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.(2)

Despite their formidable titles, these themes can be represented with activities at the K-2 grade level which involve purposeful play and simple analysis. These five themes are discussed in the paragraphs below.

Activities involving systematic listing, counting, and reasoning can be done very concretely at the K-2 grade level. For example, dressing cardboard teddy bears with different outfits (consisting of, say, shirt and shorts) becomes a mathematical activity when the task is to make a list of all possible outfits and count them. Similarly, playing tic-tac-toe becomes a mathematical activity when children 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 K-2 level, children can recognize graphs and use life-size models of graphs in various ways, for example, by finding a way to get from one island to another by crossing exactly four bridges.

Children can recognize and work with repetitive patterns and processes involving numbers and shapes, using objects in the classroom and in the world around them. For example, children at the K-2 level can create a patterns of tiles to cover a section of the floor (this is called a "tessellation"), can start with a number and repeatedly add three, or can observe how the pattern of branches is repeated by the pattern of veins on leaves.

Children at the K-2 grade levels should investigate ways of sorting items according to attributes like color or shape or size, and ways of arranging data by developing relationships like family trees and building charts and tables. For example, they can use sort attribute blocks or stuffed animals by color or species, organize their families into family trees, and tabulate the number of children who have birthdays in each month by organizing themselves into a bar chart.

Finally, at the K-2 grade levels, children should be able to follow and describe simple procedures and in simple cases determine and discuss what is the best solution to a problem. For example, they should discuss various routes they might take from the classroom to the nurse's office and different ways of dividing a pile of snacks, and should determine the shortest path from one site to another on a map laid out on the classroom floor.

Experiences will be such that all students in grades K-2:

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

• Students use teddy bear cut-outs with, for example, 2 colors of shirts and 3 colors of shorts and decide how many different looking outfits can be made.
• Students make four squares on a geoboard with each square sharing at least one side with one of the other three squares. How many different shapes have been made?
• Each group of students receives a container in which there are 2 red, 1 black and 1 green beads. (Different groups may have similar or different containers.) Students take turns drawing a bead from the container, recording its color, and replacing it in the container. After 20 beads are drawn, the group should make a bar graph and use it to explain the results. As a follow-up activity, students should draw from a container with an unknown mixture of beads and try to guess (and justify) how many of each color is in the container.
• Students (who are beginning to grasp addition) work in groups to figure out the rules of placement and addition that are used to pass from one row to the next in the diagram below:

  1
  1 2 1
  1 3 3 1
  1 4 6 4 1
  

  Can you figure out what the next row should be? (In this diagram, called Pascal's triangle, each number in a row (except the 1s) is below and between two numbers in the row above.)

• Students cut out "coins" labeled 1 cent, 2 cents, 4 cents , 8 cents, and perhaps also 16 cents, 32 cents, etc. depending on ability in addition. For each number in the counting sequence 1, 2, 3, 4, 5, ... (as far as is appropriate for your students), students can determine how to obtain that amount of money using a combination of coins which does not repeat any denomination.
• Students play simple games like tic-tac-toe and discuss why they make the moves they do.

• Students create "human graphs" whose vertices are themselves and whose edges are pieces of yarn or rope; if each child is holding two strings, ask the children to untangle themselves into one or more large loops; if each child is holding three strings, ask the children if they can stand so that no two ropes cross.
• Students create "human graphs" whose vertices are pie plates and whose edges are children, each holding on to two pie plates, one with each hand; you might ask them to create a graph where exactly three children are holding on to each plate, or where each child is holding on to plates of different colors, etc.
• Students walk through a pattern of islands and bridges on the classroom floor to see if they can get from one island to another by crossing exactly four bridges, or to see if they can visit each island exactly once, or to see if they can cross each bridge exactly one, or to see how to get from one island to another by crossing the smallest number of bridges.
• Students color the countries in maps so that adjacent countries have different colors; you can use, for example, the diagram on the next page, giving names of fictitious countries to the seven regions created by the various lines.

• Students use a calculator to create a sequence of ten numbers each of which is three more than the previous one.
• Students "tessellate" the plane, by using groups of triangles (for example, from sets of pattern blocks) to completely cover a sheet of paper without overlapping; they also tessellate the plane using other shapes (or groups of shapes).
• Students collect leaves and tree branches, observe how the pattern of the branches is repeated by the pattern of the veins of the leaves, and explain their observations to the class.
• Students listen for rhythmic patterns in musical selections and use clapping to simulate those patterns.
• Students make "Sierpinski triangles" by starting with a large equilateral triangle (prepared by the teacher), finding and connecting the approximate midpoints of the three sides, and then coloring the triangle in the middle (see first picture). They can then repeat this to get the section picture, and repeat this again to get the third picture.

       [Graphic Not Available]
  

• Students sort themselves by month of birth, and then within each group by height or birthdate.(3)
• Each student is given a card with a number on it, with different students having different numbers. Students are lined up in a row and asked to put the numbers in numerical order by exchanging cards, one at a time, with adjacent children.
• Students sort stuffed animals in various ways and explain why they sorted them as they did.
• Students use two hula hoops (or large circles drawn on paper so that their interiors overlap) to assist in sorting attribute blocks according to two attributes: all red items go inside hoop #1 and all others on the outside; all square items go inside hoop #2 and all others on the outside. What is inside both hoops? What is inside only the first hoop?
• Students develop a sequence of blocks where each block differs from the previous one in only one attribute.

• Students follow directions for a trip within the classroom -- e.g., if you start at a given point, take three steps forward, turn left, go two steps, turn right, and go three more steps, where will you be.
• Students follow oral directions for going from the classroom to the nurse's office, and represent these directions with a diagram.
• Working in groups, students devise and explain a fair way of sharing a bagful of similar candies or cookies (see also the vignette entitled "Sharing A Snack" in Chapter 1).
• Students fill a box (e.g., of dimensions 4"x4"x5") with rectangular blocks (e.g., 10 of dimension 1"x2"x4").
• Students use LOGO to create a procedure to get through a series of mazes.
• Using diagrams like the one below, students can look for the shortest route from SCHOOL to HOME by counting the circles along various routes.

       [Graphic Not Available]