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Learning Standards for Grades 9–10

Number Sense and Operations
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
Understand meanings of operations and how they relate to one another
Compute fluently and make reasonable estimates Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
10.N.1 Identify and use the properties of operations on real numbers, including the associative, commutative, and distributive properties; the existence of the identity and inverse elements for addition and multiplication; the existence of nth roots of positive real numbers for any positive integer n; and the inverse relationship between taking the nth root of and the nth power of a positive real number.
10.N.2 Simplify numerical expressions, including those involving positive integer exponents or the absolute value, e.g., 3(24 – 1) = 45, 4|3 – 5| + 6 = 14; apply such simplifications in the solution of problems.
10.N.3 Find the approximate value for solutions to problems involving square roots and cube roots without the use of a calculator, e.g., .
10.N.4 Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers.

Patterns, Relations, and Algebra
Understand patterns, relations, and functions
Represent and analyze mathematical situations and structures using algebraic symbols
Use mathematical models to represent and understand quantitative relationships
Analyze change in various contexts

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
10.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g., Fibonnacci Numbers), linear, quadratic, and exponential functional relationships.
10.P.2 Demonstrate an understanding of the relationship between various representations of a line. Determine a line’s slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the “point-slope” or “slope y-intercept” formulas. Explain the significance of a positive, negative, zero, or undefined slope.
10.P.3 Add, subtract, and multiply polynomials. Divide polynomials by monomials.
10.P.4 Demonstrate facility in symbolic manipulation of polynomial and rational expressions by rearranging and collecting terms; factoring (e.g., a2 – b2 = (a + b)(a – b), x2 + 10x + 21 = (x + 3)(x + 7), 5x4 + 10x3 – 5x2 = 5x2 (x2 + 2x – 1)); identifying and canceling common factors in rational expressions; and applying the properties of positive integer exponents.
10.P.5 Find solutions to quadratic equations (with real roots) by factoring, completing the square, or using the quadratic formula. Demonstrate an understanding of the equivalence of the methods.
10.P.6 Solve equations and inequalities including those involving absolute value of linear expressions (e.g., |x - 2| > 5) and apply to the solution of problems.
10.P.7 Solve everyday problems that can be modeled using linear, reciprocal, quadratic, or exponential functions. Apply appropriate tabular, graphical, or symbolic methods to the solution. Include compound interest, and direct and inverse variation problems. Use technology when appropriate.
10.P.8 Solve everyday problems that can be modeled using systems of linear equations or inequalities. Apply algebraic and graphical methods to the solution. Use technology when appropriate. Include mixture, rate, and work problems.

Geometry
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
Specify locations and describe spatial relationships using coordinate geometry and other representational systems
Apply transformations and use symmetry to analyze mathematical situations
Use visualization, spatial reasoning, and geometric modeling to solve problems

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
10.G.1 Identify figures using properties of sides, angles, and diagonals. Identify the figures’ type(s) of symmetry.
10.G.2 Draw congruent and similar figures using a compass, straightedge, protractor, and other tools such as computer software. Make conjectures about methods of construction. Justify the conjectures by logical arguments.
10.G.3 Recognize and solve problems involving angles formed by transversals of coplanar lines. Identify and determine the measure of central and inscribed angles and their associated minor and major arcs. Recognize and solve problems associated with radii, chords, and arcs within or on the same circle.
10.G.4 Apply congruence and similarity correspondences (e.g., ?ABC ? ?XYZ) and properties of the figures to find missing parts of geometric figures, and provide logical justification.
10.G.5 Solve simple triangle problems using the triangle angle sum property and/or the Pythagorean theorem.
10.G.6 Use the properties of special triangles (e.g., isosceles, equilateral, 30º–60º–90º, 45º–45º–90º) to solve problems.
10.G.7 Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems.
10.G.8 Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the “point-slope” form of the equation.
10.G.9 Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solutions of problems.
10.G.10 Demonstrate the ability to visualize solid objects and recognize their projections and cross sections.
10.G.11 Use vertex-edge graphs to model and solve problems.

