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.

Traditionally, by grades 5 and 6, teachers are devoting relatively little time to student modeling with manipulatives since they have begun to concentrate on symbolic and abstract approaches to content. It is fairly common for teachers at this level to think that once initial notions of number and shape have been established with concrete materials in the lower grades, such materials are no longer necessary and a more symbolic approach is preferable. Research shows, however, that concrete materials and the modeling of mathematical operations and concepts is just as useful in these grades as it is for younger students. The content being modeled is, of course, different and so the models are different - but they are no less important.

Fifth- and sixth-graders can use a great variety of materials, including colored rods, base-ten blocks, pattern blocks, fraction strips and circles, and tangrams, to develop very rich notions of rational numbers and the operations associated with them. Initial fraction notions are well-modeled with colored rods. Students use different length rods to represent different units and then decide on the fractional or mixed number value of the other rods. Base-ten blocks, with the values 1, 0.1, 0.01, and 0.001 assigned to the different sizes, model operations with decimals just as well as the values normally associated with base-ten blocks model whole number computation. Pattern blocks and tangram pieces are comfortable and familiar tools with which to begin to explore notions of ratio and proportion.

Geometry models, both two- and three-dimensional, are an important part of learning about geometry and development of spatial sense in students of this age. Students can use geoboards to develop procedures for finding the areas of polygons or irregular shapes. They can also use construction materials like pipe cleaners and straws or cut-out cardboard faces to make complex three-dimensional geometric figures which can be studied directly. It is much easier to determine the number of faces or edges in a figure from such a model than from a two-dimensional drawing of the figure.

Fifth- and sixth-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 two-colored counters to represent positive and negative integers in initial explorations of addition and subtraction of signed numbers. They should be able to use a variety of measurement tools to measure and record the data in a science experiment. They might use play money to concretely construct solutions to coin problems or to riddles that ask how much an individual actually profited or lost in some complex business dealing.

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 children to construct mathematics that is meaningful to them.

There are many appropriate uses for calculators in these grade levels. In his article, Using Calculators inthe 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 at these grade levels. For another powerful example of the first category, consider using an ordinary four-function calculator to explore and begin to describe the relationships between common fractions and decimals. Entering 2/3 into the calculator by pressing 2, then the division key, and then 3 gives a result of 0.6666667. Discussion of this result, attempts to create other similar results, and working out some of the problems by hand lead to discoveries about terminating and non-terminating decimals, repeating decimals, and fraction-decimal equivalence. Such explorations also should be used to highlight the limitations of the calculator, which does not always give the answer 1 when 1/3 is added three times.

Computers are a valuable resource for students in fifth and sixth 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 problem solutions.

In terms of specific mathematics education software, there are many good choices. Logo, of course, can be used effectively by students at these grade 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. Tesselmania!, the Teaching and Learning with Computers series, Elastic Lines, and the Geometry Workshop all allow students to make geometric constructions on the computer screen and then transform them in a variety of ways in order to experiment with the effects of the transformations.

Graph Power, 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 fifth- and sixth-graders as they retrieve and share information. Specifically, in these grades, they might look for demographic data about geographic locations in which they are interested, summaries of the vote totals for different precincts in local elections, and home pages from other schools in this country and abroad.

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 5 and 6.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 5-6 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 work through the Pizza Possibilities lesson that is described in the First Four Standards of this Framework. They use a variety of manipulatives to help them visualize and solve the problem.

• Having used paper-and-pencil to develop an interesting geometric shape which tessellates the plane, the students go to the computer to use Tesselmania! to reproduce it and then to tile the computer screen with it. They color the printout from the program to produce a unique piece of artwork which is then posted in a class display.

• Students engage in the four activities of Target Games: Estimation is Essential! in The New Jersey Calculator Handbook. In these activities, students learn the role that estimation plays in effective calculator use and learn to identify reasonable and unreasonable answers in the calculator display.

• Students explore the rich source of problems at the Math Forum World Wide Web site at Swarthmore College (http://forum.swarthmore.edu).

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

• Using standard base ten blocks as a model, where the four blocks, as usual, represent one, ten, one hundred, and one thousand, students demonstrate their understanding of place value on an assessment by writing in their journals a verbal description (accompanied by drawings) of what a ten-thousand block and a one-hundredth block might look like.

• Students use gumdrops and toothpicks to build a variety of polyhedra. Using these models, they try to generalize a relationship among the faces, edges, and vertices that works for all solids. (There is one! It's called Euler's formula after its discoverer and is: F + V = E + 2.)

• Students read How Much is a Million by David Schwartz. The book describes how tall a stack of a million children standing on each other's shoulders would be, how long it would take to count to a million, and so on. The students pick some object of their own and try to determine how big a space would be needed to contain a million of them. Typical objects to inquire about include blades of grass, pennies, and dollar bills.

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

• Students are asked to enter the number 6561 into their calculators and then to keep pressing the square root key to try to discover what it does. After more experimentation with the key, the students are asked to predict what would have to be entered into the calculator's display if they wanted to press the square root key 6 times and wind up with the number 4.

• Students work through the Two-Toned Towers lesson that is described in the First Four Standards of this Framework. Students use manipulatives to determine how many towers can be built which are 4 cubes tall and use no more than 2 colors, and then discuss the pattern that results when the length of the towers can be 4,5,6, or a larger number of cubes. They also relate their answers to the solution of the Pizza Possibilities lesson on the following page in the First Four Standards.

