Virginia State Standards for Mathematics:

K.1. The student, given two sets, each containing 10 or fewer concrete objects, will identify and describe one set as having more, fewer, or the same number of members as the other set, using the concept of one-to-one correspondence.

K.2. The student, given a set containing 15 or fewer concrete objects, will

K.2.a) tell how many are in the set by counting the number of objects orally.

K.2.b) write the numeral to tell how many are in the set.

K.2.c) select the corresponding numeral from a given set of numerals.

K.3. The student, given an ordered set of ten objects and/or pictures, will indicate the ordinal position of each object, first through tenth, and the ordered position of each object.

K.4. The student will

K.5. The student will identify the parts of a set and/or region that represent fractions for halves and fourths.

K.6. The student will model adding and subtracting whole numbers, using up to 10 concrete objects.

K.7. The student will recognize a penny, nickel, dime, and quarter and will determine the value of a collection of pennies and/or nickels whose total value is 10 cents or less.

K.8. The student will identify the instruments used to measure length (ruler), weight (scale), time (clock: digital and analog; calendar: day, month, and season), and temperature (thermometer).

K.9. The student will tell time to the hour, using analog and digital clocks.

K.10. The student will compare two objects or events, using direct comparisons or nonstandard units of measure, according to one or more of the following attributes: length (shorter, longer), height (taller, shorter), weight (heavier, lighter), temperature (hotter, colder). Examples of nonstandard units include foot length, hand span, new pencil, paper clip, and block.

K.11. The student will

K.12. The student will describe the location of one object relative to another (above, below, next to) and identify representations of plane geometric figures (circle, triangle, square, and rectangle) regardless of their positions and orientations in space.

K.13. The student will gather data by counting and tallying.

K.14. The student will display gathered data in object graphs, picture graphs, and tables, and will answer questions related to the data.

K.15. The student will sort and classify objects according to attributes.

K.16. The student will identify, describe, and extend repeating patterns.

K.17. The student will sort and classify objects according to similar attributes (size, shape, and color).

K.18. The student will identify, describe, and extend a repeating relationship (pattern) found in common objects, sounds, and movements.

1.1. The student will

1.2. The student will count forward by ones, twos, fives, and tens to 100 and backward by ones from 30.

1.3. The student will identify the parts of a set and/or region that represent fractions for halves, thirds, and fourths and write the fractions.

1.4. The student, given a familiar problem situation involving magnitude, will

1.5. The student will recall basic addition facts with sums to 18 or less and the corresponding subtraction facts.

1.6. The student will create and solve one-step story and picture problems using basic addition facts with sums to 18 or less and the corresponding subtraction facts.

1.7. The student will

1.7.a) identify the number of pennies equivalent to a nickel, a dime, and a quarter.

1.7.b) determine the value of a collection of pennies, nickels, and dimes whose total value is 100 cents or less.

1.8. The student will tell time to the half-hour, using analog and digital clocks.

1.9. The student will use nonstandard units to measure length, weight/mass, and volume.

1.10. The student will compare, using the concepts of more, less, and equivalent,

1.10.a) the volumes of two given containers.

1.10.b) the weight/mass of two objects, using a balance scale.

1.11. The student will use calendar language appropriately (e.g., names of the months, today, yesterday, next week, last week).

1.12. The student will identify and trace, describe, and sort plane geometric figures (triangle, square, rectangle, and circle) according to number of sides, vertices, and right angles.

1.13. The student will construct, model, and describe objects in the environment as geometric shapes (triangle, rectangle, square, and circle) and explain the reasonableness of each choice.

1.14. The student will investigate, identify, and describe various forms of data collection (e.g., recording daily temperature, lunch count, attendance, favorite ice cream), using tables, picture graphs, and object graphs.

1.15. The student will interpret information displayed in a picture or object graph, using the vocabulary more, less, fewer, greater than, less than, and equal to.

1.16. The student will sort and classify concrete objects according to one or more attributes, including color, size, shape, and thickness.

1.17. The student will recognize, describe, extend, and create a wide variety of growing and repeating patterns.

1.18. The student will demonstrate an understanding of equality through the use of the equal sign.

1.19. The student will interpret information displayed in a picture or object graph, using the vocabulary more, less, fewer, greater than, less than, and equal to.

1.20. The student will sort and classify concrete objects according to one or more attributes, including color, size, shape, and thickness.

1.21. The student will recognize, describe, extend, and create a wide variety of patterns, including rhythmic, color, shape, and numerical. Patterns will include both growing and repeating patterns. Concrete materials and calculators will be used by students.

2.1. The student will

2.1.a) read, write, and identify the place value of each digit in a three-digit numeral, using numeration models.

2.1.b) round two-digit numbers to the nearest ten.

2.2. The student will

2.3. The student will

2.4. The student will

2.5. The student will recall addition facts with sums to 20 or less and the corresponding subtraction facts.

2.5.a) Count forward by twos, fives, and tens to 100, starting at various multiples of 2, 5, or 10, using mental mathematics, paper and pencil, hundred chart, calculators, and/or concrete objects, as appropriate;

2.5.b) Count backward by tens from 100;

2.5.c) Group objects by threes and fours; and

2.5.d) Recognize even and odd numbers, using objects.

2.6. The student, given two whole numbers whose sum is 99 or less, will

2.7. The student, given two whole numbers, each of which is 99 or less, will

2.7.a) estimate the difference.

2.7.b) find the difference, using various methods of calculation.

2.8. The student will create and solve one- and two-step addition and subtraction problems, using data from simple tables, picture graphs, and bar graphs.

2.8.a) Estimate the difference; and

2.8.b) Find the difference, using various methods of calculation (mental computation, concrete materials, and paper and pencil).

2.9. The student will recognize and describe the related facts that represent and describe the inverse relationship between addition and subtraction.

2.10. The student will

2.11. The student will estimate and measure

2.11.a) length to the nearest centimeter and inch.

2.11.b) weight/mass of objects in pounds/ounces and kilograms/grams, using a scale.