Measurement
Understand measurable attributes of objects and the units, systems, and processes of measurement
Apply appropriate techniques, tools, and formulas to determine measurements

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
10.M.1 Calculate perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles.
10.M.2 Given the formula, find the lateral area, surface area, and volume of prisms, pyramids, spheres, cylinders, and cones, e.g., find the volume of a sphere with a specified surface area.
10.M.3 Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing the radius or height of a cylinder affects its surface area or volume.
10.M.4 Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements.

Data Analysis, Statistics, and Probability
Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
Select and use appropriate statistical methods to analyze data
Develop and evaluate inferences and predictions that are based on data
Understand and apply basic concepts of probability

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
10.D.1 Select, create, and interpret an appropriate graphical representation (e.g., scatterplot, table, stem-and-leaf plots, box-and-whisker plots, circle graph, line graph, and line plot) for a set of data and use appropriate statistics (e.g., mean, median, range, and mode) to communicate information about the data. Use these notions to compare different sets of data.
10.D.2 Approximate a line of best fit (trend line) given a set of data (e.g., scatterplot). Use technology when appropriate.
10.D.3 Describe and explain how the relative sizes of a sample and the population affect the validity of predictions from a set of data.

Selected Problems or Classroom Activities for Grades 9–10, Algebra I, and Geometry
Note: The parentheses contain the code number(s) for the corresponding standard(s) in the single-subject courses.

Refers to standards 10.N.2, 10.N.3, and 10.N.4 (AI.N.2, AI.N.3, and AI.N.4)†
As high school students’ understanding of numbers grows, they should learn to consider operations in general ways, rather than in only particular computations. The questions in the figure below call for reasoning about the properties of the numbers involved rather than for following procedures to arrive at exact answers. Such reasoning is important in judging the reasonableness of results. Although the questions can be approached by substituting approximate values for the numbers represented by a through h, teachers should encourage students to arrive at and justify their conclusions by thinking about properties of numbers. For example, to determine the point whose coordinate is closest to ab, a teacher might suggest considering the sign of ab and whether the magnitude of ab is greater or less than that of b. Likewise, students should be able to explain why, if e is positioned as given in the figure, the magnitude of ?e is greater than that of e. Listening to students explain their reasoning gives teachers insights into the sophistication of their arguments as well as their conceptual understanding.

Refers to standard 10.N.3†
Locating square roots on the number line

?27 is a little more than 5 because 52 = 25 and 62 = 36
?99 is a little less than 10 because 92 = 81 and 102 = 100.

Refers to standards 10.P.1 and 10.P.7 (AI.P.1 and AI.P.11)
Research the changes in the number of cellular phones and personal computers in the United States between 1980 and 2000. First estimate, then use graphing calculators to decide whether the linear, quadratic, or exponential model is appropriate in each case. Compare growth rates and predict future changes in the use of each item. [The discussion may lead to topics in history and social studies related to growth and use of technology, including mathematical models to represent the changes.]

Refers to standard 10.P.1 (AI.P.1)†
These two graphs represent different relationships in a cellular telephone company’s pricing scheme.

Refers to standards 10.P.1 and 10.P.7 (AI.P.1 and AI.P.11)†
Consider rectangles with a fixed area of 36 square units. The width (W) of the rectangles varies in relation to the length (L) according to the formula W = 36/L. Make a table showing the widths for all the possible whole-number lengths for these rectangles up to L = 36.

Solution:
Length 1 2 3 4 5 6 7 8 9 10 11 … 36
Width 36 18 12 9 7.2 6 5.14 4.5 4 3.6 3.27 … 1

Look at the table and examine the pattern of the difference between consecutive entries for the length and the width. As the length increases by 1, the width decreases, but not at a constant rate. What do you expect the graph of the relationship between L and W to look like? Will it be a straight line? Why or why not?

Refers to standard 10.G.3 (G.G.6)
Your shot put circle was washed out in a storm. There is only a portion left. You can redraw the circle if you know its center. Explain how you could use a geometric construction and the properties of circles to find the center of the original circle.

