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.

Seventh- and eighth-graders can use a variety of manipulatives to enhance their mathematical understanding and problem solving ability. For example, new approaches to the teaching of elementary 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 best understand issues of projection, perspective, and shadow by actually building concrete constructions out of blocks or cubes and viewing them from a variety of directions and in different ways. Slicing clay models of various three-dimensional figures convinces students of the resulting planar shape that is the cross-section. Using pipe cleaners and straws, students can build their own version of a Sierpinski tetrahedron.

Seventh- and eighth-graders 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. 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 the 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, Using Calculators in the Middle Grades, in The New Jersey Calculator Handbook, David Glatzer suggests that there are three major categories of calculator use in the middle grades:

To explore, develop, and extend concepts - for example, when the students use the square and square root keys to try to understand these functions and their relationship to each other.

As a problem solving tool - for example, to see how increasing each number in a set by 15 increases the mean of the set.

To learn and apply calculator-specific skills - for example, to learn how to use the memory function of a calculator to efficiently solve a multi-step problem.

These three categories provide a good framework for thinking about calculators in the seventh and eighth grade. For another powerful example of the first category, consider the question of compounding interest. When asked how much a bank account might accumulate after 10 years with an initial balance of $1000 and a simple annual interest rate of 6 percent, most students would first calculate the interest for the first year, add it to the initial balance to get the new balance, multiply that by 0.06 to get the interest for the second year, add that to the previous year's balance, and so forth. After discussion of that iteration, most seventh- and eighth-graders are able to understand that each year's balance is the product of the previous year's balance times 1.06, so to find the balance after three years, one could simply use the formula: $1000 x 1.06 x 1.06 x 1.06. After still more discussion, most students will transform this into the standard formula, which is easy to apply with a calculator: $1000 x 1.06ⁿ. These concepts develop nicely in a classroom where all of the students have calculators and can do the computations easily and quickly. In a traditional classroom without calculators, the progression takes much longer and the resulting formula is much less believable to students.

Students at this level should also have some experience with graphing calculators. Although these tools will be most useful in the high school curriculum, middle school students should be exploring graphs of linear functions and other simple graphs and should be making use of the statistical capabilities of most graphing calculators. They should also be exploring the use of Calculator Based Laboratories (CBL) which enables them to gather data and display the data graphically in the viewing window.

Computers are also an essential resource for students in seventh and eighth grade, and the software tools available for them are more like adult tools than those available for younger children. 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 solutions to problems.

In terms of specific mathematics education software, there are also many good choices. Logo, of course, can be used effectively by students at these levels to explore computer programming and geometry concepts at the same time. It is an ideal tool to learn about one of the critical cumulative progress indicators for Standard 14 (Discrete Mathematics) for grades 5-8: the use of iterative and recursive processes. Oregon Trail II is a very popular CD-ROM program that effectively integrates mathematics applications with social studies. How the West Was One + Three x Four also uses an old west theme to work on arithmetic operations.

A variety of computer golf games allow students to play a competitive game while sharpening their estimation ability with angle and length measure. The Geometric Supposer and Pre-Supposer series is one of the most popular geometry construction tools for students of this age. With it, 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.

Graph Power, the Graphing and Probability Workshop, AppleWorks, TableTop, Graphers, and MacStat are some of the many tools available that include database, spreadsheet, or graphing facilities written for students at this age. Many other valuable pieces of software are available.

The World Wide Web can be an exciting and eye-opening tool for seventh- and eighth-graders as they retrieve and share information. Specifically, in these grades, they might look for good math problems from the Web bulletin boards, biographical data about famous mathematicians, and census data for local towns.

The cumulative progress indicators for grade 8 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 7 and 8.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 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.

• In problem solving situations, students are no longer provided with instructions concerning which tool to use, but rather are expected to select the appropriate tool from the array of manipulatives, calculators, computers and other tools that are always available to them.

• Students continue with activities used in previous grades and use problems like those of Target Games: Estimation is Essential! in The New Jersey Calculator Handbook, in which students learn to identify reasonable and unreasonable answers in the calculator display.

