Manchester Academic Standards for Mathematics 9-12

(updated 5-30-14)

The Manchester Academic Standards for Mathematics in Grades 9-12 are organized into six categories to align with competencies: Numbers and Operations; Algebraic Expressions; Equations and Inequalities; Functions; Data, Statistics, and Probability; and Geometry. Standards with (+) are designated as mathematics that students should learn in order to take advanced mathematics courses such as calculus or advanced statistics.

Sources from among the following: Common Core State Standards (© Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.); Massachusetts Mathematics Curriculum Framework (November 2000); Pennsylvania Common Core Standards for Mathematics (2013); K-12 Mathematics New Hampshire Curriculum Framework (2006); Content and Performance Standards for Alaska Students (2006); Indiana Academic Standards (2011).

Category: Numbers and Operations

References

Competency

Numbers and Operations (NO): Students will demonstrate the ability to use properties of the complex number system and to reason quantitatively when analyzing, representing, and solving problems.

NHDOE Mathematics Competencies

MAS:M:HS:NO:1

Use the definition of rational exponents and properties of exponents to rewrite and evaluate expressions involving radicals and rational exponents.

CCSS:N-RN.1

CCSS:N-RN.2

MAS:M:HS:NO:2

Use properties of rational and irrational numbers to perform arithmetic operations on irrational numbers.

CCSS:N-RN.3

MAS:M:HS:NO:3

Choose appropriate units for scales in graphs, data displays, and formulas.

CCSS:N-Q.1,2

MAS:M:HS:NO:4

Interpret the units used in graphs, data displays, and formulas.

CCSS:N-Q.1

MAS:M:HS:NO:5

Choose an appropriate level of accuracy in measurements.

CCSS:N-Q.3

MAS:M:HS:NO:6

Use the definition and apply the properties of complex numbers to perform arithmetic operations; (+) use conjugates to find quotients of complex numbers.

CCSS:N-CN.1,2,3

(cluster)

MAS:M:HS:NO:7

(+) Convert between polar and rectangular coordinates, and equations.

CCSS:N-CN.4,5

MAS:M:HS:NO:8

(+) Graph equations in the polar coordinate plane.

CCSS:N-CN.4,5

MAS:M:HS:NO:9

(+) Find products, quotients, powers, and roots of complex numbers in polar form.

CCSS:N-CN.4,5

MAS:M:HS:NO:10

(+) Use and apply the algebraic and geometric properties of vectors to perform operations and solve problems.

CCSS:N-VM:1-5

MAS:M:HS:NO:11

(+) Use matrices to represent and organize data.

CCSS:N-VM:6

MAS:M:HS:NO:12

(+) Use and apply the properties of matrices to perform operations and solve problems.

CCSS:N-VM:6-10

(cluster)

Category: Algebraic Expressions

References

Competency

Algebraic Expressions (AE): Students will demonstrate the ability to simplify, evaluate, and solve problems using algebraic expressions.

MSD

Mathematics Competencies

MAS:M:HS:AE:1

Interpret the structure of an expression such as terms, factors, and coefficients.

CCSS:A-SSE.1

MAS:M:HS:AE:2

Use properties of algebra to rewrite and factor expressions in order to solve problems 

CCSS:A-SSE.2,3

CCSS:A-APR.4

CCSS:A-APR.6

MAS:M:HS:AE:3

Perform arithmetic operations on polynomial expressions.

CCSS:A-APR.1

MAS:M:HS:AE:4

(+) rewrite and factor expressions containing complex numbers

CCSS:N-CN.8

MAS:M:HS:AE:5

(+) Perform arithmetic operations on rational expressions.

CCSS:A-APR.7

MAS:M:HS:AE:6

(+) Write and find the coefficients of specified terms in binomial expansions

CCSS:A-APR.5

Category: Equations and Inequalities

References

Competency

Equations and Inequalities (EQ): Students will demonstrate the ability to solve equations and inequalities, algebraically and graphically; and to create and use algebraic models to solve problems.