Refers to standard 10.G.11 (G.G.17)†
A vertex-edge graph depicting the lengths of roads between towns

Refers to standards 10.G.4, 10.G.5, and 10.M.1 (G.G.5, G.G.7, and G.M.1)†
A geometric problem requiring deduction and proof

Refers to standards 10.D.1 and 10.D.2 (AI.D.1 and AI.D.2)
Use an almanac to find the winning times for the women’s 400-meter freestyle swim for the Olympics from 1924-1984.
1. On graph paper, using 1920 as the base year, plot (year, time).
2. Construct a best-fit line.
3. What is the slope and what does it mean?
4. Write the equation of the line. Use the line to predict what the times might has been if the Olympics had been held in 1940 and 1944.
5. Is it reasonable to use this line to predict the winning time for the 1988 Summer Games. Why or why not?
6. Look up the winning time for the 400-meter freestyle swim in the 1988 Summer Games and compare it to the time predicted by the best-fit line.

Exploratory Concepts and Skills for Grades 9–10 and Single-Subject Courses

Number Sense and Operations
? Analyze relationships among the various subsets of the real numbers (whole numbers, integers, rationals, and irrationals).
? Explore higher powers and roots using technology.
? Explore the system of complex numbers and find complex roots of quadratic equations.

Patterns, Relations, and Algebra
? Explore matrices and their operations. Use matrices to solve systems of linear equations.
? Investigate recursive function notation.

Geometry
? Apply properties of chords, tangents, and secants to solve problems.
? Use deduction to establish the validity of geometric conjectures and to prove theorems in Euclidean geometry.

Measurement
? Explore the scientific use of different systems of measurement, e.g., centimeter-gram-second (CGS), Scientific International (SI).

Data Analysis, Statistics, and Probability
? Explore designs of surveys, polls, and experiments to assess the validity of their results and to identify potential sources of bias; identify the types of conclusions that can be drawn.
? Describe the differences between the theoretical probability of simple events and the experimental outcome from simulations.

Learning Standards for Grades 11–12

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
12.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns such as Pascal’s Triangle.
12.P.2 Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the general term and sum recursively and explicitly.
12.P.3 Demonstrate an understanding of the binomial theorem and use it in the solution of problems.
12.P.4 Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions.
12.P.5 Perform operations on functions, including composition. Find inverses of functions.
12.P.6 Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, exponential, or trigonometric.
12.P.7 Find solutions to quadratic equations (with real coefficients and real or complex roots) and apply to the solutions of problems.
12.P.8 Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic, and trigonometric functions; expressions involving absolute values; trigonometric relations; and simple rational expressions.
12.P.9 Use matrices to solve systems of linear equations. Apply to the solution of everyday problems.
12.P.10 Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions. Also use technology where appropriate. Describe the relationships among the methods.
12.P.11 Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G(m1m2)/d2) variation, and periodic processes.
12.P.12 Relate the slope of a tangent line at a specific point on a curve to the instantaneous rate of change. Identify maximum and minimum values of functions in simple situations. Apply these concepts to the solution of problems.
12.P.13 Describe the translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af (b(x + c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, logarithmic, and trigonometric functions.

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
12.G.1 Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.
12.G.2 Derive and apply basic trigonometric identities (e.g., sin2? + cos2? = 1, tan2? + 1 = sec2?) and the laws of sines and cosines.
12.G.3 Use the notion of vectors to solve problems. Describe addition of vectors and multiplication of a vector by a scalar, both symbolically and geometrically. Use vector methods to obtain geometric results.
12.G.4 Relate geometric and algebraic representations of lines, simple curves, and conic sections.
12.G.5 Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.

Measurement
Understand measurable attributes of objects and the units, systems, and processes of measurement
Apply appropriate techniques, tools, and formulas to determine measurements

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
12.M.1 Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular, problems involving angular velocity and acceleration.
12.M.2 Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense.

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
12.D.1 Design surveys and apply random sampling techniques to avoid bias in the data collection.
12.D.2 Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data.
12.D.3 Apply regression results and curve fitting to make predictions from data.
12.D.4 Apply uniform, normal, and binomial distributions to the solutions of problems.
12.D.5 Describe a set of frequency distribution data by spread (i.e., variance and standard deviation), skewness, symmetry, number of modes, or other characteristics. Use these concepts in everyday applications.
12.D.6 Use combinatorics (e.g., “fundamental counting principle,” permutations, and combinations) to solve problems, in particular, to compute probabilities of compound events. Use technology as appropriate.
12.D.7 Compare the results of simulations (e.g., random number tables, random functions, and area models) with predicted probabilities.

