Standards for Mathematical Practice

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?)

Plan a solution pathway

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?

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?)

Monitor Progress, change course if needed

maintain sight of the goal.

(what others of your own belong here?)

MP 1

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

Evaluating teachers on MP-1 Make sense of problems and persevere in solving them.

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

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

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.

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.

Allows for multiple entry points and solution paths.
Requires students to defend and justify their solution by comparing multiply solution paths.

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

Make sense of the quantities and their relationships in problem situations

Using a variety of concrete and visual representations to highlight quantities, relationships between quantities, and the underlying mathematical structure of a problem situation

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?)

De-contextualize

Abstracting and representing a problem situation symbolically and manipulating those symbols without attending to their referents

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?)

Contextualize

Recalling and considering the referents for the symbols you are manipulating

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?)

MP 2

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

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.

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.

Evaluating teachers on MP-2: Reason abstractly and quantitatively.

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

Has realistic context.
Requires students to frame solutions in a context.
Have solutions that can be expressed with multiple representations.

Expects students to interpret and model using multiple representations.
Provides structure for students to connect algebraic procedures to contextual meaning.
Links mathematical solution with a question’s answer.

Has relevant realistic context.

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.

Math Speak

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.

(what others of your own belong here?)

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?

Use Previous Results

Students build on what they know - vocabulary, processes, mathematical properties, to extend their thinking by making conjectures and exploring them logically.

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?)

Listen and Critique

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.

(what others of your own belong here?)

MP 3

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They 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. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

The standard lists the practices of Mathematically proficient students

Evaluating teachers on MP-3: Construct viable arguments and critique the reasoning of others.

Understand and use stated assumptions, definitions, and previously established results in constructing arguments.
Make conjectures and build a logical progression of statements to explore the truth of their conjectures.
Are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.
Justify their conclusions, communicate them to others, and respond to the arguments of others.
Reason inductively about data, making plausible arguments that take into account the context from which the data arose.
Are also able to:
compare the effectiveness of two plausible arguments,
distinguish correct logic or reasoning from that which is flawed,
and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions.
Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.
Later, students learn to determine domains to which an argument applies.
- listen or read the arguments of others
- decide whether they make sense
- ask useful questions to clarify or improve the arguments

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

Avoids single steps or routine algorithms.

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.

Select a model that interacts with the underlying math

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.

Some ideas, but by no means is this an exhaustive list:

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?

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)

Simplify Complexity

Students look for ways to simplify problems into those that can be managed by models.

(what others of your own belong here?)

(what others of your own belong here?)

(what others of your own belong here?)

MP 4

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

Evaluating teachers on MP-4: Model with mathematics.

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.

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

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.

Asks questions to help students identify appropriate variables and procedures.
Facilitates discussions in evaluating the appropriateness of model.

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.

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.

Use Estimation Strategically
Added by JB

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.

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?)

Select a tool that interacts with the underlying math

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.

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?)

Select and Apply Technology Tools

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.

Use Internet and Online Resources Strategically

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

description Mathematically proficient students are able to identify relevant external mathematical resources, such as web sites, and use them to pose or solve problems.

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?

MP 5

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

Evaluating teachers on MP-5: Use appropriate tools strategically.

consider the available tools when solving a mathematical problem.
- 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.

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

Lends itself to multiple learning tools.
Gives students opportunity to develop fluency in mental computations.

Chooses appropriate learning tools for student use.
Models error checking by estimation.

Requires multiple learning tools (i.e., graph paper, calculator, manipulatives).
Requires students to demonstrate fluency in mental computations.

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

Practice 6: Attend to precision.

Use specific language/Communication

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.

(what others of your own belong here?)

Writing

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.

Evaluating writing

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

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?

Are the explanations of mathematical concepts and examples easily understood by the intended audience?
Can the reader readily follow the paper's development? (or a solution)

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

Is the paper is free of grammatical, typographical, and spelling errors?
Is the mathematical content is formatted and referenced appropriately?

Calculate Accurately

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.

(what others of your own belong here?)

(what others of your own belong here?)

MP 6

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

Evaluating teachers on MP-6: Attend to precision.

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.

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

Has realistic context.
Requires students to frame solutions in a context.
Has solutions that can be expressed with multiple representations.

Expects students to interpret and model using multiple representations.
Provides structure for students to connect algebraic procedures to contextual meaning.
Links mathematical solution with a question’s answer.

Has relevant realistic context.

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.

Consider behavior of calculations

Noticing consistencies and relationships among objects, the results of calculations, and how they behave

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?)

Surface underlying structure

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

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?)

Chunk

Interpreting complicated things (e.g. expressions) by viewing one or more of their parts as a single entity

Connect seemingly disparate objects or processes

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

Looking for and specifying structural similarities. First steps include justifying the equivalence of rules and articulating the underlying math structure of a word problem

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?)

MP 7

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

Evaluating teachers on MP-7: Look for and make use of structure.

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.

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

Avoids single steps or routine algorithms.

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.

Look for repetition in calculations

Noticing when the same calculation is being done over and over again
seeing a “rhythm” in the operations.
Looking for a calculation short cut

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?)

Look for general methods and short cuts

Describing a rule or shortcut that grows out of a set of repeated calculations or a repeated process

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?)

Maintain oversight of the process while simultaneously attending to details

Organizing work and/or labeling the process so as not to get lost in the details

Evaluate the reasonableness of intermediate results

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

Checking results along the way to ensure that they make sense for the problem situation and/or given algebraic properties at play

Does this value make sense?
Does this calculation or expression make sense?
(What others of your own belong here?)

MP 8

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

Evaluating teachers on MP-8: Look for and express regularity in repeated reasoning.

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.

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

Reviews prior knowledge and requires cumulative understanding.
Lends itself to developing a pattern or structure.

Connects concept to prior and future concepts to help students develop an understanding of procedural shortcuts.
Demonstrates connections between tasks.