 ## Algebra 2

### Number and Quantity The Real Number System

Use properties of rational and irrational numbers.
3a

3a. Use properties and operations to understand the different forms of rational and irrational numbers.

Perform all four arithmetic operations and apply properties to generate equivalent forms of rational numbers and square roots.

3b

3b. Categorize the sum or product of rational or irrational numbers.
• The sum and product of two rational numbers is rational.
• The sum of a rational number and an irrational number is irrational.
• The product of a nonzero rational number and an irrational number is irrational.
• The sum and product of two irrational numbers could be either rational or irrational.

### Quantities

Reason quantitatively and use units to solve problems.
1

1. Select quantities and use units as a way to:
i) interpret and guide the solution of multi-step problems;
ii) choose and interpret units consistently in formulas; and
iii) choose and interpret the scale and the origin in graphs and data displays.

3

3. Choose a level of accuracy appropriate to limitations on measurement and context when reporting quantities.

### Seeing Structure in Expressions

Interpret the structure of expressions
1

1. Interpret expressions that represent a quantity in terms of its context.

a. Write the standard form of a given polynomial and identify the terms, coefficients, degree, leading coefficient, and constant term.
b. Interpret expressions by viewing one or more of their parts as a single entity.
2

2. Recognize and use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to reveal their characteristics.
3

3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

c. Use the properties of exponents to rewrite exponential expressions.

### Arithmetic with Polynomials and Rational Expressions

Perform arithmetic operations on polynomials
1

1. Add, subtract, and multiply polynomials and recognize that the result of the operation is also a polynomial. This forms a system analogous to the integers.

Understand the relationship between zeros and factors of polynomials
3

3. Identify zeros of polynomial functions when suitable factorizations are available.

### Creating Equations

Create equations that describe numbers or relationships
1

1. Create equations and inequalities in one variable to represent a real-world context.

2

2. Create equations and linear inequalities in two variables to represent a real-world context.

3

3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

4

4. Rewrite formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

### Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning
1a

1a. Explain each step when solving a linear or quadratic equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Solve equations and inequalities in one variable
3

3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

4

Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Understand that the quadratic formula is a derivative of this process.

i) inspection,

ii) taking square roots,

iii) factoring,

iv) completing the square,

vi) graphing.

Solve systems of equations
6a

6a. Solve systems of linear equations in two variables both algebraically and graphically.

7a

7a. Solve a system, with rational solutions, consisting of a linear equation and a quadratic equation (parabolas only) in two variables algebraically and graphically.

Represent and solve equations and inequalities graphically
10

10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

11

11. Given the equations y = f(x) and y = g(x):

i) recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x);

ii) find the solutions approximately using technology to graph the functions or make tables of values; and

iii) interpret the solution in context.

12

12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

### Interpreting Functions

Understand the concept of a function and use function notation
1

1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

2

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

3

3. Recognize that a sequence is a function whose domain is a subset of the integers.

Interpret functions that arise in applications in terms of the context
4

4. For a function that models a relationship between two quantities:

i) interpret key features of graphs and tables in terms of the quantities; and

ii) sketch graphs showing key features given a verbal description of the relationship.

5

5. Determine the domain of a function from its graph and, where applicable, identify the appropriate domain for a function in context.

6

6. Calculate and interpret the average rate of change of a function over a specified interval.

Analyze functions using different representations
7

7. Graph functions and show key features of the graph by hand and by using technology where appropriate.

a. Graph linear, quadratic, and exponential functions and show key features.
b. Graph square root, and piecewise-defined functions, including step functions and absolute value functions and show key features.
8

8. Write a function in different but equivalent forms to reveal and explain different properties of the function.

a. For a quadratic function, use an algebraic process to find zeros, maxima, minima, and symmetry of the graph, and interpret these in terms of context.

9

9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

### Building Functions

Build a function that models a relationship between two quantities
NUMBER

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

a. Determine a function from context.
Define a sequence explicitly or steps for calculation from a context.
Build new functions from existing functions
3a

3a. Using f(x) + k, k f(x), and f(x + k):

i) identify the effect on the graph when replacing f(x) by f(x) + k, k f(x), and f(x + k) for specific values of k (both positive and negative);

ii) find the value of k given the graphs;

iii) write a new function using the value of k; and

iv) use technology to experiment with cases and explore the effects on the graph

### Linear, Quadratic, and Exponential Models

Construct and compare linear, quadratic, and exponential models and solve problems
1

1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

a. Justify that a function is linear because it grows by equal differences over equal intervals, and that a function is exponential because it grows by equal factors over equal intervals.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another, and therefore can be modeled linearly.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another, and therefore can be modeled exponentially.

2

2. Construct a linear or exponential function symbolically given:

i) a graph;

ii) a description of the relationship;

iii) two input-output pairs (include reading these from a table).