2.12. The student will tell and write time to the nearest five minutes, using analog and digital clocks.

2.13. The student will

2.14. The student will read the temperature on a Celsius and/or Fahrenheit thermometer to the nearest 10 degrees.

2.15. The student will

2.16. The student will identify, describe, compare, and contrast plane and solid geometric figures (circle/sphere, square/cube, and rectangle/rectangular prism).

2.17. The student will use data from experiments to construct picture graphs, pictographs, and bar graphs.

2.18. The student will use data from experiments to predict outcomes when the experiment is repeated.

2.18.a) Use calendar language appropriately (e.g., months, today, yesterday, next week, last week);

2.18.b) Determine past and future days of the week;

2.18.c) Identify specific dates on a given calendar.

2.19. The student will analyze data displayed in picture graphs, pictographs, and bar graphs.

2.20. The student will identify, create, and extend a wide variety of patterns.

2.21. The student will solve problems by completing numerical sentences involving the basic facts for addition and subtraction. The student will create story problems, using the numerical sentences.

2.22. T

2.23. The student will read, construct, and interpret a simple picture and bar graph.

2.24. The student will record data from experiments, using spinners and colored tiles/cubes, and use the data to predict which of two events is more likely to occur if the experiment is repeated.

2.25. The student will identify, create, and extend a wide variety of patterns, using numbers, concrete objects, and pictures.

2.26. The student will solve problems by completing a numerical sentence involving the basic facts for addition and subtraction. Examples include: 3 + __ = 7, or 9 - __ = 2. Students will create story problems, using the numerical sentences.

3.1. The student will

3.2. The student will recognize and use the inverse relationships between addition/subtraction and multiplication/division to complete basic fact sentences. The student will use these relationships to solve problems.

3.3. The student will

3.4. The student will estimate solutions to and solve single-step and multistep problems involving the sum or difference of two whole numbers, each 9,999 or less, with or without regrouping.

3.5. The student will recall multiplication facts through the twelves table, and the corresponding division facts.

3.5.a) Divide regions and sets to represent a fraction;

3.5.b) Name and write the fractions represented by a given model (area/region, length/measurement, and set). Fractions (including mixed numbers) will include halves, thirds, fourths, eighths, and tenths.

3.6. The student will represent multiplication and division, using area, set, and number line models, and create and solve problems that involve multiplication of two whole numbers, one factor 99 or less and the second factor 5 or less.

3.7. The student will add and subtract proper fractions having like denominators of 12 or less.

3.8. The student will determine, by counting, the value of a collection of bills and coins whose total value is $5.00 or less, compare the value of the bills and coins, and make change.

3.9. The student will estimate and use U.S. Customary and metric units to measure

3.10. The student will

3.11. The student will

3.12. The student will identify equivalent periods of time, including relationships among days, months, and years, as well as minutes and hours.

3.13. The student will read temperature to the nearest degree from a Celsius thermometer and a Fahrenheit thermometer. Real thermometers and physical models of thermometers will be used.

3.14. The student will identify, describe, compare, and contrast characteristics of plane and solid geometric figures (circle, square, rectangle, triangle, cube, rectangular prism, square pyramid, sphere, cone, and cylinder) by identifying relevant characteristics, including the number of angles, vertices, and edges, and the number and shape of faces, using concrete models.

3.14.a) Length-inches, feet, yards, centimeters, and meters;

3.14.b) Liquid volume-cups, pints, quarts, gallons, and liters;

3.14.c) Weight/mass-ounces, pounds, grams, and kilograms.

3.15. The student will identify and draw representations of points, line segments, rays, angles, and lines.

3.16. The student will identify and describe congruent and noncongruent plane figures.

3.17. The student will

3.18. The student will investigate and describe the concept of probability as chance and list possible results of a given situation.

3.19. The student will recognize and describe a variety of patterns formed using numbers, tables, and pictures, and extend the patterns, using the same or different forms.

3.20. The student will

3.21. The student, given grid paper, will

3.21.a) Collect and organize data on a given topic of his/her choice, using observations, measurements, surveys, or experiments; and

3.21.b) Construct a line plot, a picture graph, or a bar graph to represent the results. Each graph will include an appropriate title and key.

3.22. The student will read and interpret data represented in line plots, bar graphs, and picture graphs and write a sentence analyzing the data.

3.23. The student will investigate and describe the concept of probability as chance and list possible results of a given situation.

3.24. The student will recognize and describe a variety of patterns formed using concrete objects, numbers, tables, and pictures, and extend the pattern, using the same or different forms (concrete objects, numbers, tables, and pictures).

3.25. The student will

3.25.a) Investigate and create patterns involving numbers, operations (addition and multiplication), and relations that model the identity and commutative properties for addition and multiplication;

3.25.b) Demonstrate an understanding of equality by recognizing that the equal sign (=) links equivalent quantities, such as 4 x 3 = 2 x 6.

4.1. The student will

4.1.a) identify orally and in writing the place value for each digit in a whole number expressed through millions.

4.1.b) compare two whole numbers expressed through millions, using symbols (>, <, or =).

4.1.c) round whole numbers expressed through millions to the nearest thousand, ten thousand, and hundred thousand.

4.2. The student will

4.2.a) compare and order fractions and mixed numbers.

4.2.b) represent equivalent fractions.

4.2.c) identify the division statement that represents a fraction.

4.3. The student will

4.4.a) estimate sums, differences, products, and quotients of whole numbers.

4.4. The student will

4.4.b) add, subtract, and multiply whole numbers.

4.4.c) divide whole numbers, finding quotients with and without remainders.

4.5. The student will

4.6. The student will

4.7. The student will

4.8. The student will

4.9. The student will determine elapsed time in hours and minutes within a 12-hour period.

4.9.a) Add and subtract with fractions having like and unlike denominators of 12 or less, using concrete materials, pictorial representations, and paper and pencil;

4.9.b) Add and subtract with decimals through thousandths, using concrete materials, pictorial representations, and paper and pencil;

4.9.c) Solve problems involving addition and subtraction with fractions having like and unlike denominators of 12 or less and with decimals expressed through thousandths, using various computational methods, including calculators, paper and pencil, mental computation, and estimation.

4.10. The student will

4.10.a) identify and describe representations of points, lines, line segments, rays, and angles, including endpoints and vertices.

4.10.b) identify representations of lines that illustrate intersection, parallelism, and perpendicularity.