• Students use Elastic Lines or another version of an electronic geoboard to construct geometric figures and then transform them through rotations, reflections, and translations. Students create a figure and its transformed image on the screen and challenge each other to describe the specific transformation that created the image.

• Students play a computer golf game where they must hit a ball into a hole. The ball and the hole are both visible on the screen, but at opposite sides. Players specify an angular orientation (where 0^o is straight up) and a number of units of length which will describe the path of the ball once it is struck. The object, just like in real golf, is to get the ball in the hole in as few strokes as possible. Good estimation of both angle measure and length are critical to success.

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

• Students divide into groups to make a scale model of their classroom by accurately measuring critical elements of the room, using a standard proportional relationship to convert the actual measurements to the model's measurements, and then measuring again to cut the modeling material (cardboard, balsa wood, or manila paper) to the correct size. Their model of the room should also contain models of the blackboard, the teacher's desk, some student desks, the shelves in the room, and so on. Each student group is responsible for different elements of the room.

• Students measure the volumes of several rectangular boxes by filling them with cubic inch blocks or cubic centimeter blocks. After some thought and discussion, they devise formulas to compute the volumes from direct measurement of the three appropriate dimensions.

• Students use ratios and proportions to determine the heights of objects that are too tall to easily measure directly. Measuring the heights of some known objects and the lengths of the shadows they cast, students determine the heights of the school building, the flagpole, and the tallest tree outside the school by measuring their shadows.

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

• Students use the New Jersey State homepage http://www.state.nj.us on the World Wide Web to gather data about the latest reported population for each county in the state and about the area of the counties. They enter the collected data into two adjacent columns in a spreadsheet and configure a third column to calculate the population density for each county (population / area). They highlight their own county in the printout of the spreadsheet to show where it stands in relationship to the others.

• 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 measure various body parts such as height, length of forearm, length of thigh, length of hands, and arm span. They enter the data into a spreadsheet and produce various graphs as well as a statistical analysis of the class. They update their data every month and discuss the change, as it relates both to individuals and to the class.

• Students conduct a survey of the population of the entire school to determine the most popular of all of the choices for school lunches. After gathering the data, they enter it into a spreadsheet and use the program to graph it in a variety of ways - as a bar graph, a circle graph, and a pictograph. They discuss which of the graphs best illustrates their data and publish the one they choose in a report distributed to all of the students in the school.

• 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."

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

• Students solve the problems posed in Logical Journey of the Zoombinis by using logic and classification and categorization skills. In it, they create Zoombinis, little creatures that have specific characteristics that allow them to accomplish specified tasks.

• Students use their knowledge of theoretical probability to predict the relative frequency of occurrence of each of the possible sums when rolling a pair of dice. They use simulation software like the Graphing and Probability Workshop to simulate the rolling of 300 pairs of dice. They examine the simulated frequencies and judge them to either be consistent or inconsistent with their predictions and reexamine their predictions if necessary.

• Students use the data they gathered earlier concerning the heights of objects and the lengths of the shadows they cast at the same time on a sunny day. They enter the data as ordered pairs (height, shadow) into a simple graphing program and notice that the resulting points all lie on a line. They use the line to predict the heights of objects whose shadows they can measure.

• Students make a pattern using square tiles to build increasingly larger squares (a 1x1, a 2x2, a 3x3, and so on). They count the number of tiles it took to build each successive square and plot the resulting ordered pairs ((1,1), (2,4), (3,9), (4,16), . . . ) on an x-y plane. The resulting parabola is a non-linear function which is easy to discuss.

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

• Students measure each of a variety of objects in both inches and centimeters. They enter the collected data into a spreadsheet as ordered pairs in two adjacent columns, measurements in inches followed by measurements in centimeters. They have the spreadsheet program graph the ordered pairs on an x-y plane. After they discover that all of the points lie on a line, they draw the line and use it to determine the customary measure of an object whose metric measure they know and vice versa.

• Students configure a simple spreadsheet to assist them in finding magic squares by automatically computing all of the sums. For example, they reserve a three-by-three array of cells for the magic square somewhere in the middle of the spreadsheet. In the cells that are at the end of the rows, they enter formulas that show the sums of the entries in the cells in each row, and enter similar formulas at the end of each column and diagonal. When proposed entries are placed in the magic square cells, their various sums are instantly provided in the adjacent cells, facilitating adjustment of the entries. The students then use their new tool to solve and create magic square puzzles.

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. Association of Mathematics Teachers of New Jersey, 1993.

Schwartz, D. How Much is a Million? New York: A Mulberry Paperback Book, 1985.

Software

AppleWorks. Apple Computer Corp.

Elastic Lines. Sunburst Communications.

Geometry Workshop. Scott Foresman.

Graph Power. Ventura Educational Systems.

Graphers. Sunburst Communications.

Graphing and Probability Workshop. Scott Foresman.

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

HyperStudio. Roger Wagner.

Logical Journey of the Zoombinis. Broderbund.

MacStat. Minnesota Educational Computing Consortium (MECC).

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

Table Top. TERC.

Teaching and Learning with Computers. International Business Machine, Inc. (IBM).

Tesselmania! Minnesota Educational Computing Consortium (MECC).

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.