• Students use a variety of materials to demonstrate their understandings of basic mathematical properties and relationships. For instance, they are able to use geoboards, dot paper, and Geometer's Sketchpad to demonstrate the Pythagorean Theorem.

• Students work through the Rod Dogs lesson that is described in The First Four Standards of this Framework. They use Cuisenaire rods to model the increase of the dimensions of an object by various scale factors, but when they realize that there are not enough rods to simulate the situation, they find other models which can be used.

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

• Students use two-colored counters to model signed numbers and integer operations. On the red side, a counter represents +1, on the white side, -1. As sets of counters are combined or separated to model the operations, students look for patterns in the answers so that they can write rules for completing the operations without counters.

• Students use the same counters and also red and white cubes, representing +x and -x, to model and solve equations. By setting up counters and cubes to represent the initial equation and then removing equal sets from both sides, students model the essential elements of solving linear equations and develop the appropriate language with which to discuss those elements. Conversion to symbolic processes comes soon after mastery is achieved with the concrete objects.

• Students make three-inch cubes of clay and then experiment to see in how many different ways they can slice the cube 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.

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

• Students work on this seemingly simple problem from the New Jersey Department of Education's Mathematics Instructional Guide: A copy machine makes 40 copies per minute. How long will it take to make 20,000 copies? A) 5 hours B) 8 hours 20 minutes C) 8 hours 33 minutes D) 10 hours. Students immediately decide to use their calculators to solve the problem, but then have an interesting discussion regarding the calculator display of the answer. When 20,000 is divided by 40, the display shows 8.33333. Which of the answer choices is that? Why?

• Students use the Geometer's Sketchpad to create a triangle on the computer screen and simultaneously place on the screen the measures of each of the angles as well as the sum of the three angles. They notice that the sum of the angles is 180 degrees. They then click and drag one of the vertices around the screen to make a whole variety of other triangles. They notice that even though the measure of each of the angles changes in this process, the sum of 180 degrees never changes, thus intuitively demonstrating the triangle sum theorem.

• Students use videotape of a person walking to model integer multiplication. The videotape shows a person walking forward with a sign that says "forward" and then walking backward with a sign that says "backward." When run forward, the video shows the "forward" walker walking forward (+ × + = +). When run backward, the video shows the "forward" walker walking backward (- × + = -). The other two possibilities also work out correctly to show all of the forms of integer multiplication.

4. Use a variety of tools to measure mathematical and physical objects in the world around them.

• Groups of students build toothpick bridges in a competition to see whose bridge can hold the most weight in the center of the span. Each group has the same materials with which to work and the bridges must all span the same distance. In the process of building the bridges, the students conduct a good deal of research into bridge designs and about factors that contribute to structural strength.

• Students work through the Sketching Similarities lesson that is described in the First Four Standard of this Framework. They use the Geometer's Sketchpad to measure the length of sides and the angles of similar figures to discover the geometric relationships between corresponding parts.

• Students explore the relationship between the height of a ramp and the length of time ittakes a matchbox car to roll down it. The teacher provides stopwatches, long wooden boards, and meter sticks. The students use a spreadsheet program to enter their data relating height and time for several different heights, and use the spreadsheet's integrated graphing program to plot the ordered pairs. They then look for the relationship between the height and the time.

• Students are challenged to answer the question: In how many ways can you measure a ball? After the obvious spatial characteristics are named (volume, diameter, circumference, and so on), students get more creative and suggest bounceability, density, fraction of its height that it loses if a five pound book is placed on it, weight, number of times it bounces when dropped from one meter, and so on. When the list is complete, different groups of students select several of the possible characteristics and develop ways to measure them.

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

• Students use a temperature probe connected to a graphing calculator to collect data about the rate of cooling of a cup of boiling water. The data is displayed in chart form and in a graph by the calculator after the experiment is performed.

• 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 gather data from their fellow students regarding the number of people in their households. They then enter the data into a graphing calculator and learn how to produce a histogram, showing the number of students with each size household, from the data on the calculator.