4.10.c) Estimate the conversion of ounces and grams and pounds and kilograms, using approximate comparisons (1 ounce is about 28 grams, or 1 gram is about the weight of a paper clip; 1 kilogram is a little more than 2 pounds).

4.11. The student will

4.11.a) investigate congruence of plane figures after geometric transformations, such as reflection, translation, and rotation, using mirrors, paper folding, and tracing.

4.11.b) recognize the images of figures resulting from geometric transformations, such as translation, reflection, and rotation.

4.11.c) Estimate the conversion of inches and centimeters, yards and meters, and miles and kilometers, using approximate comparisons (1 inch is about 2.5 centimeters, 1 meter is a little longer than 1 yard, 1 mile is slightly farther than 1.5 kilometers, or 1 kilometer is slightly farther than half a mile).

4.12. The student will

4.12.a) define polygon.

4.12.b) identify polygons with 10 or fewer sides.

4.12.c) Estimate the conversion of quarts and liters, using approximate comparisons (1 quart is a little less than 1 liter, 1 liter is a little more than 1 quart).

4.13. The student will

4.13.a) predict the likelihood of an outcome of a simple event.

4.13.b) represent probability as a number between 0 and 1, inclusive.

4.14. The student will collect, organize, display, and interpret data from a variety of graphs.

4.15. The student will recognize, create, and extend numerical and geometric patterns.

4.15.a) Identify and draw representations of points, lines, line segments, rays, and angles, using a straightedge or ruler;

4.15.b) Describe the path of shortest distance between two points on a flat surface.

4.16. The student will

4.17. The student will

4.17.a) Analyze and compare the properties of two-dimensional (plane) geometric figures (circle, square, rectangle, triangle, parallelogram, and rhombus) and three-dimensional (solid) geometric figures (sphere, cube, and rectangular solid [prism]);

4.17.b) Identify congruent and noncongruent shapes; and

4.17.c) Investigate congruence of plane figures after geometric transformations such as reflection (flip), translation (slide) and rotation (turn), using mirrors, paper folding, and tracing.

4.18. The student will identify the ordered pair for a point and locate the point for an ordered pair in the first quadrant of a coordinate plane.

4.19. The student will

4.19.a) Predict the likelihood of outcomes of a simple event, using the terms certain, likely, unlikely, impossible; and

4.19.b) Determine the probability of a given simple event, using concrete materials.

4.20. The student will collect, organize, and display data in line and bar graphs with scale increments of one or greater than one and use the display to interpret the results, draw conclusions, and make predictions.

4.21. The student will recognize, create, and extend numerical and geometric patterns, using concrete materials, number lines, symbols, tables, and words.

4.22. The student will recognize and demonstrate the meaning of equality, using symbols representing numbers, operations, and relations [e.g., 3 + 5 = 5 + 3 and 15 + (35 + 16) = (15 + 35) + 16].

5.1. The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth.

5.1.a) Read, write, and identify the place values of decimals through thousandths;

5.1.b) Round decimal numbers to the nearest tenth or hundredth;

5.1.c) Compare the values of two decimals through thousandths, using the symbols >, <, or =.

5.2. The student will

5.2.a) recognize and name fractions in their equivalent decimal form and vice versa.

5.2.b) compare and order fractions and decimals in a given set from least to greatest and greatest to least.

5.3. The student will

5.4. The student will create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers.

5.5. The student will

5.6. The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form.

5.7. The student will evaluate whole number numerical expressions, using the order of operations limited to parentheses, addition, subtraction, multiplication, and division.

5.8. The student will

5.9. The student will identify and describe the diameter, radius, chord, and circumference of a circle.

5.10. The student will determine an amount of elapsed time in hours and minutes within a 24-hour period.

5.11. The student will measure right, acute, obtuse, and straight angles.

5.11.a) Length-part of an inch (1/2, 1/4, and 1/8), inches, feet, yards, miles, millimeters, centimeters, meters, and kilometers;

5.11.b) Weight/mass-ounces, pounds, tons, grams, and kilograms;

5.11.c) Liquid volume-cups, pints, quarts, gallons, milliliters, and liters;

5.11.d) Area-square units; and

5.11.e) Temperature-Celsius and Fahrenheit units.

5.12. The student will classify

5.13. The student, using plane figures (square, rectangle, triangle, parallelogram, rhombus, and trapezoid), will

5.14. The student will make predictions and determine the probability of an outcome by constructing a sample space.

5.15. The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots and line graphs.

5.15.a) Recognize, identify, describe, and analyze their properties in order to develop definitions of these figures;

5.15.b) Identify and explore congruent, noncongruent, and similar figures;

5.15.c) Investigate and describe the results of combining and subdividing shapes;

5.15.d) Identify and describe a line of symmetry;

5.15.e) Recognize the images of figures resulting from geometric transformations such as translation (slide), reflection (flip), or rotation (turn).

5.16. The student will

5.17. The student will describe the relationship found in a number pattern and express the relationship.

5.17.a) Solve problems involving the probability of a single event by using tree diagrams or by constructing a sample space representing all possible results;

5.17.b) Predict the probability of outcomes of simple experiments, representing it with fractions or decimals from 0 to 1, and test the prediction;

5.17.c) Create a problem statement involving probability and based on information from a given problem situation. Students will not be required to solve the created problem statement.

5.18. The student will

5.19. The student will investigate and recognize the distributive property of multiplication over addition.

5.20. The student will analyze the structure of numerical and geometric patterns (how they change or grow) and express the relationship, using words, tables, graphs, or a mathematical sentence. Concrete materials and calculators will be used.

5.21. The student will

5.21.a) Investigate and describe the concept of variable;

5.21.b) Use a variable expression to represent a given verbal quantitative expression involving one operation ; and

5.21.c) Write an open sentence to represent a given mathematical relationship, using a variable.

5.22. The student will create a problem situation based on a given open sentence using a single variable.

6.1. The student will describe and compare data, using ratios, and will use appropriate notations, such as a/b, a to b, and a:b.

6.2. The student will

6.3. The student will

6.3.a) identify and represent integers.

6.3.b) order and compare integers.

6.3.c) identify and describe absolute value of integers.

