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.(4)

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 3-4 grade levels in discrete mathematics presupposes that corresponding activities have taken place at the K-2 grade levels. Hence 3-4 grade teachers should review the K-2 discussion of discrete mathematics and use activities similar to those described there before introducing these activities.

Activities involving systematic listing, counting, and reasoning can be done very concretely at the 3-4 grade levels, building on similar activities at the K-2 grade levels. For example, the children could systematically list and count the number of possible combinations of dessert and beverage that can be chosen from a fixed menu. Similarly, playing games like Nim, dots, dominos, and sprouts becomes a mathematical activity when children systematically reflect on the moves they make in the game.

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 3-4 levels, children can recognize use models of graphs in various ways, for example, by finding a way to get from one island to another by crossing exactly four bridges, or by finding a route for a city mail carrier which uses each street once, or by constructing a friendship graph for the class which describes who is friends with who. A special kind of graph is called a "tree"; at the 3-4 grade levels, students can draw a family tree and recognize that as an example of a graph.

Children can recognize and work with repetitive patterns and processes involving numbers and shapes, with classroom objects and in the world around them. Children at the 3-4 grade levels 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. This sequence of numbers turns up in petals of flowers, in the growth of populations (see activity involving rabbits), in pineapples and pine cones, and lots of other places. Although this sequence starts with small numbers, the numbers in this sequence become large very quickly (a million rabbits appear in no time). Another important sequence to introduce at this age is the doubling sequence 1, 2, 4, 8, 16, 32, ... and to discuss different situations in which it appears.

Children at the 3-4 grade levels should investigate ways of sorting items according to attributes like color or shape, or by quantitative information like size, arranging data by developing relationships like family trees and building charts and tables, and recovering hidden information in games and encoded messages. For example, they can sort letters into zip code order or sort the class alphabetically, create bar charts based on information obtained experimentally (such as soda drink preferences of the class), and play games like hangman to discover concealed messages.

Children at the 3-4 grade levels should describe and discuss simple algorithmic procedures such as providing and following directions from one location to another, and should in simple cases determine and discuss what is the best solution to a problem. For example, they might follow a recipe to make a cake or to assemble a simple toy from its component parts. Or they might find the best way of playing tic-tac-toe or the shortest route that can be used to get from one location to another.

Experiences will be such that all students in grades 3-4, building upon the K-2 expectations:

A. play and explore a variety of puzzles, games, and counting problems.

• Students determine the number of possible combinations of dessert and beverage that could be chosen from a fixed menu, and how many of those combinations could be ordered if they only have $2.
• Students find the number of different rows that can be made in a "garden" if each row has four flowers which can be red or yellow. They can model this with Unifix cubes and explain how they know that all combinations have been obtained.
• Students determine the number of zip codes that are possible (using five digits). Students also 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 are assigned.
• Students determine the number of different ways any three items (or four, or more, items) can be arranged in order, and use the factorial notation 3! to summarize the result.
• Students generate additional rows of Pascal's triangle (see K-2 activities), coloring odd entries with one color and even entries with another color, explain the patterns that result, and project their conclusions to a still larger version of Pascal's triangle.
• Students determine what amounts of postage can and cannot be made using only 3 and 5 stamps.
• Students make a table indicating which stamps of the denominations 1 cent, 2 cents, 4 cents, 8 cents, 16 cents, 32 cents would be used (with no repeats) to obtain each amount of postage from 1 cent to 63 cents, and describe a pattern which could be used if additional stamps (like 64 cents and 128 cents) were used.
• Students play games like Nim, dots, dominos, and sprouts, and systematically reflect on the moves they make in the game.(5) In Nim, you start with a number of piles of counters -- for example, you could start with three piles which have respectively 1, 2, and 3 counters. Two children alternative moves, and each move consists of taking some or all of the counters from a single pile; the child who takes the last counters wins the game. In dots, you start with a square (or rectangular) array of dots, and two children alternate drawing a line which joins two adjacent dots. Whenever all sides of a square have been drawn, the child puts her or his initial in the square and draws another line; the person with initials in the larger number of squares wins the game.

