This standard addresses the use of calculators, computers and manipulatives in the teaching and learning of mathematics. These tools of mathematics can and should play a vital role in the development of mathematical thought in students of all ages.

High school students can use a variety of manipulatives to enhance their mathematical understanding and problem solving ability. For example, new approaches to the teaching of concepts of algebra incorporate concrete materials at many levels. Two-colored counters are used to represent positive and negative integers as students build a sense of operations with integers. Algebra tiles are used to represent variables and polynomials in operations involving literal expressions. Concrete approaches to equation solving are becoming more and more popular as students deal meaningfully with such mathematical constructs as equivalence, inequality, and balance.

In geometry, students can create solids of revolution by cutting plane figures out of cardboard, attaching a rubber band along an axis of rotation, winding it up, and then letting it unwind by itself, creating a vision of the solid as it does so. Students can use Miras to do reflective geometry - to find the center of a circle or a perpendicular bisector of a line segment. They might build models of a pyramid with a square base and a cube with the same size base to use in an investigation of the relationship of their volumes. Using pipe cleaners and straws, students can build their own version of a Sierpinski tetrahedron.

High school students should also be in the habit of using a variety of materials to help them model problem situations in other areas of the mathematics curriculum. They might use spinners or dice to simulate a variety of real-life events in a probability experiment. Suppose, for instance, they had statistics about the frequency of occurrence of a particular genetic trait in fruit flies and were interested in the probability that they would see it in a given population. A simulation using dice or a spinner might be a useful approach to the problem. They should be able to use a variety of measurement tools to measure and record the data in a science experiment. They might use counters to represent rabbits as they simulate Fibonacci's famous question about rabbit populations.

This list is, of course, not intended to be exhaustive. Many more suggestions for materials to use and ways to use them are given in the other sections of this Framework. The message in this section is a very simple one - concrete materials help students to construct mathematics that is meaningful to them.

There are many appropriate uses for calculators in these grade levels as well. In his article, "Technology and Mathematics Education: Trojan Horse or White Knight?" in The New Jersey Calculator Handbook, Ken Wolff suggests that the availability of calculators, especially graphing calculators, has presented a unique opportunity for secondary mathematics educators. He asserts: "Tedium can be replaced with excitement and wonder. Memorization and mimicry can be replaced with opportunities to explore and discover." Wolff offers some challenging problems for students to try with their calculators to illustrate the scope of what is now possible in secondary classrooms:

What happens when we continually square a number close to the value 1? Try continually taking the square root of a number. Does it matter what number you start with?

Enter the radian measure of an angle and continually take the sine of the resulting values. What happens? Can you explain why it happens? Replace the sine operator with the tangent and repeat the experiment.

Where does the graph of y = 2 sin (3x) intersect the graph of y = - 4x + 3?

He suggests that these are but a few of the problems that students will gladly try if they have a calculator, but would be very reluctant to do without one. Many secondary teachers have had similar experiences with graphing calculators. As graphing calculators pervade the world of secondary mathematics, what we teach and how we teach it will dramatically change. Nowhere more than in these classrooms will the educational impact of this technology be felt. A sample unit on finding regression lines using graphing calculators can be found in Chapter 17 of this Framework.

High school students should be using Calculator Based Laboratories (CBL) in conjunction with their graphing calculators, to generate, analyze, and display data obtained using a variety of probes; discussions of these activities should be coordinated with activities in their science classrooms.

Computers are also an essential resource for students in high school, and the software tools available for them are very much like adult tools. The standard computer productivity tools - word processors, spreadsheets, graphing utilities, and databases - can all be used as powerful tools in problem solving situations, and students should begin to rely on them to help in finding and conveying problem solutions.

In terms of specific mathematics education software, there are also many good choices. The Geometric Supposer and Pre-Supposer series, Geometer's Sketchpad, and Cabri Geometry are all popular geometry construction tools for students of this age. With them, students construct geometric figures on the screen, measure them, transform them, and identify a variety of geometric properties of their creations. Discovery-oriented lessons using these types of software are easy to create and very engaging and useful for students.

Algebra tools include Derive, Maple, and Mathematica. These tools all can manipulate algebraic symbols and equations, solve a variety of equations, do two- and three-dimensional plotting, and much more. The programs offer significantly more power than the graphing calculators, but are also more expensive. They can be used very effectively for classroom presentations with a projection viewing device.

There is also a good variety of algebra learning programs. The Function Supposer, Green Globs and Graphing Equations, and The Algebra Sketchbook are all popular pieces of software that deal with functions and their graphs. Many other valuable pieces of software are available.

The World Wide Web can be an exciting and eye-opening tool for ninth- through twelfth-graders as they retrieve and share information. Specifically, in these grades, they might look for information about colleges in which they might be interested, the history of mathematics, or ecology experiments in which students are gathering and contributing local data.

