Building on the work in Integrated Mathematics Course 1 with logic and its application in geometric, algebraic, and probabilistic arenas, this second course opens with a look at probability distributions and reasoning from data. The binomial expansion theorem, an example of an algebraic representation of a discrete distribution, provides a bridge to algebraic topics. Quadratic functions and equations follow, with the introduction of rational exponents, roots, and complex numbers opening the way for the solution of all quadratic equations. The concepts and techniques applied to quadratic functions are extended to the study of power and polynomial functions in the final unit of this course.

Throughout Integrated Mathematics Course 2, technology is an important tool for data analysis as well as for graphical visualization and deepening understanding of function relationships. If computer or calculator-based algebra manipulation is available, there are many opportunities in this course to use such technology for exploration. Care should be taken, however, to parallel any such use of technology with development of by-hand algebraic skills. Testing with and without technological tools is recommended.

How a particular subject is taught influences not only the depth and retention of the content of a course but also the development of skills in inquiry, problem solving, and critical thinking. Every opportunity should be taken to link the concepts of Integrated Mathematics Course 2 to those encountered in Integrated Mathematics Course 1 and middle school mathematics as well as to other disciplines. Students should be encouraged to be creative and innovative in their approach to problems, to be productive and persistent in seeking solutions, and to use multiple means to communicate their insights and understanding.

The Major Concepts below provide the focus for this second course in a three-course integrated mathematics sequence. They should be taught using a variety of methods and applications so that students attain a deep understanding of these concepts.

• Reasoning from Data
• Applying Exponents
• Fundamentals of Logic
• Quadratic Functions and Equations with Real Zeros/Roots
• Quadratic Functions and Equations with Complex Zeros/Roots
• Power and Polynomial Functions and Expressions

Maintenance Concepts should have been taught previously and are important foundational concepts that will be applied in this course. Continued facility with and understanding of the Maintenance Concepts is essential for success in the Major Concepts defined for this course.

• Elementary Data Analysis
• Integer Exponents and Roots
• Real Numbers
• Integer Exponents
• Linear and Proportional Relationships
• Fundamental Logic, Reasoning, and Proof

A. Reasoning from Data

Integrated Mathematics Course 1 ended with a look at probability and its applications, and this course begins by extending those concepts to probability distributions and the information they convey that leads to rational, reasoned decision-making. Since issues of precision and number comprehension often affect decisions, a short section on those topics is included as well.

Successful students will:

A1 Describe key characteristics of a distribution.

Key characteristics include measures of center and spread.

1. Identify and distinguish between discrete and continuous probability distributions.
2. Calculate and use the mean and standard deviation to describe the characteristics of a distribution.
3. Reason from empirical distributions of data to make assumptions about their underlying theoretical distributions.

A2 Know and use the chief characteristics of the normal distribution.

The normal (or Gaussian) distribution is actually a family of mathematical distributions that are symmetric in shape with scores more concentrated in the middle than in the tails. They are sometimes described as bell-shaped. Normal distributions may have differing centers (means) and scale (standard deviation). The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. In normal distributions, approximately 68% of the data lie within one standard deviation of the mean and 95% within two.

1. Identify examples that demonstrate that the mean and standard deviation of a normal distribution can vary independently of each other (e.g., that two normal distributions with the same mean can have different standard deviations).
2. Identify common examples that fit the normal distribution (height, weight) and examples that do not (salaries, housing prices, size of cities) and explain the distinguishing characteristics of each.

A3 Apply probability to make and communicate informed decisions.

1. Apply probability to practical situations.

   Examples: Communicate an understanding of the inverse relation of risk and return; explain the benefits of diversifying risk.

2. Calculate the expected value of a random variable having a discrete probability distribution and interpret the results.

A4 Interpret and apply numbers used in practical situations.

1. Interpret and compare extreme numbers.

   Examples: Lottery odds, national debt, astronomical distances.

2. Determine a reasonable degree of precision in a given situation.
3. Assess the amount of error resulting from estimation and determine whether the error is within acceptable tolerance limits.
4. Choose appropriate techniques and tools to measure quantities in order to achieve specified degrees of precision, accuracy, and error (or tolerance) of measurements.

   Example: Humans have a reaction time to visual stimuli of approximately 0.1 sec. Thus, it is reasonable to use hand-activated stopwatches that measure tenths of a second.