Selected Problems or Classroom Activities for
Grades 11–12, Geometry, Algebra II, and Precalculus
Note: The parentheses contain the code number(s) for the corresponding standard(s) in the single-subject courses.

Refers to standard 12.P.1 (AII.P.1)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

1. Construct the first 10 rows.
2. Identify different families or sets of numbers in the diagonals.
3. Relate the numbers in the triangle to the row numbers.
4. Examine sums of rows. Relate row sums to the row numbers.
5. For each row, form two sums by adding every other number. Compare sums within and between rows. Describe the patterns that emerge and why they occur.
6. Describe how the triangle is developed recursively.

Refers to standard 12.P.3 (AII.P.3) (TIMSS)
Problem: Brighto soap powder is packed in cube-shaped cartons that measure 10 cm on each side. The company decides to increase the length of each side by 10%. How much does the volume increase?

Solution: (10 + 1)3 – 103 = (103 x 10 + 3 x 102 x 11 + 3 x 101 x 12 + 100 x 13) – 103 = 331, therefore the volume increases by 331cm3.

Refers to standards 12.P.1, 12.P.11, and 12.P.12 (AII.P.1, AII.P.11, and PC.P.9)†

1. State the relationship between the position of car A and that of car B at t = 1 hr. Explain.
2. State the relationship between the velocity of car A and that of car B at t = 1 hr. Explain.
3. State the relationship between the acceleration of car A and that of car B at t = 1 hr. Explain.
4. How are the positions of the two cars related during the time interval between t = 0.75 hr. and t = 1 hr.? (That is, is one car pulling away from the other?) Explain.

Refers to standards 12.P.8, 12.P.11, and 12.P.12 (AII.P.8, AII.P.11, and AII.P.12)
A stone is thrown straight up into the air with initial velocity v0 = 10 feet per second. If one neglects the effects of air resistance, after t seconds the height of the stone is (until the stone hits the ground), where g ? 32 feet per second squared (the gravitational acceleration at the Earth’s surface). What is the greatest height that the stone reaches, and when does it reach that height?

Refers to standards 12.P.8, 12.P.11, and 12.G.2 (AII.P.8, AII.P.11, and AII.G.2)
A stabilizing wire (guy wire) runs from the top of a 60 foot tower to a point 15 feet down the hill (measured on the slant) from the base of the tower. If the hill is inclined 11 degrees from the horizontal, how long does the wire need to be?

Refers to standards 12.P.8, 12.P.11, and 12.G.1 (AII.P.8, AII.P.11, and AII.G.1)
Students replicate the experiment in which Eratosthenes calculated the circumference of the earth and got a remarkably good answer. They locate some schools roughly due north or south of their school and connect with students in those schools through electronic mail. Students in each school agree that on a given day, at high noon, they will measure the shadow cast by a vertical stick on level ground. After sharing the measurements of the stick and the shadow, students use trigonometric ratios to determine the angle of the sun’s rays. Using this information, along with the approximate distance between the schools, students use proportions to find an approximation of the earth’s circumference. This example can be extended to sharing data with students from other states and countries.

Refers to standards 12.P.8, 12.P.11, and 12.G.1 (AII.P.8, AII.P.11, and AII.G.1)
How far from the horizontal must a sheet of plywood 4 feet wide be rotated to fit through a doorway 30 inches wide?

Refers to standard 12.G.3 (G.G.18)†
A simple vector sum.

Refers to standard 12.M.1 (PC.M.1)
In one hour, the minute hand on a clock moves through a complete circle, and the hour hand moves through 1/12 of a circle. Through how many radians do the minute and the hour hand move between 1:00 p.m. and 6:45 p.m. on the same day?

