Students should learn to recognize examples of discrete mathematics in familiar settings, and should explore and solve a variety of problems for which discrete techniques have proved useful. These ideas should be pursued throughout the school years. Students should start with many of the basic ideas in concrete settings, including games and general play, and progressively develop these ideas in more complicated settings and more abstract forms. Five major themes of discrete mathematics should be addressed at all K-12 grade levels -- systematic listing, counting, and reasoning; discrete mathematical modeling using graphs and trees; repetitive patterns and processes; organizing and processing information; and finding the best solution to problems using algorithms.(8)

Despite their formidable titles, these themes can be represented with activities at the elementary grade levels which involve purposeful play and simple analysis. 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 discussions of discrete mathematics and use activities similar to those described there before introducing these activities.

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 three letters followed by three numbers, and make a determination as to whether this total provides an adequate number of license plates for New Jersey drivers. They should also 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 permutations; for example if the three children are Amy (A), Bethamy (B), and Coriander (C), the six different permutations can be described as ABC, ACB, BAC, BCA, CAB, and CBA. Similarly, the number of different ways in which three students out of a class of thirty can be arranged at the blackboard is altogether 30x29x28.

An important discrete mathematical model is that of a 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".) 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 the notion 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. and be able to solve problems by using graphs to model them, and then finding suitable paths.

Children can recognize and work with repetitive patterns and processes involving numbers and shapes, with classroom objects and in the world around them. Building on these explorations, children at the 5-6 grade levels can also recognize and work with iterative and recursive processes. They can explore iteration using LOGO, where they can 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 children, 5-6 graders are fascinated with the Fibonacci sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... where every number is the sum of the previous two numbers (see 3-4 activities), and can now also begin to understand this and other sequences recursively -- where each term of the sequence can be described in terms of preceding terms.

Children at the 5-6 grade levels should investigate sorting items using Venn diagrams, continue their explorations of recovering hidden information by decoding messages, and 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 the 5-6 grade levels, children 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, fairly dividing a cake, planning a tournament schedule, and planning layouts for a class newspaper.

Experiences will be such that all students in grades 5-6, building upon the K-4 expectations:

F. use systematic listing, counting, and reasoning in a variety of different contexts.

• Students determine the number of different sandwiches or hamburgers that can be made at the local eateries.
• Students determine the number of ways any five items (or six, or more items) can be arranged in order, understand the concept of permutation, and explain and evaluate n! for various numbers.
• Students determine the number of ways there are of asking three different students in the class to write their solutions to three homework problems on the blackboard.
• Students determine the number of possible telephone numbers with a given area code and the number of possible license plates with three letters followed by three numbers. They can also investigate why several years ago the telephone company introduced a new area code (908) in New Jersey and why the state license bureau tried (unsuccessfully) to introduce license plates with seven characters.
• Students look for patterns in the diagonals of Pascal's triangle, and the differences of consecutive terms in these diagonals.
• Beth wins the game whenever the two dice give an even total, and Hobart wins whenever the two dice give an odd total. Students should analyze this game and others like it, both experimentally and theoretically, to see whether the game is fair, and if not which player is more likely to win.
• Students make a table which indicates how many of each of the coins of the fictitious country "Ternamy" -- in denominations of 81, 27, 9, 3, and 1 -- are needed to make up any amount from 1 to 200. This table can be used to introduce base 3 ("ternary") numbers, and then numbers in other bases.

• Students color maps (e.g., the fifty states) so that adjacent states have different colors, with as few colors as possible, and use the associated graph (whose vertices correspond to countries) to explain why the map cannot be colored with fewer colors.
• Students convert a street map to a graph where vertices on the graph correspond to intersections on the street map, and use this graph to determine whether a garbage truck can complete its sector without repeating any streets.
• Students play Lewis Carroll's map coloring game, where Player A draws a fictitious map and Player B tries to color the map (adjacent countries must have different colors) using as few colors as possible; Player A's goal is to force Player B to use as many colors as possible.
• Students find paths in graphs which pass through each vertex once, or which pass through each edge once.
• Students play games using graphs -- for example, in the strolling game, two players start with any graph and stroll down a path: Bob starts at any vertex and strolls down an edge to another vertex, then Carol strolls from that vertex along an edge to another vertex, the Bob strolls from that vertex along an edge to another vertex, etc. If each player is only permitted to stroll down unused edges (although vertices may be revisited), and the last stroller wins, who will win the game? How does it depend on the graph that you're using?
• Students write graph problems based on their own real life situations.
• Students plan emergency evacuation routes at school or at home using graphs.

• Students write a description for solving the Tower of Hanoi problem: There are three pegs, on one of which is stacked a number of disks, each smaller than the ones below; the problem is to move the entire stack to another peg, moving them one at a time from one 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 with 5 disks? with 6 disks? How long would it take to do this with 64 disks? (An ancient legend predicts that when this task is completed, the world will end; should we worry?)
• Students use iteration in LOGO to draw checkerboards, stars, and other designs -- iterating the construction of a simple component of the pattern (such as a square) to recreate (for example) an entire checkerboard design.
• Students explore their surroundings to find rectangular objects whose ratio of length to width is the "golden ratio" -- approximately the ratio of two successive Fibonacci numbers; to assist in these explorations, they might cut a rectangular peephole of dimensions 21mm x 34 mm out of a piece of cardboard.
• Students construct "Koch snowflakes of stage 5" by adding 1/9" triangles to the Koch snowflakes of stage 4 described in the grade level 3-4 expectations.
• Students mark a long string midway between the two ends and at one end, and then continue marking it 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. Have students investigate the relationship of the lengths of the segments between marks; what number of marks are possible in this process if it is assume that the marks take up no space on the string, and what happens if you lay the string down in a spiral so that each of the marks line up in a straight line through the center of the spiral. What happens if the rule is changed to "make a new mark midway between the last two marks"?
• Students study the patterns of patchwork quilts, and make one of their own.

• 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 (e.g., 5 x 8 - 2 / 7) can be evaluated unambiguously; they can then experiment with calculators to discover their built-in rules.
• Students explore modular arithmetic through applications involving clocks, calendars, and binary codes.
• Students assign each letter in the alphabet a numerical value (possibly negative) and then look for with words worth a specified number of points.
• Students send and decode messages in which letters of the message are systematically replaced by other letters.
• Students use Venn diagrams to sort and then report on their findings in a survey. For example, then 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?"
• Students bring in zip codes of their relatives, paste them to a map of the United States, and look for patterns which reveal how zip codes may be assigned. They then compare their conclusions with post office materials to see whether they are consistent with the way that zip codes are actually assigned.
• Students use LOGO to make a rectangle procedure that uses variables so that they can create a graphic scene and use their procedure to make objects, such as buildings, of varying sizes.

     [Graphic Not Available]

• Students find the shortest route from one location in their town to another.
• Students use flowcharts to represent visually the instructions for carrying out a complex project.
• Students discuss 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).
• 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 where each team plays each other team once.
• Students plan and develop efficient and attractive layouts for a class newspaper.