# Standards for Mathematical Practice

## Practice 1: Make sense of problems and persevere in solving them.

Big Idea 1 Identify entry points to a solution
 Student behavior Students start understanding a problem by explaining to themselves what a problem is asking, what relevant information is being provided, and identify entry points toward its solution.
 Questions to ask - students or yourself What is the meaning of the problem? Have I seen a similar problem? What are the givens, constraints, relationships, or goals? (what others of your own belong here?)
Big Idea 2 Plan a solution pathway
 Student behavior Students generate examples, and non-examples, and look for what successful examples have in common. They make conjectures about the solution form and meaning. How can we make sense of the problem? Do we need to create a table, graph, diagram to identify what we know and what we don't know, and what we want to find out? What strategies will put us on a path to the solution? How will we know that we are still on the path once we start to solve the problem? Will be able to recognize when we are off the path and not leading to a solution? When we are done, can we check our answers? Can we apply our solution path to solve other problems? If we get stuck, or off a solution path, can we use what we already know to adjust the path?
 Questions to ask - students or yourself Is it clear why this problem requires a given operation, such as division? Does this answer make sense based on what we are given in the problem? Is it clearly communicated why we worked on a problem in a particular way? How do you know when you are finished? (what others of your own belong here?)
Big Idea 3 Monitor Progress, change course if needed
 Student behavior maintain sight of the goal.
 Questions to ask - students or yourself (what others of your own belong here?)
Really nice poster
by:
MP 1
Interpreting the Math Practice
 The Standard of Mathematical Practice as written: Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. The standard lists the practices of Mathematically proficient students Mathematically proficient students: start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution They plan a solution pathway rather than simply jumping into a solution attempt They consider analogous problems, They try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. They can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Older students might, depending on the context of the problem: transform algebraic expressions change the viewing window on their graphing calculator to get the information they need
Evaluating teachers on MP-1 Make sense of problems and persevere in solving them.

A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.

PROFICIENT
(teacher mostly models)
• Is cognitively demanding.
• Has more than one entry point.
• Requires a balance of procedural fluency and conceptual understanding.
• Requires students to check solutions for errors using one other solution path.

Teacher
• Allows ample time for all students to struggle with task.
• Expects students to evaluate processes implicitly.
• Models making sense of the task (given situation) and the proposed solution.
EXEMPLARY
(students take ownership)
• Allows for multiple entry points and solution paths.
• Requires students to defend and justify their solution by comparing multiply solution paths.

Teacher
• Differentiates to keep advanced students challenged during work time.
• Integrates time for explicit meta-cognition.
• Expects students to make sense of the task and the proposed solution.

## Practice 2: Reason abstractly and quantitatively

Big Idea 1 Make sense of the quantities and their relationships in problem situations
 Student behavior Using a variety of concrete and visual representations to highlight quantities, relationships between quantities, and the underlying mathematical structure of a problem situation
 Questions to ask - students or yourself How can I capture important information in a diagram or model? What relationships do I see? What solution path does this diagram or model imply? (what others of your own belong here?)
Big Idea 2 De-contextualize
 Student behavior Abstracting and representing a problem situation symbolically and manipulating those symbols without attending to their referents
 Questions to ask - students or yourself How do I represent this in symbols / graphs / tables? What can I figure out about these numbers / quantities? (I don’t need to think about the problem context right now.) (what others of your own belong here?)
Big Idea 3 Contextualize
 Student behavior Recalling and considering the referents for the symbols you are manipulating
 Questions to ask - students or yourself What do my variables stand for? What does my answer for x represent in the problem context? Does my answer make sense given the problem context? (what others of your own belong here?)
Really nice poster
by:
MP 2
Interpreting the Math Practice
 The Standard of Mathematical Practice as written: Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. The standard lists the practices of Mathematically proficient students Mathematically proficient students: Make sense of quantities and their relationships in problem situations. Bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents; the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Evaluating teachers on MP-2: Reason abstractly and quantitatively.

A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.

PROFICIENT
(teacher mostly models)
• Has realistic context.
• Requires students to frame solutions in a context.
• Have solutions that can be expressed with multiple representations.

Teacher
• Expects students to interpret and model using multiple representations.
• Provides structure for students to connect algebraic procedures to contextual meaning.
EXEMPLARY
(students take ownership)
• Has relevant realistic context.

Teacher
• Expects students to interpret, model, and connect multiple representations.
• Prompts students to articulate connections between algebraic procedures and contextual meaning.

