High School Math Minnesota Standards

Data Analysis & Probability

Geometry & Measurement

Algebra

Understand the concept of function, and identify important features of functions and other relations using symbolic and graphical methods where appropriate.

Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain.

Distinguish between functions and other relations defined symbolically, graphically or in tabular form.

Find the domain of a function defined symbolically, graphically or in a real-world context.

Obtain information and draw conclusions from graphs of functions and other relations.

Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f(x) = ax² + bx + c, in the form f(x) = a(x – h)² + k, or in factored form.

Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function.

Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods.

Make qualitative statements about the rate of change of a function, based on its graph or table of values.

Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations.

Recognize linear, quadratic, exponential and other common functions in real-world and mathematical situations; represent these functions with tables, verbal descriptions, symbols and graphs; solve problems involving these functions, and explain results in the original context.

Represent and solve problems in various contexts using linear and quadratic functions.

Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth.

Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions.

Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively.

Recognize and solve problems that can be modeled using finite geometric sequences and series, such as home mortgage and other compound interest examples. Know how to use spreadsheets and calculators to explore geometric sequences and series in various contexts.

Sketch the graphs of common non-linear functions such as f(x)= vx , f(x)= |x|, f(x)= 1/x, f(x) = x³, and translations of these functions, such as f(x)= <img src="http://purl.org/ASN/resources/images/D100036B/pg36.9.2.2.6.gif" align="middle" alt="pg36.9.2.2.6.gif" /> + 4. Know how to use graphing technology to graph these functions.

Generate equivalent algebraic expressions involving polynomials and radicals; use algebraic properties to evaluate expressions.

Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains.

Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree.

Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares.

Add, subtract, multiply, divide and simplify algebraic fractions.

Check whether a given complex number is a solution of a quadratic equation by substituting it for the variable and evaluating the expression, using arithmetic with complex numbers.

Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving n^th roots.

Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables.

Represent real-world and mathematical situations using equations and inequalities involving linear, quadratic, exponential and n^th root functions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context.

Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.

Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations.

Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.

Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines.

Solve linear programming problems in two variables using graphical methods.

Represent relationships in various contexts using absolute value inequalities in two variables; solve them graphically.

Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods.

Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context.

Calculate measurements of plane and solid geometric figures; know that physical measurements depend on the choice of a unit and that they are approximations.

Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate.

Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures.

Understand that quantities associated with physical measurements must be assigned units; apply such units correctly in expressions, equations and problem solutions that involve measurements; and convert between measurement systems.

Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k² and k³, respectively.

Make reasonable estimates and judgments about the accuracy of values resulting from calculations involving measurements.

Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other results in geometry.

Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments.

Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive.

Assess the validity of a logical argument and give counterexamples to disprove a statement.

Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations.

Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets.

Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry

Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results.

Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results.

Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results.

Apply the Pythagorean Theorem and its converse to solve problems and logically justify results.

Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results.

Know and apply properties of congruent and similar figures to solve problems and logically justify results.

Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results.

Know and apply properties of a circle to solve problems and logically justify results.

Solve real-world and mathematical geometric problems using algebraic methods.

Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle.

Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios.

Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts.

Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments.

Know the equation for the graph of a circle with radius r and center (h, k), (x – h)² + (y – k)² = r², and justify this equation using the Pythagorean Theorem and properties of translations.

Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid.

Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure.

Display and analyze data; use various measures associated with data to draw conclusions, identify trends and describe relationships.

Describe a data set using data displays, including box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics.

Analyze the effects on summary statistics of changes in data sets.

Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions.

Use the mean and standard deviation of a data set to fit it to a normal distribution (bell-shaped curve) and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve.

Explain the uses of data and statistical thinking to draw inferences, make predictions and justify conclusions.

Evaluate reports based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed. Show how graphs and data can be distorted to support different points of view. Know how to use spreadsheet tables and graphs or graphing technology to recognize and analyze distortions in data displays.

Identify and explain misleading uses of data; recognize when arguments based on data confuse correlation and causation.

Design simple experiments and explain the impact of sampling methods, bias and the phrasing of questions asked during data collection.

Calculate probabilities and apply probability concepts to solve real-world and mathematical problems.

Select and apply counting procedures, such as the multiplication and addition principles and tree diagrams, to determine the size of a sample space (the number of possible outcomes) and to calculate probabilities.

Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes.

Understand that the Law of Large Numbers expresses a relationship between the probabilities in a probability model and the experimental probabilities found by performing simulations or experiments involving the model.

Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform probability simulations and to introduce fairness into decision making.

Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems.

Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets.

Understand and use simple probability formulas involving intersections, unions and complements of events.

Apply probability concepts to real-world situations to make informed decisions.

Use the relationship between conditional probabilities and relative frequencies in contingency tables.

Patterns and Relationships: Represent and connect mathematical patterns and relationships using verbal descriptions, generalizations, tables and graphs. Use representations to generate questions, make predictions and solve mathematical problems.

Equivalence and Relational Thinking: Use concepts and properties of equivalence and relational thinking to represent and compare numerical expressions, proportional relationships, algebraic expressions and equations.

Number Relationships: Describe/Interpret and use quantities, relationships between and representations of quantities and number systems. Describe and relate operations. Use strategies and procedures accurately, efficiently and flexibly. Assess the reasonableness of the results.

Patterns and Relationships

Geometry: Analyze characteristics of geometric shapes to make mathematical arguments and justifications about geometric relationships. Use visualization and geometric modeling to compare, solve problems and communicate ideas.

Measurement: Investigate measurement using a variety of tools, units, systems, processes and techniques in various cultures. Explain and reason with attributes, estimations and formulas to communicate measurement(s) and relationships effectively. Justify decisions and consider the reasonableness of the measurement.

Spatial Reasoning

Calculate and make decisions to obtain a desired outcome (maximum or minimum value or maximum or minimum probability) based on expected value and theoretical or experimental probabilities in scenarios such as games of chance or finance. (MP6, MP7) $ ☼

Calculate and make decisions based on expected values using theoretical or experimental probabilities in scenarios, including finance or games of chance. (MP1, MP6) $ μ ☼

Compare the recursion to create the nth row of Pascal’s triangle, the recursion to expand (𝑥𝑥 + 𝑦𝑦)𝑛𝑛, the recursion to generate all sequences of heads and tails on n coin flips and the recursion to generate all subsets of a set with n elements. Establish connections among the various representations. (MP8)

Construct and interpret two-way frequency tables when two categories are associated with each object being classified. Use the two-way table to decide if events are independent and to calculate conditional probabilities. (MP4, MP5) ☼

Calculate experimental probabilities by repeatedly performing simulations or experiments involving a probability model and by using relative frequencies of outcomes. Recognize that as the number of trials increases, the closer the experimental probability approaches the theoretical probability. (MP4)

Apply probability concepts, such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve situations. (MP4) # ☼

Describe events as subsets of a sample space using characteristics of the outcomes. Use Venn diagrams, tables or lists to depict events and compute event sizes. Create representations to find unions, intersections and complements of events. (MP1) #

Select and apply counting procedures such as the multiplication and addition principles, permutations, combinations and tree diagrams to determine the elements and size of a sample space. (MP1, MP7)

Chance and Uncertainty: Apply and explain the concepts of probability to interpret data, generate questions, predict and make informed decisions to solve problems and communicate ideas.

Data Sciences: Identify, formulate and investigate statistical questions by collecting data considering cultural perspectives, analyzing and interpreting data and communicating the results.

Data Analysis

Formulate statistical investigative questions and pose hypotheses. These include questions about variation or the differences between groups, associations between quantitative and categorical variables or pairing together multiple analyses. (MP1, MP3) $ μ ☼ ✜

Create and analyze data displays, including scatter plots, histograms and boxplots using technology. (MP1) ☼

Identify, create and compare statistical models with linear and exponential functions, including linear regression. Assess the reasonableness of model fit using residuals and correlation coefficients. (MP4) # μ

Examine and discuss competing explanations for data trends observed such as confounding variables. Respond to competing arguments or interpretations of the data of different community groups, paying careful attention to what conclusions the data supports. (MP3, MP6) $ ☼

Analyze and interpret data using various measures, such as difference in shapes, center and spread to draw conclusions, identify trends and describe relationships, accounting for possible effects of extreme data points (outliers). (MP1) $

Evaluate reports based on published data by identifying the source of the data, the design of the study and the way the data are analyzed and displayed. (MP3) $ ☼

Identify and explain misleading uses of data along with how to use spreadsheets, tables or graphing technology to recognize and analyze distortions in data displays. Use interactive data visualizations to support and influence different points of view. (MP3) $ #

Explain how choices concerning data collection methods can affect the quality, size, speed, accessibility and cost of the data. (MP3) $ ☼

Analyze issues of bias by considering data collection methods and cultural perspectives. (MP3) ☼

