The Mathematics Content Standards Workbook Workbook Final.pdfThe Mathematics Content Standard Workbook is designed to help you effectively and fully study the standards through knowledge,

www.abcte.org 877 669 2228

The

Mathematics ContentStandards Workbook What is the Mathematics Content Standards Workbook? Practice tests and quizzes are just a subset of the standards you are expected to know on test day. The actual test will be a partially different subset from the questions you have already answered, so it is fundamentally important to know ALL the math content standards. You want to know them anyway right? After all you don’t want to be a mediocre teacher.

The Mathematics Content Standard Workbook is designed to help you effectively and fully study the standards through knowledge, comprehension, analysis, application, and synthesis so that on test day you will be prepared for the evaluation – yes, that was Bloom’s Taxonomy.

There is one page for each exam standard, which is designed to help you identify the key concepts, define terms, know how to recognize concepts and make sure you understand how to apply those concepts to a classroom setting. Finally, There is a tally sheet to make sure you know how much practice you are getting.

It is too easy to get discouraged when studying for the mathematics exam if you don’t have a plan. This book gives you the ability to pace yourself in a structured way. There are 166 mathematics standards, and it may seem like a lot, but if you know how many sheets a day you need to finish to complete all the standards, 3-5 sheets a day, this will give you a sense of accomplishment without being overwhelmed.

Remember the end goal here, it is not to study, nor to pass the test. It isn’t even to get a certificate. Your goal is to be a GREAT teacher so that you will make a difference in a student’s life.

Helpful Tips Pace yourself. You should focus on how well you understand the standards and not just

completing the page. This workbook has a dual purpose. 1) is to keep your study focused and organized. This lets you

consolidate your formulas and your questions 2) it allows you to tally how much practice you have had using the Item Tally Sheet in the back.

Use extra paper to expand your notes as needed and print pages off as you complete the different

topics and domains. If you are creating a study binder, print the Workbook pages that match the standard for which

you are currently studying and file them in your binder once you successfully define and apply all key terms.

There are many ways to organize your studies. For one example of a study binder, view this video

(http://www.youtube.com/watch?v=RBru0w8l-Ro&feature=bulletin) of ABCTE participant Kira Brown explaining how she organized her materials.

The point is to be able to apply and use these concepts in a classroom so keep studying until you

get there.

http://www.youtube.com/watch?v=RBru0w8l-Ro&feature=bulletin�

http://www.youtube.com/watch?v=RBru0w8l-Ro&feature=bulletin�

www.abcte.org/exams 877 669 2228

Mathematics Content Standard Workbook Keeping your study focused and organized

Date Started Date Completed

How much practice am I getting?

Unlike scenario tests like the PTK, Math lends itself to mastery through sheer practice. The more you practice the better you understand how the concept is applied. To get a good idea about how much practice you have finished in each area, we have included a Standards summary sheet at the back of this workbook to tally your practice. Each time you identify the standard in a practice item, put a tally mark in the box (eg. IIII II ) next to the corresponding category.

How well do you understand the standard? 1 I am unfamiliar with the concept 3 I can solve different forms of these problems

2 I have learned this concept before 4 I can apply and use this in a classroom

1.1.01: Represent real numbers as points on the line and as decimals (possibly infinite).

Domain 1: Number Sense Topic 1: Number Sense

Terms to Define: Define any concept or term with which you are unfamiliar.

Relevant Formulas or Theorems:

Create a sample problem: Construct a problem and its solution using the standard based on problems you have practiced. Use the back of the sheet as necessary. Being able to create your own problems demonstrates mastery of the concept and application of the standard.

If you cannot solve problems based on the standard or apply and use it in a classroom, you need to go back and study the appropriate resources. If you can solve problems and apply the standard, check this box and continue to the next standard: I can apply and use this in a classroom

It is incumbent upon you to use the recommended resources for Mathematics to study if you cannot combine this concept with others to solve complex problems in a classroom setting








1.1.02: Express rational numbers as fractions, finite or repeating decimals, or percents; convert between these.














1.1.03: Understand the density of rational and irrational numbers in real numbers, i.e., between every two real numbers there are rational and irrational numbers.














1.1.04: Identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.














1.1.05: Compare and order rational numbers in various forms including integers, percents, and positive and negative fractions and decimals.














1.1.06: Know, understand and use the standard algorithms for addition, subtraction, multiplication and division of whole numbers and decimals.














1.1.07: Add, subtract, multiply, and divide rational numbers in problem situations.














1.1.08: Know and apply the Fundamental Theorem of Arithmetic.














1.1.09: Understand number theoretic concepts of primes, factors, and multiples; be able to calculate least common multiple and greatest common factor.














1.1.10: Understand “absolute value” for real numbers, its interpretation as distance from the zero on the number line, and recognize that Mathematics √(a²) = |a| for both positive and negative numbers a.














1.1.11: Use algebraic techniques to solve rate, ratio and proportion problems, such as problems involving percent mixtures, rates of work, unit rates, rates of change, speed, density, pressure, scale drawings, and similar triangles.














