Algebraic understanding is necessary for all students regardless of the structure of the 9-12 program. Students involved in a tech prep program or a general math program should have similar experiences. All students should learn the same basic ideas. All students benefit from instructional methods which provide context for the content. Such an approach makes algebra more understandable and motivating.

Manipulative techniques, such as evaluating expressions, remain important, especially for students who will continue into a calculus program. These can be woven into the curriculum or they might all be combined into a separate program which students who intend to pursue a mathematics-related career take in their final year. No matter how this instruction is organized, however, instruction must produce students who understand the logic and purposes of algebraic procedures.

Students should be comfortable with their solving equations, by whatever means they find most appropriate. They should understand the relationship between the graphs of functions and their equations. Prior to high school, they have focused predominantly on linear functions. In high school, students should gain more familiarity with nonlinear functions. They should develop the ability to solve equations and inequalities using appropriate paper-and-pencil techniques as well as technology. They should recognize that the methods they use can be generalized to be used when functions look different but are actually composite functions using a basic type (e.g., sin²x + 2 sin x + 1 = 0 is like x² + 2x + 1 = 0); this method is sometimes called "chunking". This use of patterns to note commonalities among seemingly different problems is an important part of algebra in the high school.

Algebraic instruction at the secondary level should provide the opportunity for students to revisit problems. Traditional school problems leave students with the impression that there is one right answer and that once an answer is found there is no need to continue to think about the problem. Since algebra is the language of generalization, instruction in this area should encourage students to ask questions such as "Why does the solution behave this way?" They should develop an appreciation of the way algebraic representation can make problems easier to understand. Algebraic instruction should be rich in problems which are meaningful to students.

Algebra is the gatekeeper for the future study of mathematics as well as science, social sciences, business, and a host of other areas. In the past, algebra has served as a filter, screening people out of these opportunities. For New Jersey to be part of a global society, it is important that 9-12 algebraic instruction be the culminating experiences of a twelve-year program that open these gates for all.

Building upon K-8 expectations, experiences in grades 9-12 will be such that all students:

N. model and solve problems that involve varying quantities with variables, expressions, equations, inequalities, absolute values, vectors, and matrices.

• Students take data involving two variables in an area of interest to them from the World Almanac, construct a scatterplot, and predict the equation of a function which would be a good model. They use a computerized statistics package or a calculator to fit a function to the data. They choose from linear, quadratic, exponential or logarithmic methods and discuss how well the model fits as well as the limitations.
• Students are asked to use matrices to represent tabular information such as the print runs below for each of two presses owned by a book company. They then calculate a third matrix that indicates the growth in production of each press from 1987 to 1988 and discuss the meaning of the data contained in it.

  1987	Textbooks	Novels	Nonfiction
  Press 1	250,000	125,000	312,000
  Press 2	60,000	48,000	90,000

  1988	Textbooks	Novels	Nonfiction
  Press 1	190,000	100,000	140,000
  Press 2	45,000	60,000	72,000

  Some students perform the operations by hand while others explain how they would do it and then use their graphing calculator or a spreadsheet or write a computer program which accomplishes the task.

• Students use vectors to find out where a plane would be that was flying on a heading of due north at 300 miles per hour while a wind was blowing from the southwest at a speed of 15 miles per hour. Some students draw a diagram, while others use an algebraic approach.

• On a test, students are asked to determine the truth value of the statement "log x > 0 for all positive real numbers x." One student remembers that log x is the exponent needed for 10 to become some number and to get a number less than 10 would require a negative exponent. Another student picks some trial points, produces a t-chart of the points and their logarithms and discovers that when x < 1, log x < 0. A third student graphs the function on his graphing calculator and sees that when x < 1, the graph of log x is below the x-axis.
• Faced with the problem of solving the inequality x² - 3x < 4, some students use the equation
• x² - 3x - 4 = 0 to determine the endpoints of the interval that satisfies the inequality. They factor the equation and find that these endpoints are -1 and 4. They place dots on the number line at those points, since they know the endpoints are included and then substitute 0 for x in the original inequality. When they find that the resulting statement is true, they shade the region connecting the two endpoints to obtain their solution. Some students determine the endpoints in the same way but lightly sketch a two-dimensional representation of the parabola related to the problem and determine that since the inequality was <, it was asking when the parabola was below the x-axis. Their graph indicates that this happens in the region between the two endpoints. Other students used their graphing calculators to graph the function
• x² - 3x and used the trace function to determine when the graph was on or below the line y=4.