6.4. The student will demonstrate multiple representations of multiplication and division of fractions.

6.5. The student will investigate and describe concepts of positive exponents and perfect squares.

6.6. The student will

6.6.a) multiply and divide fractions and mixed numbers.

6.6.b) estimate solutions and then solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.

6.7. The student will solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimals.

6.8. The student will evaluate whole number numerical expressions, using the order of operations.

6.9. The student will make ballpark comparisons between measurements in the U.S. Customary System of measurement and measurements in the metric system.

6.9.a) Length-part of an inch (1/2, 1/4, and 1/8), inches, feet, yards, miles, millimeters, centimeters, meters, and kilometers;

6.9.b) Weight/mass-ounces, pounds, tons, grams, and kilograms;

6.9.c) Liquid volume-cups, pints, quarts, gallons, milliliters, and liters;

6.9.d) Area-square units.

6.10. The student will

6.11. The student will

6.12. The student will determine congruence of segments, angles, and polygons.

6.12.a) Solve problems involving the circumference and/or area of a circle when given the diameter or radius;

6.12.b) Derive approximations for pi from measurements for circumference and diameter, using concrete materials or computer models.

6.13. The student will describe and identify properties of quadrilaterals.

6.13.a) Estimate angle measures, using 45 degrees, 90 degrees, and 180 degrees as referents, and use the appropriate tools to measure the given angles;

6.13.b) Measure and draw right, acute, and obtuse angles and triangles.

6.14. The student, given a problem situation, will

6.15. The student will

6.16. The student will

6.17. The student will identify and extend geometric and arithmetic sequences.

6.18. The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.

6.18.a) Line, bar, and circle graphs;

6.18.b) Stem-and-leaf plots; and

6.18.c) Box-and-whisker plots.

6.19. The student will investigate and recognize

6.20.a) Make a sample space for selected experiments and represent it in the form of a list, chart, picture, or tree diagram;

6.20. The student will graph inequalities on a number line.

6.20.b) Determine and interpret the probability of an event occurring from a given sample space and represent the probability as a ratio, decimal or percent, as appropriate for the given situation.

6.21. The student will investigate, describe, and extend numerical and geometric patterns, including triangular numbers, patterns formed by powers of 10, and arithmetic sequences.

6.22. The student will investigate and describe concepts of positive exponents, perfect squares, square roots, and, for numbers greater than 10, scientific notation. Calculators will be used to develop exponential patterns.

6.23. The student will

6.23.a) Model and solve algebraic equations, using concrete materials;

6.23.b) Solve one-step linear equations in one variable, involving whole number coefficients and positive rational solutions;

6.23.c) Use the following algebraic terms appropriately: variable, coefficient, term, and equation.

7.1. The student will

7.2. The student will describe and represent arithmetic and geometric sequences, using variable expressions.

7.3. The student will

7.3.a) model addition, subtraction, multiplication, and division of integers.

7.3.b) add, subtract, multiply, and divide integers.

7.3.c) The additive and multiplicative identity properties;

7.3.d) The additive and multiplicative inverse properties; and

7.3.e) The multiplicative property of zero.

7.4. The student will solve single-step and multistep practical problems, using proportional reasoning.

7.4.a) Solve practical problems using rational numbers (whole numbers, fractions, decimals) and percents;

7.4.b) Solve consumer-application problems involving tips, discounts, sales tax, and simple interest.

7.5. The student will

7.6. The student will determine whether plane figures--quadrilaterals and triangles--are similar and write proportions to express the relationships between corresponding sides of similar figures.

7.7. The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid.

7.7.a) Estimate and find the area of polygons by subdividing them into rectangles and right triangles; and

7.7.b) Apply perimeter and area formulas in practical situations.

7.8. The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane.

7.9. The student will investigate and describe the difference between the experimental probability and theoretical probability of an event.

7.10. The student will determine the probability of compound events, using the Fundamental (Basic) Counting Principle.

7.11. The student, given data for a practical situation, will

7.12. The student will represent relationships with tables, graphs, rules, and words.

7.13. The student will

7.14. The student will

7.15. The student will

7.16. The student will apply the following properties of operations with real numbers:

7.17. The student, given a problem situation, will collect, analyze, display, and interpret data, using a variety of graphical methods, including

7.17.a) Frequency distributions;

7.17.b) Line plots;

7.17.c) Histograms;

7.17.d) Stem-and-leaf plots;

7.17.e) Box-and-whisker plots; and

7.17.f) Scattergrams.

7.18. The student will make inferences, conjectures, and predictions based on analysis of a set of data.

7.19. The student will represent, analyze, and generalize a variety of patterns, including arithmetic sequences and geometric sequences, with tables, graphs, rules, and words in order to investigate and describe functional relationships.

7.20. The student will write verbal expressions as algebraic expressions and sentences as equations.

7.21. The student will use the following algebraic terms appropriately: equation, inequality, and expression.

7.22. The student will

7.22.a) Solve one-step linear equations and inequalities in one variable with strategies involving inverse operations and integers, using concrete materials, pictorial representations, and paper and pencil;

7.22.b) Solve practical problems requiring the solution of a one-step linear equation.

8.1. The student will

8.1.a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers.

8.1.b) compare and order decimals, fractions, percents, and numbers written in scientific notation.

8.1.c) Compare and order decimals, fractions, percents, and numbers written in scientific notation.

8.2. The student will describe orally and in writing the relationships between the subsets of the real number system.

8.3. The student will

8.4. The student will apply the order of operations to evaluate algebraic expressions for given replacement values of the variables.

8.5. The student will

8.6. The student will

8.7. The student will

8.8. The student will

8.9. The student will construct a three-dimensional model, given the top or bottom, side, and front views.

8.10. The student will

8.10.a) verify the Pythagorean Theorem.

8.10.b) apply the Pythagorean Theorem.

8.11. The student will solve practical area and perimeter problems involving composite plane figures.

8.12. The student will determine the probability of independent and dependent events with and without replacement.

8.13. The student will

8.14. The student will make connections between any two representations (tables, graphs, words, and rules) of a given relationship.

