STANDARD 14 - DISCRETE MATHEMATICS
All students will apply the concepts and methods of discrete
mathematics to model and explore a variety of practical
Standard 14 - Discrete Mathematics - Grades 7-8
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 7-8 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 7-8 grade levels
in discrete mathematics presupposes that corresponding
activities have taken place at the K-6 grade levels. Hence 7-8 grade
teachers should review the K-2, 3-4, and 5-6 grade level
discussions of discrete mathematics and might use activities similar
to those described there before introducing the activities for
this grade level.
Students in 7th and 8th grade should be able to use permutations
and combinations and other counting strategies in a wide
variety of contexts. In addition to working with permutations,
where the order of the items is important (see Grades 5-6 Overview and
Activities), they should also be able to work with combinations, where
the order of the items is irrelevant. For example, the number of
different three digit numbers that can be made using three different
digits is 10x9x8 because each different ordering of the three digits
results in a different number. However, the number of different
pizzas that can be made using three of ten available toppings is
(10x9x8)/(3x2x1) because the order in which the toppings are
added is irrelevant; the division by 3x2x1 eliminates the
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.")
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. Students in the 7th and 8th grades should be able to use
graphs to model situations and solve problems using the
model. For example, students should be able to use graphs
to schedule a school's extracurricular activities so that, if at
all possible, no one is excluded because of conflicts. This can be
done by creating a graph whose vertices are the activities, with two
activities joined by an edge if they have a person in common, so that
the activities should be scheduled for different times. Coloring the
vertices of the graph so that adjacent vertices have different colors,
using a minimum number of colors, then provides an efficient solution
to the scheduling problem - a separate time slot is needed for
each color, and two activities are scheduled for the same time slot if
they have the same color.
Students can recognize and work with iterative and recursive
processes, extending their earlierexplorations of repetitive
patterns and procedures. In the 7th and 8th grade, they can
combine their understanding of exponents and iteration to solve
problems involving compound interest with a calculator or spreadsheet.
Topics which before were viewed iteratively - arriving at the
present situation by repeating a procedure n times - can
now be viewed recursively - arriving at the present situation by
modifying the previous situation. They can apply this understanding
to Fibonacci numbers, to the Tower of Hanoi puzzle, to programs in
Logo, to permutations and to other areas.
Students in the 7th and 8th grades should 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. Students should
investigate and report about various codes that are commonly used,
such as binary codes, UPCs (universal product codes) on grocery items,
and ISBN numbers on books. They should also explore how
information is processed. A useful metaphor is how a waiting line
or queue is handled (or "processed") in various situations;
at a bank, for example, the queue is usually processed in
first-in-first-out (FIFO) order, but in a supermarket or restaurant
there is usually a pre-sorting into smaller queues done by the
shoppers themselves before the FIFO process is activated.
In the 7th and 8th grade, students should be able to use
algorithms to find the best solution in a number of
situations - including the shortest route from one city to
another on a map, the cheapest way of connecting sites into a network,
the fastest ways of alphabetizing a list of words, the optimal route
for a class trip (see the Short-Circuiting Trenton lesson in
the Introduction to this Framework), or optimal work schedules
for employees at a fast-food restaurant.
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. Teachers of grades 7-8
would also find useful the textbook Discrete Mathematics
Standard 14 - Discrete Mathematics - Grades 7-8
Indicators and Activities
The cumulative progresses 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 7 and
Building upon knowledge and skills gained in the preceding grades,
experiences will be such that all students in grades 7-8:
6. Use systematic listing, counting, and reasoning in a
variety of different contexts.
- Students determine the number of possible different
sandwiches or hamburgers that can be created at local eateries using a
combination of specified ingredients. They find the number of pizzas
that can be made with three out of eight available toppings and relate
the result to the numbers in Pascal's triangle.
- Students determine the number of dominoes in a set
that goes up to 6:6 or 9:9, the number of candles used throughout
Hannukah, and the number of gifts given in the song "The Twelve
Days of Christmas," and connect the results through discussion
of the triangular numbers. (Note that in a 6:6 set of dominoes there
is exactly one domino with each combination of dots from 0 to 6.)
