2. Number
Core Skill 1: Compare numbers and make sense of their magnitude.Include positive and negative numbers expressed as fractions, decimals, powers, and roots. Limit to square and cube roots. Include very large and very small numbers and the use of scientific notation. Example Task 1 2. Number
Core Skill 2: Know when and how to use standard algorithms, and perform them flexibly, accurately and efficiently.This aligns with the concept of procedural fluency as in the National Research Council report Adding it up: Helping children learn mathematics. Specifically, "Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently" (p. 121). Example Task 1 2. Number
Core Skill 3: Use mental strategies and technology to formulate, represent and solve problems.This aligns with the concept of strategic competence as described in Adding it up. "Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them" (p. 124). Example Task 1 2. Number
Core Skill 5: Use estimation and approximation to solve problems.Include evaluating answers for their reasonableness, detecting errors, and giving answers to an appropriate level of precision. Example Task 1 2. Number
Core Skill 4: Solve multi-step problems involving fractions and percentages.Include situations such as simple interest, tax, markups/markdowns, gratuities and commissions, fees, percent increase or decrease, percent error, expressing rent as a percentage of take-home pay, and so on. Example Task 1 3. Quantity
Core Skill 1: Know when and how to convert units in computations.Include the addition and subtraction of quantities of the same type expressed in different units; averaging data given in mixed units; converting units for derived quantities such as density and speed. 3. Quantity
Core Skill 2: Use and interpret quantities and units correctly in algebraic formulas.Include specifying units when defining variables and attending to units when writing expressions and equations. 3. Quantity
Core Skill 3: Use and interpret quantities and units correctly in graphs and data displays.Include function graphs, data tables, scatterplots and other visual displays of dimensioned data. 3. Quantity
Core Skill 4: Use units as a way to understand problems and to guide the solution of multi-step problems.Include examples such as acceleration; currency conversions; people-hours; social science measures, such as deaths per 100,000; and general rates, such as points per game. Example Task 1 4. Expressions
Core Skill 1: See structure in expressions.For example, recognize: that the expressions x^{4} − y^{4} and (x + y)^{2} - (x - y)^{2} are differences of squares; that there are different ways to rewrite the latter expression, e.g., by expanding and collecting like terms or by factoring as a difference of squares; that p is a common factor in p + 0.025p; that an expression in the form (x − 3)^{2} + 14 reveals its minimum value.
Example Task 1 4. Expressions
Core Skill 2: Manipulate simple expressions. Show procedural fluency in the following cases: factoring out common terms; factoring expressions with quadratic structure; writing in standard form sums, differences, and products of polynomials. Include completing the square and rewriting in standard form sums, differences, products, and quotients of simple rational expressions; rewriting expressions with negative exponents and those involving square or cube roots of a single term involving exponents. Example Task 1 4. Expressions
Core Skill 3: Define variables and write an expression to represent a quantity in a problem.Include contextual problems. Example Task 1 4. Expressions
Core Skill 4: Interpret an expression that represents a quantity in terms of the context.Include interpreting parts of an expression, such as terms, factors and coefficients. Example Task 1 5. Equations
Core Skill 1: Understand a problem and formulate an equation to solve it.Extend to inequalities and systems. Example Task 1 5. Equations
Core Skill 2: Solve equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.Solve linear equations with procedural fluency. For quadratic equations, include solution by inspection, by factoring, or by using the quadratic formula. Understand that the quadratic formula comes from completing the square. Include simple absolute value equations solvable by direct inspection and by understanding the interpretation of absolute value as distance. 5. Equations
Core Skill 3: Rearrange formulas to isolate a quantity of interest.Exclude cases that require extraction of roots or inverse functions. 5. Equations
Core Skill 4: Solve systems of equations.Focus on pairs of simultaneous linear equations in two variables. Include algebraic techniques, graphical techniques and solving by inspection. Example Task 1 5. Equations
Core Skill 5: Solve linear inequalities in one variable and graph the solution set on a number line.Emphasize solving the associated equality and determining on which side of the solution of the associated equation the solutions to the inequality lie. 5. Equations
Core Skill 6: Graph the solution set of a linear inequality in two variables on the coordinate plane.Emphasize graphing the associated equation, using a dashed or solid line as appropriate and shading to indicate the half-plane on which the solutions to the inequality lie. 6. Functions
Core Skill 1: Recognize proportional relationships and solve problems involving rates and ratios.Include being able to express proportional relationships as functions. Example Task 1 6. Functions
Core Skill 2: Describe the qualitative behavior of common types of functions using graphs and tables.Identify: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Use technology to explore the effects of parameter changes on the graphs of linear, power, quadratic, polynomial, simple rational, exponential, logarithmic, sine and cosine, absolute value and step functions. Example Task 1 6. Functions
Core Skill 3: Analyze functions using symbolic manipulation.Include slope-intercept and point-slope form of linear functions; vertex form of quadratic functions to identify symmetry and find maximums and minimums; factored form to find zeros. Use manipulations as described under Expressions. Example Task 1 6. Functions
Core Skill 4: Use the families of linear and exponential functions to solve problems.For linear functions f(x) = mx + b, understand b as the intercept or initial value and m as the slope or rate of change. For exponential functions f(x) = ab^{x}, understand a as the intercept or initial value and b as the growth factor.