## Practice 3: Construct viable arguments and critique the reasoning of others.

Big Idea 1 Math Speak
 Student behavior Students read and understand arguments while learning to construct them. Proofs and methods of construction should appear throughout algebra and geometry. Problems should ask students to critique and repair common errors in reasoning, or to read and analyze an argument between students.
 Questions to ask - students or yourself Families of Talk Moves Press for reasoning Say more about that. What did you mean by [x]? Can you say that again? Who can repeat? Who can put that in their own words? What else can say it again? What do you think about that? Do you agree or disagree…and why? Does that idea make sense to you? Who can add on to that idea? (what others of your own belong here?)
Big Idea 2 Use Previous Results
 Student behavior Students build on what they know - vocabulary, processes, mathematical properties, to extend their thinking by making conjectures and exploring them logically.
 Questions to ask - students or yourself Do my conjectures hold truth? Are there valid counterexamples? Given two or more plausible arguments which one is the most effective? (what others of your own belong here?)
Big Idea 3 Listen and Critique
 Student behavior Students listen to the statements of others and ask clarifying or probing questions. They explore beyond what is presented to uncover the truth of what is being presented or argued and provide examples to support arguments or counterexamples to contrast arguments.
 Questions to ask - students or yourself (what others of your own belong here?)
Really nice poster
by:
MP 3
Interpreting the Math Practice
Evaluating teachers on MP-3: Construct viable arguments and critique the reasoning of others.

A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.

PROFICIENT
(teacher mostly models)
• Avoids single steps or routine algorithms.

Teacher
• Identifies students’ assumptions.
• Models evaluation of student arguments.
• Asks students to explain their conjectures.
EXEMPLARY
(students take ownership)
Teacher
• Helps students differentiate between assumptions and logical conjectures.
• Prompts students to evaluate peer arguments.
• Expects students to formally justify the validity of their conjectures.

## Practice 4: Model with mathematics.

Big Idea 1 Select a model that interacts with the underlying math
 Student behavior description Students apply what has been uncovered with understanding a problem and select a model to represent the problem to then generate a pathway toward the solution. Students identify when a model is appropriate and can determine its limitations.
 Questions to ask - students or yourself Does the model organize information? Does the model demonstrate a concept? Does the model parallel a common algorithm? Does the model reveal a relationship? Does the model fall short of helping to frame the problem? (what others of your own belong here?)
 What are models? Some ideas, but by no means is this an exhaustive list: Tables Diagrams Graphs (line graphs, bar charts, circle graphs/pie charts, box-plots) Equations, functions Physical models using pattern blocks, toothpicks, other manipulatives (or virtual manipulatives)
Big Idea 2 Simplify Complexity
 Student behavior Students look for ways to simplify problems into those that can be managed by models.
 Questions to ask - students or yourself (what others of your own belong here?)
Big Idea 3 Big Idea
 Student behavior description
 Questions to ask - students or yourself (what others of your own belong here?)
Big Idea 4 Big Idea
 Student behavior description
 Questions to ask - students or yourself (what others of your own belong here?)
Really nice poster
by:
MP 4
Interpreting the Math Practice
 The Standard of Mathematical Practice as written: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely 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. The standard lists the practices of Mathematically proficient students Mathematically proficient students: Can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. Can analyze those relationships mathematically to draw conclusions. Routinely 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. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. In high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another.
Evaluating teachers on MP-4: Model with mathematics.

A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.

PROFICIENT
(teacher mostly models)
• Requires students to identify variables, compute and interpret results, and report findings using a mixture of representations.
• Illustrates the relevance of the mathematics involved.
• Requires students to identify extraneous or missing information.

Teacher
• Asks questions to help students identify appropriate variables and procedures.
• Facilitates discussions in evaluating the appropriateness of model.
EXEMPLARY
(students take ownership)
• Requires students to identify variables, compute and interpret results, report findings, and justify the reasonableness of their results and procedures within context of the task.

Teacher
• Expects students to justify their choice of variables and procedures.
• Gives students opportunity to evaluate the appropriateness of model.