8.14.a) Describe and represent relations and functions, using tables, graphs, and rules; and

8.14.b) Relate and compare tables, graphs, and rules as different forms of representation for relationships.

8.15. The student will

8.16. The student will graph a linear equation in two variables.

8.17. The student will identify the domain, range, independent variable, or dependent variable in a given situation.

8.18. The student will use the following algebraic terms appropriately: domain, range, independent variable, and dependent variable.

A.1. The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

A.2. The student will perform operations on polynomials, including

A.3. The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form.

A.4. The student will solve multistep linear and quadratic equations in two variables, including

A.5. The student will solve multistep linear inequalities in two variables, including

A.6. The student will graph linear equations and linear inequalities in two variables, including

A.7. The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including

A.8. The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically.

A.9. The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores.

A.10. The student will compare and contrast multiple univariate data sets, using box-and-whisker plots.

A.11. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve real-world problems, using mathematical models. Mathematical models will include linear and quadratic functions.

A.12. The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations.

A.13. The student will express the square root of a whole number in simplest radical form and approximate square roots to the nearest tenth.

A.14. The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.

A.15. The student will, given a rule, find the values of a function for elements in its domain and locate the zeros of the function both algebraically and with a graphing calculator. The value of f(x) will be related to the ordinate on the graph.

A.16. The student will, given a set of data points, write an equation for a line of best fit and use the equation to make predictions.

A.17. The student will compare and contrast multiple one- variable data sets, using statistical techniques that include measures of central tendency, range, and box-and whisker graphs.

A.18. The student will analyze a relation to determine whether a direct variation exists and represent it algebraically and graphically, if possible.

G.1. The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include

G.1.a) identifying the converse, inverse, and contrapositive of a conditional statement.

G.1.b) translating a short verbal argument into symbolic form.

G.1.c) using Venn diagrams to represent set relationships.

G.1.d) using deductive reasoning.

G.2. The student will use the relationships between angles formed by two lines cut by a transversal to

G.2.a) determine whether two lines are parallel.

G.2.b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs.

G.2.c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.

G.3. The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include

G.4. The student will construct and justify the constructions of

G.5. The student, given information concerning the lengths of sides and/or measures of angles in triangles, will

G.5.a) order the sides by length, given the angle measures.

G.5.b) order the angles by degree measure, given the side lengths.

G.6. The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.

G.7. The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs.

G.8. The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry.

G.8.a) Investigate and identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and diagonals;

G.8.b) Prove these properties of quadrilaterals, using algebraic and coordinate methods as well as deductive reasoning; and

G.8.c) Use properties of quadrilaterals to solve practical problems.

G.9. The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems.

G.10. The student will solve real-world problems involving angles of polygons.

G.11. The student will use angles, arcs, chords, tangents, and secants to

G.12. The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle.

G.13. The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.

G.14. The student will use similar geometric objects in two- or three-dimensions to

G.14.a) compare ratios between side lengths, perimeters, areas, and volumes.

G.14.b) determine how changes in one or more dimensions of an object affect area and/or volume of the object.

AII.1. The student, given rational, radical, or polynomial expressions, will

AII.2. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include sigma and a sub n.

AII.3. The student will perform operations on complex numbers, express the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the complex numbers.

AII.3.a) Add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents; and

AII.3.b) Write radical expressions as expressions containing rational exponents and vice versa.

AII.4. The student will solve, algebraically and graphically,

AII.5. The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. Graphing calculators will be used as a tool to visualize graphs and predict the number of solutions.

AII.6. The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions.

AII.7. The student will investigate and analyze functions algebraically and graphically. Key concepts include

AII.8. The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

AII.9. The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions.

AII.10. The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations.

AII.11. The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.

AII.12. The student will compute and distinguish between permutations and combinations and use technology for applications.

AII.13. The student will solve practical problems, using systems of linear inequalities and linear programming, and describe the results both orally and in writing. A graphing calculator will be used to facilitate solutions to linear programming problems.

AII.14. The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the number of solutions.

AII.15. The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions.

AII.16. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include S and an.

AII.17. The student will perform operations on complex numbers and express the results in simplest form. Simplifying results will involve using patterns of the powers of i.

AII.18. The student will identify conic sections (circle, ellipse, parabola, and hyperbola) from his/her equations. Given the equations in (h, k) form, the student will sketch graphs of conic sections, using transformations.

AII.19. The student will collect and analyze data to make predictions and solve practical problems. Graphing calculators will be used to investigate scatterplots and to determine the equation for a curve of best fit. Models will include linear, quadratic, exponential, and logarithmic functions.

AII.20. The student will identify, create, and solve practical problems involving inverse variation and a combination of direct and inverse variations.

T.1. The student, given a point other than the origin on the terminal side of an angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.

T.2 The student, given the value of one trigonometric function, will find the values of the other trigonometric functions. Properties of the unit circle and definitions of circular functions will be applied.

T.3. The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.

T.4. The student will find, with the aid of a calculator, the value of any trigonometric function and inverse trigonometric function.

T.5. The student will verify basic trigonometric identities and make substitutions, using the basic identities.

T.6. The student, given one of the six trigonometric functions in standard form, will

T.6.a) State the domain and the range of the function;

T.6.b) Determine the amplitude, period, phase shift, and vertical shift; and

T.6.c) Sketch the graph of the function by using transformations for at least a one-period interval.

T.7. The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. Restrictions on the domains of the inverse trigonometric functions will be included.

T.8. The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic trigonometric inequalities.

T.9. The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

AII/T.1. The student, given rational, radical, or polynomial expressions, will

AII/T.2. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include sigma and a sub n.

AII/T.3. The student will perform operations on complex numbers, express the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the complex numbers.

AII/T.3.a) Add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents; and

AII/T.3.b) Write radical expressions as expressions containing rational exponents and vice versa.

AII/T.4. The student will solve, algebraically and graphically,

AII/T.5. The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. Graphing calculators will be used as a tool to visualize graphs and predict the number of solutions.

AII/T.6. The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions.

AII/T.7. The student will investigate and analyze functions algebraically and graphically. Key concepts include

AII/T.8. The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.

AII/T.9. The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions.

AII/T.10. The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations.

AII/T.11. The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.