Refers to standard 12.M.2 (PC.M.2)†
High school students should be able to make reasonable estimates and sensible judgments about the precision and accuracy of the values they report. Teachers can help students understand that measurements of continuous quantities are always approximations. For example, suppose a situation calls for determining the mass of a bar of gold bullion in the shape of a rectangular prism whose length, width, and height are measured as 27.9 centimeters, 10.2 centimeters, and 6.4 centimeters, respectively. Knowing that the density is 19,300 kilograms per cubic meter, students might compute the mass as follows:

The students need to understand that reporting the mass with this degree of precision would be misleading because it would suggest a degree of accuracy far greater than the actual accuracy of the measurement. Since the lengths of the edges are reported to the nearest tenth of a centimeter, the measurements are precise only to 0.05 centimeter. That is, the edges could actually have measures in the intervals 27.9 ± 0.05, 10.2 ± 0.05, and 6.4 ± 0.05. If students calculate the possible maximum and minimum mass, given these dimensions, they will see that at most one decimal place in accuracy is justified. As suggested by the example above, units should be reported along with numerical values in measurement computations.

Refers to standard 12.D.6 (AII.D.2) (EDC, Inc)
There are 9 points on a paper. No three are on the same line. How many different triangles can be drawn with vertices on these points?

Refers to standard 12.D.6 (AII.D.2)
There are eight McBride children, three girls and five boys. How many different ways are there of forming groups of McBride children containing at least two of the three girls?

Refers to standard 12.D.6 (AII.D.2)
Some services that involve electronic access require clients to choose a six-digit password. In an effort to increase security of the passwords, clients cannot use combinations that correspond to actual dates, nor can they use two identical digits in succession, nor passwords with one digit appearing three or more times. How many “secure” passwords are available?

Refers to standard PC.P.1
The goal of the Towers of Hanoi game pictured below is to move the tower on the left to either of the other two poles. At the end, the tower must look exactly as it does in the beginning, with the largest disk on the bottom and progressively smaller disks on top. You are allowed to move only one disk at a time. You are not allowed to place a larger disk on top of a smaller one. You must always move the top disk from a given tower.

You can play the game beginning with two or more disks in the tower. Could you use reasoning by mathematical induction to convince yourself that the object of the game is attainable no matter how many disks there are? Does induction suggest a particular recursive pattern for solving the problem as more disks are added? Can you determine the minimum number of moves as a function of the number of starting disks?

Refers to standards 12.P.8 and 12.P.11 (AII.P.8 and AII.P.11)
A solution’s pH depends on the concentration of hydrogen ions per liter of the solution. The formula for determining a pH is where H+ is the number of gram atoms of hydrogen ions per liter. The pH of neutral water is 7. Acidic solutions have a pH that is lower than 7, basic solutions have a pH that is higher than 7.
1. What is the pH of a solution with 4.231 x 10-5 gram atoms of hydrogen ions per liter?
2. A certain juice has a pH of 3.9. Find the concentration of hydrogen ions of the juice.

Exploratory Concepts and Skills for Grades 11 – 12 and Single-Subject Courses

Number Sense and Operations
? Investigate special topics in number theory, e.g., the use of prime numbers in cryptography.
? Use polar-coordinate representations of complex numbers (i.e., a + bi = r(cos? + isin?)) and DeMoivre’s theorem to multiply, take roots, and raise numbers to a power.
? Plot complex numbers using both rectangular and polar coordinate systems.

Patterns, Relations, and Algebra
? Prove theorems using mathematical induction.
? Investigate parametrically defined curves and recursively defined functions, including applications to dynamic systems.

Geometry
? Investigate and compare the axiomatic structures of Euclidean and non-Euclidean geometries.
? Explore the use of conic sections in engineering, design, and other applications.
? Investigate the notion of a fractal.
? Use graphs (networks) to investigate probabilistic processes and optimization problems.

Data Analysis, Statistics, and Probability
? Use technology to perform linear, quadratic, and exponential regression on a set of data.
? Design surveys and apply random sampling techniques to avoid bias in the data collection.

Learning Standards for Algebra I

Note: The parentheses at the end of a learning standard contain the code number(s) for the corresponding standard(s) in the two-year grade spans.

