##### Rick Hobbs, Mission College, Santa Clara, CA

Mr. Hobbs' Math Corner Math 903 main page Math 3A main page

# COMMON CORE STATE STANDARDS

Develop your full potential!

### 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?

Mr. Hobbs' Math Corner Math 903 main page Math 3A main page

 Instructor: Rick Hobbs Email: rick.hobbs@wvm.edu Phone/voicemail: (408) 855-5325 Office hours: Click here

last update: 1/16/13