The five major themes of discrete mathematics, as discussed in the K-12 Overview, are systematic listing, counting, and reasoning; discrete mathematical modeling using graphs (networks) and trees; iterative (that is, repetitive) patterns and processes; organizing and processing information; and following and devising lists of instructions, called "algorithms," and using them to find the best solution to real-world problems. Two important resources on discrete mathematics for teachers at all levels are the 1991 NCTM Yearbook Discrete Mathematics Across the Curriculum K-12 and the 1997 DIMACS Volume Discrete Mathematics in the Schools: Making an Impact.

Despite their formidable titles, these five themes can be addressed with activities at the 5-6 grade level which involve both the purposeful play and simple analysis suggested for elementary school students and experimentation and abstraction appropriate at the middle grades. Indeed, teachers will discover that many activities that they already are using in their classrooms reflect these themes. These five themes are discussed in the paragraphs below.

The following discussion of activities at the 5-6 grade levels in discrete mathematics presupposes that corresponding activities have taken place at the K-4 grade levels. Hence 5-6 grade teachers should review the K-2 and 3-4 grade level discussions of discrete mathematics and might use activities similar to those described there before introducing the activities for this grade level.

Activities involving systematic listing, counting, and reasoning at K-4 grade levels can be extended to the 5-6 grade level. For example, they might determine the number of possible license plates with two letters followed by three numbers followed by one letter, and decide whether this total number of license plates is adequate for all New Jersey drivers. They need to become familiar with the idea of permutations, that is, the different ways in which a group of items can be arranged. Thus, for example, if three children are standing by the blackboard, there are altogether six different ways, call permutations, in which this can be done; for example, if the three children are Amy (A), Bethany (B), and Coriander (C), the six different permutations can be described as ABC, ACB, BAC, BCA, CAB, and CBA. Similarly, the total number of different ways in which three students out of a class of thirty can be arranged at the blackboard is altogether 30x29x28, or 24,360 ways, an amazing total!

An important discrete mathematical model is that of a network or graph, which consists of dots and lines joining the dots; the dots are often called vertices (vertex is the singular) and the lines are often called edges. (This is different from other mathematical uses of the term "graph"; the two terms "network" and "graph" are used interchangeably for this concept.) An example of a graph with 24 vertices and 38 edges is given at the left. Graphs can be used to represent islands and bridges, or buildings and roads, or houses and telephone cables; wherever a collection of things are joined by connectors, the mathematical model used is that of a graph. At the 5-6 level, students should be familiar with thenotion of a graph and recognize situations in which graphs can be an appropriate model. For example, they should be familiar with problems involving routes for garbage pick-ups, school buses, mail deliveries, snow removal, etc.; they should be able to model such problems by using graphs, and be able to solve such problems by finding suitable paths in these graphs, such as in the town whose street map is the graph above.

Students should recognize and work with repetitive patterns and processes involving numbers and shapes, with objects found in the classroom and in the world around them. Building on these explorations, fifth- and sixth-graders should also recognize and work with iterative and recursive processes. They explore iteration using Logo software, where they recreate a variety of interesting patterns (such as a checkerboard) by iterating the construction of a simple component of the pattern (in this case a square). As with younger students, 5th and 6th graders are fascinated with the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... where every number is the sum of the previous two numbers. Although the Fibonacci sequence starts with small numbers, the numbers in the sequence become large very quickly. Students can now also begin to understand the Fibonacci sequence and other sequences recursively - where each term of the sequence is described in terms of preceding terms.

Students in the 5th and 6th grade should investigate sorting items using Venn diagrams, and continue their explorations of recovering hidden information by decoding messages. They should begin to explore how codes are used to communicate information, by traditional methods such as Morse code or semaphore (flags used for ship-to-ship messages) and also by current methods such as zip codes, which describe a location in the United States by a five-digit (or nine-digit) number. Students should also explore modular arithmetic through applications involving clocks, calendars, and binary codes.

Finally, at grades 5-6, students should be able to describe, devise, and test algorithms for solving a variety of problems. These include finding the shortest route from one location to another, dividing a cake fairly, planning a tournament schedule, and planning layouts for a class newspaper.

Two important resources on discrete mathematics for teachers at all levels is the 1991 NCTM Yearbook Discrete Mathematics Across the Curriculum K-12 and the 1997 DIMACS Volume Discrete Mathematics in the Schools: Making An Impact. Another important resource for 5-6 teachers is This Is MEGA-Mathematics!