## Practice 5: Use appropriate tools strategically.

Big Idea 1 Use Estimation Strategically
 Student behavior Reformulation: changes the numbers that are used to ones that are easy and quick to work with. Compensation: make adjustments that lead to closer estimates. These may be done during or after the initial estimation.
 Questions to ask - students or yourself Do I need an exact answer? When do we estimate? Is there enough information to provide an exact answer? What are some of the errors that may be introduced by rounding? (what others of your own belong here?)
Big Idea 2 Select a tool that interacts with the underlying math
 Student behavior description Students apply what has been uncovered with understanding a problem and select a tool that can represent or interpret the problem to then generate a pathway toward the solution. Tools can be used to allow students to experiment with mathematical objects, and apply approximation and estimation strategies.
 Questions to ask - students or yourself Does the tool organize information? Does the tool demonstrate a concept? Does the tool parallel a common algorithm? Does the tool reveal a relationship? Does the tool fall short of helping to frame the problem? (what others of your own belong here?)
Big Idea 3 Select and Apply Technology Tools
 Student behavior Students can choose among tools that will help them identify a solution path, and are able to use technological tools to explore and deepen their understanding of concepts.
 Questions to ask - students or yourself Have I selected the correct representation for the problem? What tools that are available will help me produced my desired output? Examples, will using a graphing calculator or spreadsheet help me represent a table better than what I can do with pen, paper, and a ruler? (what others of your own belong here?)
Big Idea 4 Use Internet and Online Resources Strategically
 Student behavior description Mathematically proficient students are able to identify relevant external mathematical resources, such as web sites, and use them to pose or solve problems.
 Questions to ask - students or yourself Where can I get real-life data that will help me prove/disprove a hypothesis? Where can I find examples of applying this in the real-world and what type of careers use these skills? Where can I find tools that may help me to represent my solution in another way to uncover additional meaning?
Really nice poster
by:
MP 5
Interpreting the Math Practice
 The Standard of Mathematical Practice as written: Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. 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. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. The standard lists the practices of Mathematically proficient students Mathematically proficient students: consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. 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. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
Evaluating teachers on MP-5: Use appropriate tools strategically.

A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.

PROFICIENT
(teacher mostly models)
• Lends itself to multiple learning tools.
• Gives students opportunity to develop fluency in mental computations.

Teacher
• Chooses appropriate learning tools for student use.
• Models error checking by estimation.
EXEMPLARY
(students take ownership)
• Requires multiple learning tools (i.e., graph paper, calculator, manipulatives).
• Requires students to demonstrate fluency in mental computations.

Teacher
• Allows students to choose appropriate learning tools.
• Creatively finds appropriate alternatives where tools are not available.

## Practice 6: Attend to precision.

Big Idea 1 Use specific language/Communication
 Student behavior description Mathematical statements are clear and unambiguous. At any moment, it is clear what is known and what is not known. (see Wu) Students critique their use of words and the implied definitions of vocabulary used by others, seek counterexamples and contradictions to deepen their understanding precise use of mathematics vocabulary and symbols.
 Questions to ask - students or yourself (what others of your own belong here?)
Big Idea 2 Writing
 Student behavior Writing, as applied in mathematics, can be evaluated for accuracy, organization, clarity, insight, and mechanics. Each of these can be a basis for evaluating works ranging from simple problems to reflective writing to research papers.
Questions to ask - students or yourself

### Evaluating writing

Accuracy
• Is the paper is free of mathematical errors?
• Does the writing conform to good practice in the use of language, notation, and symbols?

Organization
• Is the paper is organized around a central idea?
• Is there is a logical and smooth progression of the content and a cohesive paragraph structure?

Clarity
• Are the explanations of mathematical concepts and examples easily understood by the intended audience?

Insight
• Does the paper demonstrate originality, depth, and independent thought?

Mechanics
• Is the paper is free of grammatical, typographical, and spelling errors?
• Is the mathematical content is formatted and referenced appropriately?
Big Idea 3 Calculate Accurately
 Student behavior Students perform operations correctly, apply rounding appropriately, analyze the units for their answers, and label tables, graphs, and charts accurately and completely. Precision is reflected in both oral and written work.
 Questions to ask - students or yourself (what others of your own belong here?)
Big Idea 4 Big Idea
 Student behavior description
 Questions to ask - students or yourself (what others of your own belong here?)
Really nice poster
by:
MP 6
Interpreting the Math Practice
 The Standard of Mathematical Practice as written: Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. The standard lists the practices of Mathematically proficient students Mathematically proficient students: try to communicate precisely to others. try to use clear definitions in discussion with others and in their own reasoning. state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
Evaluating teachers on MP-6: Attend to precision.

A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.

PROFICIENT
(teacher mostly models)
• Has realistic context.
• Requires students to frame solutions in a context.
• Has solutions that can be expressed with multiple representations.

Teacher
• Expects students to interpret and model using multiple representations.
• Provides structure for students to connect algebraic procedures to contextual meaning.
EXEMPLARY
(students take ownership)
• Has relevant realistic context.

Teacher
• Expects students to interpret, model, and connect multiple representations.
• Prompts students to articulate connections between algebraic procedures and contextual meaning.

