# Mathematical Practices – Common Core CA

The Common Core State Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students in Grades K-12.  Below are the standards and a few examples of how these Eight Practices may be integrated into problem solving tasks.

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

Solve real world problems through the application of arithmetic, algebraic, statistical and geometric concepts. These problems involve addition, multiplication, subtraction, division, ratio, rate, area, volume and statistics.   —-   Seek the meaning of a problem and look for efficient ways to represent and solve it.  —  You may check your thinking by asking, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.   —  Explain the relationships between equations, verbal descriptions, tables and graphs. —   When mathematically proficient check answers to problems using a different method.

 How can this information be used?
 What other information may be needed?
 Why did I choose that operation?
 What is another way to solve that problem?
 What did I do first? Why?
 What can I do if I don’t know how to solve a problem?
 Have I solved a problem similar to this one?
 When will I realize my first method may not work for this problem?
 How do I know my answer makes sense?

2. Reason abstractly and quantitatively.

Represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities.   —  Analyze to understand the meaning of the number or variable as related to the problem — Manipulate symbolic representations by applying properties of operations.

 What is a situation that could be represented by this equality?
 What operation can I use to represent the situation?
 Why does that operation represent the situation?
 What properties did I use to find the answer?
 How do I know my answer is reasonable?

3. Construct viable arguments and critique the reasoning of others.
Construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). — Further refine mathematical communication skills through mathematical discussions in which I critically evaluate my own thinking and the thinking of other students.  —  Pose questions like “How did I get that?”, “Why is that true?” “Does that always work?” —  Explain my  thinking to others and respond to others’ thinking.

 Will that method always work?
 How do I know?
 What do I think about what was said?
 Who can tell me about a different method?
 What do I think will happen if …?
 When would that not be true?
 Why do I agree/disagree with what was said?
 How does that drawing support my work?

4. Model with mathematics.
Model problem situations symbolically, graphically, with tables, and contextually. —  You form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations.  —  You begin to explore co-variance and represent two quantities simultaneously. —  You use number lines to compare numbers and represent inequalities.  —  You use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data sets. —  You need many opportunities to connect and explain the connections between the different representations.  —  You should be able to use all of these representations as appropriate to a problem context.

 Why is that a good model for this problem?
 How can I use a simpler problem to help me find the answer?
 What conclusions can be made from this model?
 How would I change my model if…?

5. Use appropriate tools strategically.

Consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, you may decide to represent figures on the coordinate plane to calculate area. —  Number lines are used to understand division and to create dot plots, histograms and box plots to visually compare the center and variability of the data. —  Additionally, you might use physical objects or applets to construct nets and calculate the surface area of three-dimensional figures.

 What could I use to help me solve this problem?
 What strategy could I use to make that calculation easier?
 How would estimation help me solve that problem?
 Why did I decide to use…?

6. Attend to precision.

In grade 6, you continue to refine their mathematical communication skills by using clear and precise language in your discussions with others and in their own reasoning. —  You use appropriate terminology when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or inequalities.

 How do I know my answer is reasonable?
 How can better I use math vocabulary in my explanation?
 How do I know those answers are equivalent?
 What does that mean?

7. Look for and make use of structure.

You routinely seek patterns or structures to model and solve problems. For instance, you recognize
patterns that exist in ratio tables recognizing both the additive and multiplicative properties. —  You apply properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality, c=6 by division property of equality). —  You compose and decompose two  and three-dimensional figures to solve real world problems involving area and volume.

 How did I discover that pattern?
 What other patterns can I find?
 What rule did I use to make this group?
 Why can I use that property in this problem?
 How is that like…?

8. Look for and express regularity in repeated reasoning.

.You use repeated reasoning to understand algorithms and make generalizations about patterns..  During multiple opportunities to solve and model problems, you may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm your generalization. —  You  connect place value and your prior work with operations to understand algorithms to fluently divide multi-digit numbers and perform all operations with multi-digit decimals.  —  You informally begin to make connections between covariance, rates, and representations showing the relationships between quantities.