5. Apply significant figures, orders of magnitude, and scientific notation when making calculations or estimations.

B. Applying Exponents

Building on the understanding of whole number exponents, students in Integrated Mathematics Course 2 will develop an understanding of the impact of a negative exponent and generalize the properties of exponents to all rational exponents. Application of the laws of exponents to numerical and algebraic monomials and their use in operations with binomials forms a foundation for important algebraic skills. Basic factoring and multiplication enable algebraic expressions to be written in various forms that provide insight and clarify information. The binomial theorem is an example of the multiplication of a binomial. Its links to the binomial distribution and to probability studied in the previous unit provide an effective bridge from the study of reasoning with data and distributions of data to the study of algebraic expressions, equations, and functions.

Successful students will:

B1 Interpret negative integer and rational exponents; use them to rewrite numeric expressions in alternative forms.

1. Convert between expressions involving negative exponents and those involving only positive ones; apply the properties as necessary.

   Examples:

2. Convert between expressions involving rational exponents and those involving roots and integral powers; apply the properties of exponents as necessary.

   Examples:

B2 Apply the properties of exponents to transform variable expressions involving integer exponents.

1. Know and apply the laws of exponents for integer exponents.

   Examples: a^m · aⁿ = a^m+n, for m, n real; ; 9^x = 3^2x; 64^5/6 = 32.

2. Factor out common factors in expressions involving integer exponents.

   Factoring transforms an expression that was written as a sum or difference into one that is written as a product.

   Examples: 6v⁷ + 12v⁵ - 8v³ = 2v³(3v⁴ + 6v² - 4); 27x^-2 = 12x^-4 + 45x = 3x^-4(9x² - 4 + 15x⁵); 3x(x + 1)² - 2(x + 1)² = (x + 1)²(3x - 2).

   Chunking is a term often used to describe treating an expression, such as the x + 1 above, as a single entity.

B3 Make regular fluent use of basic algebraic identities such as
(a + b)² = a² + 2ab + b²; (a - b)² = a² - 2ab + b²; and (a + b)(a - b) = a² - b².

1. Use the distributive law to derive each of these formulas.

   Examples: (a + b)(a - b) = (a + b)a - (a + b)b = (a² + ab) - (ab + b²) = a² + ab - ab - b² = a² - b²; applying this to specific numbers, 37 · 43 = (40 - 3)(40 + 3) = 1,600 - 9 = 1,591.

2. Use geometric representations to illustrate these formulas.

   Example: Use a partitioned square or tiles to provide a geometric representation of (a + b)² = a² + 2ab + b².

B4 Know and use the binomial expansion theorem.

1. Relate the expansion of (a + b)ⁿ to the possible outcomes of a binomial experiment and the nth row of Pascal’s triangle.

B5 Convert between forms of numerical expressions involving roots and perform operations on numbers expressed in radical form.

Example: Convert and use the understanding of this conversion to perform similar calculations and to compute with numbers in radical form.

B6 Solve linear and simple nonlinear equations involving several variables for one variable in terms of the others; use fractional exponents and roots as needed to express the solution.

Example: Solve A = πr²h for h or for r.

C. Quadratic Functions and Equations with Real Zeros/Roots

The study of quadratic functions and equations builds on the work with algebraic identities and forms begun in the last unit. Early work with quadratic functions and equations should focus on those with real zeros/roots.

Successful students will:

C1 Identify quadratic functions expressed in multiple forms; identify the specific information each form clarifies.

1. Express a quadratic function as a polynomial, f(x) = ax² + bx + c, where a, b and c are constants with a ≠ 0, and identify its graph as a parabola that opens up when a > 0 and down when a < 0; relate c to where the graph of the function crosses the y-axis.
2. Express a quadratic function in factored form, f(x) = (x – r)(x – s), when r and s are integers; relate the factors to the solutions of the equation (x – r)(x – s) = 0 (x = r and x = s) and to the points ((r, 0) and (s, 0)) where the graph of the function crosses the x-axis.

C2 Transform quadratic functions and relate their symbolic and graphical forms.

