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.
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- 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?
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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.
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- 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?
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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.
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- 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?
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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?"
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- 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?
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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.
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- 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?
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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.
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- 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?
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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.
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- 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?
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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.
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- 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?
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