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 will be such that all students in grades 5-6:

6. Use systematic listing, counting, and reasoning in a variety of different contexts.

• Students determine the number of different sandwiches or hamburgers that can be created at local eateries using a combination of specific ingredients.

• Students find the number of different ways to make a row of flowers each of which is red or yellow, if the row has 1, 2, 3, 4, or 5 flowers. Modeling this with Unifix cubes, they discover that adding an additional flower to the row doubles the number of possible rows, provide explanations for this, and generalize to longer rows. Similar activities can be found in the Pizza Possibilities and Two-Toned Towers lessons that are described in the First Four Standards of this Framework.

• Students find the number of ways of asking three different students in the class to write three homework problems on the blackboard.

• Students understand and use the concept of permutation. They determine the number of ways any five items can be arranged in order, justify their conclusion using a tree diagram, and use factorial notation, 5!, to summarize the result.

• Students find the number of possible telephone numbers with a given area code and investigate why several years ago the telephone company introduced a new area code (908) in New Jersey, and why additional area codes are being introduced in 1997. Is the situation the same with zip codes?

• Students estimate and then calculate the number of possible license plates with two letters followed by three numbers followed by one letter. They investigate why the state license bureau tried to introduce license plates with seven characters and why this attempt might have been unsuccessful.

• Students explore the sequence of triangular numbers 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, ... which represent the number of dots in the triangular arrays below, and find the location of the triangular numbers in Pascal's triangle.
  

• Students look for patterns in the various diagonals of Pascal's triangle, and in the differences between consecutive terms in these diagonals. Patterns in Pascal's Triangle Poster is a nice resource for introducing these ideas.

• Students analyze simple games like the following: Beth wins the game whenever the two dice give an even total, and Hobart wins whenever the two dice give an odd total. They play the game a number of times, and using experimental evidence, decide whether the game is fair, and, if not, which player is more likely to win. They then try to justify their conclusions theoretically, by counting the number of combinations of dice that would result in a win for each player.

• Students create a table in the form of a grid which indicates how many of each of the coins of the fictitious country "Ternamy" - in denominations of 1, 3, 9, 27, and 81 "terns" - are needed to make up any amount from 1 to 200. They list the denominations in the columns at the top of the table and the amounts they are trying to make in the rows at the left. They write the number of each coin needed to add up to the desired amount in the appropriate squares in that row. The only "rule" to be followed is that the least number of coins must be used; for example, three 1's should always be replaced by one 3. This table can be used to introduce base 3 ("ternary") numbers, and then numbers in other bases.

7. Recognize common discrete mathematical models, explore their properties, and design them for specific situations.

• Students experiment with drawing make-believe maps which can be colored with two, three, and four colors (where adjacent countries must have different colors), and explain why their fictitious maps, and real maps like the map of the 50 states, cannot be colored with fewer colors. Note that it was proven in 1976 that no map can be drawn on a flat surface which requires more than four colors. The Mathematician's Coloring Book contains a variety of map-coloring activities, as well as historical background on the map coloring problem.

• Students play games using graphs. For example, in the strolling game, two players stroll together on a path through the graph which never repeats itself; they alternate in selecting edges for the path, and the winner is the one who selects the last edge on the path. Who wins? In the game below, Charles and Diane both start at V, Charles picks the first edge (marked 1) and they both stroll down that edge. Then Diane picks the second edge (marked 2) and the game continues. Diane has won this play of the game since the path cannot be continued after the sixth edge without repeating itself. Does Diane have a way of always winning this game, or does Charles have a winning strategy? What if there was a different starting point? What if a different graph was used? What if the path must not cross itself (instead of requiring that it not repeat itself)? Students should try to explain in each case why a certain player has a winning strategy.
  

• Students find paths in graphs which utilize each edge exactly once; a path in a graph is asequence of edges each of which begins where the previous one ends. They apply this idea by converting a street map to a graph where vertices on the graph correspond to intersections on the street map, and by using this graph to determine whether a garbage truck can complete its sector without repeating any streets. See the segment Snowbound: Euler Circuits on the videotape Geometry: New Tools for New Technologies; the module Drawing Pictures With One Line provides a strong background for problems of this kind.

• Students plan emergency evacuation routes at school or from home using graphs.

