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.

Despite their formidable titles, these five themes can be addressed with activities at the 3-4 grade level which involve purposeful play and simple analysis. 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 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 grade level discussion of discrete mathematics and use might use activities similar to those described there before introducing the activities for this grade level.

Activities involving systematic listing, counting, and reasoning should 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 total number of possible combinations of dessert and beverage that can be selected from pictures of those two types of foods they have cut out of magazines or that can be selected from a restaurant menu. Similarly, playing games like Nim, dots and boxes, and dominoes becomes a mathematical activity when children systematically reflect on the moves they make in the game and use those reflections to decide on the next move.

An important discrete mathematical model is that of a graph, which is used whenever a collection of things are joined by connectors - such as buildings and roads, islands and bridges, or houses and telephone cables - or, more abstractly, whenever the objects have some defined relationship to each other; this kind of model is described in the K-2 Overview. At the 3-4 grade levels, children can recognize and 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 collaboration graph for the class which describes who has worked with whom during the past week. A special kind of graph is called a "tree." Three views of the same tree are pictured in the diagram below; the first suggests a family tree, the second a tree diagram, and the third a "real" tree.

At the 3-4 grade levels, students can use a tree diagram to organize the six ways that three people can bearranged in order. (See the Grades 3-4 Indicators and Activities for an example.)

Students 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 the activity involving rabbits), in pineapples and pine cones, and in lots of other places in nature. 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.

Students 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 using tree diagrams 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 hidden messages.

Students 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.

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. Another important resource for 3-4 teachers is This Is MEGA-Mathematics!

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 3 and 4.

Building upon knowledge and skills gained in the preceding grades, experiences will be such that all students in grades 3-4:

1. Explore a variety of puzzles, games, and counting problems.

• Students read One Hundred Hungry Ants by Elinor Pinczes and then illustrate and write their own story books (perhaps titled 18 Ailing Alligators or 24 Furry Ferrets) in a style similar to the book using as many different arrangements of the animals as possible in creating their books. They read their books to students in the lower grades.

• Students count the number of squares of each size (1x1, 2x2, 3x3, 4x4, 5x5) that they can find on a geoboard, and in larger square or rectangular grids.

• Students determine the number of possible combinations of dessert and beverage that could be selected from pictures of those two types of foods they have cut out of magazines. Subsequently, they determine the number of possible combinations of dessert and beverage that could be chosen from a restaurant menu, and how many of those combinations could be ordered if they only have $4.

• Students find the number of different ways to make a row of four flowers each of which could be red or yellow. They can model this with Unfix cubes and explain how they know that all combinations have been obtained.

• Students determine the number of different ways any three people can be arranged in order, and use a tree diagram to organize the information. The tree diagram below represents the six ways that Barbara (B), Maria (M), and Tarvanda (T), can be arranged in order. The three branches emerging from the "start" position represent the three people who could be first; each path from left to right represents the arrangement of the three people listed to the right.
  

• Each student uses four squares to make designs where each square shares an entire side with at least one of the other three squares. Geoboards, attribute blocks or Linker cubescan be used. How many different shapes can be made? These shapes are called "tetrominoes."

• Each group of students receives a bag containing four colored beads. One group may be given 1 red, 1 black and 2 green beads; other groups may have the same four beads or different ones. Students take turns drawing a bead from the bag, recording its color, and replacing it in the bag. After 20 beads are drawn, each group makes a bar graph illustrating the number of beads drawn of each color. They make another bar graph illustrating the number of beads of each color actually in the bag, and compare the two bar graphs. As a follow-up activity, students should draw 20 or more times from a bag containing an unknown mixture of beads and try to guess, and justify, how many beads of each color are in the container.

• Students determine what amounts of postage can and cannot be made using only 3 cent and 5 cent stamps.

• Students generate additional rows of Pascal's triangle (at right). They color all odd entries one color and all even entries another color. They examine the patterns that result, and try to explain what they see. They discuss whether their conclusions apply to a larger version of Pascal's triangle.

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

• 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 cent. For the table, they list the available denominations across the top and the postage amounts from 1 cent to 63 cents at the left; they put a checkmark in the appropriate spot if they need the stamp for that amount, and leave it blank otherwise. They try to find a pattern which could be used to decide which amounts of postage could be made if additional stamps (like 64 cents and 128 cents) were used.

• Students play games like Nim and reflect on the moves they make in the game. (See Math for Girls and Other Problem Solvers, by D. Downie et al., for other games for this grade level.) In Nim, you start with a number of piles of objects - for example, you could start with two piles, one with five buttons, the other with seven buttons. Two students alternate moves, and each move consists of taking some or all of the buttons from a single pile; the child who takes the last button off the table wins the game. Once they master this game, students can try Nim with three piles, starting with three piles which have respectively 1, 2, and 3 buttons.

