Patterns, relationships, and functions continue to provide a unifying theme for the study of mathematics in high school. Pattern-based thinking throughout the earlier grades, as described in the K-12 Overview, and the informal investigations begun in the middle grades have prepared students to make extensive use of both the concept of a function and functional notation. Students should describe the relationships found in concrete situations with algebraic expressions, formulas and equations, as well as with tables of input-output values, with graphs, and with written statements.

Students in high school construct, recognize, and extend patterns as they encounter new areas of the mathematics curriculum. For example, students in algebra recognize patterns when multiplying binomials, and students in geometry utilize patterns in similar triangles. Students in high school should also analyze a variety of different types of sequences, including both arithmetic and geometric sequences, and express their behavior using functional notation.

High school students continue to categorize and classify objects, especially in the context of learning new mathematics. For example, in studying geometry, they classify various lines and line segments as chords or secants or tangents to a given circle. In studying algebra, they distinguish linear relationships from non-linear relationships.

The function concept is one of the most fundamental unifying ideas of modern mathematics. Students begin their study of functions in the primary grades, as they observe and study patterns in nature and create patterns using concrete models. As students grow and their ability to abstract matures, students investigate patterns using concrete models, and then abstract them to form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, students move to expand their knowledge of functions as a natural outcome of the earlier discussion of patterns and relationships. Concepts such as domain and range are formalized and the f(x) notation is introduced as a natural extension of initial informal experiences.

Students frequently have difficulty with the concept of a function, possibly because of its many interpretations. The formal ordered-pair definition of a function, while perhaps the most familiar to many teachers, is also the least understood and possibly the most abstract way of approaching functions (Wagner and Parker, 1993). Looking at functions as correspondences between two sets seems to be more easily grasped while facilitating the introduction of the concepts of domain and range. Visualizing functions as graphs which satisfy the vertical line test provides an extremely accessible way of representing functions, especially when graphing calculators and computers are used. Students entering high school should already be familiar with functions as input-output processes through the use of function machines. They should also have encountered functions given by rules or formulas involving independent and dependent variables. Students moving on to calculus also need to view functions as objects of study in themselves.

The correspondence between all of these interpretations of the concept of a function may not be very clear to students, and so attention should be drawn explicitly to the different ways of understanding functions, and how together they provide a more complete understanding of the concept. For example, while discussing sequences, students should explore how they can be considered as functions using the correspondence model, the rule model, the input-output model, the graph model, and the ordered pairs model.

High school students should spend considerable time in analyzing relationships involving two variables,

and should understand how dependent and independent variables are used. Beginning with concrete situations (possibly involving social studies or science concepts), students should collect and graph data (often using graphing calculators or computers), discover the relationship between the two variables, and express this relationship symbolically. Students need to have experiences with situations involving linear, quadratic, polynomial, trigonometric, exponential, and rational functions as well as piecewise-defined functions and relationships that are not functions at all.

High school students should use functions extensively in solving problems. They should frequently be asked to analyze a real-world situation by using patterns and functions. They should extend their understanding of relationships involving two variables to using functions with several dependent variables in mathematical modeling.

Throughout high school, students continue to work with patterns by collecting and organizing data in tables, by graphing the relationships among variables, and by discovering and describing these relationships in formal, written, and symbolic form.

Reference

Wagner, S., and S. Parker. "Advancing Algebra" in Research Ideas for the Classroom: High School Mathematics, P. Wilson, Ed. New York: Macmillan Publishing Company, 1993.

The cumulative progress indicators for grade 12 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 9, 10, 11, and 12.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

14. Analyze and describe how a change in the independent variable can produce a change in a dependent variable.

• Students investigate the relationship between stopping distance and speed of travel in a car. The students gather data from the driver's education manual, graph the values they have found, note that the relationship is linear, and look for an equation that fits the data.

• Students investigate the effect on the perimeters of given shapes if each side is doubled or tripled. They summarize their findings in writing and symbolically.

