Number, Number Sense and Operations Standard

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

A. Demonstrate that vectors and matrices are systems having some of the same properties of the real number system.

1. Determine what properties hold for matrix addition and matrix multiplication; e.g., use examples to show addition is commutative and when multiplication is not commutative.

2. Determine what properties hold for vector addition and multiplication, and for scalar multiplication.

B. Develop an understanding of properties of and representations for addition and multiplication of vectors and matrices.

1. Determine what properties hold for matrix addition and matrix multiplication; e.g., use examples to show addition is commutative and when multiplication is not commutative.

2. Determine what properties hold for vector addition and multiplication, and for scalar multiplication.

5. Model using the coordinate plane, vector addition and scalar multiplication.

Number, Number Sense and Operations Standard

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

C. Apply factorials and exponents, including fractional exponents, to solve practical problems.

3.       Use factorial notation and computations to represent and solve problem situations involving arrangements.  (Grade 10)

8. Use fractional and negative exponents as optional ways of representing and finding solutions for problem situations; e.g., .    

2. Apply combinations as a method to create coefficients for the Binomial Theorem, and make connections to everyday and workplace problem situations.

D. Demonstrate fluency in operations with real numbers, vectors and matrices, using mental computation or paper and pencil calculations for simple cases, and technology for more complicated cases.

4. Use matrices to represent given information in a problem situation.

6. Compute sums, differences and products of matrices using paper and pencil calculations for simple cases, and technology for more complicated cases.

9. Use vector addition and scalar multiplication to solve problems.

Number, Number Sense and Operations Standard

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

E. Represent and compute with complex numbers.

3. Represent complex numbers on the complex plane.

7. Compute sums, differences, products and quotients of complex numbers.

1. Determine what properties (closure, identity, inverse, commutative and associative) hold for operations with complex numbers.

Geometry and Spatial Sense

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

A.     Use trigonometric relationships to verify determine solutions in problem situations.

4.       Use trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines.

2.       Derive and apply the basic trigonometric identities; i.e., angle addition, angle subtraction, and double angle.

B.       Represent transformations within a coordinate system using vectors and matrices.

1.       Use polar coordinates to specify locations on a plane.

2.       Represent translations using vectors.

3.       Describe multiplication of a vector and a scalar graphically and algebraically, and apply to problem situations.

1.       Use matrices to represent translations, reflections, rotations, dilations and their compositions.

Geometry and Spatial Sense

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

Note: This is an extension of the following benchmarks in grades 8-10 for more complex figures.

A.     Formally define geometric figures.

D.     Use coordinate geometry to represent and examine the properties of geometric figures.

E.       Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools, such as straightedge, compass and technology.

5.       Identify, sketch and classify the cross sections of three-dimensional objects.

Note: This is an extension of benchmark H in grades 11-12 in Mathematical Processes.

H.     Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations.

3.       Relate graphical and algebraic representations of lines, simple curves and conic sections.

Geometry and Spatial Sense

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

Note: This is closely related to benchmark H in grades 11-12 in Mathematical Processes.

H.      Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations.

4.       Recognize and compare specific shapes and properties in multiple geometries; e.g., plane, spherical and hyperbolic.

Patterns, Functions and Algebra

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

A.       Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

3.       Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e.g., general shape, number of roots, domain and range, asymptotic behavior.

4.       Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.

5.       Identify families of functions with graphs that have rotation symmetry or reflection symmetry about the y-axis, x-axis or y = x.

6.       Represent the inverse of a function symbolically and graphically as a reflection about y = x.

10.   Describe the characteristics of the graphs of conic sections.

11.   Describe how a change in the value of a constant in an exponential, logarithmic or radical equation affects the graph of the equation.

1.       Analyze the behavior of arithmetic and geometric sequences and series as the number of terms increases.

2.  Translate between the numeric and symbolic form of a sequence or series.

3.       Describe and compare the characteristics of transcendental and periodic functions; e.g., general shape, number of roots, domain and range, asymptotic behavior, extrema, local and global behavior.

4.       Represent the inverse of a transcendental function symbolically.

6.       Make arguments about mathematical properties using mathematical induction.

7.       Make mathematical arguments using the concepts of limit.

9.       Translate freely between polar and Cartesian coordinate systems.

10.   Use the concept of limit to find instantaneous rate of change for a point on a graph as the slope of a tangent at a point.