MSD

Mathematics Competencies

MAS:M:HS:EQ:1

Explain each step in solving an equation citing properties of equality

CCSS:A-REI.1

MAS:M:HS:EQ:2

Solve linear equations and inequalities in one variable

CCSS:A-REI.3

MAS:M:HS:EQ:3

Rearrange formulas and literal equations to solve for a specified variable

CCSS:A-CED.4

MAS:M:HS:EQ:4

Solve quadratic equations, with real or complex solutions, and inequalities in one variable.

CCSS:N-CN:7

CCSS:A-REI.4

MAS:M:HS:EQ:5

Solve equations and inequalities involving absolute value expressions.

CCSS:A-REI.3

CCSS:A-REI.4

MAS:M:HS:EQ:6

Solve systems of equations in two variables, algebraically and graphically; (+) use matrices to solve systems of equations.

CCSS:A-REI.5,6

CCSS:A-REI.8,9

MAS:M:HS:EQ:7

Solve a system consisting of a linear equation and a quadratic equation, algebraically and graphically.

CCSS:A-REI.7

MAS:M:HS:EQ:8

Write and graph an equation in two variables.

CCSS:A-REI.10

MAS:M:HS:EQ:9

Graph a linear inequality in two variables.

CCSS:A-REI.12

MAS:M:HS:EQ:10

Solve rational equations in one variable.

CCSS:A-REI.2

MAS:M:HS:EQ:11

Solve radical equations in one variable.

CCSS:A-REI.2

MAS:M:HS:EQ:12

Model a real-world situation by writing and solving an equation or inequality in one variable

CCSS:A-CED.1

MAS:M:HS:EQ:13

Model a real-world situation by writing and solving an equation or inequality in two or more variables

CCSS:A-CED.2

MAS:M:HS:EQ:14

Model a real-world situation by writing and solving a system of  equations or inequalities in two or more variables

CCSS:A-CED.3

MAS:M:HS:EQ:15

Derive the formulas for the sums of arithmetic and geometric series, and use the formulas to solve problems.

CCSS:A-SSE.4

CCSS:A-CED.4

Category: Functions

References

Competency

Functions and Relations (FR): Students will demonstrate the ability to identify, represent, interpret, analyze, and use functions and relations to solve problems.

MSD

Mathematics Competencies

MAS:M:HS:FR:1

Identify a function using the definition.

CCSS:F-IF.1

MAS:M:HS:FR:2

Use function notation, evaluate functions, and interpret statements that use function notation in terms of a context.

CCSS:F-IF.2

MAS:M:HS:FR:3

Recognize that sequences are functions.

CCSS:F-IF.3

MAS:M:HS:FR:4

Interpret key features of functions modeled in graphs and tables

CCSS:F-IF.4

MAS:M:HS:FR:5

Sketch graphs of functions, displaying key features, given a description of a relationship.

CCSS:F-IF.4

MAS:M:HS:FR:6

Define the domain and the range of a given function.

CCSS:F-IF.5

MAS:M:HS:FR:7

Calculate and interpret the average rate of change of a function, presented algebraically, graphically, or numerically over a specified interval.

CCSS:F-IF.6

MAS:M:HS:FR:8

Graph linear, quadratic, polynomial, (+) rational, square root, cube root, exponential, logarithmic, piecewise-defined, step, and absolute value functions showing key features of each graph to include intercepts, maxima, minima, zeros, asymptotes, and end behavior.

CCSS:F-IF.7

MAS:M:HS:FR:9

Write a function defined by an expression in equivalent forms to show and explain the relationship between zeros, roots, and factors of polynomials.

CCSS:F-IF.8

CCSS:A-APR.2,3

CCSS:N-CN:9

MAS:M:HS:FR:10

Analyze and compare functions algebraically, graphically, and numerically to find key features

CCSS:F-IF.8,9

MAS:M:HS:FR:11

Write a function rule that describes a relationship between two quantities.

CCSS:F-BF.1

MAS:M:HS:FR:12

Define and use arithmetic and geometric sequences, both recursively and explicitly, to model situations and translate between the two forms.

CCSS:F-BF.2

CCSS:F-IF.3

MAS:M:HS:FR:13

Interpret the effects of transformations on functions.