• Students investigate how the area of a parallelogram changes as the length of the base is doubled, or the height is doubled, or both are doubled. They repeat the experiment for tripling and quadrupling each measurement. They discuss their findings and represent them symbolically.

• Students compare two fare structures for taxis: one in which the taxi charges $2.75 for the first 1/4 mile and $.50 for each additional 1/4 mile, and one in which $4.25 is charged for the first 1/4 mile and $.20 for each additional 1/8 mile. They develop tables, graph specific points, and generate equations to describe each situation. They find which trips cost more for each fare structure and when both will result in the same cost.

• Students investigate patterns of growth, such as compound interest or bacterial growth, with a calculator. They make a table showing how much money is in a savings account (if none is withdrawn) after one quarter, two quarters, and so on, for ten years. They represent their findings graphically, note that this is not a linear relationship (although simple interest is linear), and write an equation describing the relationship between the amount P deposited initially, the interest rate r, the number n of times that interest is paid each year, the number of years y, and the total T available at the end of that time period: T = (1 + r/n)^ny(P).

15. Use polynomial, rational, trigonometric, and exponential functions to model real world phenomena.

• Students model population growth and decline of people, animals, bacteria and decay of radioactive materials, using the appropriate exponential functions.

• Students use a sound probe and a graphing calculator or computer to collect data on sound waves or voice patterns, and graph these data noting that these patterns are represented by trigonometric functions. They use a light probe to collect data on the relationship between the brightness of the light and its distance from the light source, and analyze the graphprovided by the calculator.

• Students use M&Ms to model decay. They spill a package of M&Ms on a paper plate and remove those with the M showing, recording the number of M&Ms removed. They put the remaining M&Ms in a cup, shake, and repeat the process until all of the M&Ms are gone. They plot the trial number versus the number of M&Ms remaining and note that the graph represents an exponential function. They try different equations until they find one that they think fits the data pretty well. They verify their results using a graphing calculator.

• Students work on this HSPT-like problem from the New Jersey Department of Education's Mathematics Instructional Guide: The Granda Theater has a special rate for groups of 10 or more people: $40 for the first 10 people and $3 for each additional person. Which of the following expressions tells the amount that a group of 10 or more will have to pay if n represents the number of people in the group, where n is at least 10: a. 40 + 3n, b. (40 + 3)n, c. 40 + 3(n + 10), or d. 40 + 3(n - 10)?

• Students learn about the Richter Scale for measuring earthquakes and about the pH measurement of a solution, noting how exponents are built into these measurements. For example, a pH of 4 is 10 times more acidic than a pH of 5 and 100 times more acidic than a pH of 6.

• Students work in groups to investigate what size square to cut from each corner of a rectangular piece of cardboard in order to make the largest possible open-top box. They make models, record the size of the square and the volume for each model, and plot the points on a graph. They note that the relationship is not linear and make a conjecture about the maximum volume, based on the graph. The students also generate a symbolic expression describing this situation and check to see if it matches their data by using a graphing calculator.

16. Recognize that a variety of phenomena can be modeled by the same type of function.

• Different groups of students work on problems with different settings but identical structures. For example, one group determines the number of collisions possible between two, three and four bumper cars at an amusement park and develops an equation to represent the number of possible collisions among n bumper cars (assuming that no two bumper cars collide more than once). Another group investigates the number of possible handshakes between 2, 3, and 4 people, and develops an equation to represent the number of handshakes for n people. A third group discusses the total number of sides and diagonals possible in a triangle, a quadrilateral, and a pentagon, and develops an equation that gives the total number of sides and diagonals for an n-sided polygon. A fourth group looks at the number of games required for a tournament if each team plays every other team only once, while a fifth considers connecting telephone lines to houses. Each group presents its problem, its approach to solving the problem, and its solution. Then the teacher leads the class in a discussion of the similarities and differences among the problems. Students note the similarities between the approaches used by the different groups and that they all came up with the general expression n(n1)/2.