6. Functions
Core Skill 5: Find and interpret rates of change.Compute the rate of change of linear functions and make qualitative observations about how the rate of change varies for nonlinear functions. Example Task 1 7. Modeling
Core Skill 1: Model numerical situations.Include readily applying the four basic operations in combination to solve multi-step quantitative problems with dimensioned quantities; making estimates to introduce numbers into a situation and get problems started; recognizing proportional or near-proportional relationships and analyzing them using characteristic rates and ratios. Example Task 1 7. Modeling
Core Skill 2: Model physical objects with geometric shapes.Include common objects that can reasonably be idealized as two- and three-dimensional geometric shapes. Identify the ways in which the actual shape varies from the idealized geometric model. Example Task 1 7. Modeling
Core Skill 3: Model situations with equations and inequalities.Include situations well described by a linear inequality in two variables or a system of linear inequalities defining a region in the plane. Example Task 1 7. Modeling
Core Skill 4: Model situations with common functions.Include situations well described by linear, quadratic or exponential functions; and situations that can be well described by inverse variation ( f(x) = k/x). Include identifying a family of functions that models features of a problem, and identifying a particular function of that family and adjusting it to fit by changing parameters. Understand the recursive nature of situations modeled by linear and exponential functions.
Example Task 1 7. Modeling
Core Skill 5: Model situations using probability and statistics.Include using simulations to model probabilistic situations; describing the shape of a distribution of values and summarizing a distribution with measures of center and variability; modeling a bivariate relationship using a trend line or a regression line. Example Task 1 7. Modeling
Core Skill 6: Interpret the results of applying a model and compare models for a particular situation.Include realizing that models seldom fit exactly and so there can be error; identifying simple sources of error and being careful not to over-interpret models. Include recognizing that there can be many models that relate to a situation, that they can capture different aspects of the situation, that they can be simpler or more complex, and that they can have a better or worse fit to the situation and the questions being asked. 8. Shape
Core Skill 1: Use multiple geometric properties to solve problems involving geometric figures.Properties include: measures of interior angles of a triangle sum to 180°; vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; measures of supplementary angles sum to 180°; two lines parallel to a third are parallel to each other; points on a perpendicular bisector of a segment are exactly those equidistant from the segment’s endpoints; and a line tangent to a circle is perpendicular to the radius meeting it. Example Task 1 8. Shape
Core Skill 2: Prove theorems, test conjectures and identify logical errors.Include theorems establishing the properties in Core Skill 1 and other theorems about angles, parallel and perpendicular lines, similarity and congruence of triangles. Example Task 1 8. Shape
Core Skill 3: Construct and interpret representations of geometric objects.Include classical construction techniques and construction techniques supported by modern technologies. Include moving between two- dimensional representations and the three-dimensional objects they represent, such as in schematics, assembly instructions, perspective drawings and multiple views. Example Task 1 8. Shape
Core Skill 4: Solve problems involving measurements.Include measurement (length, angle measure, area, surface area, and volume) of a variety of figures and shapes in two- and three-dimensions. Compute measurements using formulas and by decomposing complex shapes into simpler ones. Example Task 1 8. Shape
Core Skill 5: Solve problems involving similar triangles and scale drawings.Include computing actual lengths, areas and volumes from a scale drawing and reproducing a scale drawing at a different scale. Example Task 1 8. Shape
Core Skill 6: Apply properties of right triangles and right triangle trigonometry to solve problems.Include using the Pythagorean theorem and properties of special right triangles, and applying sine, cosine and tangent to determine lengths and angle measures of right triangles. Use right triangles and their properties to solve real-world problems. Limit angle measures to degrees. Example Task 1 9. Coordinates
Core Skill 1: Translate fluently between lines in the coordinate plane and their equations.Include predicting visual features of lines by inspection of their equations, determining the equation of the line through two given points, and determining the equation of the line with a given slope passing through a given point. Example Task 1 9. Coordinates
Core Skill 2: Identify the correspondence between parameters in common families of equations and the location and appearance of their graphs.Include common families of equations—the graphs of Ax + By = C, y = mx + b and x = a are straight lines; the graphs of y = a(x – h)^{2} + k and y = Ax^{2} + Bx + C are parabolas; and the graph of (x – h)^{2} + (y – k)^{2} = r^{2} is a circle.