1.1.12: Solve consumer problems involving sales tax, tips, interest, discounts, compound interest, markups, commissions, percent increase, and percent decrease using whole numbers, fractions, decimals and percents.














1.1.13: Understand the associative, distributive and commutative properties and use them to simplify calculations.














1.1.14: Approximate the values of irrational numbers as they arise from problem situations (e.g., Mathematics, √2 and their rational multiples).














1.1.15: Estimate squares and square roots, cubes and cube roots, and understand the concept of the nth root.














1.1.16: Give answers to numerical problems to a specified degree of accuracy.














1.1.17: Express numbers in scientific notation, including negative exponents, in appropriate problem situations and perform computations with numbers in scientific notation.














1.1.18: Understand and use the rules of exponents.














1.1.19: Understand and use the inverse relationship between exponents and logarithms.














1.1.20: Deduce and use simple laws of logarithms.














1.1.21: Use the definition of logarithms to translate between logarithms in different bases. Use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.














1.1.22: Use addition, subtraction, multiplication and division to solve problems involving monomials, binomials, polynomials, and algebraic fractions and mixed expressions.














1.1.23: Use the field properties of the rational and real number systems to prove or disprove mathematical statements concerning arithmetic and algebraic combinations of numbers.














1.1.24: Know how real and complex numbers are related both arithmetically and graphically. In particular, plot complex numbers as points in the plane.














1.1.25: Add, subtract, multiply, and divide complex numbers.














2.1.01: Understand the concepts of relation and function.

Domain 2: Algebra and Functions Topic 1: Algebra and Functions













2.1.02: Represent relations and functions through tables, graphs, verbal rules, or symbolic rules.














2.1.03: Know that a function is not defined if its domain is not explicitly given; recognize that functions incompletely defined by formulas or rules alone are generally construed as having as domain all the numbers to which the formula or rule can apply, and be














2.1.04: Determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.














2.1.05: Derive and use the formulas for general term and sum of arithmetic series and both finite and infinite geometric series.














2.1.06: Derive the summation formulas for arithmetic series and for both finite and infinite geometric series.














2.1.07: Use and interpret formulas to answer questions about quantities and their relationships, e.g., the relationship between distance, speed, and time.














2.1.08: Find compositions, inverses and algebraic combinations of given functions.














2.1.09: Solve linear equations such as (3/2)(2x-3) - 4(x-2) = 12, and be able to explain how and why the procedures which succeed in this case fail for the equations: 2(2x-3) - 4(x-2) = 12 and 2(2x-3) -4(x-2) = 2.














2.1.10: Given an equation of a line, determine whether a particular point lies on it.














2.1.11: Graph a linear equation in the plane, e.g., 2x+6y-4 = 0; find the intercepts and slope of that line; and explain what sort of line would occur if one or more of the three coefficients were changed to zero.














2.1.12: Sketch the region in the plane defined by one or more linear inequalities (e.g., the region defined by 2x+6y < 4 and x-y < 8).














2.1.13: Find an equation of a line given sufficient geometric information, such as two points on the line, or one point and the slope.














2.1.14: Understand the concepts of parallel lines and perpendicular lines and how their slopes are related and why.














2.1.15: Solve equations and inequalities involving absolute values.














2.1.16: Solve systems of linear equations (in two or three variables) by substitution, with graphs, and with matrices.














2.1.17: Be proficient in the four operations on polynomials, including “long division” with remainder.














2.1.18: Factor polynomials representing the difference of squares, perfect squares of binomials and the sum and difference of two cubes.














2.1.19: Simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.














2.1.20: Add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions.














2.1.21: Solve quadratic equations by factoring, completing the square, and the quadratic formula.














2.1.22: Apply quadratic equations to problems such as those involving the motion of an object under the force of gravity.














2.1.23: Know the binomial coefficients and use the binomial theorem to expand positive integer powers of binomials.














2.1.24: Graph quadratic functions and determine their maxima, minima, and zeros.














2.1.25: Determine how the graph of a parabola changes as a, b, and c vary in the equation: y = a(x-b) 2 + c.














2.1.26: Given a quadratic equation of the form ax 2 + by 2 + cx + dy + e = 0, use the method of completing the square to put the equation into standard form; recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola; and graph the e














2.1.27: When a conic section is described by a quadratic equation in standard form, explain how its geometry (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the equation.














2.1.28: Know the laws of fractional exponents and deduce them from the laws of integral exponents.














2.1.29: Understand exponential functions, and use these functions in problems involving exponential growth and decay.














2.1.30: Solve quadratic equations in the complex number system.














2.1.31: Know the statement of the Fundamental Theorem of Algebra and deduce that every polynomial of degree n is a product of n complex linear factors.














3.1.01: Identify symmetry in common objects as examples of point, line, or rotational symmetry.

Domain 3: Geometry and Measurement Topic 1: Geometry and Spatial Reasoning













3.1.02: Classify two or three dimensional solid objects and see relationships among them according to their geometric attributes (e.g., shape, number of corners, number and shape of faces, edges, and vertices).














