Students in high school construct, recognize, and extend patterns as they encounter new areas of the mathematics curriculum. For example, students in algebra look at the patterns that they find when multiplying binomials and students in geometry look for the patterns that they find in similar triangles. Students in high school should also analyze a variety of different types of sequences, including both arithmetic and geometric ones.

High school students also continue to categorize and classify objects, especially in the context of learning new mathematics. In studying geometry, they classify objects in circles as chords or secants or tangents, for example. In studying algebra, it is critical that students differentiate linear relationships from non-linear relationships.

The function concept is one of the most fundamental unifying ideas of modern mathematics and yet is also one of the most misunderstood concepts at virtually all grade levels. 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 looking at functions as a natural outcome of the earlier discussion of patterns and relationships. Concepts such as domain and range is formalized and the f(x) notation is introduced as natural extensions of initial informal experiences.

Perhaps one of the difficulties with the function concept is its many possible meanings. The formal ordered-pair definition of a function, while perhaps the most familiar to many teachers, is also the least understood and possibly most abstract way of approaching functions (Wagner & 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. Graphs (the good old vertical line test) provide an extremely accessible way of representing functions, especially when graphing calculators and computers are used. Students entering high school should already have encountered functions as a process in the guise of function machines and functions expressed as formulas involving dependent variables, although the correspondence between these two meanings may not be very clear to them. Students need additional experiences in discovering a rule to help them better understand the notion of a dependent variable. Those students moving on to calculus also need to view functions as objects of study in themselves.

High school students should spend considerable time in analyzing relationships among two variables. Beginning with concrete situations (perhaps involving science concepts), students should collect and graph data (perhaps 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 further 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 among 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 written and symbolic form.

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

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

M. analyze and describe how a change in the independent variable can produce a change in a dependent variable.

• Students investigate how increasing the temperature measured in degrees Celsius affects the temperature measured in degrees Fahrenheit and vice versa. Students collect data using water, ice, and a burner. They graph the values they have generated, note that the relationship is linear, and find an equation that fits those values, using a graphing calculator to check how well their equation fits the data.
• 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 of changing the radius of a circle upon its circumference by measuring the radius and the circumference of circular objects. They graph the values they have generated, notice that it is close to a straight line, and develop an equation that describes that relationship.
• 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 make models of cubes using blocks or other manipulatives, and investigate how the volume changes if the length, width, and height are all multiplied by the same constant.
• 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 system and when both fare structures will cost the same amount.

• Students model population growth, decline and decay of people, animals, bacteria and 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, noting that these are trigonometric (wave or periodic) functions.
• 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 removed and note that the graph represents an exponential function. Some of the students try out different equations until they find one that they think fits pretty well. They verify their results using a graphing calculator.
• Students use a graphing calculator, together with a light probe, to develop the relationship between brightness of a light and distance from it. They do this by collecting data with the probe on the brightness of a light bulb at increasing distances and analyzing the graph generated on the calculator.
• Students learn about the Richter Scale for measuring earthquakes, focussing on its representation as an exponential function.
• 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.

• 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 collisions among n bumper cars (ignoring the limitations of physical reality). 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 number of diagonals possible in a triangle, a quadrilateral, and a pentagon, and develops an equation that gives the number of diagonals for an n-sided plane figure. 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 that all of the groups used a similar approach and came up with the general equation n(n-1)/2.
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• Students investigate a number of situations involving the equation y = 2^x. They look at how much money you might earn by starting out with a penny on the first day and doubling the amount on each successive day. They discuss what happens if you 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, money earned for hours worked, the relationship between distance and time if the rate is kept constant, and profit with respect to price per ticket and ticket sales.

• Students look at a graph that shows the height of a flag on its pole at various times and explain what was happening to the flag. They use the computer program Interpreting Graphs to match up other graphs with appropriate problem situations.
• 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, the height of your head from the ground as you ride a ferris wheel as a function of the amount of time since you got on, the time it takes to travel 100 miles as a function of average speed, the cost of mailing a first-class letter based on its weight in ounces.
• Students can use a 30 inch string (constant length) and list all possible lengths and widths of rectangles with integral sides which have this size 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 match each graph, determine an appropriate range of values for each variable, and then graph the functions using graphing calculators or computers. The maximum area, a square, does not have integral values but can be found using the trace function or algebraic procedures. Students also investigate what happens if a circle is made with the string rather than a rectangle.
• Students determine the maximum volume of an open box with various sizes of uniform squares (by half inch intervals from 1/2 to 4 inches) cut from the corners of a 9" x 12" rectangular piece of paper. They begin by actually cutting and folding the paper into a box. Once the data and/or the equation is graphed, students trace the function to find the maximum. They also look for boxes with specific volumes, such as 40 cubic inches, finding two dramatically different boxes with this volume.
• Students take on the role of "forensic mathematicians," trying to determine how tall a person was whose femur is 17 inches long. They measure their own femurs and their heights, entering this 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.

• Students look at the effects of changing the coefficients of a quadratic equation on the graph. For example, how is the graph of y = 4x^2 different from that of y = x^2? How is y = .2x^2 different from y = x^2? How are y = x^2 + 4, y = x^2 - 4, y = x^2 - 4x, and y = x^2 - 4x + 4 each different from y = x^2? Students use graphing calculators to look at the graphs and summarize their conjectures in writing.
• 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 study the behavior of functions of the form y = ax^n. 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 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 understanding.
• 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 computer programs 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 y-value. Using this y-value as the new x-value, they then calculate its corresponding y-value, and so on. (The resulting values are .6, .66, .666, .6666, etc.--an approximation to the decimal value of 2/3!) Students investigate using other starting values for the same function; the results are surprising! They use other equations to repeat the procedure. They graph the results and investigate the behavior of the resulting functions, using a calculator to reduce the computational burden.