J. Algebra

Because major areas of study at postsecondary institutions have different prerequisites, certain mathematics benchmarks are marked with an asterisk (*). These asterisked benchmarks represent content that is recommended for all students, but is required for those students who plan to take calculus in college, a requisite for mathematics and many mathematics-intensive majors. The high school graduate can:

J1. Perform basic operations on algebraic expressions fluently and accurately:

J1.1. Understand the properties of integer exponents and roots and apply these properties to simplify algebraic expressions.

Example:

Simplify the expression to obtain either or .

J1.2.

* Understand the properties of rational exponents and apply these properties to simplify algebraic expressions.

Example:

Explain why for any non-negative number x.

J1.3. Add, subtract and multiply polynomials; divide a polynomial by a low-degree polynomial.

Example: Divide x³ – 8 by x – 2 to obtain x² + 2x + 4;

divide x⁴ – 5x³ – 2x by x² to obtain .

Example: Divide x³ – x² + x – 2 by x² + 1 to obtain and understand that also means that (x² + 1)(x – 1) – 1 = x³ – x² + x – 2.

J1.4. Factor polynomials by removing the greatest common factor; factor quadratic polynomials.

Example:

Remove the greatest common factor 3x³y from 12x³y² + 9x⁴y + 6x⁵y³ to obtain the factorization 3x³y(4y + 3x + 2x²y²).

Example:

Factor x² – 36, 4x² + 12xy + 9y² and x² – 5x – 6 to obtain (x + 6)(x – 6), (2x +3y)² and (x – 6)(x + 1) respectively.

J1.5. Add, subtract, multiply, divide and simplify rational expressions.

(Associated Workplace Task: #1)

(Associated Postsecondary Assignments: #1 and 2)

Example:

Express as a single fraction to obtain .

Example:

Simplify to obtain to obtain 3a (a – b).

J1.6. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified values of their variables.

J1.7. * Derive and use the formulas for the general term and summation of finite arithmetic and geometric series; find the sum of an infinite geometric series whose common ratio, r, is in the interval (–1, 1).

Example:

Derive the formula for the sum S of the first N terms of a geometric series whose first term is 1 and common ratio is r to obtain .

Example:

Determine the 126th term of the arithmetic sequence whose third term is 5 and seventh term is 29.

J2. Understand functions, their representations and their properties:

J2.1. Recognize whether a relationship given in symbolic or graphical form is a function.

J2.2. * Determine the domain of a function represented in either symbolic or graphical form.

Example:

Determine that the domain of the function can be written in interval form as and the domain of the function contains all real numbers except 3 and -3.

J2.3. Understand functional notation and evaluate a function at a specified point in its domain.

(Associated Postsecondary Assignment: #1)

J2.4. * Combine functions by composition, as well as by addition, subtraction, multiplication and division.

J2.5. * Identify whether a function has an inverse and when functions are inverses of each other; explain why the graph of a function and its inverse are reflections of one another over the line y = x.

J2.6. * Know that the inverse of an exponential function is a logarithm, prove basic properties of a logarithm using properties of its inverse and apply those properties to solve problems.

J3. Apply basic algebraic operations to solve equations and inequalities:

J3.1. Solve linear equations and inequalities in one variable including those involving the absolute value of a linear function.

Example:

The length L of a spring in centimeters is given by , where F is the applied force in dynes. What force F will produce a spring length of 14 centimeters?

Example: A pipe is to be cut to a length of 5 meters accurate to within a tenth of a centimeter. Recognize that an acceptable length (in meters) of the pipe satisfies the inequality .

J3.2. Solve an equation involving several variables for one variable in terms of the others.

(Associated Postsecondary Assignment: #2)

Example:

If C represents the temperature in degrees Celsius and F represents the temperature in degrees Fahrenheit, then . Solve this equation for F to obtain .

Example:

Newton's law of gravitation says that the force F exerted by a body of mass m on a body of mass M is, where G is the gravitational constant and r is the distance between the bodies. Solve this equation for r to obtain .

J3.3. Solve systems of two linear equations in two variables.

J3.4. * Solve systems of three linear equations in three variables.

(Associated Postsecondary Assignment: #1)

J3.5. Solve quadratic equations in one variable.