3.1.03: Represent three-dimensional objects using two-dimensional projections or perspective drawings.














3.1.04a: Know, whether as axioms or as deduced from other axioms: sufficient conditions for the congruence of triangles














3.1.04b: Know, whether as axioms or as deduced from other axioms: properties of angles formed by transversals to parallels














3.1.04c: Know, whether as axioms or as deduced from other axioms: that the angles of a triangle sum to 180 degrees.














3.1.05: Know that when three parallels are cut by two transversals, the ratio of the segments formed on one transversal is equal to the ratio of the corresponding segments formed on the other.














3.1.06: Know the relationship of ratio to similarity for rectilinear figures, and be able to write numerical proportions related to given similar figures.














3.1.07a: Know the following terms, and be able to construct illustrations for them by straight-edge and compass constructions: adjacent, vertical, complementary and supplementary angles














3.1.07b: Know the following terms, and be able to construct illustrations for them by straight-edge and compass constructions: angle bisectors














3.1.07c: Know the following terms, and be able to construct illustrations for them by straight-edge and compass constructions: perpendicular bisector of a segment














3.1.07d: Know the following terms, and be able to construct illustrations for them by straight-edge and compass constructions: a line through a given point, perpendicular to a given line














3.1.07e: Know the following terms, and be able to construct illustrations for them by straight-edge and compass constructions: a line through a given point, parallel to a given line














3.1.07f: Know the following terms, and be able to construct illustrations for them by straight-edge and compass constructions: a partition of a given segment into n equal subsegments, for any whole number n.














3.1.08: Understand what skew lines in space are, and make an illustrative drawing.














3.1.09a: Use definitions and theorems to solve problems related to convex polygons: determine when they are similar or congruent














3.1.09b: Use definitions and theorems to solve problems related to convex polygons: compute the sum of the interior angles














3.1.09c: Use definitions and theorems to solve problems related to convex polygons:compute the sum of the exterior angles














3.1.09d: Use definitions and theorems to solve problems related to convex polygons: in the case of quadrilaterals of special types, such as parallelograms and trapezoids, determine the relationships of the diagonals, sides and angles.














3.1.10a: Know the vocabulary related to special triangles and their properties (right, isosceles, equilateral), and know the meaning of altitude, base, median, interior angle, exterior angle, and the fundamental theorems on their relationships.














3.1.10b: Know the vocabulary related to special triangles and their properties (right, isosceles, equilateral), and use the definition of area for a rectangle to deduce by partitions and congruence the area for parallelograms, triangles, and trapezoids.














3.1.11: Know and be able to prove theorems concerning arcs, chords, radii, secants and tangents of a circle, the angles they form and the ratios of certain of their lengths.














3.1.12: Prove the Pythagorean theorem and its converse, making explicit the axioms and preceding theorems used in the proofs.














3.1.13: Obtain the equation of a circle in a coordinate plane, given its center and radius; conversely, given the equation, find the radius and the coordinates of the center.














3.1.14: Be able to prove simple Euclidean theorems using coordinate systems and algebra, e.g., that the diagonals of a parallelogram bisect each other.














3.2.01: Convert between US customary and metric systems of measurement, length, area, volume, weight, time and temperature and convert composite measures within and across systems, e.g., feet per second vs. meters per hour.

Domain 3: Geometry and Measurement Topic 2: Measurement













3.2.02: Use composite measures, such as grams per cubic centimeters and person-hours.














3.2.03: Compute areas of triangles and other polygons.














3.2.04: Know and derive formulas for and solve problems involving the perimeter, circumference, and area of simple two dimensional figures.














3.2.05: Understand the relationships between linear, square, and cubic measures and determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.














3.2.06: Know the formulas, and be able to compute the volumes and surface areas of prisms, pyramids, cylinders and cones.














3.2.07: Compute arc lengths and the area of sectors of a circle.














3.2.08: Interpret scale drawings, maps and other diagrams.














4.1.01a: Solve problems using: the Pythagorean theorem, e.g., find the length of the missing side of a right triangle and the lengths of other line segments

Domain 4: Trigonometry Topic 1: Trigonometry













4.1.01b: Solve problems using: basic trigonometric ratios, e.g., sine and cosine for angles between 0 and 180°














4.1.01c: Solve problems using: angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.














4.1.02: Understand the notion of angle and how to measure it in both degrees and radians, and convert between the two.














4.1.03: Know the law of sines and the law of cosines and apply those laws to solve problems; e.g., determine the area of a triangle, given one angle and the two adjacent sides.














4.1.04: Understand the definitions of sine and cosine as the y and x coordinates of the appropriate point on the unit circle.














4.1.05: Know the identity cos 2 (x) + sin 2 (x) = 1; prove this identity by using the Pythagorean theorem, and conversely prove the Pythagorean theorem as a consequence of this identity.














4.1.06: Prove trigonometric identities and simplify expressions by using the identity














: cos 2 (x) + sin 2 (x) = 1.














