Mathematical Practices by Standards
*each poster is downloadable for classroom printing by clicking on the icon*
*the questioning document can be downloaded here*
Standard 1: Make sense of problems and persevere in solving them
Students...
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Analyze and explain the meaning of the problem​
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Actively engage in problem solving (Develop, carry out, and refine a plan)
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Show patience and positive attitudes
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Ask if their answers make sense
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Check their answers with a different method
Because Teachers...
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Pose rich problems and/or ask open ended questions​
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Provide wait-time for processing/finding solutions
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Circulate to pose probing questions and monitor student progress
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Provide opportunities and time for cooperative problem solving and reciprocal teaching
Questions to Develop Mathematical Thinking...
How would you describe the problem in your own words?
How would you describe what you are trying to find?
What do you notice about...?
What information is given in the problem?
Describe the relationship between the quantities.
Describe what you have already tried. What might you change?
How else might you organize...represent... show...?
Talk me through the steps you’ve used to this point.
What steps in the process are you most confident about?
What are some other strategies you might try?
What are some other problems that are similar to this one?
How might you use one of your previous problems to help
you begin?
Standard 2: Reason abstractly and quantitatively
Students...
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Represent a problem with symbols
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Explain their thinking
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Use numbers flexibly by applying properties of operations and place value
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Examine the reasonableness of their answers/calculations
Because Teachers...
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Ask students to explain their thinking regardless of accuracy
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Highlight flexible use of numbers
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Facilitate discussion through guided questions and representations
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Accept varied solutions/representations
Questions to Develop Mathematical Thinking...
What do the numbers used in the problem represent?
What is the relationship of the quantities?
How is _______ related to ________?
What is the relationship between ______and ______?
What does_______mean to you? (e.g. symbol, quantity, diagram)
What properties might we use to find a solution?
How did you decide in this task that you needed to use...?
Could we have used another operation or property to solve this task? Why or why not?
Standard 3: Construct Viable Arguments and Critique the Reasoning of Others
Students...
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Make reasonable guesses to explore their ideas
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Justify solutions and approaches
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Listen to the reasoning of others, compare arguments, and decide if the arguments of others makes sense
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Ask clarifying and probing questions
Because Teachers...
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Provide opportunities for students to listen to or read the conclusions of arguments of others
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Establish and facilitate a safe environment for discussion
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Ask clarifying and probing questions
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Avoid giving too much assistance (e.g., providing answers or procedures)
Questions to Develop Mathematical Thinking...
What mathematical evidence would support your solution?
How can we be sure that...? / How could you prove that...?
Will it still work if...?
What were you considering when...?
How did you decide to try that strategy?
How did you test whether your approach worked?
How did you decide what the problem was asking you to
find? (What was unknown?)
Did you try a method that did not work? Why didn’t it
work? Would it ever work? Why or why not?
What is the same and what is different about...?
How could you demonstrate a counter-example?
Standard 4: Model with Mathematics
Students...
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Apply prior knowledge to new problems and reflect
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Use representations to solve real life problems
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Apply formules and equations when appropriate
Because Teachers...
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Pose problems connected to previous concepts
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Provide a variety of real world contexts
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Use intentional representations
Questions to Develop Mathematical Thinking...
What number model could you construct to represent the problem?
What are some ways to represent the quantities?
What is an equation or expression that matches the diagram, number line.., chart..., table..?
Where did you see one of the quantities in the task in your equation or expression?
How would it help to create a diagram, graph, table...?
What are some ways to visually represent...?
What formula might apply in this situation?
Standard 5: Use Appropriate Tools Strategically
Students...
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Select and use tools strategically (and flexibly) to visualize, explore, and compare information
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Use technological tools and resources to solve problems and deepen understanding
Because Teachers...
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Make appropriate tools available for learning (calculators, concrete models, digital resources, pencil/paper, compass, protractor, etc.)
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Use tools with their instruction
Questions to Develop Mathematical Thinking...
What mathematical tools could we use to visualize and represent the situation?
What information do you have?
What do you know that is not stated in the problem?
What approach are you considering trying first?
What estimate did you make for the solution?
In this situation would it be helpful to use...a graph...,
number line..., ruler..., diagram..., calculator..., manipulative?
Why was it helpful to use...?
What can using a ______ show us that _____may not?
In what situations might it be more informative or
helpful to use...?
Standard 6: Attend to Precision
Students...
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Calculate accurately and efficiently
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Explain their thinking using mathematices vocabulary
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Use appropriate symbols and specify units of measure
Because Teachers...
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Recognize and model efficient strategies for computation
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Use (and challenge students to use) mathematics vocabulary precisely and consistently
Questions to Develop Mathematical Thinking...
What mathematical terms apply in this situation?
How did you know your solution was reasonable?
Explain how you might show that your solution answers the problem.
What would be a more efficient strategy?
How are you showing the meaning of the quantities?
What symbols or mathematical notations are important in
this problem?
What mathematical language...,definitions..., properties can
you use to explain...?
How could you test your solution to see if it answers the problem?
Standard 7: Look For and Make Use of Structure
Students...
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Look for, develop, and generalize relationships and patterns
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Apply reasonable thoughts about patterns and propeties to new situations
Because Teachers...
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Provide time for applying and discussing properties
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Ask questions about the applications of patterns
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Highlight different approaches for solving problems
Questions to Develop Mathematical Thinking...
What observations do you make about...?
What do you notice when...?
What parts of the problem might you eliminate..., simplify...?
What patterns do you find in...?
How do you know if something is a pattern?
What ideas that we have learned before were useful in
solving this problem?
What are some other problems that are similar to this one?
How does this relate to...?
In what ways does this problem connect to other
mathematical concepts?
Standard 8: Look For and Express Regularity in Repeated Reasoning
Students...
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Look for methods and shortcuts in patterns and repeated calculations
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Evaluate the reasonableness of results and solutions
Because Teachers...
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Provide tasks and problems with patterns
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Ask about answers before and reasonableness after computations
Questions to Develop Mathematical Thinking...
Explain how this strategy work in other situations?
Is this always true, sometimes true or never true?
How would we prove that...?
What do you notice about...?
What is happening in this situation?
What would happen if...?
Is there a mathematical rule for...?
What predictions or generalizations can this pattern support?
What mathematical consistencies do you notice ?