AII/T.12. The student will compute and distinguish between permutations and combinations and use technology for applications.

AII/T.13. The student, given a point other than the origin on the terminal side of an angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.

AII/T.14. The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions and properties of the trigonometric functions.

AII/T.15. The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.

AII/T.16. The student will find, with the aid of a calculator, the value of any trigonometric function and inverse trigonometric function.

AII/T.17. The student will verify basic trigonometric identities and make substitutions, using the basic identities.

AII/T.18. The student, given one of the six trigonometric functions in standard form, will

AII/T.19. The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. Restrictions on the domains of the inverse trigonometric functions will be included.

AII/T.20. The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic trigonometric inequalities.

AII/T.21. The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

AII/T.22. The student, given the value of one trigonometric function, will find the values of the other trigonometric functions. Properties of the unit circle and definitions of circular functions will be applied.

AII/T.23. The student will find without the aid of a calculating utility the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting radians to degrees and vice versa.

AII/T.24. The student will find with the aid of a calculator the value of any trigonometric function and inverse trigonometric function.

AII/T.25. The student will verify basic trigonometric identities and make substitutions, using the basic identities.

AII/T.26. The student, given one of the six trigonometric functions in standard form [e.g., y = A sin (Bx + C) + D, where A, B, C, and D are real numbers], will (The graphing calculator will be used to investigate the effect of changing A, B, C, and D on the graph of a trigonometric functions.)

AII/T.26.a) State the domain and the range of the function;

AII/T.26.b) Determine the amplitude, period, phase shift, and vertical shift; and

AII/T.26.c) Sketch the graph of the function by using transformations for at least a one-period interval.

AII/T.27. The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. Restrictions on the domains of the inverse trigonometric functions will be included.

AII/T.28. The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic trigonometric inequalities. Graphing utilities will be used to solve equations, check for reasonableness of results, and verify algebraic solutions.

AII/T.29. The student will identify, create, and solve practical problems involving triangles.

COM.1. The student will apply programming techniques and skills to solve practical real-world problems in mathematics arising from consumer, business, and other applications in mathematics. Problems will include opportunities for students to analyze data in charts, graphs, and tables and to use their knowledge of equations, formulas, and functions to solve these problems.

COM.2. The student will design, write, test, debug, and document a program. Programming documentation will include preconditions and postconditions of program segments, input/output specifications, the step-by-step plan, the test data, a sample run, and the program listing with appropriately placed comments.

COM.3. The student will write program specifications that define the constraints of a given problem. These specifications will include descriptions of preconditions, postconditions, the desired output, analysis of the available input, and an indication as to whether or not the problem is solvable under the given conditions.

COM.4. The student will design a step-by-step plan (algorithm) to solve a given problem. The plan will be in the form of a program flowchart, pseudo code, hierarchy chart, and/or data-flow diagram.

COM.5. The student will divide a given problem into manageable sections (modules) by task and implement the solution. The modules will include an appropriate user-defined function, subroutines, and procedures. Enrichment topics might include user-defined libraries (units) and object-oriented programming.

COM.6. The student will design and implement the input phase of a program, which will include designing screen layout and getting information into the program by way of user interaction, data statements, and/or file input. The input phase will also include methods of filtering out invalid data (error trapping).

COM.7. The student will design and implement the output phase of a computer program, which will include designing output layout, accessing a variety of output devices, using output statements, and labeling results.

COM.8. The student will design and implement computer graphics, which will include topics appropriate for the available programming environment as well as student background. Students will use graphics as an end in itself, as an enhancement to other output, and as a vehicle for reinforcing programming techniques.

COM.9. The student will define simple variable data types that include integer, real (fixed and scientific notation), character, string, and Boolean.

COM.10. The student will use appropriate variable data types, including integer, real (fixed and scientific notation), character, string, and Boolean. This will also include variables representing structured data types.

COM.11. The student will describe the way the computer stores, accesses, and processes variables, including the following topics: the use of variables versus constants, variables' addresses, pointers, parameter passing, scope of variables, and local versus global variables.

COM.12. The student will translate a mathematical expression into a computer statement, which involves writing assignment statements and using the order of operations.

COM.13. The student will select and implement built-in (library) functions in processing data.

COM.14. The student will implement conditional statements that include ''if/then'' statements, ''if/then/else'' statements, case statements, and Boolean logic.

COM.15. The student will implement loops, including iterative loops. Other topics will include single entry point, single exit point, preconditions, and postconditions.

COM.16. The student will select and implement appropriate data structures, including arrays (one-dimensional and/or multidimensional), files, and records. Implementation will include creating the data structure, putting information into the structure, and retrieving information from the structure.

COM.17. The student will implement pre-existing algorithms, including sort routines, search routines, and simple animation routines.

COM.18. The student will test a program, using an appropriate set of data. The set of test data should be appropriate and complete for the type of program being tested.

COM.19. The student will debug a program, using appropriate techniques (e.g., appropriately placed controlled breaks, the printing of intermediate results, other debugging tools available in the programming environment), and identify the difference between syntax errors and logic errors.

COM.20. The student will design, write, test, debug, and document a complete structured program that requires the synthesis of many of the concepts contained in previous standards.

PS.1. The student will analyze graphical displays of univariate data, including dotplots, stemplots, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers. Appropriate technology will be used to create graphical displays.

PS.2. The student will analyze numerical characteristics of univariate data sets to describe patterns and departures from patterns, using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers.

PS.3. The student will compare distributions of two or more univariate data sets, analyzing center and spread (within group and between group variations), clusters and gaps, shapes, outliers, or other unusual features.

PS.4. The student will analyze scatterplots to identify and describe the relationship between two variables, using shape; strength of relationship; clusters; positive, negative, or no association; outliers; and influential points.

PS.5. The student will find and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plots to assess linearity.

PS.6. The student will make logarithmic and power transformations to achieve linearity.

PS.7. The student, using two-way tables, will analyze categorical data to describe patterns and departure from patterns and to find marginal frequency and relative frequencies, including conditional frequencies.

PS.8. The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting.

PS.9. The student will plan and conduct a survey. The plan will address sampling techniques (e.g., simple random, stratified) and methods to reduce bias.