1. Write a quadratic function in polynomial or standard form, f(x) = ax² + bx + c, to identify the y-intercept of the function’s parabolic graph or the x-coordinate of its vertex, .
2. Write a quadratic function in factored form, f(x) = a(x – r)(x – s), to identify the zeros of the function.
3. Write a quadratic function in vertex form, f(x) = a(x – h)² + k, to identify the vertex and axis of symmetry of the function’s parabolic graph.
4. Describe the effect that changes in the leading coefficient or constant term of f(x) = ax² + bx + c have on the shape, position, and characteristics of the graph of f(x).

   Examples: If a and c have opposite signs, then the roots of the quadratic must be real and have opposite signs; varying c varies the y-intercept of the graph of the parabola; if a is positive, the parabola opens up, if a is negative, it opens down; as |a| increases, the graph of the parabola is stretched vertically, i.e., it looks narrower.

5. Determine domain and range, intercepts, axis of symmetry, and maximum or minimum.

C3 Solve and graph quadratic equations having real solutions using a variety of methods.

1. Solve quadratic equations having real solutions by factoring, by completing the square, and by using the quadratic formula.
2. Estimate the real zeros of a quadratic function from its graph; identify quadratic functions that do not have real zeros by the behavior of their graphs.
3. Use a calculator to approximate the roots of a quadratic equation and as an aid in graphing.

D. Quadratic Functions and Equations with Complex Zeros/Roots

This unit begins with the definition of complex numbers. Extension of the real number system to the complex number system permits solution of all quadratic equations. Students should be comfortable using a variety of solutions methods for quadratic equations and in identifying and interpreting their graphs. These techniques should then be applied to solving and graphing quadratic inequalities and transforming quadratic expressions and equations, including those that are not functions, to extract information.

Successful students will:

D1 Know that if a and b are real numbers, expressions of the form a + bi are called complex numbers and explain why every real number is a complex number.

Every real number, a, is a complex number because it can be expressed as a + 0i. The imaginary unit, sometimes represented as , is a solution to the equation x² = −1.

1. Explain why every real number is a complex number.

   Every real number, a, is a complex number because it can be expressed as a + 0i.

2. Express the square root of a negative number in the form bi, where b is real.

   Just as with square roots of positive numbers, there are two square roots for negative numbers; in , 2i is taken to be the principal square root based on both the Cartesian and trigonometric representations of complex numbers.

   Examples: Determine the principal square root for each of the following:

3. Identify complex conjugates.

   The conjugate of a complex number a + bi is the number a – bi.

D2 Solve and graph quadratic equations having complex roots and find those roots.

1. Use the quadratic formula to solve any quadratic equation and write it as a product of linear factors.
2. Use the discriminant D = b² – 4ac to determine the nature of the roots of the equation ax² + bx + c = 0.
3. Know that complex solutions of quadratic equations with real coefficients occur in conjugate pairs and show that multiplying factors related to conjugate pairs results in a quadratic equation having real coefficients.

   Example: The complex numbers are the roots of the equation = x² - 6x + 14 = 0 whose coefficients are real.

D3 Recognize and solve practical problems that can be expressed using simple quadratic equations; interpret their solutions in terms of the context of the situation.

Examples: Determine the height of an object above the ground t seconds after it has been thrown upward from a platform d feet above the ground at an initial velocity of v₀ feet per second; find the area of a rectangle with perimeter 120 in terms of the length, L, of one side.

1. Create, interpret, and apply mathematical models to solve problems arising from contextual situations that involve quadratic relationships; distinguish relevant from irrelevant information, identify missing information, and find what is needed or make appropriate estimates and apply problem solving heuristics.
2. Select and explain a method of solution (e.g., exact vs. approximate) that is effective and appropriate to a given problem.

D4 Solve and graph quadratic inequalities in one or two variables.

Example: Solve (x – 5)(x + 1) > 0 and relate the solution to the graph of (x – 5)(x + 1) > y.

D5 Manipulate quadratic equations to extract information.

Example: Use completing the square to determine the center and radius of a circle from its equation given in general form.

E. Power and Polynomial Functions and Expressions

Power and polynomial functions are natural extensions of the work done in this course with quadratic functions. The majority of work in this unit involves recognizing power and polynomial functions, identifying some of their characteristics, and applying them to contextual situations. Manipulation of polynomial and rational expressions completes the unit.