## Practice 7: Look for and make use of structure.

Big Idea 1 Consider behavior of calculations
 Student behavior Noticing consistencies and relationships among objects, the results of calculations, and how they behave
 Questions to ask - students or yourself Are these two calculations the same? How can I compute without calculating? How can I use properties to calculate quickly? (what others of your own belong here?)
Big Idea 2 Surface underlying structure
 Student behavior Using properties and rules of operations to uncover form and structure. For example, delaying calculations to see form and intentionally complicating expressions to see properties at play
 Questions to ask - students or yourself How can “complicate” this expression to see surface some hidden meaning? If I don’t simplify this step, will I be able to see some underlying structure? (what others of your own belong here?)
Big Idea 3 Chunk
 Student behavior Interpreting complicated things (e.g. expressions) by viewing one or more of their parts as a single entity
 Questions to ask - students or yourself How do I want to use this expression and what form will help me? How can I make sense of this expression by considering its chunks? How can I use this calculation as a placeholder in another calculation? (what others of your own belong here?)
Big Idea 4 Connect seemingly disparate objects or processes
 Student behavior Looking for and specifying structural similarities. First steps include justifying the equivalence of rules and articulating the underlying math structure of a word problem
 Questions to ask - students or yourself How can I change the form of this expression to reveal additional meaning? What type of problem is this? Does this problem remind me of another problem I’ve worked on? Is this situation behaving like another problem context I’ve seen /I know? (what others of your own belong here?)
Really nice poster
by:
MP 7
Interpreting the Math Practice
 The Standard of Mathematical Practice as written: Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. The standard lists the practices of Mathematically proficient students Mathematically proficient students: look closely to discern a pattern or structure. Young students , for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. Older students in the expression x2 + 9x + 14, can see the 14 as 2 × 7 and the 9 as 2 + 7. Recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. Can step back for an overview and shift perspective. Can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Evaluating teachers on MP-7: Look for and make use of structure.

A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.

PROFICIENT
(teacher mostly models)
• Avoids single steps or routine algorithms.

Teacher
• Identifies students’ assumptions.
• Models evaluation of student arguments.
• Asks students to explain their conjectures.
EXEMPLARY
(students take ownership)
Teacher
• Helps students differentiate between assumptions and logical conjectures.
• Prompts students to evaluate peer arguments.
• Expects students to formally justify the validity of their conjectures.

## Practice 8: Look for and express regularity in repeated reasoning.

Big Idea 1 Look for repetition in calculations
 Student behavior Noticing when the same calculation is being done over and over again seeing a “rhythm” in the operations. Looking for a calculation short cut
 Questions to ask - students or yourself Each time I check a guess, what is the set of calculations I’m doing? This applies to the guess-check-generalize strategy. Am I doing the same calculations over and over again? Is there a pattern in the calculation? Can I find a short cut? How can I find the answer without doing all those steps? (What others of your own belong here?)
Big Idea 2 Look for general methods and short cuts
 Student behavior Describing a rule or shortcut that grows out of a set of repeated calculations or a repeated process
 Questions to ask - students or yourself I keep doing the same process to check my guess? How can I describe that repetition with a rule/expression/equation? How can I generalize this process?? (What others of your own belong here?)
Big Idea 3 Maintain oversight of the process while simultaneously attending to details
 Student behavior Organizing work and/or labeling the process so as not to get lost in the details
 Questions to ask - students or yourself Wait, what am I doing? What does (a specific number) represent? Where am I in the process? Do I have a generalizable process yet? Have I included every step? (What others of your own belong here?)
Big Idea 4 Evaluate the reasonableness of intermediate results
 Student behavior Checking results along the way to ensure that they make sense for the problem situation and/or given algebraic properties at play
 Questions to ask - students or yourself Does this value make sense? Does this calculation or expression make sense? (What others of your own belong here?)
Really nice poster
by:
MP 8
Interpreting the Math Practice
 The Standard of Mathematical Practice as written: Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. The standard lists the practices of Mathematically proficient students Mathematically proficient students: notice if calculations are repeated, and look both for general methods and for shortcuts. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. middle school students might abstract the equation (y – 2)/(x – 1) = 3 by paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead students to the general formula for the sum of a geometric series.
Evaluating teachers on MP-8: Look for and express regularity in repeated reasoning.

A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns.

PROFICIENT
(teacher mostly models)
• Reviews prior knowledge and requires cumulative understanding.
• Lends itself to developing a pattern or structure.

Teacher
• Connects concept to prior and future concepts to help students develop an understanding of procedural shortcuts.
EXEMPLARY
(students take ownership)
• Addresses and connects to prior knowledge in a non-routine way.
• Requires recognition of pattern or structure to be completed.

Teacher