CCSS:F-BF.3

MAS:M:HS:FR:14

Find and graph the inverse of an invertible function

CCSS:F-BF.4

MAS:M:HS:FR:15

(+) Verify by composition that one function is the inverse of another

CCSS:F-BF.4

MAS:M:HS:FR:16

(+) Create an invertible function from a non-invertible function by restricting the domain

CCSS:F-BF.4

MAS:M:HS:FR:17

(+) Use inverse relationships of exponents and logarithms to solve equations and problems

CCSS:F-BF.5

MAS:M:HS:FR:18

Construct and compare linear, quadratic, and exponential models and solve problems while interpreting the parameters in terms of the context

CCSS:F-LE.1,2,3,4,5

(cluster)

Competency

Trigonometric Functions (TF): Students will demonstrate the ability to identify, represent, interpret, analyze, and use trigonometric functions to solve problems.

 MSD

Mathematics Competencies

MAS:M:HS:TF:1

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.    

CCSS:F-TF.1

MAS:M:HS:TF:2

Use the unit circle to evaluate trigonometric functions.

CCSS:F-TF.2

MAS:M:HS:TF:3

Use special triangles to determine the values of sine, cosine, and tangent.

CCSS:F-TF.3

MAS:M:HS:TF:4

Use the unit circle to explain symmetry and periodicity of trigonometric functions.

CCSS:F-TF.4

MAS:M:HS:TF:5

Use trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

CCSS:F-TF.5

MAS:M:HS:TF:6

(+) Find inverses of trigonometric functions by restricting domains when necessary.

CCSS:F-TF.6

MAS:M:HS:TF:7

(+) Use inverse trigonometric functions to solve equations and problems.

CCSS:F-TF.7

MAS:M:HS:TF:8

Prove the Pythagorean identity and use it to solve problems.

CCSS:F-TF.8

MAS:M:HS:TF:9

(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

CCSS:F-TF.9

Category: Data, Statistics, and Probability

References

Competency

Data and Statistics (DS): Students will demonstrate the ability to apply statistical methods or reasoning to summarize, represent, interpret, and justify conclusions using categorical and quantitative data.

MAS:M:HS:DS:1

Represent data with plots on the real number line (dot plots, histograms, and box plots).

CCSS:S-ID.1

MAS:M:HS:DS:2

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

CCSS:S-ID.2

MAS:M:HS:DS:3

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

CCSS:S-ID.3

MAS:M:HS:DS:4

Use the mean and standard deviation of a data set to fit it to a normal distribution 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.

CCSS:S-ID.4

MAS:M:HS:DS:5

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

CCSS:S-ID.6

MAS:M:HS:DS:6

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

CCSS:S-ID.7

MAS:M:HS:DS:7

Compute (using technology) and interpret the correlation coefficient of a linear fit.

CCSS:S-ID.8

MAS:M:HS:DS:8

Distinguish between correlation and causation.

CCSS:S-ID.9

MAS:M:HS:DS:9

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

CCSS:S-IC.1-6

(cluster)

Competency

Probability (PR): Students will demonstrate the ability to apply the rules of probability to determine the likelihood of a given outcome.

MAS:M:HS:PR:1

Use independence and conditional probability to interpret data.

CCSS:S-CP:1-5

(cluster)

MAS:M:HS:PR:2

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

CCSS:S-CP:6,7

MAS:M:HS:PR:3

(+) Apply the general multiplication rule in a uniform probability model and interpret the answer in terms of the model

CCSS:S-CP:8

MAS:M:HS:PR:4

(+) Use combinations and permutations to compute probabilities and solve problems

CCSS:S-CP:9

MAS:M:HS:PR:5

(+) Calculate expected values and use them to solve problems.

CCSS:S-MD:1-4

(cluster)

MAS:M:HS:PR:6

(+) Use probability to evaluate outcomes of decisions.

CCSS:S-MD:5-7

(cluster)

Category: Geometry

References

Competency

Congruence (GC): Students will demonstrate the ability to use reasoning to construct and apply viable arguments about congruence.