Patterns, Functions and Algebra

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

B.      Use the quadratic formula to solve quadratic equations that have complex roots.

8.       Solve equations involving radical expressions and complex roots.

C.      Use recursive functions to model and solve problems; e.g., home mortgages, annuities.

1.       Identify and describe problem situations involving an iterative process that can be represented as a recursive function; e.g., compound interest.

2.       Translate a recursive function into a closed form expression or formula for the nth term to solve a problem situation involving an iterative process; e.g., find the value of an annuity after 7 years.

8.       Compare estimates of the area under a curve over a bounded interval by partitioning the region with rectangles; e.g., make successive estimates using progressively smaller rectangles. 

D.      Apply algebraic methods to represent and generalize problem situations involving vectors and matrices.

7.       Model and solve problems with matrices and vectors.

9.       Solve 3 by 3 systems of linear equations by elimination and using technology, and interpret graphically what the solution means (a point, line, plane, or no solution).

5.       Set up and solve systems of equations using matrices and graphs, with and without technology.

Data Analysis & Probability

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

A.     Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators.

4.       Create a scatterplot of bivariate data, identify trends, and find a function to model the data.

5.        Use technology to find the Least Squares Regression Line, the regression coefficient, and the correlation coefficient for bivariate data with a linear trend, and interpret each of these statistics in the context of the problem situation.

7.       Describe the standard normal curve and its general properties, and answer questions dealing with data assumed to be normal.

8.       Analyze and interpret univariate and bivariate data to identify patterns, note trends, draw conclusions, and make predictions.

10.   Understand and use the concept of random variable, and compute and interpret the expected value for a random variable in simple cases.

2.       Transform bivariate data so it can be modeled by a function; e.g., use logarithms to allow nonlinear relationship to be modeled by linear function.

4.       Apply the concept of a random variable to generate and interpret probability distributions, including binomial, normal and uniform.

Data Analysis & Probability

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

B.       Use descriptive statistics to analyze and summarize data, including measures of center, dispersion, correlation and variability.

3.       Describe how a linear transformation of univariate data affects range, mean, mode, and median.

5.       Use technology to find the Least Squares Regression Line, the regression coefficient, and the correlation coefficient for bivariate data with a linear trend, and interpret each of these statistics in the context of the problem situation.

6.       Use technology to compute the standard deviation for a set of data, and interpret standard deviation in relation to the context or problem situation.

8.       Analyze and interpret univariate and bivariate data to identify patterns, note trends, draw conclusions, and make predictions.

3.       Describe the shape and find all summary statistics for a set of univariate data, and describe how a linear transformation affects shape, center and spread.

Data Analysis & Probability

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

C.      Design and perform a statistical experiment, simulation or study; collect and interpret data; and use descriptive statistics to communicate and support predictions and conclusions.

1.        Design a statistical experiment, survey or study for a problem; collect data for the problem; and interpret the data with appropriate graphical displays, descriptive statistics, concepts of variability, causation, correlation and standard deviation.

2.        Describe the role of randomization in a well-designed study, especially as compared to a convenience sample, and the generalization of results from each.

9.       Evaluate validity of results of a study based on characteristics of the study design, including sampling method, summary statistics and data analysis techniques.

1.       Identify and use various sampling methods (voluntary response, convenience sample, random sample, stratified random sample, census) in a study.

5.       Use sampling distributions as the basis for informal inference.

Data Analysis & Probability

Grades 11-12

Ohio Benchmarks

Grades 11-12

Grade-Level Indicators

Grade 11

Grade-Level Indicators

Grade 12

D.     Connect statistical techniques to applications in workplace and consumer situations.

1.       Design a statistical experiment, survey or study for a problem; collect data for the problem; and interpret the data with appropriate graphical displays, descriptive statistics, concepts of variability, causation, correlation and standard deviation.

2.       Describe the role of randomization in a well-designed study, especially as compared to a convenience sample, and the generalization of results from each.

9.       Evaluate validity of results of a study based on characteristics of the study design, including sampling method, summary statistics and data analysis techniques.

11.   Examine statements and decisions involving risk; e.g., insurance rates and medical decisions.

6.       Use theoretical or experimental probability, including simulations, to determine probabilities in real-world problem situations involving uncertainty, such as mutually exclusive events, complementary events and conditional probability.