Explain the purposes of and differences among sample surveys, experiments and observational studies. Explain whether randomization in each allows for conclusions of causation and/or generalization of a population. (MP3) ☼

Analyze and explain when arguments based on data confuse correlation and causation. (MP3)

Compute using technology or estimate the correlation coefficient of a linear model. Interpret the linear model in the context of the data. (MP5, MP6) $

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Identify data sets for which such a procedure is not appropriate. Tools can include calculators, spreadsheets, apps or tables. (MP5, MP6) $ ☼

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples or simulated samples of the same size to gauge the variation in estimates or predictions. (MP1, MP4) # ☼

Use statistics appropriate to the shape of the data distribution to compare the center and spread of two or more data sets. (MP4)

Apply the Pythagorean Theorem and distance formula to compute perimeters of polygons and areas of right triangles and rectangles on coordinate planes. (MP1) ☼

Apply properties of 45°-45°-90° and 30°-60°-90° triangles to solve situations and logically justify results. (MP1, MP4) ☼

Use trigonometric ratios and the Pythagorean Theorem and its converse to solve contextual situations, including right triangles or polygons that can be decomposed into triangles. (MP4, MP7) ☼

Develop, justify and use decomposition to determine the formula for surface area and volume of various three-dimensional figures. (MP7, MP8) ✜ # ☼

Use the formulas for surface area and volume of various three-dimensional figures to solve multi- step contextual situations. (MP4, MP5) ✜ ☼

Use units of measure and dimensional analysis to solve multi-step situations. Interpret units consistently in formulas. Interpret the scale and the origin in graphs and data displays. (MP1, MP4) $ ☼ ✜

Compute the unit cost and total cost to cover different areas and volumes in applications, such as painting, carpeting and gardening. Optimize costs for various models, including linear and quadratic. Use technology, including spreadsheets. (MP4, MP5) $ μ ☼

Use similarity to determine the side ratios in right triangles as properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. (MP8)

Apply dilations of scale factor k on length, area and volume and recognize the effect is multiplication by 𝑘𝑘, 𝑘𝑘2, 𝑘𝑘3, respectively. (MP2) μ ☼

Apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve situations and logically justify results. (MP1, MP3)

Apply properties of similar figures to solve situations and logically justify results. (MP3) ✜

Show that two triangles are similar using the AA, SAS and SSS similarity criteria. (MP7)

Make geometric constructions with a variety of tools, including dynamic geometric technology, to examine theorems, make and test conjectures, represent transformations and develop mathematical reasoning skills in multi-step situations. (MP5)

Create and compare rigid and non-rigid transformations, and connect the preimage and image to congruence. (MP2)

Describe the steps used to show various sequences of transformations, applying reflections, rotations, translations and/or dilations of geometric figures that map one figure onto the other. (MP8) # ✜

Apply geometric methods to solve design situations, including designing an object or structure to satisfy physical constraints or optimize resources. (MP4) $ μ

Apply concepts of density based on area and volume in modeling contextual situations, including population density, BTUs per cubic foot and data storage. (MP4)

Apply properties of equilateral, isosceles and scalene triangles to solve situations and logically justify results. (MP1, MP3) ✜ ☼

Apply properties such as interior and exterior angles and parallel and perpendicular sides to define, classify and solve situations involving regular and irregular polygons. (MP1) ✜ ☼

Apply properties of congruent figures to solve situations and logically justify results. (MP1, MP3) ✜ ☼

Accurately interpret and use words and phrases such as “if ... then,” “if and only if,” “all” and “not.” Recognize the logical relationships between an “if ... then” statement and its inverse, converse and contrapositive. (MP6, MP7) ✜ # ☼

Analyze the validity of a logical argument and give counterexamples to disprove a statement. (MP1, MP3) ✜ # ☼

Construct logical arguments using axioms, definitions, theorems and postulates that clearly justify the reasoning. (MP3, MP6) #

Identify, describe and solve using relationships among inscribed angles, circumscribed angles, radii and chords in circles. (MP5) ✜ ☼

Describe and solve using the relationships of angles formed outside the circle for both secants and tangent lines. (MP7) ☼

Add, subtract, multiply and divide numbers in scientific notation. (MP6, MP7) ✜ ☼

Compute the time it takes to pay off a loan given the interest rate, loan amount and monthly payment. Determine the change in cost by the variation in loan amount, fixed or variable interest rates and payment amounts. Use technology, including spreadsheets. (MP1, MP5) ✜ $ ☼