• Students decide to resolve the debate that two of them were having about which of their favorite baseball players was the better hitter. They find a great deal of numerical data on their respective teams' World Wide Web homepages regarding the number of at bats, singles, doubles, triples, homeruns, and walks each batter had accumulated in his career. The class decides what weight to attribute to each type of hit and then computes a weighted score for each player to decide who the winner is.

• 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."1150

6. Use a variety of technologies to evaluate and validate problem solutions, and to investigate the properties of functions and their graphs.

• Students use Geometer's Sketchpad to work out a solution to this problem from the New Jersey Department of Education's Mathematics Instructional Guide: Two of the opposite sides of a square are increased by 20% and the other two sides are decreased by 10%. What is the percent of change in the area of the original square to the area of the newly formed rectangle? Explain the process you used to solve the problem. In their solution attempts, they construct a square on the screen with known sides, and then a rectangle with the sides indicated in the parameters of the problem. The program calculates the areas of the two figures and the students are close to a solution.

• Students use their calculators to solve this problem from the New Jersey Department of Education's Mathematics Instructional Guide:

A set of test scores in Mrs. Ditkof's class of 20 students is shown below.

62	77	82	88	73	64	82	85	90	75
74	81	85	89	96	69	74	98	91	85

Determine the mean, median, mode, and range for the data.

Suppose each student completes an extra-credit assignment worth 5 points, which is then added to his/her score. What is the mean of the set of scores now if each student received the extra five points? Explain how you calculated your answer.

• Students play Green Globs and Graphing Equations, a computer game in which they score points for writing the equations of lines that will pass through several green globs splattered on the x-y plane.

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

• Students use a simple spreadsheet/graphing program to solve this problem from the New Jersey Department of Education's Mathematics Instructional Guide:

  VOTING RESULTS

  Class Colors	Number of Votes
  red and white	10
  green and gold	12
  blue and orange	5
  black and yellow	9

  Rather than using the tools the problem suggests (protractor, compass, and straight edge), the students enter the data into a spreadsheet and construct a circle graph from the spreadsheet.

• Students measure the temperatures of a variety of differently heated and cooled liquids in both Fahrenheit and Celsius. They then enter the collected data into a spreadsheet as ordered pairs in two adjacent columns, measurements in Fahrenheit followed by measurements in Celsius. They have the spreadsheet program graph the pairs on an x-yplane. After they discover that all of the points lie on a line, they draw the line and use it to determine the Fahrenheit temperature for a given Celsius temperature and vice versa.

• Students configure a spreadsheet to act as an order-processing form for a stationery store (or some other retail operation). They decide on the five items they'd like to sell, enter the prices they'll charge, and then program all of the surrounding cells to compute the prices for the quantities of items ordered, add the tax, and compute the final charge.

  Items	Price	Quantity Ordered	Cost
  Pencils	.05
  Pens	.29
  Paper Pads	.59
  Tape	.49
  Scissors	1.39
  		Tax:
  		Total:

References

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

Glatzer, D. "Using Calculators in the Middle Grades," in The New Jersey Calculator Handbook. New Jersey: Association of Mathematics Teachers of New Jersey, 1993.

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

Software

AppleWorks. Apple Computer Corp.

Geometer's Sketchpad. Key Curriculum Press.

Geometric Pre-Supposer. Sunburst Communications.

Geometric Supposer. Sunburst Communications.

Geometry Workshop. Scott Foresman.

Graph Power. Ventura Educational Systems.

Graphers. Sunburst Communications.

Graphing and Probability Workshop. Scott Foresman.

Green Globs and Graphing Equations. Sunburst Communications.

How the West Was One + Three x Four. Sunburst Communications.

HyperStudio. Roger Wagner.

Logo. Many versions of Logo are commercially available.

MacStat. Minnesota Educational Computing Consortium (MECC).

Oregon Trail II. Minnesota Educational Computing Consortium (MECC).

TableTop. TERC.

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.