• All of the students together create a "human graph" where each child in the class is holding two strings, one in each hand. This can be accomplished by placing in the center of the room a number of pieces of yarn (each six feet long) equal to the number of students, and having each student take the ends of two strings. The children are asked to untangle themselves, and discuss or write about what happens.

• Students play the game of Sprouts, in which two students take turns in building a graph until one of them (the winner!) completes the graph. The rules are: start the game with two or three vertices; each person adds an edge (it can be a curved line!) joining two vertices, and then adds a new vertex at the center of that edge; no more than three edges can occur at a vertex; edges may not cross. In the sample game below, the second player (B) wins because the first player (A) cannot draw an edge connecting the only two vertices that have degree less than three without crossing an existing edge.
  

8. Experiment with iterative and recursive processes, with the aid of calculators and computers.

• Students develop a method for solving the Tower of Hanoi problem: There are three pegs, on the first of which is stacked five disks, each smaller than the ones underneath it (see diagram below); the problem is to move the entire stack to the third peg, moving disks, one at a time, from any peg to either of the other two pegs, with no disk ever placed upon a smaller one. How many moves are required to do this?
  

• Students use iteration in Logo software to draw checkerboards, stars, and other designs. For example, they iterate the construction of a simple component of a pattern, such as a square, to recreate an entire checkerboard design.

• Students use paper rabbits (prepared by the teacher) with which to simulate Fibonacci's 13th century investigation into the growth of rabbit populations: If you start with one pairof baby rabbits, how many pairs of rabbits will there be a year later? Fibonacci's assumption was that each pair of baby rabbits results in another pair of baby rabbits two months later - allowing a month for maturation and a month for gestation. Once mature, each pair has baby rabbits monthly. (Each pair of students should be provided with 18 cardboard pairs each of baby rabbits, not-yet-mature rabbits, and mature rabbits.) The Fascinating Fibonaccis by Trudi Garland illustrates the rabbit problem and a number of other interesting Fibonacci facts. In Mathematics Mystery Tour by Mark Wahl, an elementary school teacher provides a year's worth of Fibonacci explorations and activities.

• Students use calculators to compare the growth of various sequences, including counting by 4's (4, 8, 12, 16, ... ), doubling (1, 2, 4, 8, 16, ... ), squaring (1, 4, 9, 16, 25, ... ), and Fibonacci (1, 1, 2, 3, 5, 8, 13, ... ).

• Students explore their surroundings to find rectangular objects whose ratio of length to width is the "golden ratio." Since the golden ratio can be approximated by the ratio of two successive Fibonacci numbers, students should cut a rectangular peephole of dimensions 21mm x 34 mm out of a piece of cardboard, and use it to "frame" potential objects; when it "fits," the object is a golden rectangle. They describe these activities in their math journals.

• Students study the patterns of patchwork quilts, and make one of their own. They might first read Eight Hands Round.

• Students make equilateral triangles whose sides are 9", 3", and 1" (or other lengths in ratio 3:1), and use them to construct "Koch snowflakes of stage 2" (as shown below) by pasting the 9" triangle on a large sheet of paper, three 3" triangles at the middle of the three sides of the 9" triangle (pointing outward), and twelve 1" triangles at the middle of the exposed sides of the twelve 3" segments (pointing outward). To get Koch snowflakes of stage 3, add forty-eight 1/3" equilateral triangles. How many 1/9" equilateral triangles would be needed for the Koch snowflake of stage 4? Fractals for the Classroom is a valuable resource for these kinds of activities and explorations.
  

• Students mark one end of a long string and make another mark midway between the two ends. They then continue marking the string by following some simple rule such as "make a new mark midway between the last midway mark and the marked end" and then repeat this instruction. Students investigate the relationship of the lengths of the segments between marks. How many marks are possible in this process if it is assumed that the marks take up no space on the string? What happens if the rule is changed to "make a new mark midway between the last two marks?"

9. Explore methods for storing, processing, and communicating information.

• After discussing possible methods for communicating messages across a football field, teams of students devise methods for transmitting a short message (using flags, flashlights, arm signals, etc.). Each team receives a message of the same length and must transmit it to members of the team at the other end of the field as quickly and accurately as possible.

• Students devise rules so that arithmetic expressions without parentheses, such as 5 x 8 - 2 / 7, can be evaluated unambiguously. They then experiment with calculators to discover the calculators' built-in rules for evaluating these expressions.