4.1.07: Graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. Also, show how the sum a cos mt + b sin mt can be rewritten in the form r cos (mt + e), exhibit














4.1.08: Know the definitions of the tangent and cotangent functions, secant and cosecant functions, and inverse trigonometric functions, and graph them.














4.1.09: Know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.














4.1.10: Compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points.














4.1.11: Understand the addition formulas for sines and cosines and their proofs; use those formulas to prove identities and to simplify expressions.














4.1.12: Know how the addition formulas imply the half-angle and double-angle formulas; use the latter to prove identities and to simplify expressions.














4.1.13: Convert between rectangular and polar coordinates of a point; obtain the polar equation of a curve given in rectangular coordinates and vice versa.














4.1.14: Understand complex numbers (e.g., represent a complex number in polar form; know how to multiply complex numbers in polar form).














4.1.15: Know DeMoivre's theorem; use it to find powers and roots of complex numbers. Relate DeMoivre's theorem to the addition formulas for sine and cosine.














5.1.01: Know that probability measures the likelihood that an event will occur by assigning a number between 0 and 1 to that event. If the probability of an event is p, then 1-p is the probability of the event not occurring.

Domain 5: Probability, Statistics, and Data Analysis Topic 1. Probability, Statistics, and Data Analysis













5.1.02: Find the probability in certain simple situations, e.g., multiple coin tosses, drawing cards from a deck, throwing dice.














5.1.03: Use fundamental counting principles to compute combinations and permutations.














5.1.04: Use combinations and permutations to compute probabilities.














5.1.05: Solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probabilities; understand the simplifications that arise with independent events.














5.1.06: Know the definition of conditional probability and use it to solve probabilities for finite sample spaces.














5.1.07: Know the definitions of mean, median, mode of a data distribution, and compute each of them in particular situations.














5.1.08: Compute the variance and the standard deviation of a distribution of data.














5.1.09: Know the meaning of, and compute, the minimum, the lower quartile, the median, the upper quartile and the maximum of a data set.














5.1.10: Organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.














5.1.11: Understand the concept of normal distribution; estimate probabilities using its graph or tabulated values.














5.1.12: Find the line of best fit to a given distribution of data by using least squares regression.














6.1.01: Solve linear equations in any number of variables.

DOMAIN 6. Linear Algebra Topic 1: Linear Algebra













6.1.02: Interpret linear systems in terms of coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix.














6.1.03: Reduce rectangular matrices to row echelon forms.














6.1.04: Perform addition of matrices and of vectors.














6.1.05: Perform matrix multiplication and multiply vectors by matrices and by scalars.














6.1.06: Determine when linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions. Understand that these are the only possibilities.














6.1.07: Understand the geometric interpretation of vectors and vector addition in the plane and in three-dimensional space.














6.1.08: Interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersec














6.1.09: Parametrize straight lines in the plane and in three-dimensional space, and use parametrization to find intersections of lines and planes.














6.1.10: Understand the notion of the inverse of a square matrix and apply that concept to solve systems of linear equations.














6.1.11: Compute the determinants of 2 x 2 and 3 x 3 matrices and be familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dim














6.1.12: Know that a square matrix is invertible if, and only if, its determinant is nonzero. Compute the inverses of 2 x 2 and 3 x 3 matrices.














6.1.13: Compute the scalar (dot) product of two vectors in n-dimensional space. Know the geometric interpretation of the dot product, and how it follows from the law of cosines.














6.1.14: Compute the cross product of two vectors in three-dimensional space and know the geometric interpretation of the cross product.














7.1.01: Know both the formal definition and the graphical interpretation of limit of values of functions, including one-sided limits, infinite limits, and limits at infinity.

DOMAIN 7: Calculus Topic 1: Calculus













7.1.02: Prove and use theorems evaluating the limits of sums, products, quotients, and compositions of functions














7.1.03: Prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.














7.1.04: Know of both the formal definition and the graphical interpretation of continuity of a function.














7.1.05: Understand and apply the intermediate value theorem and the extreme value theorem.














7.1.06a: Know the formal definition of the derivative of a function at a point and the notion of differentiability including: the derivative of a function as the slope of the tangent line to the graph of the function














7.1.06b: Know the formal definition of the derivative of a function at a point and the notion of differentiability including: the interpretation of the derivative as an instantaneous rate of change in order to solve a variety of problems from physics, chemistry, e














7.1.06c: Know the formal definition of the derivative of a function at a point and the notion of differentiability including: the relation between differentiability and continuity














7.1.06d: Know the formal definition of the derivative of a function at a point and the notion of differentiability including: deriving the formulas for derivatives of sums, products and quotients of differentiable functions, and applying them














7.1.07: Know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.














7.1.08: Find the derivatives of parametrically defined functions, and use “implicit differentiation” in applied problems.














7.1.09: Compute derivatives of higher orders.














7.1.10: Know and apply Rolle's theorem, the mean value theorem, and L'Hospital's rule.