Number Sense and Operations
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
Understand meanings of operations and how they relate to one another
Compute fluently and make reasonable estimates Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
AI.N.1 Identify and use the properties of operations on real numbers, including the associative, commutative, and distributive properties; the existence of the identity and inverse elements for addition and multiplication; the existence of nth roots of positive real numbers for any positive integer n; the inverse relationship between taking the nth root of and the nth power of a positive real number; and the density of the set of rational numbers in the set of real numbers. (10.N.1)
AI.N.2 Simplify numerical expressions, including those involving positive integer exponents or the absolute value, e.g., 3(24 – 1) = 45, 4|3 – 5| + 6 = 14; apply such simplifications in the solution of problems. (10.N.2)
AI.N.3 Find the approximate value for solutions to problems involving square roots and cube roots without the use of a calculator, e.g., . (10.N.3)
AI.N.4 Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers. (10.N.4)

Patterns, Relations, and Algebra
Understand patterns, relations, and functions
Represent and analyze mathematical situations and structures using algebraic symbols
Use mathematical models to represent and understand quantitative relationships
Analyze change in various contextsStudents engage in problem solving, communicating, reasoning, connecting, and representing as they:
AI.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g., Fibonnacci Numbers), linear, quadratic, and exponential functional relationships. (10.P.1)
AI.P.2 Use properties of the real number system to judge the validity of equations and inequalities, to prove or disprove statements, and to justify every step in a sequential argument.
AI.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent, and independent variables of functions.
AI.P.4 Translate between different representations of functions and relations: graphs, equations, point sets, and tabular.
AI.P.5 Demonstrate an understanding of the relationship between various representations of a line. Determine a line’s slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the “point-slope” or “slope y-intercept” formulas. Explain the significance of a positive, negative, zero, or undefined slope. (10.P.2)
AI.P.6 Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the “point-slope” form of the equation. (10.G.8)
AI.P.7 Add, subtract, and multiply polynomials. Divide polynomials by monomials. (10.P.3)
AI.P.8 Demonstrate facility in symbolic manipulation of polynomial and rational expressions by rearranging and collecting terms, factoring (e.g., a2 – b2 = (a + b)(a - b), x2 + 10x + 21 = (x + 3)(x + 7), 5x4 + 10x3 – 5x2 = 5x2 (x2 + 2x – 1)), identifying and canceling common factors in rational expressions, and applying the properties of positive integer exponents. (10.P.4)
AI.P.9 Find solutions to quadratic equations (with real roots) by factoring, completing the square, or using the quadratic formula. Demonstrate an understanding of the equivalence of the methods. (10.P.5)
AI.P.10 Solve equations and inequalities including those involving absolute value of linear expressions (e.g., |x - 2| > 5) and apply to the solution of problems. (10.P.6)

Patterns, Relations, and Algebra
(continued)

AI.P.11 Solve everyday problems that can be modeled using linear, reciprocal, quadratic, or exponential functions. Apply appropriate tabular, graphical, or symbolic methods to the solution. Include compound interest, and direct and inverse variation problems. Use technology when appropriate. (10.P.7)
AI.P.12 Solve everyday problems that can be modeled using systems of linear equations or inequalities. Apply algebraic and graphical methods to the solution. Use technology when appropriate. Include mixture, rate, and work problems. (10.P.8)

Data Analysis, Statistics, and Probability
Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
Select and use appropriate statistical methods to analyze data
Develop and evaluate inferences and predictions that are based on data
Understand and apply basic concepts of probability

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
AI.D.1 Select, create, and interpret an appropriate graphical representation (e.g., scatterplot, table, stem-and-leaf plots, circle graph, line graph, and line plot) for a set of data and use appropriate statistics (e.g., mean, median, range, and mode) to communicate information about the data. Use these notions to compare different sets of data. (10.D.1)
AI.D.2 Approximate a line of best fit (trend line) given a set of data (e.g., scatterplot). Use technology when appropriate. (10.D.2)
AI.D.3 Describe and explain how the relative sizes of a sample and the population affect the validity of predictions from a set of data. (10.D.3)

Learning Standards for Geometry

Note: The parentheses at the end of a learning standard contain the code number for the corresponding standard in the two-year grade spans.