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:

1^*. Select and use calculators, software, manipulatives, and other tools based on their utility and limitations and on the problem situation.

• Students have a variety of tools available to them in the well-equipped mathematics classroom: a bank of computers loaded with algebraic symbol manipulation and function-plotting programs, spreadsheet and graphing programs, and geometry construction programs; a set of graphing calculators for relevant explorations and computations; and manipulative materials related to the content studied. The students easily move from one type of tool to another, understanding both their strengths and limitations.

• Students work through the Making Rectangles lesson that is described in The First Four Standards of this Framework. They use algebra tiles organized in rectangular form to help them develop procedures for re-writing binomial expressions as multiplication problems (factoring).

• Students can use algebra tiles, Hands-On Equations materials, and a variety of equation-manipulating software to simplify and solve equations. They understand and can demonstrate the relationship between various manipulations of tiles or pawns and the corresponding symbolic actions in the software solution procedure.

• Students use both graphing calculator techniques and paper-and-pencil techniques for solving systems of equations. Depending on the complexity of the system and on the degree of accuracy needed in the answer, they may try to locate the intersection of two graphs by tracing and zooming on the calculator screen, by calculating the solution with matrices, or by using a simple addition or substitution paper-and-pencil method.

2^*. Use physical objects and manipulatives to model problem situations, and to develop and explain mathematical concepts involving number, space, and data.

• Students use a process described in Algebra in a Technological World to construct cones from a circular piece of paper by cutting a wedge-shaped sector from it and then taping together the edges. They then try to find the cone constructed in this manner that has the largest volume. A similar activity is described in The Ice Cones lesson in The First Four Standards of this Framework.

• Students use molds to make cones of clay and then experiment to see in how many different ways they can slice the cone with a plane to produce different cross-sections. Drawings of the cross sections and a description of the cuts that created them are displayed on a poster in the classroom.

• In one of the units of the Interactive Mathematics Program, students read The Pit and the Pendulum by Edgar Allan Poe and then work in groups to investigate the properties and behavior of pendulums. The ultimate goal, after a good deal of measurement and statistical manipulations of their data, is to determine how much time the prisoner in the story has to escape from the 30-foot, razor-sharp, descending pendulum.

3. Use a variety of technologies to discover number patterns, demonstrate number sense, and visualize geometric objects and concepts.

• Students play Green Globs and Graphing Equations, a computer game in which they score points for writing the equations of functions that will pass through several green globs splattered on the x-y plane. As they gain experience with the game, their ability to hit the targets with more and more creative functions improves.

• Students use a set of spherical materials like the Lenart Sphere to study a non-Euclidean geometry. With these materials, students make geometric constructions on the surface of a sphere to realize that, in some geometries, a triangle can have three right angles, and to find the spherical equivalent of the line that is the shortest distance between two points.

• Students use calculators to investigate interesting number patterns. For example, they try to determine why this old trick always works: Enter any three-digit number into the calculator. Without clearing the display, enter the same three digit number again so that you have a six-digit number. Divide the number by 7. Then divide the result by 11. Then divide that result by 13. What is in the display?

5^*. Use technology to gather, analyze, and display mathematical data and information.

• Students use a simulation program to check their predictions regarding the answer to this problem from the New Jersey Department of Education's Mathematics Instructional Guide: Two standard dice are rolled. What is the probability that the sum of the two numbers rolled will be less than 5? A) 1/3 B) 1/6 C) 1/9 D) 1/12. After determining the probability theoretically, they use a simulation program for 1000 rolls of two dice and check the outcome data to see if their predicted probability was in the right ballpark.

• Students use HyperStudio to create the reports they write about biographies of mathematicians, about how mathematics is used in real life, or about solutions to problems they've solved. The software allows them to create true multimedia presentations.

• Students explore the great wealth of mathematical information available at the University of St. Andrews' History of Mathematics World Wide Web site (http://www.groups.dcs.st-and.ac.uk/~history/).

• Students use Algebra Animator software to simulate and manipulate the motion of a variety of objects such as cars, projectiles, and even planets. They gather data about the motion and directly visualize both the functions that describe the motion and their graphs.

• Working in small groups, students use a distance probe connected to a graphing calculator to collect data about the rate of approach of a classmate walking toward the calculator. After the walk is finished, the calculator plots the student's position relative to the calculator as a function of time. The group then presents the finished graph to the rest of the class and challenges them to describe the walk that was taken: What rate of progress was made? Was it steady progress? Where did the student stop? Was there ever any backward walking?

• Students explore the rich links suggested on the Cornell University Math and Science Gateway World Wide Web Site (http://www.tc.cornell.edu/Edu/MathSciGateway).