7.1.11: Use differentiation to help sketch by hand the graphs of functions; identify maxima, minima, inflection points, and intervals of increase, decrease, and convexity (up or down).














7.1.12: Know Newton's method for approximating the zeros of a function.














7.1.13: Use differentiation to solve optimization (maximum-minimum problems) in pure and applied contexts.














7.1.14: Use differentiation to solve related rate problems in pure and applied contexts.














7.1.15: Know the definition of the definite integral by using Riemann sums and use this definition to approximate integrals.














7.1.16: Apply the derivative and integral to model problems in physics and economics.














7.1.17: Know the Fundamental Theorem of Calculus and its proof; use this theorem to interpret integrals as antiderivatives and to find integrals in closed form.














7.1.18: Use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.














7.1.19: Compute, by hand, the integrals of functions by using techniques of integration such as substitution, integration by parts, and trigonometric substitution; combine these techniques when appropriate.














7.1.20: Compute, by hand, the integrals of rational functions by combining the techniques above with the algebraic techniques of partial fractions and completing the square.














7.1.21: Know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.














7.1.21: Compute the integrals of trigonometric functions by using the techniques noted above.














7.1.22: Understand and use Simpson's rule and the trapezoidal rule.














7.1.23: Understand improper integrals as limits of definite integrals .














7.1.24: Know the techniques of solution of selected elementary differential equations and their applications, including growth-and-decay problems and oscillating springs or linear circuits.














7.1.25: Understand the definitions of convergence and divergence of sequences and series of real numbers; determine whether a series converges.














7.1.26: Understand and compute the radius (interval) of convergence of power series.














7.1.27: Differentiate and integrate the terms of a power series in order to form new series.














7.1.28: Calculate Taylor polynomials and Taylor series of basic functions, including the remainder term. Use Taylor polynomials for approximations to a desired degree of accuracy.














7.1.29: Know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.







Mathematics Content Standards Tally Sheet How to use: Knowing standards in Mathematics is not enough, you need practice. The question is if you have practiced enough. The Tally Sheet is one way to do that. Everytime you practice and review, take a moment to make a mark on this sheet to see if you have practiced questions in all of the standards. There should be few unmarked areas when you are ready for the exam.

Domain 1: Number Sense

Topic 1: Number Sense

Number Standard Subject Matter

Quizzes Practice Tests Supplementary

Quizzes Math Quiz

Bank Additional Resources

1.1.01

Represent real numbers as points on the line and as decimals (possibly infinite).

1.1.02

Express rational numbers as fractions, finite or repeating decimals, or percents; convert between these.

1.1.03

Understand the density of rational and irrational numbers in real numbers, i.e., between every two real numbers there are rational and irrational numbers.

1.1.04

Identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.

1.1.05

Compare and order rational numbers in various forms including integers, percents, and positive and negative fractions and decimals.

1.1.06

Know, understand and use the standard algorithms for addition, subtraction, multiplication and division of whole numbers and decimals.

1.1.07

Add, subtract, multiply, and divide rational numbers in problem situations.

1.1.08 Know and apply the Fundamental Theorem of Arithmetic.



Quizzes Math Quiz


1.1.09

Understand number theoretic concepts of primes, factors, and multiples; be able to calculate least common multiple and greatest common factor.

1.1.10

Understand “absolute value” for real numbers, its interpretation as distance from the zero on the number line, and recognize that Mathematics √(a²) = |a| for both positive and negative numbers a.

1.1.11

Use algebraic techniques to solve rate, ratio and proportion problems, such as problems involving percent mixtures, rates of work, unit rates, rates of change, speed, density, pressure, scale drawings, and similar triangles.

1.1.12

Solve consumer problems involving sales tax, tips, interest, discounts, compound interest, markups, commissions, percent increase, and percent decrease using whole numbers, fractions, decimals and percents.

1.1.13

Understand the associative, distributive and commutative properties and use them to simplify calculations.

1.1.14

Approximate the values of irrational numbers as they arise from problem situations (e.g., Mathematics, √2 and their rational multiples).



Quizzes Math Quiz


.1.15

Estimate squares and square roots, cubes and cube roots, and understand the concept of the nth root.

1.1.16

Give answers to numerical problems to a specified degree of accuracy.

1.1.17

Express numbers in scientific notation, including negative exponents, in appropriate problem situations and perform computations with numbers in scientific notation.

1.1.18 Understand and use the rules of exponents.

1.1.19

Understand and use the inverse relationship between exponents and logarithms.

1.1.20 Deduce and use simple laws of logarithms.

1.1.21

Use the definition of logarithms to translate between logarithms in different bases. Use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.

1.1.22

Use addition, subtraction, multiplication and division to solve problems involving monomials, binomials, polynomials, and algebraic fractions and mixed expressions.

1.1.23

Use the field properties of the rational and real number systems to prove or disprove mathematical statements concerning arithmetic and algebraic combinations of numbers.



Quizzes Math Quiz


1.1.24

Know how real and complex numbers are related both arithmetically and graphically. In particular, plot complex numbers as points in the plane.

