each standard begins with the grade level, followed by the domain code, the cluster letter, and the number of the standard. For example, 3.NF.A. 1 would be the first standard (1), in the first cluster (A), in the domain of Number and Operations-Fractions (NF) of the standards for grade 3.

Reading the Mathematical Content Standards in High School

The Arizona Mathematics high school standards are organized into three main courses and also include additional and extended or plus ( P ) standards that could be included in a 4th^ credit mathematics course or to extend instruction in Algebra 1, Geometry or Algebra 2. The Arizona Mathematics high school courses are organized into Algebra 1, Algebra 2, and Geometry courses. Districts can use a variety of pathways including traditional or integrated pathways in which students can master the high school mathematics standards over the path of four courses.

In high school, the Arizona Mathematics Standards are organized by course and then by conceptual category and domains (clusters of standards that address “big ideas” and support connections of topics across the grades), clusters (groups of related standards inside domains), and the standards (what students should know, understand. and be able to do). The standards do not dictate curriculum or pedagogy. The numbering of standards within a grade level does not imply an instructional sequence (A1.F-LE.A. 2 is not required to be taught before A1.F-LE.A. 3 ).

The high school mathematics standards are organized by Conceptual Category, Domain, Cluster and Standard. There are six conceptual categories for high school.

• N: Number and Quantity
• A: Algebra
• F: Functions
• G: Geometry
• S: Statistics and Probability
• CM: Contemporary Mathematics

Conceptual Categories portray a coherent view of higher mathematics. All conceptual categories are in bold, italicized , centered and larger font. Domains are intended to convey coherent groupings of content. All domains are bold and centered.

3.NF.A.

Domain Cluster Standard

Grade

Additional High School Standards - Plus

The Arizona Mathematics Standards contain an additional set of standards that are found outside the limits of a high school Algebra 1, Geometry, or Algebra 2 minimum course of study as outlined by the Arizona Mathematics Standards. The high school conceptual category Contemporary Mathematics (CM) includes the domain Discrete Mathematics (DM). This additional conceptual category and standards could be included in a fourth credit math course. These additional high school standards are represented with the code P. The plus ( P ) standards are standards that are found outside the limits of a high school Algebra 1, Algebra 2, or Geometry minimum course of study as outlined by the standards. The Plus standards are represented with the code P. The plus standards are intended to be included in honors, accelerated, advanced courses, fourth credit courses, as well as extensions of the regular courses (Algebra 1, Algebra 2, and Geometry).

Key Considerations for Standards Implementation

Addition/Subtraction and Multiplication/Division Problem Types or Situations

There are important distinctions among different types of addition/subtraction and multiplication/division problems that are reflected in the ways that children think about and solve them.^4 Table 1 and Table 2 describe different problem types that provide a structure for selecting problems for instruction and interpreting how children solve them. This is a critical consideration for standards implementation. When planning instruction, educators must provide all students with the opportunity to learn and experience all different problem types associated with a given standard. Without the opportunity to learn and experience different problem types, students cannot truly master and apply the grade level standards in future mathematical tasks and experiences.

Table 1 and 2 can be found in the Introduction, Glossary and at the end of grade level standards that would utilize Table 1 and/or Table 2.

Table 1: Addition and Subtraction Situations Table 1 provides support to clarify the varied problem structures necessary to build student conceptual understanding of addition and subtraction, focusing on developing student flexibility. In order to fully implement the standards, students must solve problems from all problem subcategories relevant to the grade level. All problem types should not be mastered at all grades in Kindergarten through fifth grade. Guidance on what problem types could be mastered at each grade level is available in the Progressions for Operations and Algebraic Thinking document.^5

Table 2: Multiplication and Division Situations Table 2 provides support to clarify the varied problem structures necessary to build student conceptual understanding of multiplication and division, focusing on developing student flexibility. In order to fully implement the standards, students must solve problems from all problem subcategories relevant to the grade level.

(^4) Carpenter, T., Fennema, E., Franke, M., Levi, L., Empson, S. (1999). Children’s Mathematics Cognitively Guided Instruction. (^5) University of Arizona Institute for Mathematics and Education. (2011). Progression on Counting and Cardinality and Operations and Algebraic Thinking.

Table 2. Common Multiplication and Division Problem Types/Situations.^1 Unknown Product Group Size Unknown (“How many in each group?” Division)

Number of Groups Unknown (“How many groups?” Division) 3 x 6 =****? 3 x? = 18 and 18 ÷ 3 =?? x 6 = 18 and 18 ÷ 6 =****?

Equal

Groups

There are 3 bags with 6 plums in each bag. How many plums are there in all?

Measurement example****.

You need 3 lengths of string, each 6 inches long. How much string will you need altogether?

If 18 plums are shared equally into 3 bags, then how many plums will be in each bag?

Measurement example****.

You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be?

If 18 plums are to be packed 6 to a bag, then how many bags are needed?

Measurement example****.

You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have?

Arrays,^2

Area^3

There are 3 rows of apples with 6 apples in each row. How many apples are there?

Area example****.

What is the area of a 3 cm by 6 cm rectangle?

If 18 apples are arranged into 3 equal rows, how many apples will be in each row?

Area example****.

A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it?

If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?

Area example****.

A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it?

Compare

A straw hat costs $6. A baseball hat costs 3 times as much as the straw hat. How much does the baseball hat cost?

Measurement example.

A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long?

A baseball hat costs $18 and that is 3 times as much as a straw hat costs. How much does a straw hat cost?

Measurement example****.

A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first?

A baseball hat costs $18 and a straw hat costs $6. How many times as much does the baseball hat cost as the straw hat?