• Students use algebra tiles to develop procedures for multiplying binomials and factoring trinomials. They summarize these procedures in their math notebooks, applying them to the solution of real-world problems.
• After many experiences with trying to determine appropriate windows for graphing functions on computers or graphing calculators, students develop an understanding of the need to know what the general behavior of a function will be before they use the technology. Students are then asked to explain how they could determine the behavior of the graph of the function below.
  

  Students factor the numerator and denominator to determine the values which make them zero and use those values to identify the x-intercepts and vertical asymptotes, respectively. They discuss the fact that the factors x and x+6 appear in both places and lead to a removal discontinuity represented by a hole in the graph. They discuss the end behavior of the function as approaching y = 1 and the behavior near the vertical asymptote of x = 6.

• Following a unit on combinations and binomial expansion, students make a journal entry discussing the power of Pascal's Triangle and combinations for expanding binomials as compared to the traditional multiplication algorithm.

• Students are asked to find the solutions to 2^x = 3x². Some students use a spreadsheet to develop a table of values. Once they find an interval of length 1 which contains a solution, they refine their numbers to develop the answer to the desired precision. Other students graph both y=2x and y=3x² using graphing calculators or computers and use the trace function to determine where they intersect. Other students graph the function y=3x²-2^x and use the trace function to find the zeroes. Other students enter the function in their graphing calculator and check the table of values.
• As a portion of a final assessment, students are given one opportunity to place a cup which is supported 6 inches off the ground in such a position as to catch a marble rolled down a ramp. They perform the roll without the cup to locate the point where it strikes the ground. They measure the distance the end of the ramp is off the ground and the distance from the point on the ground directly beneath the end to the point where the marble struck the ground. They generate the quadratic function which models the path of the marble. Several students use different methods to ensure they have the correct function. Then they decide where they will have to place the cup by substituting 6 for the function value and determining the corresponding x-value. Some students solve the equation by hand using the quadratic formula, while others use their trace function on the graphing calculator.

• Students use a graphic representation of the mapping resulting from a multiplication by a negative number (such as -2 as shown below) to explain why the order of the inequality reverses when both sides are multiplied by that negative number.
  They explain that while the original sequence (the inputs) is ascending ( {-2,-1,0,1,2} ), the images are descending ( {4,2,0,-2,-4} ).
• Students explain in their journals how the identity matrix is like the number one.

• Students investigate the characteristics of the linear functions. For example, in y = kx, how does a change in "k" affect the graph? In y = mx + b, what does "b" do? Does "k" in the first equation serve the same purpose as "m" in the second? Students use the graphing calculator to investigate and verify their conclusions.
• Students investigate the effects of a dilation and/or a horizontal or vertical shift on the coefficients of a quadratic function. For example, how does moving the graph up 3 units affect the equation?
• Students look at the effects of changing the coefficients of a quadratic equation on the graph. For example, how is the graph of y = 4x² different from that of y = x²? How is y = .2x² different from y = x²? How are y = x² + 4, y = x² - 4, y = x² - 4x, and y = x² -4x + 4 each different from y = x²? Students use graphing calculators to look at the graphs and summarize their conjectures in writing.
• Students study the behavior of functions of the form y = axⁿ. They investigate the effect of "a" on the curve and the characteristics of the graph when n is even or odd. They use the graphing calculator to assist them and write a sentence summarizing their discoveries.
• Students who have developed the ability to use transformations geometrically and the ability to perform matrix operations are shown how transformations can be represented as matrices and composites of transformations can be found by multiplying the matrices. As part of an assessment during this unit, students are asked to draw the polygon represented by the matrix below and its image under a size change of 1/2. They are further required to develop a matrix which can perform the size change and demonstrate the validity of their matrix to do so.
  

• Students are asked to consider the following situation:

  A landscaping contractor uses a combination of two brands of fertilizers, each containing a different amount of phosphates and nitrates. In a package, brand A has 4 lb. of phosphates and 2 lb. of nitrates. Brand B contains 6 lb. of phosphates and 5 lb. of phosphates. On her current job, the lawn requires at least 24 lb. of phosphates and at least 16 lb. of nitrates. How much of each fertilizer does the contractor need?

  Students represent the given conditions as inequalities and use the intersection of their regions as the set of feasible answers.

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New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition