|
|
|
|
Content Area: Math
Index: 4.3C Grade 12 CPI 1
Standard: 4.3 - Patterns and Algebra
Strand: C - Modeling
Cumulative Progress Indicator: 1 - The student will use functions to model real-world phenomena and solve problems that involve varying quantities · Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years) · Direct and inverse variation · Absolute value · Expressions, equations and inequalities · Same function can model variety of phenomena · Growth/decay and change in the natural world · Applications in mathematics, biology, and economics (including compound interest)
Grade: 12
Sample Activities:
· 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 = 2x. They look at how much money would be earned by starting out with a penny on the first day and doubling 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.
· Students use a graphing calculator, together with a light probe, to examine 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 then analyzing the graph generated on the calculator to see what kind of graph it is. They use other CBL probes to investigate the kinds of functions used to model a variety of real-world situations.
· Students learn about the Richter Scale for measuring earthquakes, focusing on its relation to logarithmic and exponential functions, and why this kind of scale is used.
· 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, start 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 and repeat the procedure. They graph the results and investigate the behavior of the resulting functions, using a calculator to reduce the computational burden.
· Students work through the Breaking the Mold lesson described in the Introduction to this Framework. They grow mold and collect data on the area of a pie plate covered by the mold. They make a graph showing the percent of increase in the area vs. the days. The students graph their data and find an equation that fits the data to their satisfaction.
|
|