• There is always math help available at the Dr. Math World Wide Web site (dr.math@forum.swarthmore.edu). In Dr. Math's words, "Tell us what you know about your problem, and where you're stuck and think we might be able to help you. Dr. Math will reply to you via e-mail, so please be sure to send us the right address. K-12 questions usually include what people learn in the U.S. from the time they're five years old through when they're about eighteen."

7. Use computer spreadsheets and graphing programs to organize and display quantitative information and to investigate properties of functions.

• Students work through the Building Parabolas lesson that is described in The First Four Standards of this Framework. They use both the Green Globs software and their graphing calculators to investigate how the various coefficients affect the graph of parabolas.

• Students use calculators, a spreadsheet, and an integrated plotter to work on this problem from Algebra in a Technological World:
  

  A new professional team is in the process of determining the optimal price for a special ticket package for its first season. A survey of potential fans reveals how much they are willing to pay for a four-game package. The data from the survey are displayed below.

  Price of the Four-Game Package	Number of Packages That Could Be Sold at That Price
  $96.25	5,000
  90.00	10,000
  81.25	15,000
  56.25	25,000
  50.00	27,016
  40.00	30,000
  21.25	35,000

  On the basis of the foregoing data, find a relationship that describes the price of a package as a function of the number sold (in thousands). Then determine the selling price which will maximize the revenue, and its number of packages likely to be sold at this price.

• Students investigate the growth of the world's population by researching estimates of the level of population at various times in history and plotting the corresponding ordered pairs in a piece of software called Data Models. They then use the software tool to find a line or curve of best fit and use the resulting graph to predict the population in the year 2100. As a last step, they find the predictions made by several social scientists and compare them to their own.

• Students work on a lesson from The New Jersey Calculator Handbook which uses graphing calculators to focus on the linear functional relationship between circumference and diameter. The students measure everyday circular objects to collect a sample of diameters and circumferences. They then enter their data into a calculator which plots a scattergram for them and finds a line of best fit. The slope of the line is, of course, an approximation of pi.

• Students use the Geometry Inventor for constructions which illustrate a proof of the Pythagorean Theorem. With the construction tool, they create a right triangle in the center of the computer screen, and a square on each of the legs. They then make a table of the areas of the three squares. As they manipulate the triangle to adjust the relationship among the lengths of the legs, they notice that the basic additive relationship of the areas of the three squares remains the same.

• Students use The Geometer's Sketchpad to create an initial polygon and then apply a series of complex transformations to it resulting in a whole sequence of transformed polygons spread out across the screen. The results are often striking colorful images that the students can preserve as evidence of the connections between geometry and modern artistic design.

8. Use calculators and computers effectively and efficiently in applying mathematical concepts and principles to various types of problems.

• Students quickly determine the appropriate window for finding the intersection of two functions by playing with the zoom and range functions on a graphing calculator.

• Students solve a variety of on-line trigonometry problems posted on the Trigonometry Explorer World Wide Web site (http://www.cogtech.com/EXPLORE).

• Having just conducted a science experiment where they collected data about the rates of cooling of a liquid in three different containers, the students quickly and efficiently enter the data into a computer spreadsheet and generate broken-line graphs to represent the three different settings.

• Students solve the following problem by writing a function that describes the volume of the box, plotting the function on a graphing calculator, and searching visually for the peak of the graph. An open-topped box is made from a six-inch square piece of paper by cutting a square out of each corner, folding up the sides and taping them together. What size square should be cut out of the corners to maximize the volume of the box that is formed?1150

References

Association of Mathematics Teachers of New Jersey. The New Jersey Calculator Handbook. 1993.

Fendel, D., D. Resek, L. Alper, and S. Fraser. Interactive Mathematics Program. Key Curriculum Press.

Heid, M.K., et al. Algebra in a Technological World. Reston, VA: National Council of Teachers of Mathematics, 1995.

Lenart Sphere. Key Curriculum Press.

New Jersey Department of Education. Mathematics Instructional Guide. D. Varygiannes, Coord. Trenton, NJ, 1996.

Wolff, K. "Technology and Mathematics Education: Trojan Horse or White Knight?" in The New Jersey Calculator Handbook. Association of Mathematics Teachers of New Jersey, 1993.

Software

Algebra Animator. Logal.

The Algebra Sketchbook. Sunburst Communications.

Cabri Geometry. IBM.

Data Models. Sunburst Communications.

Derive. Soft Warehouse.

The Function Supposer. Sunburst Communications.

Geometer's Sketchpad. Key Curriculum Press.

Geometric Pre-Supposer. Sunburst Communications.

Geometric Supposer. Sunburst Communications.

Geometry Inventor. Logal.

Green Globs and Graphing Equations. Sunburst Communications.

HyperStudio. Roger Wagner.

Maple. Brooks/Cole Publishing Co.

Mathematica. Wolfran Research.

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-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.