SUMMARY OF STANDARDS FOR MATHEMATICAL PRACTICE 
QUESTIONS TO DEVELOP MATHEMATICAL THINKING 
1. Make sense of problems and persevere in solving them.
Mathematically proficient students:
 Interpret and make meaning of the problem to find a starting point. Analyze what information is given in order to understand the meaning of the problem.
 Analyze what information is given, constraints and goals in order to understand the meaning of the problem.
 Make conjectures about the form and meaning of the solution attempt.
 Consider similar problems; try special cases or simpler forms of the original problem.
 Monitor and evaluate their progress and change course as necessary.
 Use a variety of strategies to solve the problem.
 Are flexible in choosing strategies for solving the problem.
 Continually ask themselves, "Does this make sense?"
 Use concrete objects or pictures to help conceptualize and solve the problem.
 Check their answers to problems by using a different method.
 Understand the approaches of others in solving complex problems.

 How would I describe the problem in my own words?
 How would I describe what I am trying to find?
 What did I notice as I proceed?
 What information is given in the problem?
 Can I describe the relationship between the different quantities?
 Can I describe what I have tried and what I might change?
 What steps am I the most confident about?
 What other strategies could I try?
 What are some problems similar to this one?
 How can I use my previous experience to solve this problem?

2. Reason abstractly and quantitatively.
Mathematically proficient students:
 Make sense of quantities and relationships.
 Decontextualize: represent a problem symbolically and manipulate the symbols.
 Contextualize: make meaning of the symbols in a problem
 Attends to the meaning of quantities, not just how to compute them.
 Create a logical representation of the problem.
 Know and flexibly use different properties of operations and objects.

 What do the numbers in the problem represent?
 What is the relationship between the quantities?
 What do the symbols, quantities or diagrams mean to me?
 What properties might I use to find a solution?
 Could I use another property or operation to solve the problem?

3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students:
 Analyze problems and use stated mathematical assumptions, definitions, and previously established results in constructing arguments.
 Justify conclusions with mathematical ideas and communicate them to others.
 Listen to the arguments of others and ask useful questions to determine if an argument makes sense.
 Ask clarifying questions or suggest ideas to improve or revise arguments.
 Compare two arguments and determine correct or flawed logic.
 Make conjectures and build logical a progression of statements to explore the truth of their conjectures.
 Analyze problems by breaking them into specific cases.
 Recognize and use counterexamples.

 What mathematical evidence would support my conclusion?
 How can I be sure or prove that my steps are correct?
 Why did I decide to use my strategy?
 How did I test my results?
 How did I decide what needed to be done in this problem?
 Did I try a method that did not work? If so, why didn't the method work?s
 Can I find any counterexamples?

4. Model with mathematics.
Mathematically proficient students:
 Apply mathematics to solve problems arising in everyday life, society and the workplace.
 Simplify complicated problems.
 Identify important quantities in a practical situation.
 Use equations, inequalities, graphs, formulas, tables and charts to identify numerical relationships.
 Analyzes numerical relationships to draw conclusions.
 Interpret mathematical results in the context of the problem.
 Reflect on whether the results make sense, possibly improving or revising the model.
 Ask themselves, "How can I represent this problem or situation mathematically?"

 What mathematical model can I construct to represent the problem?
 What are some ways to represent the quantiies?
 What equation, graph, formula, table or chart can I use to model the problem?

5. Use appropriate tools strategically.
Mathematically proficient students:
 Consider the use of available tools recognizing the strengths and limitations of each.
 Use estimation and other mathematical knowledge to detect possible errors.
 Identify relevant external mathematical resources to pose and solve problems.
 Use technological tools to deepen their understanding of mathematics.
 Are familiar with tools appropriate for their course to make sound decisions about when to use each tool.
 Know that technology can enable them to visualize and verify results.

 What mathematical tools could I use to visualize or represent the problem?
 What information do I have?
 Can I estimate the solution?
 Does it make sense to use a ruler, calculator, graphing utility, algebra utility, graph paper or other too?
 Did using a tool help?

6. Attend to precision.
Mathematically proficient students:
 Communicate precisely with others and try to use clear mathematical language when discussing their reasoning.
 Understand the meaning of symbols used in mathematics and can label quantities with appropriate units.
 Express numerical answers with a degree of precision appropriate for the problem context.
 Perform computations efficiently and accurately.
 Use mathematical symbols correctly.
 Use clear definitions and appropriate vocabulary in their reasoning and in discussions with others.
 Label diagrams, charts, tables and graphs appropriately.

 What mathematical terminology applies in this problem?
 Is my solution reasonable? If so, how do I know?
 Did I include appropriate units in my answers?
 How am I communicating the meaning of quantities?
 What mathematical symbols and notation are important in this problem?
 How can I test the accuracy of my solution?
 Can I explain my solution using correct mathematical vocabulary?

7. Look for and make use of structure.
Mathematically proficient students:
 Apply general mathematics to specific situations.
 Look for the overall structure and patterns in mathematics.
 Se complicated things, such as algebraic expressions, as single objects composed of several objects.
 Connect new mathematics concepts with ideas previously learned.

 What observations can I make?
 How does this problem seem like other problems?
 What patterns to I see?
 How does this problem relate to current or past topics in my mathematics classes?

8. Look for and express regularity in repeated reasoning.
Mathematically proficient students:
 Notice if calculations are repeated.
 Look for both general methods and shortcuts.
 Maintain oversight of the process, while attending to details.
 Continually evaluate reasonableness of intermediate results.
 Understand the broader application of patterns and see the structure in similar situations.

 Would this strategy work in other situations? If so, how?
 How could I prove my conjecture?
 Is there a mathematical property involved in this problem?
 What predications or generalizations can this pattern support?
 What mathematical consistencies do I notice?