1.1.25 Add, subtract, multiply, & divide complex #s

Domain 2: Algebra and Functions

Topic 1: Algebra and Functions



Quizzes Math Quiz


2.1.01 Understand the concepts of relation and function.

2.1.02

Represent relations and functions through tables, graphs, verbal rules, or symbolic rules.

2.1.03

Know that a function is not defined if its domain is not explicitly given; recognize that functions incompletely defined by formulas or rules alone are generally construed as having as domain all the numbers to which the formula or rule can apply, and be able to determine these in common cases; understand the terms “dependent variable” and “independent variable”.

2.1.04

Determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.

2.1.05

Derive and use the formulas for general term and sum of arithmetic series and both finite and infinite geometric series.

2.1.06

Derive the summation formulas for arithmetic series and for both finite and infinite geometric series.

2.1.07

Use and interpret formulas to answer questions about quantities and their relationships, e.g., the relationship between distance, speed, and time.



Quizzes Math Quiz


2.1.08

Find compositions, inverses and algebraic combinations of given functions.

2.1.09

Solve linear equations such as (3/2)(2x-3) - 4(x-2) = 12, and be able to explain how and why the procedures which succeed in this case fail for the equations: 2(2x-3) - 4(x-2) = 12 and 2(2x-3) -4(x-2) = 2.

2.1.10

Given an equation of a line, determine whether a particular point lies on it.

2.1.11

Graph a linear equation in the plane, e.g., 2x+6y-4 = 0; find the intercepts and slope of that line; and explain what sort of line would occur if one or more of the three coefficients were changed to zero.

2.1.12

Sketch the region in the plane defined by one or more linear inequalities (e.g., the region defined by 2x+6y < 4 and x-y < 8).

2.1.13

Find an equation of a line given sufficient geometric information, such as two points on the line, or one point and the slope.

2.1.14

Understand the concepts of parallel lines and perpendicular lines and how their slopes are related and why.

2.1.15 Solve equations and inequalities involving absolute values.

2.1.16

Solve systems of linear equations (in two or three variables) by substitution, with graphs, and with matrices.



Quizzes Math Quiz


2.1.17

Be proficient in the four operations on polynomials, including “long division” with remainder.

2.1.18

Factor polynomials representing the difference of squares, perfect squares of binomials and the sum and difference of two cubes.

2.1.19

Simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

2.1.20

Add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions.

2.1.21

Solve quadratic equations by factoring, completing the square, and the quadratic formula.

2.1.22

Apply quadratic equations to problems such as those involving the motion of an object under the force of gravity.

2.1.23

Know the binomial coefficients and use the binomial theorem to expand positive integer powers of binomials.

2.1.24

Graph quadratic functions and determine their maxima, minima, and zeros.

2.1.25

Determine how the graph of a parabola changes as a, b, and c vary in the equation: y = a(x-b) 2 + c.



Quizzes Math Quiz


2.1.26

Given a quadratic equation of the form ax 2 + by 2 + cx + dy + e = 0, use the method of completing the square to put the equation into standard form; recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola; and graph the equation.

2.1.27

When a conic section is described by a quadratic equation in standard form, explain how its geometry (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the equation.

2.1.28

Know the laws of fractional exponents and deduce them from the laws of integral exponents.

2.1.29

Understand exponential functions, and use these functions in problems involving exponential growth and decay.

2.1.30

Solve quadratic equations in the complex number system.

2.1.31

Know the statement of the Fundamental Theorem of Algebra and deduce that every polynomial of degree n is a product of n complex linear factors.

Domain 3: Geometry and Measurement

Topic 1: Geometry and Spatial Reasoning



Quizzes Math Quiz


3.1.01

Identify symmetry in common objects as examples of point, line, or rotational symmetry.

3.1.02

Classify two or three dimensional solid objects and see relationships among them according to their geometric attributes (e.g., shape, number of corners, number and shape of faces, edges, and vertices).

3.1.03

Represent three-dimensional objects using two-dimensional projections or perspective drawings.

3.1.04

Know, whether as axioms or as deduced from other axioms, sufficient

conditions for the congruence of triangles

properties of angles formed by transversals to parallels

that the angles of a triangle sum to 180 degrees.

3.1.05

Know that when three parallels are cut by two transversals, the ratio of the segments formed on one transversal is equal to the ratio of the corresponding segments formed on the other.

3.1.06

Know the relationship of ratio to similarity for rectilinear figures, and be able to write numerical proportions related to given similar figures.



Quizzes Math Quiz


3.1.07

Know the following terms, and be able to construct illustrations for them by straight-edge and compass constructions: adjacent,

vertical, complementary and supplementary angles

angle bisectors

perpendicular bisector of a segment

a line through a given point, perpendicular to a given line

a line through a given point, parallel to a given line

a partition of a given segment into n equal subsegments, for any whole number n.

3.1.08

Understand what skew lines in space are, and make an illustrative drawing.