3

3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Interpret expressions for functions in terms of the situation they model
5

5. Interpret the parameters in a linear or exponential function in terms of a context.

### Interpreting Categorical and Quantitative Data

Summarize, represent, and interpret data on a single count or measurement variable
1

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

2

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

3

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

Summarize, represent, and interpret data on two categorical and quantitative variables
5

5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

6

6. Represent bivariate data on a scatter plot, and describe how the variables’ values are related.

a. Fit a function to real-world data; use functions fitted to data to solve problems in the context of the data.
Interpret linear models
7

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

8

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

9

9. Distinguish between correlation and causation.

### Congruence

Experiment with transformations in the plane
1

1. Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc as these exist within a plane.

2

2. Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not.

3

3. Given a regular or irregular polygon, describe the rotations and reflections (symmetries) that carry the polygon onto itself.

4

4. Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments.

5

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions
6

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

7

7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

8

8. Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS and HL (Hypotenuse Leg)) follow from the definition of congruence in terms of rigid motions.

Prove geometric theorems
9

9. Prove and apply theorems about lines and angles.

10

10. Prove and apply theorems about triangles.

11

11. Prove and apply theorems about parallelograms.

Make geometric constructions
12

12. Make, justify, and apply formal geometric constructions.

13

13. Make and justify the constructions for inscribing an equilateral triangle, a square and a regular hexagon in a circle.

### Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations
1

1. Verify experimentally the properties of dilations given by a center and a scale factor.

a. Verify experimentally that dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. Explain using similarity transformations that similar triangles have equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

3

3. Use the properties of similarity transformations to establish the AA~, SSS~, and SAS~ criterion for two triangles to be similar.

Prove theorems involving similarity
4

4. Prove and apply similarity theorems about triangles.

5

5. Use congruence and similarity criteria for triangles to:
a. Solve problems algebraically and geometrically.
b. Prove relationships in geometric figures.

Define trigonometric ratios and solve problems involving right triangles
6

6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of sine, cosine and tangent ratios for acute angles.

7

7. Explain and use the relationship between the sine and cosine of complementary angles.

8

8. Use sine, cosine, tangent, the Pythagorean Theorem and properties of special right triangles to solve right triangles in applied problems.

Apply trigonometry to general triangles
9

9. Justify and apply the formula A= 12ab sin (C) to find the area of any triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

### Circles

Understand and apply theorems about circles
1

1. Prove that all circles are similar.

2a

2a. Identify, describe and apply relationships between the angles and their intercepted arcs of a circle.

2b

2b. Identify, describe and apply relationships among radii, chords, tangents, and secants of a circle.

Find arc lengths and areas of sectors of circles
5

5. Using proportionality, find one of the following given two others; the central angle, arc length, radius or area of sector.

### Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section
1a

1a. Derive the equation of a circle of given center and radius using the Pythagorean Theorem. Find the center and radius of a circle, given the equation of the circle.

1b

1b. Graph circles given their equation.

Use coordinates to prove simple geometric theorems algebraically
4

4. On the coordinate plane, algebraically prove geometric theorems and properties.

5

5. On the coordinate plane:

a. Explore the proof for the relationship between slopes of parallel and perpendicular lines;

b. Determine if lines are parallel, perpendicular, or neither, based on their slopes; and

c. Apply properties of parallel and perpendicular lines to solve geometric problems.

6

6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

7

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

### Geometric Measurement and Dimension

Explain volume formulas and use them to solve problems
1

1. Provide informal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

3

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects
4

4. Identify the shapes of plane sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

### Modeling with Geometry

Apply geometric concepts in modeling situations
1

1. Use geometric shapes, their measures, and their properties to describe objects.

2

2. Apply concepts of density based on area and volume of geometric figures in modeling situations.

3

3. Apply geometric methods to solve design problems.

### Number and Quantity The Real Number System

Extend the properties of exponents to rational exponents.
1

1. Explore how the meaning of rational exponents follows from extending the properties of integer exponents.

2

2. Convert between radical expressions and expressions with rational exponents using the properties of exponents.

### Number and Quantity The Complex Number System

Perform arithmetic operations with complex numbers.
1

1. Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.

2

2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

### Seeing Structure in Expressions

Write expressions in equivalent forms to reveal their characteristics.
2

2. Recognize and use the structure of an expression to identify ways to rewrite it.

3

3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

3a

a. Factor quadratic expressions including leading coefficients other than 1 to reveal the zeros of the function it defines.

3c

c. Use the properties of exponents to rewrite exponential expressions.

### Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomials.
2

2. Apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

3

3. Identify zeros of polynomial functions when suitable factorizations are available.

Rewrite rational expressions.
6

6. Rewrite rational expressions in different forms: Write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x).

### Creating Equations

Create equations that describe numbers or relationships.
1

1. Create equations and inequalities in one variable to represent a real-world context.

### Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning.
1b

1b. Explain each step when solving rational or radical equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

2

2. Solve rational and radical equations in one variable, identify extraneous solutions, and explain how they arise.

Solve equations and inequalities in one variable.
4b

4. Solve quadratic equations in one variable.

i) inspection,

ii) taking square roots,

iii) factoring,

iv) completing the square,

vi) graphing.