PS.10. The student will plan and conduct an experiment. The plan will address control, randomization, and measurement of experimental error.

PS.11. The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive.

PS.12. The student will find probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the Law of Large Numbers concept, the addition rule, and the multiplication rule.

PS.13. The student will develop, interpret, and apply the binomial probability distribution for discrete random variables, including computing the mean and standard deviation for the binomial variable.

PS.14. The student will simulate probability distributions, including binomial and geometric.

PS.15. The student will identify random variables as independent or dependent and find the mean and standard deviations for sums and differences of independent random variables.

PS.16. The student will identify properties of a normal distribution and apply the normal distribution to determine probabilities, using a table or graphing calculator.

PS.17. The student, given data from a large sample, will find and interpret point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, and difference between two means (independent and paired).

PS.18. The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large sample tests for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared tests for goodness of fit, homogeneity of proportions, and independence.

PS.19. The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between two sample proportions, and a difference between two sample means.

PS.20. The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between two sample proportions, and a difference between two sample means.

PS.21. The student will identify properties of a t-distribution and apply t-distributions to single-sample and two-sample (independent and matched pairs) t-procedures, using tables or graphing calculators.

DM.1. The student will model problems, using vertex-edge graphs. The concepts of valence, connectedness, paths, planarity, and directed graphs will be investigated. Adjacency matrices and matrix operations will be used to solve problems (e.g., food chains, number of paths).

DM.2. The student will solve problems through investigation and application of circuits, cycles, Euler Paths, Euler Circuits, Hamilton Paths, and Hamilton Circuits. Optimal solutions will be sought using existing algorithms and student-created algorithms.

DM.3. The student will apply graphs to conflict-resolution problems, such as map coloring, scheduling, matching, and optimization. Graph coloring and chromatic number will be used.

DM.4. The student will apply algorithms, such as Kruskal's, Prim's, or Dijkstra's, relating to trees, networks, and paths. Appropriate technology will be used to determine the number of possible solutions and generate solutions when a feasible number exists.

DM.5. The student will use algorithms to schedule tasks in order to determine a minimum project time. The algorithms will include critical path analysis, the list-processing algorithm, and student-created algorithms.

DM.6. The student will solve linear programming problems. Appropriate technology will be used to facilitate the use of matrices, graphing techniques, and the Simplex method of determining solutions.

DM.7. The student will analyze and describe the issue of fair division (e.g., cake cutting, estate division). Algorithms for continuous and discrete cases will be applied.

DM.8. The student will investigate and describe weighted voting and the results of various election methods. These may include approval and preference voting as well as plurality, majority, runoff, sequential run-off, Borda count, and Condorcet winners.

DM.9. The student will identify apportionment inconsistencies that apply to issues such as salary caps in sports and allocation of representatives to Congress. Historical and current methods will be compared.

DM.10. The student will use the recursive process and difference equations with the aid of appropriate technology to generate

DM.10.a) Compound interest;

DM.10.b) Sequences and series;

DM.10.c) Fractals;

DM.10.d) Population growth models; and

DM.10.e) The Fibonacci sequence.

DM.11. The student will describe and apply sorting algorithms and coding algorithms used in sorting, processing, and communicating information. These will include

DM.11.a) Bubble sort, merge sort, and network sort; and

DM.11.b) ISBN, UPC, Zip, and banking codes.

DM.12. The student will select, justify, and apply an appropriate technique to solve a logic problem. Techniques will include Venn diagrams, truth tables, and matrices.

DM.13.a) The Fundamental (Basic) Counting Principle;

DM.13. The student will apply the formulas of combinatorics in the areas of

DM.13.b) Knapsack and bin-packing problems;

DM.13.c) Permutations and combinations; and

DM.13.d) The pigeonhole principle.

MA.1. The student will investigate and identify the characteristics of polynomial and rational functions and use these to sketch the graphs of the functions. This will include determining zeros, upper and lower bounds, y-intercepts, symmetry, asymptotes, intervals for which the function is increasing or decreasing, and maximum or minimum points. Graphing utilities will be used to investigate and verify these characteristics.

MA.2. The student will apply compositions of functions and inverses of functions to real-world situations. Analytical methods and graphing utilities will be used to investigate and verify the domain and range of resulting functions.

MA.3. The student will investigate and describe the continuity of functions, using graphs and algebraic methods.

MA.4. The student will expand binomials having positive integral exponents through the use of the Binomial Theorem, the formula for combinations, and Pascal's Triangle.

MA.5. The student will find the sum (sigma notation included) of finite and infinite convergent series, which will lead to an intuitive approach to a limit.

MA.6. The student will use mathematical induction to prove formulas and mathematical statements.

MA.7. The student will find the limit of an algebraic function, if it exists, as the variable approaches either a finite number or infinity. A graphing utility will be used to verify intuitive reasoning, algebraic methods, and numerical substitution.

MA.8. The student will investigate and identify the characteristics of conic section equations in (h, k) and standard forms. Transformations in the coordinate plane will be used to graph conic sections.

MA.9. The student will investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions and solve equations and real-world problems. This will include the role of e, natural and common logarithms, laws of exponents and logarithms, and the solution of logarithmic and exponential equations.

MA.10. The student will investigate and identify the characteristics of the graphs of polar equations, using graphing utilities. This will include classification of polar equations, the effects of changes in the parameters in polar equations, conversion of complex numbers from rectangular form to polar form and vice versa, and the intersection of the graphs of polar equations.

MA.11. The student will perform operations with vectors in the coordinate plane and solve real-world problems, using vectors. This will include the following topics: operations of addition, subtraction, scalar multiplication, and inner (dot) product; norm of a vector; unit vector; graphing; properties; simple proofs; complex numbers (as vectors); and perpendicular components.

MA.12. The student will use parametric equations to model and solve application problems.

MA.13. The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

APC.1. The student will define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these functions, using a graphing calculator. Properties of functions will include domains, ranges, combinations, odd, even, periodicity, symmetry, asymptotes, zeros, upper and lower bounds, and intervals where the function is increasing or decreasing.