Geometry
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
Specify locations and describe spatial relationships using coordinate geometry and other representational systems
Apply transformations and use symmetry to analyze mathematical situations
Use visualization, spatial reasoning, and geometric modeling to solve problems

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
G.G.1 Recognize special types of polygons (e.g., isosceles triangles, parallelograms, and rhombuses). Apply properties of sides, diagonals, and angles in special polygons; identify their parts and special segments (e.g., altitudes, midsegments); determine interior angles for regular polygons. Draw and label sets of points such as line segments, rays, and circles. Detect symmetries of geometric figures.
G.G.2 Write simple proofs of theorems in geometric situations, such as theorems about congruent and similar figures, parallel or perpendicular lines. Distinguish between postulates and theorems. Use inductive and deductive reasoning, as well as proof by contradiction. Given a conditional statement, write its inverse, converse, and contrapositive.
G.G.3 Apply formulas for a rectangular coordinate system to prove theorems.
G.G.4 Draw congruent and similar figures using a compass, straightedge, protractor, or computer software. Make conjectures about methods of construction. Justify the conjectures by logical arguments. (10.G.2)
G.G.5 Apply congruence and similarity correspondences (e.g., ?ABC ? ?XYZ) and properties of the figures to find missing parts of geometric figures, and provide logical justification. (10.G.4)
G.G.6 Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.
G.G.7 Solve simple triangle problems using the triangle angle sum property, and/or the Pythagorean theorem. (10.G.5)
G.G.8 Use the properties of special triangles (e.g., isosceles, equilateral, 30º–60º–90º, 45º–45º–90º) to solve problems. (10.G.6)
G.G.9 Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.
G.G.10 Apply the triangle inequality and other inequalities associated with triangles (e.g., the longest side is opposite the greatest angle) to prove theorems and solve problems.
G.G.11 Demonstrate an understanding of the relationship between various representations of a line. Determine a line’s slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the “point-slope” or “slope y-intercept” formulas. Explain the significance of a positive, negative, zero, or undefined slope. (10.P.2)
G.G.12 Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems. (10.G.7)
G.G.13 Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the “point-slope” form of the equation. (10.G.8)
G.G.14 Demonstrate an understanding of the relationship between geometric and algebraic representations of circles.
G.G.15 Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solution of problems. (10.G.9)
G.G.16 Demonstrate the ability to visualize solid objects and recognize their projections and cross sections. (10.G.10)
G.G.17 Use vertex-edge graphs to model and solve problems. (10.G.11)
G.G.18 Use the notion of vectors to solve problems. Describe addition of vectors and multiplication of a vector by a scalar, both symbolically and pictorially. Use vector methods to obtain geometric results. (12.G.3)

Learning Standards for Measurement

Understand measurable attributes of objects and the units, systems, and processes of measurement
Apply appropriate techniques, tools, and formulas to determine measurements

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
G.M.1 Calculate perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles. (10.M.1)
G.M.2 Given the formula, find the lateral area, surface area, and volume of prisms, pyramids, spheres, cylinders, and cones, e.g., find the volume of a sphere with a specified surface area. (10.M.2)
G.M.3 Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing the radius or height of a cylinder affects its surface area or volume. (10.M.3)
G.M.4 Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. (10.M.4)
G.M.5 Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. (12.M.2)

Learning Standards for Algebra II

Note: The parentheses at the end of a learning standard contain the code number for the corresponding standard in the two-year grade spans.

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
AII.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns such as Pascal’s Triangle. (12.P.1)
AII.P.2 Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the formula for the general term and the sum, recursively and explicitly. (12.P.2)
AII.P.3 Demonstrate an understanding of the binomial theorem and use it in the solution of problems. (12.P.3)
AII.P.4 Demonstrate an understanding of the exponential and logarithmic functions.
AII.P.5 Perform operations on functions, including composition. Find inverses of functions. (12.P.5)
AII.P.6 Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, or exponential. (12.P.6)
AII.P.7 Find solutions to quadratic equations (with real coefficients and real or complex roots) and apply to the solutions of problems. (12.P.7)
AII.P.8 Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, and logarithmic functions; expressions involving the absolute values; and simple rational expressions. (12.P.8)
AII.P.9 Use matrices to solve systems of linear equations. Apply to the solution of everyday problems. (12.P.9)
AII.P.10 Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions. Also use technology where appropriate. Describe the relationships among the methods. (12.P.10)
AII.P.11 Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, and step functions, absolute values and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; logistic growth; joint (e.g., I = Prt, y = k(w1 + w2)), and combined (F = G(m1m2)/d2) variation. (12.P.11)
AII.P.12 Identify maximum and minimum values of functions in simple situations. Apply to the solution of problems. (12.P.12)
AII.P.13 Describe the translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af (b(x + c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, and logarithmic functions. (12.P.13)