• Students play games like dots and boxes and systematically think about the moves they make in the game. In dots and boxes, you start with a square (or rectangular) array of dots, and two students alternate drawing a line which joins two adjacent dots. Whenever all four sides of a square have been drawn, the student puts her or his initial in the square and draws another line; the person with initials in more squares wins the game.

2. Use networks and tree diagrams to represent everyday situations.

• Students make a collaboration graph for the members of the class which describes who has worked with whom during the past week.

• Students draw specified patterns on the chalkboard without retracing, such as those below. Alternatively, they may trace these patterns in a small box of sand, as done historically in African cultures. (See Ethnomathematics, Drawing Pictures With One Line, or Insides, Outsides, Loops, and Lines.) Alternatively, on a pattern of islands and bridges laid out on the floor with masking tape, students might try to take a walk which involves crossing each bridge exactly once (leaving colored markers on bridges already crossed); note that for some patterns this may not be possible. The patterns given here can be used, but students can develop their own patterns and try to take such a walk for each pattern that they create.

• Students create "human graphs" where they themselves are the vertices and they use pieces of yarn (several feet long) as edges; each piece of yarn is held by two students, one at each end. They might create graphs with specified properties; for example, they might create a human graph with four vertices of degree 2, or, as in the figure at the right, with six vertices of which four have degree 3 and two have degree 2. (The degree of a vertex is the total number of edges that meet at the vertex.) They might count the number of different shapes of human graphs they can form with four students (or five, or six).

• Students use a floor plan of their school to map out alternate routes from their classroom to the school's exits, and discuss whether the fire drill route is in fact the shortest route to an exit.

• Students draw graphs of their own neighborhoods, with edges representing streets and vertices representing locations where roads meet. 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. The class could then share a NJ cake frosted accordingly. (See The Mathematician's Coloring Book.)

• Students recognize and understand family trees in social and historical studies, and instories that they read. Where appropriate, they create their own family trees.

3. Identify and investigate sequences and patterns found in nature, art, and music.

• Students read A Cloak for a Dreamer by A. Friedman, and make outlines of cloaks or coats like those worn by the sons of the tailor in the book by tracing their upper bodies on large pieces of paper. Students could use pattern blocks or pre-cut geometric shapes to cover (tessellate) the paper cloaks with patterns like those in the book or try to make their own cloth designs.

• Students read Sam and the Blue Ribbon Quilt by Lisa Ernst, and by rotating, flipping, or sliding cut-out squares, rectangles, triangles, etc., create their own symmetrical designs on quilt squares similar to those found in the book. The designs from all the members of the class are put together to make a patchwork class quilt or to form the frame for a math bulletin board.

• Students take a "pattern walk" through the neighborhood, searching for patterns in the trees, the houses, the buildings, the manhole covers (by the way, why are they always round?), the cars, etc.; the purpose of this activity is to create an awareness of the patterns around us. By Nature's Design is a photographic journey with an eye for many of these natural patterns.

• Students "tessellate" the plane using squares, triangles, or hexagons to completely cover a sheet of paper without overlapping. They also tessellate the plane using groups of shapes, like hexagons and triangles as in the figure at the right.

• Students might ask if their parents would be willing to give them a penny for the first time they do a particular chore, two pennies for the second time they do the chore, four pennies for the third time, eight pennies for the fourth time, and so on. Before asking, they should investigate, perhaps using towers of Unifix cubes that keep doubling in height, how long their parents could actually afford to pay them for doing the chore.

• Students cut a sheet of paper into two halves, cut the resulting two pieces into halves, cut the resulting four pieces into halves, etc. If they do this a number of times, say 12 times, and stacked all the pieces of paper on top of each other, how high would the pile of paper be? Students estimate the height before performing any calculations.

• 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? Why, or why not?

• Students count the number of rows of bracts on a pineapple or pine cone, or rows of petals on an artichoke, or rows of seeds on a sunflower, and verify that these numbers all appear in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, ... of Fibonacci numbers, where each number is the sum of the two previous numbers on the list. Students find other pictures depicting Fibonacci numbers as they arise in nature, referring, for example, to Fibonacci Numbersin Nature. In Mathematical Mystery Tour by Mark Wahl, an elementary school teacher provides a year's worth of Fibonacci explorations and activities.

• Using a large equilateral triangle provided by the teacher, students find and connect the approximate midpoints of the three sides, and then color the triangle in the middle. (See Stage 1 picture.) They then repeat this procedure with each of the three uncolored triangles to get the Stage 2 picture, and then repeat this procedure again with each of the nine uncolored triangles to get the Stage 3 picture. These are the first three stages of the Sierpinski triangle; subsequent stages become increasingly intricate. How many uncolored triangles are there in the Stage 3 picture? How many would there be in the Stage 4 picture if the procedure were repeated again?
  