• Students explore binary arithmetic and arithmetic for other bases through applications involving clocks (base 12), days of the week (base 7), and binary (base 2) codes.

• Students assign each letter in the alphabet a numerical value (possibly negative) and then look for words worth a specified number of points.

• Students send and decode messages in which letters of the message are systematically replaced by other letters. The Secret Code Book by Helen Huckle shows these coding systems as well as others.

• Students use Venn diagrams to sort and then report on their findings in a survey. For example, they can seek responses to the question, When I grow up I want to be a) rich and famous, b) a parent, c) in a profession I love, where respondents can choose more than one option. The results can be sorted into a Venn diagram like that at the right, where entries "m" and "f" are used for male and female students. The class can then determine answers to questions like Are males or females in our class more likely to have a single focus? Tabletop, Jr. software can be used to sort and explore data using Venn diagrams.

10. Devise, describe, and test algorithms for solving optimization and search problems.

• Students use a systematic procedure to find the total number of routes from one location in their town to another, and the shortest such route. (See Problem Solving Using Graphs.)

• In Turtle Math, students use Logo commands to go on a treasure hunt, and look for the shortest route to complete the search.

• Students discuss and write about various methods of dividing a cake fairly, such as the "divider/chooser method" for two people (one person divides, the other chooses) and the "lone chooser method" for three people (two people divide the cake using the divider/chooser method, then each cuts his/her half into thirds, and then the third person takes one piece from each of the others). Fair Division: Getting Your Fair Share can be used to explore methods of fairly dividing a cake or an estate.

• Students conduct a class survey for the top ten songs and discuss different ways to use the information to select the winners.

• Students devise a telephone tree for disseminating messages to all 6th grade students and their parents.

• Students schedule the matches of a volleyball tournament in which each team plays each other team once.

• Students use flowcharts to represent visually the instructions for carrying out a complex project, such as scheduling the production of the class newspaper.

• Students develop an algorithm to create an efficient layout for a class newspaper.

References

Bennett, S., et al. Fair Division: Getting Your Fair Share. Consortium for Mathematics and Its Applications (COMAP). Module #9, 1987.

Casey, Nancy, and Mike Fellows. This is MEGA-Mathematics! - Stories and Activities for Mathematical Thinking, Problem-Solving, and Communication. Los Alamos, CA: Los Alamos National Laboratories, 1993. (A version is available online at http://www.c3.lanl.gov/mega-math)

Chavey, D. Drawing Pictures With One Line. Consortium for Mathematics and Its Applications (COMAP). Module #21, 1992.

Cozzens, M., and R. Porter. Problem Solving Using Graphs. Consortium for Mathematics and Its Applications (COMAP). Module #6, 1987.

Francis, R. The Mathematician's Coloring Book. Consortium for Mathematics and Its Applications (COMAP). Module #13.

Garland, Trudi. The Fascinating Fibonaccis. Palo Alto, CA: Dale Seymor Publications, 1987.

Huckle, Helen. The Secret Code Book. Dial Books.

Kenney, M. J., Ed. Discrete Mathematics Across the Curriculum K-12. 1991 Yearbook of the National Council of Teachers of Mathematics (NCTM). Reston, VA, 1991.

Paul, A. Eight Hands Round. New York: Harper Collins, 1991.

Peitgen, Heinz-Otto, et al. Fractals for the Classroom: Strategic Activities Volume One & Two. Reston, VA: NCTM and New York: Springer- Verlag, 1992.

Rosenstein, J. G., D. Franzblau, and F. Roberts, Eds. Discrete Mathematics in the Schools: Making an Impact. Proceedings of a 1992 DIMACS Conference on "Discrete Mathematics in the Schools." DIMACS Series on Discrete Mathematics and Theoretical Computer Science. Providence, RI: American Mathematical Society (AMS), 1997. (Available online from this chapter in http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/.)

Wahl, Mark. Mathematical Mystery Tour: Higher-Thinking Math Tasks. Tucson, AZ: Zephyr Press, 1988.

Software

Logo. Many versions of Logo are commercially available.

Tabletop, Jr. Broderbund, TERC.

Turtle Math. LCSI.

Video

Geometry: New Tools for New Technologies, videotape by the Consortium for Mathematics and Its Applications (COMAP). Lexington, MA, 1992.

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.