Example Task 1Example Task 2 Example Task 3 9. Coordinates
Core Skill 3: Use coordinates to solve geometric problems.Include proving simple theorems algebraically, using coordinates to compute perimeters and areas for triangles and rectangles, finding midpoints of line segments, finding distances between pairs of points and determining when two lines are parallel or perpendicular. Example Task 1 10. Probability
Core Skill 1: Compute theoretical probabilities by systematically counting points in the sample space.Make use of symmetry and equally likely outcomes. Include permutation and combination problems as long as small numbers are involved or technology is used, so that formulas are not required. Example Task 1 10. Probability
Core Skill 2: Interpret probabilities of compound events using concepts of independence and conditional probability.Include reading conditional probabilities from two-way tables. Example Task 1 10. Probability
Core Skill 3: Compute probabilities of compound events.Make use of the additive and multiplicative laws of probability, tree diagrams and frequency or relative frequency tables in real contexts. Do not emphasize fluency with the related formulas. Example Task 1 10. Probability
Core Skill 4: Estimate probabilities empirically.Include using data from simulations carried out with technology to estimate probabilities. Example Task 1 10. Probability
Core Skill 5: Identify and explain common misconceptions regarding probability.Include misconceptions about long-run versus short-run behavior of relative frequencies (the law of large numbers). Include attention to the use and misuse of probability in the media, especially in terms of interpreting charts and tables and in the contextual meaning of terms connected to probability, such as ‘odds’ or ‘risk.’ Example Task 1 10. Probability
Core Skill 6: Adapt probability models to solve real-world problems.Include the use of conditional probability to assess subsets of data (e.g., what does the data say about males and females separately). Include the use of independence as a simplifying assumption (e.g., find the probability that two students both contract the disease this year). Example Task 1 11. Statistics
Core Skill 1: Formulate questions that can be addressed with data.Identify the relevant data, collect and organize it to respond to the question. Include determining whether a question can best be addressed through a sample survey, randomized experiment or observational study. Include unbiased selection for a sample and randomization of assignment to treatment for an experiment. Example Task 1 11. Statistics
Core Skill 2: Use appropriate displays and summary statistics for data.Include univariate, bivariate, categorical and quantitative data. Include the thoughtful selection of displays and measures of center and spread to summarize data. Example Task 1 11. Statistics
Core Skill 3: Interpret data displays and summaries critically; draw conclusions and develop recommendations.Include paying attention to the context of the data, interpolating or extrapolating judiciously, and examining the effects of extreme values of the data on summary statistics of center and spread. Include data sets that follow a normal distribution. Include observing and interpreting linear trends in bivariate quantitative data. Example Task 1 11. Statistics
Core Skill 4: Draw statistical conclusions involving population means or proportions using sample data.Conclusions should be based on simulations or other informal techniques, rather than formulas. Example Task 1 11. Statistics
Core Skill 5: Evaluate reports based on data.Include looking for bias or flaws in the way the data were gathered or presented, as well as unwarranted conclusions, such as claims that confuse correlation with causation. Example Task 1 |