4. Investigate ways to represent and classify data according to attributes, such as shape or color, and relationships, and discuss the purpose and usefulness of such classification.

• Students are provided with a set of index cards on each of which is written a word (or a number). Working in groups, students put the cards in alphabetical (or numerical) order, explain the methods they used to do this, and then compare the various methods that were used.

• Students bring to class names of cities and their zip codes where their relatives and friends live, paste these at the appropriate locations on a map of the United States, and look for patterns which might explain how zip codes are assigned. Then they compare their conclusions with post office information to see whether they are consistent with the way that zip codes actually are assigned.

• Students send and decode messages in which each letter has been replaced by the letter which follows it in the alphabet (or occurs two letters later). Students explore other coding systems described in Let's Investigate Codes and Sequences by Marion Smoothey.

• Students collect information about the soft drinks they prefer and discuss various ways of presenting the resulting information, such as tables, bar graphs, and pie charts, displayed both on paper and on a computer.

• Students play the game of Set in which participants try to identify three cards from those on display which, for each of four attributes (number, shape, color, and shading), all share the attribute or are all different. Similar ideas can be explored using Tabletop, Jr. software.

5. Follow, devise, and describe practical lists of instructions.

• Students follow a recipe to make a cake or to assemble a simple toy from its component parts, and then write their own versions of those instructions.

• Students give written and oral directions for going from the classroom to another room in the school, and represent these directions with a diagram drawn approximately to scale.

• Students read Anno's Mysterious Multiplying Jar by Mitsumasa Anno. During a second reading they devise a method to record and keep track of the increasing number of items in the book and predict how that number will continue to grow. Each group explains its method to the class.

• Students write step-by-step directions for a simple task like making a peanut butter and jelly sandwich, and follow them to prove that they work.

• 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 the shortest route from school to home on a map (see figure at right), where each edge has a specified numerical length in meters; students modify lengths to obtain a different shortest route.

• Students write a program which will create specified pictures or patterns, such as a house or a clown face or a symmetrical design. Logo software is well-suited to this activity. In Turtle Math, students use Logo commands to go on a treasure hunt, and look for the shortest route to complete the search.

• Working in groups, students create and explain a fair way of sharing a bagful of similar candies or cookies. (See also the vignette entitled Sharing A Snack in the Introduction to this Framework.) For example, if the bag has 30 brownies and there are 20 children, then they might suggest that each child gets one whole brownie and that the teacher divide each of the remaining brownies in half. Or they might suggest that each pair of children figure out how to share one brownie. What if there were 30 hard candies instead of brownies? What if there were 25 brownies? What if there were 15 brownies and 15 chocolate chip cookies? The purpose of this activity is for students to brainstorm possible solutions in the situations where there may be no solution that everyone perceives as fair.

• Students devise a strategy for never losing at tic-tac-toe.

• Students find different ways of paving just 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. In "muddy city" none of the roads are paved, so that whenever it rains all streets turn to mud. The mayor has asked the class to propose different ways of paving the roads so that a person can get from any one location to any other location on paved roads, but so that the fewest number of roads possible are paved.

• Students divide a collection of Cuisenaire rods of different lengths into two or three groups whose total lengths are equal (or as close to equal as possible).

References

Anno, M. Anno's Mysterious Multiplying Jar. Philomel Books, 1983.

Asher, M. Ethnomathematics. Brooks/Cole Publishing Company, 1991.

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, Darrah. Drawing Pictures with One Line: Exploring Graph Theory. Consortium for Mathematics and Its Applications (COMAP), Module #21, 1992.

Downie, D., T. Slesnick, and J. Stenmark. Math for Girls and Other Problem Solvers. EQUALS. Lawrence Hall of Science, 1981.

Ernst, L. Sam Johnson and the Blue Ribbon Quilt. Mulberry Paperback Book, 1992.

Francis, R. The Mathematician's Coloring Book. Consortium for Mathematics and Its Applications (COMAP), Module #13, 1989.

Fibonacci Numbers in Nature. Dale Seymour Publications

Friedman, A. A Cloak for a Dreamer. Penguin Books. Scholastics.

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.

Kohl, Herbert. Insides, Outsides, Loops, and Lines. New York: W. H. Freeman, 1995.

Murphy, P. By Nature's Design. San Francisco, CA: Chronicle Books, 1993.

Pinczes, E. J. One Hundred Hungry Ants. Houghton Mifflin Company, 1993.

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/.)

Set. Set Enterprises.

Smoothey, Marion. Let's Investigate Codes and Sequences. New York: Marshall Cavendish Corporation, 1995.

Tompert, Ann. Grandfather Tang's Story. Crown Publishing, 1990.

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 Software. TERC.

Turtle Math. LCSI.

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.