APC.2. The student will define and apply the properties of limits of functions. Limits will be evaluated graphically and algebraically. This will include

APC.2.a) Limits of a constant;

APC.2.b) Limits of a sum, product, and quotient;

APC.2.c) One-sided limits; and

APC.2.d) Limits at infinity, infinite limits, and non-existent limits. AP Calculus BC will include l'Hopital's Rule, which will be used to find the limit of functions whose limits yield the indeterminate forms: 0/0 and 8 / 8.

APC.3. The student will use limits to define continuity and determine where a function is continuous or discontinuous. This will include

APC.3.a) Continuity in terms of limits;

APC.3.b) Continuity at a point and over a closed interval;

APC.3.c) Application of the Intermediate Value Theorem and the Extreme Value Theorem; and

APC.3.d) Geometric understanding and interpretation of continuity and discontinuity.

APC.4. The student will investigate asymptotic and unbounded behavior in functions. This will include

APC.4.a) Describing and understanding asymptotes in terms of graphical behavior and limits involving infinity; and

APC.4.b) Comparing relative magnitudes of functions and their rates of change.

APC.5. The student will investigate derivatives presented in graphic, numerical, and analytic contexts and the relationship between continuity and differentiability. The derivative will be defined as the limit of the difference quotient and interpreted as an instantaneous rate of change.

APC.6. The student will investigate the derivative at a point on a curve. This will include

APC.6.a) Finding the slope of a curve at a point, including points at which the tangent is vertical and points at which there are no tangents;

APC.6.b) Using local linear approximation to find the slope of a tangent line to a curve at the point;

APC.6.c) Defining instantaneous rate of change as the limit of average rate of change; and

APC.6.d) Approximating rate of change from graphs and tables of values.

APC.7. The student will analyze the derivative of a function as a function in itself. This will include

APC.7.a) Comparing corresponding characteristics of the graphs of f, f'', and f'';

APC.7.b) Defining the relationship between the increasing and decreasing behavior of f and the sign of f ';

APC.7.c) Translating verbal descriptions into equations involving derivatives and vice versa;

APC.7.d) Analyzing the geometric consequences of the Mean Value Theorem;

APC.7.e) Defining the relationship between the concavity of f and the sign of f ''; and

APC.7.f) Identifying points of inflection as places where concavity changes and finding points of inflection.

APC.8. The student will apply the derivative to solve problems. This will include

APC.8.a) Analysis of curves and the ideas of concavity and monotonicity;

APC.8.b) Optimization involving global and local extrema;

APC.8.c) Modeling of rates of change and related rates;

APC.8.d) Use of implicit differentiation to find the derivative of an inverse function;

APC.8.e) Interpretation of the derivative as a rate of change in applied contexts, including velocity, speed, and acceleration; and

APC.8.f) Differentiation of non-logarithmic functions, using the technique of logarithmic differentiation. This will include AP Calculus BC will also apply the derivative to solve problems. This will include

APC.8.f.a) Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors;

APC.8.f.b) Numericalal solution of differential equations, using Euler's method;

APC.8.f.c) l'Hopital's Rule to test the convergence of improper integrals and series; and

APC.8.f.d) Geometric interpretation of differential equations via slope fields and the relationship between slope fields and the solution curves for the differential equations.

APC.9. The student will apply formulas to find derivatives. This will include

APC.9.a) Derivatives of algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions;

APC.9.b) Derivations of sums, products, quotients, inverses, and composites (chain rule) of elementary functions;

APC.9.c) Derivatives of implicitly defined functions; and

APC.9.d) Higher order derivatives of algebraic, trigonometric, exponential, and logarithmic, functions. AP Calculus BC will also include finding derivatives of parametric, polar, and vector functions.

APC.10. The student will use Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, graphically, and by a table of values and will interpret the definite integral as the accumulated rate of change of a quantity over an interval interpreted as the change of the quantity over f'(x)dx = f(b) - f(a). Riemann sums will use left, right, and midpoint evaluation points over equal subdivisions.

APC.11. The student will find antiderivatives directly from derivatives of basic functions and by substitution of variables (including change of limits for definite integrals). AP Calculus BC will also include finding antiderivatives by parts and simple partial fractions (nonrepeating linear factors only), and finding improper integrals as limits of definite integrals. AP Calculus BC will also solve logistic differential equations and use them in modeling.

APC.12. The student will identify the properties of the definite integral. This will include additivity and linearity, the definite integral as an area, and the definite integral as a limit of a sum as well as the fundamental theorem: d/dx[integral f(t) x d(t)] = f(x)

APC.13. The student will use the Fundamental Theorem of Calculus to evaluate definite integrals, represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined.

APC.14. The student will find specific anti-derivatives, using initial conditions (including applications to motion along a line). Separable differential equations will be solved and used in modeling (in particular, the equation y'=ky and exponential growth).

APC.15. The student will use integration techniques and appropriate integrals to model physical, biological, and economic situations. The emphasis will be on using the integral of a rate of change to give accumulated change or on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. Specific applications will include

APC.15.a) The area of a region;

APC.15.b) The volume of a solid with known cross-section;

APC.15.c) The average value of a function; and

APC.15.d) The distance traveled by a particle along a line. AP Calculus BC will include finding the area of a region (including a region bounded by polar curves) and finding the length of a curve (including a curve given in parametric form).

APC.16. The student will define a series and test for convergence of a series in terms of the limit of the sequence of partial sums. This will include

APC.16.a) Geometric series with applications;

APC.16.b) Harmonic series;

APC.16.c) Alternating series with error bound;

APC.16.d) Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series; and

APC.16.e) Ratio test for convergence and divergence. For those students who are enrolled in AP Calculus BC.

APC.17. The student will define, restate, and apply Taylor series. This will include

APC.17.a) Taylor polynomial approximations with graphical demonstration of convergence;

APC.17.b) Maclaurin series and the general Taylor series centered at x = a;

APC.17.c) Maclaurin series for the functions ex, sin x, cos x, and 1/(1 - x);

APC.17.d) Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, anti-differentiation, and the formation of new series from known series;

APC.17.e) Functions defined by power series;

APC.17.f) Radius and interval of convergence of power series; and

APC.17.g) Lagrange error bound of a Taylor polynomial. For those students who are enrolled in AP Calculus BC.