Compute the total payments to pay off a loan given the interest rate, loan amount and monthly payment. Compare the total payments for various monthly payments and various interest rates. Use technology, including spreadsheets. (MP1, MP5) ✜ $ ☼

Compare different types of retirement plans, considering after tax or pretax contributions, employer match contributions, the benefit of starting contributions early and the consequences of use prior to retirement. (MP1) $

Compare the definition of rational exponents and properties of radicals. Explain how the definition of rational exponents follows from extending the properties of integer exponents, allowing for a notation for radicals in terms of rational exponents. (MP6)

Apply knowledge of number systems extending from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers and from real numbers to complex numbers to solve equations. (MP2, MP7)

Use matrices to represent and manipulate data and interpret the results in context. Add, subtract and multiply matrices of appropriate dimensions. Multiply matrices by scalars. (MP4, MP5) #

Estimate and verify the cost of an item, including multiple discounts and taxes. Show an understanding of the order of operations. (MP6, MP8) $

Assess the reasonableness of a solution in its given context, including financial literacy applications. Compare the solution to appropriate graphical or numerical estimates. Interpret a solution in the original context. (MP3) $ ☼

Use the structure of an expression, equation and/or formula to create an equivalent form that is more helpful given the situation. Rearrange formulas to highlight a quantity of interest, using the same reasoning in solving equations. (MP6) $

Use the structure of an expression to write it in multiple ways. (MP7)

Find and evaluate the composition of multiple functions. (MP7)

Add, subtract and multiply polynomials. (MP1, MP7)

Reason abstractly to compare general forms of quadratics, including vertex form, general form, factored form and the graph. Develop procedures to convert from one form to another. (MP2, MP7) μ

Choose and produce an equivalent form of a quadratic function, using symbolic and graphical methods, to identify the vertex, line of symmetry and intercepts of the parabola. (MP8)

Factor common monomial factors from polynomials, quadratic polynomials and the difference of two squares. (MP7, MP8)

Solve quadratic equations by appropriate methods using factoring, completing the square, graphing or the quadratic formula. Find non-real complex roots when they exist. (MP2, MP7)

Compare the equation of a circle, the Pythagorean Theorem and the Distance Formula. Complete the square to find the center and radius of a circle given an equation. (MP2, MP8)

Solve situations involving relationships which are inversely proportional in various contexts, including rates. (MP4) ✜ $ ☼

Apply the properties of rational exponents and radicals to generate equivalent algebraic expressions. (MP2, MP7)

Represent and solve situations in various contexts, including financial literacy, using systems of linear equations, systems of linear inequalities and exponential and quadratic functions. (MP4) $ # ☼

Use the concept of a function as a connection between inputs and outputs to find function values and use function notation. (MP2)

Translate between graphs of quadratic, exponential and other functions (including absolute value, rational and polynomial), tables and symbolic representations. Sketch graphs and use graphing technology to graph functions. (MP5) $

Determine how vertical/ horizontal reflecting, translating and scaling affect the symbolic and graphical forms of a function. Use graphing technology to examine transformations. (MP3)

Express the terms in an arithmetic or geometric sequence recursively and by giving an explicit formula. (MP8) #

Express recursive patterns using recursive formulas. Calculate sequences defined by recursive formulas. (MP8) $

Find the domain and range of functions defined symbolically, graphically or in a context, including piecewise and step functions. Express solutions and recognize that some answers obtained may not be valid, including cases where the function inputs are discrete instead of continuous. (MP4) ✜ $ ☼

Describe the graph of a function using key features such as intercepts, maxima/minima, intervals of increase and decrease and end behavior. Draw conclusions from graphs of functions and other relations. (MP3) $ ☼

Define the compounding of interest n times per year according to a recursive formula. Compare the recursive definition of interest to the recursive definition of a geometric sequence 𝑡𝑡(𝑛𝑛) = 𝑟𝑟(𝑡𝑡(𝑛𝑛 − 1)). Compare the interest formula 𝐴𝐴 = 𝑛𝑛𝑛𝑛 �1 + 𝑟𝑟� to the general form of an exponential 𝑛𝑛 function 𝑦𝑦 = 𝑎𝑎(𝑏𝑏)𝑥𝑥. Explain the purpose of each part of the interest formula. (MP4, MP5) $ # μ

Find the inverse of a given function and justify the results using tables, graphs or algebra. (MP4, MP6)