Write complex solutions in a + bi form.

Solve systems of equations.
7b

7b. Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Represent and solve equations and inequalities graphically.
11

11. Given the equations y = f(x) and y = g(x):

i) recognize that each x-coordinate of the intersection(s) is the solution to the equation f(x) = g(x);

ii) find the solutions approximately using technology to graph the functions or make tables of values;

iii) find the solution of f(x) < g(x) or f(x) ≤ g(x) graphically; and

iv) interpret the solution in context.

### Interpreting Functions

Understand the concept of a function and use function notation.
3

3. Recognize that a sequence is a function whose domain is a subset of the integers.

Interpret functions that arise in applications in terms of the context.
4

4. For a function that models a relationship between two quantities:

i) interpret key features of graphs and tables in terms of the quantities; and

ii) sketch graphs showing key features given a verbal description of the relationship.

6

6. Calculate and interpret the average rate of change of a function over a specified interval.

Analyze functions using different representations.
7

7. Graph functions and show key features of the graph by hand and using technology when appropriate.

7c

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

7e

e. Graph cube root, exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.

Analyze functions using different representations.
8

8. Write a function in different but equivalent forms to reveal and explain different properties of the function.

b. Use the properties of exponents to interpret exponential functions, and classify them as representing exponential growth or decay.

9

9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

### Building Functions

Build a function that models a relationship between two quantities.
1

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

1a

a. Determine a function from context.
Determine an explicit expression, a recursive process, or steps for calculation from a context.

1b

b. Combine standard function types using arithmetic operations.

2

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Build new functions from existing functions.
3b

3b. Using f(x) + k, k f(x), f(kx), and f(x + k):

i) identify the effect on the graph when replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative);

ii) find the value of k given the graphs;

iii) write a new function using the value of k; and

iv) use technology to experiment with cases and explore the effects on the graph.

Include recognizing even and odd functions from their graphs.

4a

4a. Find the inverse of a one-to-one function both algebraically and graphically.

5a

5a. Understand inverse relationships between exponents and logarithms algebraically and graphically.

6

6. Represent and evaluate the sum of a finite arithmetic or finite geometric series, using summation (sigma) notation.

7

7. Explore the derivation of the formulas for finite arithmetic and finite geometric series. Use the formulas to solve problems.

### Linear, Quadratic, and Exponential Models

Construct and compare linear, quadratic, and exponential models and solve problems.
2

2. Construct a linear or exponential function symbolically given:

i) a graph;

ii) a description of the relationship; and

iii) two input-output pairs (include reading these from a table).

4

4. Use logarithms to solve exponential equations, such as abct = d (where a, b, c, and d are real numbers and b > 0) and evaluate the logarithm using technology.

Interpret expressions for functions in terms of the situation they model.
5

5. Interpret the parameters in a linear or exponential function in terms of a context.

### Trigonometric Functions

Extend the domain of trigonometric functions using the unit circle.
1

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

2

2. Apply concepts of the unit circle in the coordinate plane to calculate the values of the six trigonometric functions given angles in radian measure.

4

4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Model periodic phenomena with trigonometric functions.
5

5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, horizontal shift, and midline.

Prove and apply trigonometric identities.
8

8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1. Find the value of any of the six trigonometric functions given any other trigonometric function value and when necessary find the quadrant of the angle.

### Interpreting Categorical and Quantitative Data

Summarize, represent, and interpret data on a single count or measurement variable.
4a

4a. Recognize whether or not a normal curve is appropriate for a given data set.

4b

4b. If appropriate, determine population percentages using a graphing calculator for an appropriate normal curve.

Summarize, represent, and interpret data on two categorical and quantitative variables.
6

6. Represent bivariate data on a scatter plot, and describe how the variables’ values are related.

6a

a. Fit a function to real-world data; use functions fitted to data to solve problems in the context of the data.

### Making Inferences and Justifying Conclusions

Understand and evaluate random processes underlying statistical experiments.
2

2. Determine if a value for a sample proportion or sample mean is likely to occur based on a given simulation.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
3

3. Recognize the purposes of and differences among surveys, experiments, and observational studies. Explain how randomization relates to each.

4

4. Given a simulation model based on a sample proportion or mean, construct the 95% interval centered on the statistic (+/- two standard deviations) and determine if a suggested parameter is plausible.

6a

6a. Use the tools of statistics to draw conclusions from numerical summaries.

6b

6b. Use the language of statistics to critique claims from informational texts. For example, causation vs correlation, bias, measures of center and spread.

### Conditional Probability and the Rules of Probability

Understand independence and conditional probability and use them to interpret data.
1

1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

4

4. Interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and calculate conditional probabilities.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.
7

7. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.