NHDOE Mathematics Competencies

MAS:M:HS:GC:1

Know precise definitions of angle, polygons, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

CCSS:G-CO:1

MAS:M:HS:GC:2

Use geometric terms, definitions, diagrams, and properties to identify relationships involving congruence between geometric figures.

CCSS:G-CO:6,7,8

MAS:M:HS:GC:3

Prove theorems about line segments, angles, congruent triangles, and parallelograms.

CCSS:G-CO:9,10,11

MAS:M:HS:GC:4

Apply theorems about congruence to prove relationships in geometric figures and solve problems.

CCSS:G-MG:3

CCSS:G.SRT.5

CCSS:G.CO.7,8

MAS:M:HS:GC:5

Use a variety of tools to construct geometric figures while applying the properties of congruence

CCSS:G-CO:12,13

Competency

Similarity (GS): Students will demonstrate the ability to use reasoning to construct and apply viable arguments about similarity.

NHDOE Mathematics Competencies

MAS:M:HS:GS:1

Use geometric terms, definitions, diagrams, and properties to identify relationships involving similarity between geometric figures.

CCSS:G-SRT:1,2,3

MAS:M:HS:GS:2

Prove theorems about similar triangles.

CCSS:G-SRT:4

MAS:M:HS:GS:3

Apply theorems about similarity to prove relationships in geometric figures and solve problems.

CCSS:G.SRT.5

CCSS:G-MG:3

MAS:M:HS:GS:4

Use the Pythagorean Theorem and apply trigonometric ratios to solve problems involving right triangles.

CCSS:G.SRT:6,7,8

(cluster)

MAS:M:HS:GS:5

(+) Apply the formula  to find the area of a triangle

CCSS:G.SRT.9

MAS:M:HS:GS:6

(+)  Prove and apply the Law of Sines and the Law of Cosines

CCSS:G.SRT.10,11

Competency

Circles (GCL): Students will demonstrate the ability to reason with and apply theorems about circles.

NHDOE Mathematics Competencies

MAS:M:HS:GCL:1

Use geometric terms, definitions, diagrams and properties to identify relationships involving circles.

CCSS:G-C-2

MAS:M:HS:GCL:2

Prove theorems about circles.

CCSS: G-C-3

MAS:M:HS:GCL:2

Apply theorems about circles to solve problems.

CCSS:G-C:1,2

MAS:M:HS:GCL:3

Use a variety of tools to construct geometric figures while applying the properties of circles

CCSS:GC:3,4

Competency

Coordinate Geometry (CG): Students will demonstrate the ability to apply algebraic models to express geometric relationships.

NHDOE Mathematics Competencies

MAS:M:HS:CG:1

Use coordinate geometry to calculate distance, midpoint, and slope

CCSS:GPE:4,6,7

MAS:M:HS:CG:2

Apply coordinate geometry to prove and identify geometric relationships algebraically; and solve problems

CCSS:GPE:4,5

MAS:M:HS:CG:3

Use geometric figures and their properties to represent transformations in the plane

CCSS:G-CO:2,3,4,5

MAS:M:HS:CG:4

Translate between the geometric descriptions and the equations for circles and parabolas

CCSS:G-GPE:1,2

(cluster)

MAS:M:HS:CG:5

(+)Translate between the geometric descriptions and the equations for ellipses and hyperbolas

CCSS:G-GPE:3

(cluster)

Competency

Measurement (GM): Students will demonstrate the ability to explain, apply, and model geometric measurement formulas.

NHDOE Mathematics Competencies

MAS:M:HS:GM:1

Use geometric terms, definitions, diagrams, and properties to identify two or three-dimensional geometric figures.

CCSS:G-GMD:1

MAS:M:HS:GM:2

Calculate perimeter, circumference, and area of two-dimensional geometric figures to solve problems

CCSS:G-GMD:1

CCSS:G-GC-5

CCSS:G-MG-3

MAS:M:HS:GM:3

Calculate surface area and volume of three-dimensional geometric figures to solve problems

CCSS:G-GMD:3

CCSS:G-MG-3

MAS:M:HS:GM:4

Identify the shapes of two-dimensional cross sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects

CCSS:G-GMD:4