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
AII.G.1 Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems. (12.G.1)
AII.G.2 Derive and apply basic trigonometric identities (e.g., sin2? + cos2? = 1, tan2? + 1 = sec2?) and the laws of sines and cosines. (12.G.2)
AII.G.3 Relate geometric and algebraic representations of lines, simple curves, and conic sections. (12.G.4)

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
AII.D.1 Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data. (12.D.2)
AII.D.2 Use combinatorics (e.g., “fundamental counting principle,” permutations, and combinations) to solve problems, in particular, to compute probabilities of compound events. Use technology as appropriate. (12.D.6)

Learning Standards for Precalculus

Note: The parentheses at the end of a learning standard contain the code number for the corresponding standard in the two-year grade spans.

Learning Standards for Number Sense and Operations
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
Understand meanings of operations and how they relate to one another
Compute fluently and make reasonable estimates Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
PC.N.1 Plot complex numbers using both rectangular and polar coordinates systems. Represent complex numbers using polar coordinates, i.e., a + bi = r(cos? + isin?). Apply DeMoivre’s theorem to multiply, take roots, and raise complex numbers to a power.

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
PC.P.1 Use mathematical induction to prove theorems and verify summation formulas, e.g., verify .
PC.P.2 Relate the number of roots of a polynomial to its degree. Solve quadratic equations with complex coefficients.
PC.P.3 Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions.
PC.P.4 Explain the identity sin2? + cos2? = 1. Relate the identity to the Pythagorean theorem.
PC.P.5 Demonstrate an understanding of the formulas for the sine and cosine of the sum or the difference of two angles. Relate the formulas to DeMoivre’s theorem and use them to prove other trigonometric identities. Apply to the solution of problems.
PC.P.6 Understand, predict, and interpret the effects of the parameters a, ?, b, and c on the graph of y = asin(?(x - b)) + c; similarly for the cosine and tangent. Use to model periodic processes. (12.P.13)
PC.P.7 Translate between geometric, algebraic, and parametric representations of curves. Apply to the solution of problems.
PC.P.8 Identify and discuss features of conic sections: axes, foci, asymptotes, and tangents. Convert between different algebraic representations of conic sections.
PC.P.9 Relate the slope of a tangent line at a specific point on a curve to the instantaneous rate of change. Explain the significance of a horizontal tangent line. Apply these concepts to the solution of problems.

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
PC.G.1 Demonstrate an understanding of the laws of sines and cosines. Use the laws to solve for the unknown sides or angles in triangles. Determine the area of a triangle given the length of two adjacent sides and the measure of the included angle. (12.G.2)
PC.G.2 Use the notion of vectors to solve problems. Describe addition of vectors, multiplication of a vector by a scalar, and the dot product of two vectors, both symbolically and geometrically. Use vector methods to obtain geometric results. (12.G.3)
PC.G.3 Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems. (12.G.5)

Measurement
Understand measurable attributes of objects and the units, systems, and processes of measurement
Apply appropriate techniques, tools, and formulas to determine measurements

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
PC.M.1 Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular problems involving angular velocity and acceleration. (12.M.1)
PC.M.2 Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. (12.M.2)

Students engage in problem solving, communicating, reasoning, connecting, and representing as they:
PC.D.1 Design surveys and apply random sampling techniques to avoid bias in the data collection. (12.D.1)
PC.D.2 Apply regression results and curve fitting to make predictions from data. (12.D.3)
PC.D.3 Apply uniform, normal, and binomial distributions to the solutions of problems. (12.D.4)
PC.D.4 Describe a set of frequency distribution data by spread (variance and standard deviation), skewness, symmetry, number of modes, or other characteristics. Use these concepts in everyday applications. (12.D.5)
PC.D.5 Compare the results of simulations (e.g., random number tables, random functions, and area models) with predicted probabilities. (12.D.7)

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