(Associated Postsecondary Assignment: #1)

Example:

Solve x² – x – 6 = 0 by recognizing that x² – x – 6 = (x – 3)(x + 2) can be factored to obtain the two solutions and x = 3 and x = –2.

Example:

Solve x² + 4x + 2 = 0 by using the quadratic formula or by completing the square.

J4. Graph a variety of equations and inequalities in two variables, demonstrate understanding of the relationships between the algebraic properties of an equation and the geometric properties of its graph, and interpret a graph:

J4.1. Graph a linear equation and demonstrate that it has a constant rate of change.

(Associated Postsecondary Assignment: #1)

J4.2. Understand the relationship between the coefficients of a linear equation and the slope and x- and y-intercepts of its graph.

(Associated Postsecondary Assignment: #3)

J4.3. Understand the relationship between a solution of a system of two linear equations in two variables and the graphs of the corresponding lines.

J4.4. Graph the solution set of a linear inequality and identify whether the solution set is an open or a closed half-plane; graph the solution set of a system of two or three linear inequalities.

Example:

Graph the solution set of the system of linear inequalities:

.

J4.5. Graph a quadratic function and understand the relationship between its real zeros and the x-intercepts of its graph.

(Associated Postsecondary Assignment: #1)

Example:

The parabola shown below has equation y = –x² + 2 and passes through the points A, B and C. What is the area of the triangle ABC, rounded to two decimal places?

J4.6. * Graph ellipses and hyperbolas whose axes are parallel to the x and y axes and demonstrate understanding of the relationship between their standard algebraic form and their graphical characteristics.

J.4.7. Graph exponential functions and identify their key characteristics.

Example: Graph the exponential function y(x) = 2^x . Recognize that y(x+1) is twice as large as y(x) since y(x+1) = 2^x+1 = 2 • 2^x = 2 • y(x).

Example: How much money must be invested at 6% annual interest if you want to have $40,000 in 20 years?

J4.8. Read information and draw conclusions from graphs; identify properties of a graph that provide useful information about the original problem. (Associated Postsecondary Assignment: #3)

Example:

The lifetime of the timing belt in your car depends on the tensioning of the belt. The manufacturer specifies 240 N as the proper tension, but the mechanic working on your car can be off by as much as 10%. Use the following graph to estimate the reduction in the life of the belt that can occur with this error in tensioning.

J5. Solve problems by converting the verbal information given into an appropriate mathematical model involving equations or systems of equations; apply appropriate mathematical techniques to analyze these mathematical models; and interpret the solution obtained in written form using appropriate units of measurement:

J5.1. Recognize and solve problems that can be modeled using a linear equation in one variable, such as time/rate/distance problems, percentage increase or decrease problems, and ratio and proportion problems.

(Associated Workplace Tasks: #1 and 2)

(Associated Postsecondary Assignment: #2)

J5.2. Recognize and solve problems that can be modeled using a system of two equations in two variables, such as mixture problems.

(Associated Postsecondary Assignment: #2)

Example:

A chemist has available two solutions of acid. The first solution contains 12% acid, and the second solution contains 20% acid. He wants to mix the two solutions to obtain a 500-milliliter mixture containing 15% acid. How many milliliters of each solution should he mix?

J5.3. Recognize and solve problems that can be modeled using a quadratic equation, such as the motion of an object under the force of gravity.

(Associated Postsecondary Assignment: #1)

Example:

A stone is dropped off a cliff 660 feet above ground. When will the stone hit the ground if its height in feet at time seconds after it is dropped is given by h(t) = 660 – 16 * t² ?

J5.4. Recognize and solve problems that can be modeled using an exponential function, such as compound interest problems.

J5.5. * Recognize and solve problems that can be modeled using an exponential function but whose solution requires facility with logarithms, such as exponential growth and decay problems.

(Associated Postsecondary Assignments: #1 and 2)

Example: How long will it take the balance in your savings account to double if you earn 1.5% compounded annually?

J5.6. Recognize and solve problems that can be modeled using a finite geometric series, such as home mortgage problems and other compound interest problems.

(Associated Workplace Task: #3)

(Associated Postsecondary Assignment: #1)

Example:

How much money will you have in a retirement fund if you deposit $1,000 each year for 20 years and the interest rate remains constant at 4%?

J6. * Understand the binomial theorem and its connections to combinatorics, Pascal's triangle and probability.