As we work together, we will employ the following eight Mathematical Practices:
 

1. Make sense of problems and persevere in solving them.

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.

 

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of the quantities and their

relationships in problem situations. Students 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 meanings of quantities, not

just how to compute them; and knowing and flexibly using different properties

of operations and objects.

 

3. Construct viable arguments and critique the reasoning of others.

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.

 

4. Model with mathematics.

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.

 

5. Use appropriate tools strategically.

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.

 

6. Attend to precision.

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.

 

7. Look for and make use of structure.

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.

 

8. Look for and express regularity in repeated reasoning.

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 reasonablenessof  their intermediate results.