Successful students will:

E1 Analyze power functions and identify their key characteristics.

Power functions include positive integer power functions such as f(x) = 3x⁴, root functions such as , and reciprocal functions such as f(x) = kx^-4.

1. Recognize that the inverse proportional function f(x) = k/x (f(x) = kxⁿ for n = –1) and the direct proportional function f(x) = kx (f(x) = kxⁿ for n = 1) are special cases of power functions.
2. Distinguish between odd and even power functions.

   Examples: When the exponent of a power function is a positive integer, then even power functions have either a minimum or maximum value, while odd power functions have neither; even power functions have reflective symmetry over the y-axis, while odd power functions demonstrate rotational symmetry about the origin.

E2 Transform the algebraic expression of power functions using properties of exponents and roots.

Example: can be more easily identified as a root function once it is rewritten as .

1. Explain and illustrate the effect that a change in a parameter has on a power function (a change in a or n for f(x) = axⁿ).

E3 Analyze polynomial functions and identify their key characteristics.

1. Know that polynomial functions of degree n have the general form f(x) = axⁿ + bx^n-1 + ... + px² + qx + r for n an integer, n ≤ 0 and a ≠ 0.

   The degree of the polynomial function is the largest power of its terms for which the coefficient is non-zero.

2. Know that a power function with an exponent that is a positive integer is a particular type of polynomial function, a monomial function, whose graph contains the origin.
3. Distinguish among polynomial functions of low degree, i.e., constant functions, linear functions, quadratic functions, or cubic functions.
4. Explain why every polynomial function of odd degree has at least one zero; identify any assumptions that contribute to your argument.

   At this level students are expected to recognize that this result requires that polynomials are connected functions without "holes." They are not expected to give a formal proof of this result.

5. Communicate understanding of the concept of the multiplicity of a root of a polynomial equation and its relationship to the graph of the related polynomial function.

   If a zero, r₁, of a polynomial function has multiplicity 3, (x - r₁)³ is a factor of the polynomial. The graph of the polynomial touches the horizontal axis at x = r₁ but does not change sign (does not cross the axis) if the multiplicity of r₁ is even; it changes sign at x = r₁ (crosses over the axis) if the multiplicity is odd.

E4 Use key characteristics to identify the graphs of simple polynomial functions.

Simple polynomial functions include constant functions, linear functions, quadratic functions or cubic functions such as f(x) = x³, f(x) = x³ – a, or f(x) = x(x – a)(x + b).

1. Decide if a given graph or table of values suggests a simple polynomial function.
2. Distinguish between the graphs of simple polynomial functions.
3. Where possible, determine the domain, range, intercepts and end behavior of polynomial functions.

   It is not always possible to determine exact horizontal intercepts.

E5 Recognize and solve problems that can be modeled using power or polynomial functions; interpret the solution(s) in terms of the context of the problem.

1. Use power or polynomial functions to represent quantities arising from numeric or geometric contexts such as length, area, and volume.

   Examples: The number of diagonals of a polygon as a function of the number of sides; the areas of simple plane figures as functions of their linear dimensions; the surface areas of simple three-dimensional solids as functions of their linear dimensions; the sum of the first n integers as a function of n.

2. Solve simple polynomial equations and use technology to approximate solutions for more complex polynomial equations.

E6 Perform operations on polynomial expressions.

1. Add, subtract, multiply, and factor polynomials.
2. Divide one polynomial by a lower-degree polynomial.

E7 Use factoring to reduce rational expressions that consist of the quotient of two simple polynomials.

E8 Perform operations on simple rational expressions.

Simple rational expressions are those whose denominators are linear or quadratic polynomial expressions.

1. Add, subtract, multiply, and divide rational expressions having monomial or binomial denominators.
2. Rewrite complex fractions composed of simple rational expressions as a simple fraction in lowest terms.

   Example:

About the Benchmarks

Elementary (K–6) Strands and Grade Levels

Secondary (7–12) Strands

Secondary Model Course Sequences

Secondary Assessments and Tasks

Correlations to the Secondary Benchmarks

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A. Reasoning from Data

B. Applying Exponents

C. Quadratic Functions and Equations with Real Zeros/Roots

D. Quadratic Functions and Equations with Complex Zeros/Roots

E. Power and Polynomial Functions and Expressions

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