Measurement example****.

A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first?

General General a x b =? a x? = p, and p ÷ a =?? x b = p, and p ÷ b =?

(^1) The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples. (^2) The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable. (^3) Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations.

Fluency in Mathematics

Wherever the word fluently appears in a content standard, the word includes efficiently, accurately, flexibly, and appropriately. Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently.^6

• Efficiency —carries out easily, keeps track of sub-problems, and makes use of intermediate results to solve the problem.
• Accuracy —reliably produces the correct answer.
• Flexibility —knows more than one approach, chooses a viable strategy, and uses one method to solve and another method to double- check.
• Appropriately —knows when to apply a particular procedure.

Please see standards 2.OA.B.2 and 3.OA.C.7 for standards related to addition and subtraction of within 20 and multiplication within 100. Both of these standards show mastery involves “from memory” as an outcome. By the end of 2nd^ and 3rd^ grade, these procedural fluency standards should be automatic recall by students.

Fluency Expectations, K- 8

Specific mathematics standards in K-6 state fluency as the intended end of grade level outcome. Some standards in grades 7-8 do not explicitly state fluently within the standard but based on the definition of fluency, we want students to efficiently, accurately, flexibly, and appropriately problem solve.

Fluency Expectations, High School

The high school standards do not always set explicit expectations for fluency but fluency is important in high school mathematics. For example, fluency in algebra can help students get past the need to manage computational details so that they can observe structure and patterns in problems. Therefore, this section makes recommendations about fluencies that can serve students well as they learn and apply mathematics.

Table 3, Fluency Expectations across All Grade Levels can also be found in the Glossary.

(^6) National Council of Teachers of Mathematics, Inc. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA. Diane Briars (2016) NCTM. [email protected]. Russell, S. J. (2000). Developing computational fluency with whole numbers. Teaching Children Mathematics, 7 (3), 154–158.

Disciplinary Literacy in Mathematics

The English Language Arts (ELA) Standards provide an integrated approach to literacy to help guide instruction throughout all disciplines. Therefore, the ELA standards in reading, writing, speaking and listening, should be integrated throughout K-12 mathematics teaching and learning.

By incorporating ELA Standards, and critical thinking in instruction, educators provide students with opportunities to develop literacy in mathematics instruction. The Standards for Mathematical Practice (MP) naturally link to the ELA Standards. By engaging in a multitude of critical thinking experiences linked to the Standards for Mathematical Practice, students will:

• Construct viable arguments through proof and reasoning.
• Critique the reasoning of others.
• Process and apply reasoning from others.
• Synthesize ideas and make connections to adjust the original argument. The goal of using literacy skills in mathematics is to foster a deeper conceptual understanding of the mathematics and provide students with the opportunity to read, write, speak, and listen within a mathematics discourse community.

Technology Integration in Mathematics

Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning. Electronic technologies – calculators and computers – are essential tools for teaching, learning, and doing mathematics. They furnish visual images of mathematical ideas, they facilitate organizing and analyzing data, and they compute efficiently and accurately. They can support student investigations in every area of mathematics, including geometry, statistics, algebra, measurement, and number. When technology tools are available, students can focus on decision making, reflection, reasoning, and problem solving. Technology should not be used as a replacement for basic understandings and intuitions; rather, it can and should be used to foster those understandings and intuitions.^7

It is the goal in teaching, learning, and assessing mathematical understanding that technology tools are used appropriately and strategically. (MP

5. Students should choose a tool, including a calculator when it is relevant and useful to the problem at hand. It is suggested that calculators in elementary grades serve as aids in advancing student understanding without replacing other calculation methods. Calculator use can promote the higher-order thinking and reasoning needed for problem solving in our information and technology based society. Their use can also assist teachers and students in increasing student understanding of and fluency with arithmetic operations, algorithms, and numerical relationships and enhancing student motivation. Strategic calculator use can aid students in recognizing and extending numeric, algebraic, and geometric

(^7) National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA.

patterns and relationships.^8 It is suggested that teaching, learning, and assessment of mathematics include calculator use in grades 5 through high school on assessments or parts of assessments.

Understand and Use Formulas

When planning instruction around the Arizona Mathematics Standards, it is important to point out that conceptual understanding is developed prior to mastery of procedural skill and fluency. This also applies to certain concepts or big ideas related to the use of formulas. Understanding an individual formula includes knowledge of each part of the equation. The formula should be developed from a foundation of conceptual understanding, and formula mastery should include this understanding as well as use of the formula in specific applied problems.

Mathematical Modeling

In the course of a student’s mathematics education, the word “model” is used in many ways. Several of these, such as manipulatives, demonstration, role modeling, and conceptual models of mathematics, are valuable tools for teaching and learning. However, these examples are different from the practice of mathematical modeling. Mathematical modeling, both in the workplace and in school, uses mathematics to answer questions utilizing real-world context.

Within the standards document, the mathematical modeling process should be used with standards that include the phrase “real-world context.”

Mathematical modeling is a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena. This process includes:

• Using the Language of mathematics to quantify real-world phenomena and analyze behaviors.
• Using math to explore and develop our understanding of real world problems.
• An iterative problem solving process in which mathematics is used to investigate and develop deeper understanding.

Students have to do the same mathematics to answer a mathematical modeling question, but they are forced to reconcile their answer with reality, making the mathematics more relevant and interesting. Making judgments about what matters and assessing the quality of a solution are components of mathematical modeling. Within the mathematical modeling process, students’ opinions matter and influence the answer to the question.

(^8) National Council of Teachers of Mathematics. (2015). Calculator Use in Elementary Grades-NCTM position statement.