• Students investigate a number of situations involving the equation y = 2^x. They look at how much money would be earned by starting out with a penny on the first day anddoubling the amount on each successive day. They discuss what happens if they start with two bacteria and the number of bacteria doubles every half hour. They consider the total number of pizzas possible as more and more toppings are added. They consider the number of subsets for a given set. They fold a sheet of paper repeatedly in half and look at how many sections are created after each fold.

• Students look for connections among problem situations involving temperature in Celsius and Fahrenheit, the relationship of the circumference of a circle to its diameter, the relationship between stopping distance and car speed, between money earned and hours worked, between distance and time if the rate is kept constant, and between profit and price per ticket.

17. Analyze and explain the general properties and behavior of functions, and use appropriate graphing technologies to represent them.

• As regular parts of their assessments, students make up graphs to represent specific problem situations, such as the cost of pencils that sell at two for a dime, the temperature of an oven as a function of the length of time since it was turned on, their height from the ground as they ride a ferris wheel as a function of the amount of time since they got on, the time it takes to travel 100 miles as a function of average speed, or the cost of mailing a first-class letter based on its weight in ounces.

• Students use a string of constant length, say 30 inches, and list all possible lengths and widths of rectangles with integral sides which have this perimeter. They determine the perimeter and area for each rectangle. Then they make three graphs from their data: length vs. width, length vs. perimeter, and length vs. area. They look for equations to describe each graph, determine an appropriate range of values for each variable, and then graph the functions using graphing calculators or computers. The rectangle of maximum area, a square, does not have integral values, but can be found using the trace function or algebraic procedures. Students also investigate the area of a circle made with the same string and compare it to the areas of the rectangles.

• Students take on the role of "forensic mathematicians," trying to determine the height of a person whose femur was 17 inches long. They measure their own femurs and their heights, entering the class data into a graphing calculator or computer and creating a scatterplot. They note that the data are approximately linear, so they use the built-in linear regression procedures to find the line of best fit and then make their prediction.

18. Analyze the effects of changes in parameters on the graphs of functions.

• Students investigate the characteristics of linear functions. For example, in y = kx, how does a change in k affect the graph? In y = mx + b, what is the role of b? 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 algebraic expression of various types of functions. For example, how does moving a graph up 3 units affect its equation?

• Students look at the effects of changing the coefficients of a quadratic equation on its 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? How is y = sin 4x different from y = 4 sin x? 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.

19. Understand the role of functions as a unifying concept in mathematics

• Students in all mathematics classes use functions, making explicit connections to what they have previously learned about functions. As students encounter a new use or meaning for functions, they relate it to their previous understandings.

• Students use recursive definitions of functions in both geometry and algebra. For example, they define n! recursively as n! = n×(n-1)! They use recursion to generate fractals in studying geometry. They may use patterns such as spirolaterals, the Koch snowflake, the Monkey's Tree curve, the Chaos Game, or the Sierpinski triangle. They may use Logo or other software to iterate patterns, or they may use the graphing calculator. In studying algebra, students consider the equation y = .1x + .6, starting with an x-value of .6, and find the resulting yvalue. Using this yvalue as the new xvalue, they then calculate its corresponding yvalue, and so on. (The resulting values are .6, .66, .666, .6666, ... - providing closer and closer approximations to the decimal value of 2/3!) Students investigate the results of iterations which are other starting values for the same function; the results are surprising! They use other equations and repeat the procedure. They graph the results with a graphing calculator, adjusting the range values to permit viewing the resulting y-values. (See Fractals for the Classroom by H.-O. Peitgen, et al.)

References

Peitgen, Heinz-Otto, et al. Fractals for the Classroom: Strategic Activities, Volume One and Two. Reston, VA: NCTM and New York: Springer- Verlag, 1992.

Wagner, S. and S. Parker. "Advancing Algebra" in Researching Ideas for the Classroom: High School Mathematics, P. Wilson, Ed. New York: Macmillan Publishing Co., 1993.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-nnspecific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.