3.1.09

Use definitions and theorems to solve problems related to convex polygons: determine

when they are similar or congruent

compute the sum of the interior angles

compute the sum of the exterior angles

in the case of quadrilaterals of special types, such as parallelograms and trapezoids, determine the relationships of the diagonals, sides and angles.



Quizzes Math Quiz


3.1.10

Know the vocabulary related to special triangles and their properties (right, isosceles, equilateral), and

1. know the meaning of altitude, base, median, interior angle, exterior angle, and the fundamental theorems on their relationships.

2. use the definition of area for a rectangle to deduce by partitions and congruence the area for parallelograms, triangles, and trapezoids.

3.1.11

Know and be able to prove theorems concerning arcs, chords, radii, secants and tangents of a circle, the angles they form and the ratios of certain of their lengths.

3.1.12

Prove the Pythagorean theorem and its converse, making explicit the axioms and preceding theorems used in the proofs.

3.1.13

Obtain the equation of a circle in a coordinate plane, given its center and radius; conversely, given the equation, find the radius and the coordinates of the center.

3.1.14

Be able to prove simple Euclidean theorems using coordinate systems and algebra, e.g., that the diagonals of a parallelogram bisect each other.

Topic 2: Measurement



Quizzes Math Quiz


3.2.01

Convert between US customary and metric systems of measurement, length, area, volume, weight, time and temperature and convert composite measures within and across systems, e.g., feet per second vs. meters per hour.

3.2.02

Use composite measures, such as grams per cubic centimeters and person-hours.

3.2.03 Compute areas of triangles and other polygons.

3.2.04

Know and derive formulas for and solve problems involving the perimeter, circumference, and area of simple two dimensional figures.

3.2.05

Understand the relationships between linear, square, and cubic measures and determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.

3.2.06

Know the formulas, and be able to compute the volumes and surface areas of prisms, pyramids, cylinders and cones.

3.2.07 Compute arc lengths and the area of sectors of a circle.

3.2.08 Interpret scale drawings, maps and other diagrams.

Domain 4: Trigonometry

Topic 1: Trigonometry



Quizzes Math Quiz


4.1.01

Solve problems using: the

Pythagorean theorem, e.g., find the length of the missing side of a right triangle and the lengths of other line segments

basic trigonometric ratios, e.g., sine and cosine for angles between 0 and 180°

angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.

4.1.02

Understand the notion of angle and how to measure it in both degrees and radians, and convert between the two.

4.1.03

Know the law of sines and the law of cosines and apply those laws to solve problems; e.g., determine the area of a triangle, given one angle and the two adjacent sides.

4.1.04

Understand the definitions of sine and cosine as the y and x coordinates of the appropriate point on the unit circle.

4.1.05

Know the identity cos 2 (x) + sin 2 (x) = 1; prove this identity by using the Pythagorean theorem, and conversely prove the Pythagorean theorem as a consequence of this identity.



Quizzes Math Quiz


4.1.06

Prove trigonometric identities and simplify expressions by using the identity cos 2 (x) + sin 2 (x) = 1.

4.1.07

Graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. Also, show how the sum a cos mt + b sin mt can be rewritten in the form r cos (mt + e), exhibiting its amplitude and phase.

4.1.08

Know the definitions of the tangent and cotangent functions, secant and cosecant functions, and inverse trigonometric functions, and graph them.

4.1.09

Know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.

4.1.10

Compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points.

4.1.11

Understand the addition formulas for sines and cosines and their proofs; use those formulas to prove identities and to simplify expressions.

4.1.12

Know how the addition formulas imply the half-angle and double-angle formulas; use the latter to prove identities and to simplify expressions.



Quizzes Math Quiz


4.1.13

Convert between rectangular and polar coordinates of a point; obtain the polar equation of a curve given in rectangular coordinates and vice versa.

4.1.14

Understand complex numbers (e.g., represent a complex number in polar form; know how to multiply complex numbers in polar form).

4.1.15

Know DeMoivre's theorem; use it to find powers and roots of complex numbers. Relate DeMoivre's theorem to the addition formulas for sine and cosine.

Domain 5: Probability, Statistics, and Data Analysis

Topic 1. Probability, Statistics, and Data Analysis



Quizzes Math Quiz


5.1.01

Know that probability measures the likelihood that an event will occur by assigning a number between 0 and 1 to that event. If the probability of an event is p, then 1-p is the probability of the event not occurring.

5.1.02

Find the probability in certain simple situations, e.g., multiple coin tosses, drawing cards from a deck, throwing dice.

5.1.03

Use fundamental counting principles to compute combinations and permutations.

5.1.04 Use combinations and permutations to compute probabilities.

5.1.05

Solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probabilities; understand the simplifications that arise with independent events.

5.1.06

Know the definition of conditional probability and use it to solve probabilities for finite sample spaces.

5.1.07

Know the definitions of mean, median, mode of a data distribution, and compute each of them in particular situations.

5.1.08

Compute the variance and the standard deviation of a distribution of data.



Quizzes Math Quiz


5.1.09

Know the meaning of, and compute, the minimum, the lower quartile, the median, the upper quartile and the maximum of a data set.

5.1.10

Organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

5.1.11

Understand the concept of normal distribution; estimate probabilities using its graph or tabulated values.

5.1.12

Find the line of best fit to a given distribution of data by using least squares regression.

DOMAIN 6. Linear Algebra

Topic 1: Linear Algebra



Quizzes Math Quiz


6.1.01 Solve linear equations in any number of variables.

6.1.02

Interpret linear systems in terms of coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix.

6.1.03 Reduce rectangular matrices to row echelon forms.

6.1.04 Perform addition of matrices and of vectors.

6.1.05

Perform matrix multiplication and multiply vectors by matrices and by scalars.

6.1.06

Determine when linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions. Understand that these are the only possibilities.

6.1.07

Understand the geometric interpretation of vectors and vector addition in the plane and in three-dimensional space.

6.1.08

Interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane.



Quizzes Math Quiz


6.1.09

Parametrize straight lines in the plane and in three-dimensional space, and use parametrization to find intersections of lines and planes.

6.1.10

Understand the notion of the inverse of a square matrix and apply that concept to solve systems of linear equations.

6.1.11

Compute the determinants of 2 x 2 and 3 x 3 matrices and be familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.

6.1.12

Know that a square matrix is invertible if, and only if, its determinant is nonzero. Compute the inverses of 2 x 2 and 3 x 3 matrices.

6.1.13

Compute the scalar (dot) product of two vectors in n-dimensional space. Know the geometric interpretation of the dot product, and how it follows from the law of cosines.

6.1.14

Compute the cross product of two vectors in three-dimensional space and know the geometric interpretation of the cross product.

DOMAIN 7: Calculus

Topic 1: Calculus



Quizzes Math Quiz


7.1.01

Know both the formal definition and the graphical interpretation of limit of values of functions, including one-sided limits, infinite limits, and limits at infinity.

7.1.02

Prove and use theorems evaluating the limits of sums, products, quotients, and compositions of functions

7.1.03

Prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.

7.1.04

Know of both the formal definition and the graphical interpretation of continuity of a function.

7.1.05

Understand and apply the intermediate value theorem and the extreme value theorem.



Quizzes Math Quiz


7.1.06

Know the formal definition of the derivative of a function at a point and the notion of differentiability including: the

derivative of a function as the slope of the tangent line to the graph of the function

the interpretation of the derivative as an instantaneous rate of change in order to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function

the relation between differentiability and continuity

deriving the formulas for derivatives of sums, products and quotients of differentiable functions, and applying them

7.1.07

Know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.

7.1.08

Find the derivatives of parametrically defined functions, and use “implicit differentiation” in applied problems.

7.1.09 Compute derivatives of higher orders.

7.1.10

Know and apply Rolle's theorem, the mean value theorem, and L'Hospital's rule.



Quizzes Math Quiz


7.1.11

Use differentiation to help sketch by hand the graphs of functions; identify maxima, minima, inflection points, and intervals of increase, decrease, and convexity (up or down).

7.1.12 Know Newton's method for approximating the zeros of a function.

7.1.13

Use differentiation to solve optimization (maximum-minimum problems) in pure and applied contexts.

7.1.14

Use differentiation to solve related rate problems in pure and applied contexts.

7.1.15

Know the definition of the definite integral by using Riemann sums and use this definition to approximate integrals.

7.1.16

Apply the derivative and integral to model problems in physics and economics.

7.1.17

Know the Fundamental Theorem of Calculus and its proof; use this theorem to interpret integrals as antiderivatives and to find integrals in closed form.

7.1.18

Use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.



Quizzes Math Quiz


7.1.19

Compute, by hand, the integrals of functions by using techniques of integration such as substitution, integration by parts, and trigonometric substitution; combine these techniques when appropriate.

7.1.20

Compute, by hand, the integrals of rational functions by combining the techniques above with the algebraic techniques of partial fractions and completing the square.

7.1.21

Know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.

7.1.21

Compute the integrals of trigonometric functions by using the techniques noted above.

7.1.22 Understand and use Simpson's rule and the trapezoidal rule.

7.1.23 Understand improper integrals as limits of definite integrals .

7.1.24

Know the techniques of solution of selected elementary differential equations and their applications, including growth-and-decay problems and oscillating springs or linear circuits.

7.1.25

Understand the definitions of convergence and divergence of sequences and series of real numbers; determine whether a series converges.



Quizzes Math Quiz


7.1.26

Understand and compute the radius (interval) of convergence of power series.

7.1.27

Differentiate and integrate the terms of a power series in order to form new series.

7.1.28

Calculate Taylor polynomials and Taylor series of basic functions, including the remainder term. Use Taylor polynomials for approximations to a desired degree of accuracy.

7.1.29

Know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.

The Mathematics Content Standards Workbook Workbook Final.pdfThe Mathematics Content Standard Workbook is designed to help you effectively and fully study the standards through knowledge,

Documents