- Students determine the number of ways of spelling
"Pascal" in the array below by following a path from top to
bottom in which each letter is directly below, and just to the right
or left of the previous letter.
- Students design different license plate systems for
different population sizes; for example, how large would the
population be before you would run out of plates which had only three
numbers, or only five numbers, or two letters followed by three
- Students find the number of different ways of making
a row of six red and yellow flowers, organize and tabulate the
possibilities according to the number of flowers of the first color,
and explain the connection with the numbers in the sixth row of
Pascal's triangle. (See also Visual Patterns in
- Students pose and act out problems involving the
number of different ways a group of people can sit around a table,
using as motivation the scene of the Mad Hatter at the tea party.
(See Mathematics, a Human Endeavor, p. 394.)
- Students count the total number of different cubes
that can be made using either red or green paper for each face. (To
solve this problem, they will have to use a "break up the problem
into cases" strategy.)
- Students determine the number of handshakes that
take place if each person in a room shakes hands with every other
person exactly once, and relate this total to the number of line
segments joining the vertices in a polygon, to the number of
two-flavor ice-cream cones, and to triangular numbers.
- Students count the number of triangles or
rectangles in a geometric design. For example, they should be able to
count systematically the number of triangles (and trapezoids) in the
figure below to the left, noting that there are triangles of three
sizes, and the number of rectangles in the 4x5 grid pictured below to
the right, listing first all dimensions of rectangles that are
7. Recognize common discrete mathematical models,
explore their properties, and design them for specific
- Students find the minimum number of colors needed to assign
colors to all vertices in a graph so that any two adjacent vertices
are assigned different colors and justify their answers. For example,
students can explain why one of the graphs below requires four colors
while for the other, three colors are sufficient.
Students use graph coloring to solve problems which
involve avoiding conflicts such as: scheduling the school's
extra curricular activities; scheduling referees for soccer games;
determining the minimum number of aquariums needed for a specified
collection of tropical fish; and assigning channels to radio stations
to avoid interference. In the graph at the right an edge between two
animals indicates that they cannot share a habitat. The videotape,
Geometry: New Tools for New Technologies has a segment
Connecting the Dots: Vertex Coloring which discusses
the minimum number of habitats required for this situation.
- Students use tree diagrams to represent and analyze
possible outcomes in counting problems, such as tossing two dice.
- Students determine whether or not a given group of dominoes
can be arranged in a line (or in a rectangle) so that the number of
dots on the ends of adjacent dominoes match. For example, the
dominoes (03), (05), (12), (14), (15), (23), (34) can be arranged as
(12), (23), (30), (05), (51), (14), (43); and if an eighth domino (13)
is added, they can be formed into a rectangle. What if instead the
eighth domino was (24) - could they then be
arranged in a rectangle or in a line?
- Students determine the minimum number of blocks that a
police car has to repeat if it must try to patrol each street exactly
once on a given map. Drawing Pictures With One Line contains
similar real-world problems and a number of related game
- Students find the best route for collecting recyclable
paper from all classrooms in the school, and discuss different ways of
deciding what is the "best." (See Drawing Pictures
With One Line.)
- Students make models of various polyhedra with
straws and string, and explore the relationship between the number of
edges, faces, and vertices.
8. Experiment with iterative and recursive processes,
with the aid of calculators and computers.
9. Explore methods for storing, processing, and
- Students conjecture which of the following (and other)
methods is the most efficient way of handing back corrected homework
papers which are already sorted alphabetically: (1) the teacher walks
around the room handing to each student individually; (2) students
pass the papers around, each taking their own; (3) students line
themselves up in alphabetical order. Students test their conjectures
and discuss the results.
- Students investigate and report about various codes
that are commonly used, such as zip codes, UPCs (universal product
codes) on grocery items, and ISBN numbers on books. (A good source for
information about these and other codes is Codes Galore by
J. Malkevitch, G. Froelich, and D. Froelich.)
- Students write a Logo procedure for making a rectangle that
uses variables, so that they can use their rectangle procedure to
create a graphic scene which contains objects, such as buildings, of
- Students are challenged to guess a secret word chosen by
the teacher from the dictionary, using at most 20 yes-no questions.