• Students make a friendship graph for the class which describes who's friends with who.
• Students count the number of different "human graphs" (see K-2 activities) that four (or five or six) of them can form.
• Students create "human graphs" (see K-2 activities) with specified properties (e.g., four vertices with degree 2, or six vertices with four of degree 3 and two of degree 2).
• Students trace specified patterns in a small box of sand (without retracing) as done historically in African cultures.(6) One simple pattern that can be drawn in a continuous line without retracing is given at the right.
• Students use the floor plan of their school to map out alternate routes from their classroom to the school's exits.
• Students draw graphs representing their neighborhoods, with edges representing streets and vertices representing crossings. Can you find a route for the mail carrier in your neighborhood which enables her to walk down each street, without repeating any streets, and which ends where it begins? Can you find such a route if she needs to walk up and down each street in order to deliver mail on both sides of the street?
• Students color maps (e.g., the 21 counties of New Jersey) so that adjacent counties (or countries) have different colors using as few colors as possible (and then together bake and eat a NJ cake frosted accordingly).
• Students create their own family trees.

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• Students might ask if their parents would be willing to give them a penny for the first time they do a pet peeve chore, two pennies for the second time they do the chore, four pennies for the third time they do the chore, eight pennies for the fourth time they do the chore, and so on. Before asking, they should investigate, perhaps using towers of Unifix cubes that keep doubling, how long their parents could actually afford to pay them for doing the chore. (A related problem involves cutting a sheet of paper into two halves, cutting the resulting two pieces into halves, cutting the resulting four pieces into halves, etc.; if you did this a number of times, say 15 times, and stacked all the pieces of paper on top of each other, how high would the pile of paper be?)
• Students use paper rabbits (prepared by the teacher) with which to simulate Fibonacci's 12th century investigation into the growth of rabbit populations: If you start with one pair of 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 -- you have to allow a month for maturation and a month for gestation. (Each pair of students should be provided with 18 cardboard pairs each of baby rabbits, not-yet-mature rabbits, and mature rabbits.)
• Students should 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 3" 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 six 1" triangles at the middle of the exposed sides of the 3" triangles (pointing outward). (Photocopying at 33% is a possibility.) To get Koch snowflakes of stage 4, add 1/3" equilateral triangles.
• Students color half a large square, then half of the remaining portion with another color, then half of the remaining portion with a third color, etc. Will the entire area ever get colored?
• Students find pictures depicting Fibonacci numbers as they arise in nature.(7)

• Groups of students alphabetize a list of words or a stack of index cards -- alphabetically if they contain words or numerically if they contain numbers.
• Students play "hangman" or Wheel of Fortune in which they try to determine an unknown word or phrase by asking about the occurrence of individual letters, and discuss why they selected the letters they did.
• Students send and decode messages in which each letter has been replaced by the letter which follows it in the alphabet (or is two away).
• Student collect information about their soft drink preferences and discuss various ways of presenting the resulting information, such as bar graphs and pie charts.

• Students give orally directions for going from the classroom to the nurse's office, and represent these directions with a diagram drawn approximately to scale.
• Students follow a recipe to make a cake or to assemble a simple toy from its component parts.
• Students find and describe the shortest path from the computer to the door or from one location in the school building to another.
• Students find different ways of paving enough streets of a "muddy city" (like the street map at the right, perhaps laid out on the floor) so that a child can walk from any one location to any other location along paved roadways.
• Students write out directions for a simple project like making a peanut butter and jelly sandwich, and act out the directions to prove that they work.
• Students write a program in LOGO which will create specified pictures, such as a house or a clown face.
• Working in groups, students devise and explain a fair way of dividing a bag containing a variety of different candies.
• Students devise a strategy for never losing at tic-tac-toe.
• Students find the shortest route from SCHOOL to HOME on a map (see K-2 activities), where each edge has a specified numerical length.
• Students divide a collection of rods of different lengths into two or three groups whose total lengths are the same (or as close to the same as possible).

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(6) See Ethnomathematics by Marcia Ascher, Brooks/Cole Pub. Co., 1991).

(7) A useful resource is the poster Fibonacci Numbers in Nature from Dale Seymour Publications.