Is this always, or only sometimes possible?
- Students use Venn diagrams to solve problems
like the following one from the New Jersey Department of
Education's Mathematics Instruction Guide (p. 7-13).
Suppose the school decided to add the springtime sport of
lacrosse to its soccer and basketball offerings for its 120
students. A follow-up survey showed that: 35 played lacrosse, 70
played soccer, 40 played basketball, 20 played both soccer and
basketball, 15 played both soccer and lacrosse, 15 played both
basketball and lacrosse, and 10 played all three sports. Using
this data, complete a Venn diagram and answer the following
questions: How many students played none of the three sports?
What percent of the students played lacrosse as their only
sport? How many students played both basketball and lacrosse, but not
- Students keep a scrapbook of different ways in which
information is stored or processed. For example a list of events is
usually stored by date, so the scrapbook might contain a picture of a
pocket calendar; a queue of people at a bank is usually processed in
first-in-first-out (FIFO) order, so the scrapbook could contain a
picture of such a queue. (How is this different from the
waiting lines in a supermarket, or at a restaurant?)
- Students determine whether it is possible to have a year in
which there is no Friday the 13th, and the maximum number of Friday
the 13th's that can occur in one calendar year.
- Students predict and then explore the frequency of letters
in the alphabet through examination of sample texts, computer
searches, and published materials.
- Students decode messages where letters are systematically
replaced by other letters without knowing the system by which letters
are replaced; newspapers and games magazines are good sources for
"cryptograms" and students can create their own. They also
explore the history of code-making and code-breaking. The videotape
Discrete Mathematics: Cracking the Code provides a good
introduction to the uses of cryptography and the mathematics behind
10.Devise, describe, and test algorithms for solving
optimization and search problems.
- Students find the shortest route from one city to another
on a New Jersey map, and discuss whether that is the best route. (See
Problem Solving Using Graphs.)
- Students write and solve problems involving distances,
times, and costs associated with going from towns on a map to other
towns, so that different routes are "best" according to
- Students use binary representations of numbers to find
winning strategy for Nim. (See Mathematical Investigations for
other mathematical games.)
- Students plan an optimal route for a
class trip. (See the Short-circuiting Trenton lesson in the
Introduction to this Framework.)
- Students devise work schedules for employees of a
fast-food restaurant which meet specified conditions yet minimize the
- Students compare strategies for alphabetizing a list
of words, and test to see which strategies are more efficient.
Students find a network of roads which
connects a number of sites and involves the smallest cost. In
the example at the right, what roads should be built so
as to minimize the total cost, where the number on each road
reflects the cost of building that road (in hundreds of
thousands of dollars)?
- Students develop a precise description of the
standard algorithm for adding two two-digit integers.
- Students devise strategies for dividing up the
work of adding a long list of numbers among the members of the
Chavey, D. Drawing Pictures with One Line. Consortium
for Mathematics and Its Applications (COMAP), Module #21, 1987.
Cozzens, M., and R. Porter. Problem Solving Using Graphs.
Consortium for Mathematics and Its Applications (COMAP), Module #6,
Crisler, N., P. Fisher, and G. Froelich, Discrete Mathematics
Through Applications. W. H. Freeman and Company, 1994.
Jacobs, H. R. Mathematics: A Human Endeavor.
W. H. Freeman and Company, 1982.
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.
Malkevitch, J., G. Froelich, and D. Froelich, Codes
Galore, Consortium for Mathematics and Its Applications
(COMAP), Module #18, 1991.
New Jersey Department of Education. Mathematics Instruction
Guide. D. Varygiannis, Coord. January 1996.
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
Seymour, D. Visual Patterns in Pascal's
Triangle. Palo Alto, CA: Dale Seymour Publications, 1986.
Souviney, R., et al. Mathematical Investigations. Book
One, Dale Seymour Publications, 1990.
Discrete Mathematics: Cracking the Code,
Consortium for Mathematics and Its Applications.
Geometry: New Tools for New Technologies,
videotape by the Consortium for Mathematics and Its Applications
(COMAP). Lexington, MA, 1992.
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