Discrete Mathematics Summer 2006
Paul Daman
Robyn Johnson
Kenny Newby
Discrete mathematics is a term that basically
describes a cross section of math problems that
are finite and non-continuous. If you understand
that definition you are already ahead of the game.
We like to think of discrete math as problem
solving. We have compiled a number of activities
to be used in the intermediate levels of the
elementary school (grades 3-5). Problems could be
worked on in groups or as individuals. Some
problems do involve the use of manipulatives. Each
activity meets an elementary standard of the
NCTM (National Council of Teachers of
Mathematics). Try one or many with your class.
Answers are provided. Math rocks baby!
Problem #1 - “A Blank Net Face”
A ‘net’ is a two-dimensional drawing of a
cube. Using the net provided, students should cut,
fold, and glue, to demonstrate how a two
dimensional drawing becomes a three dimensional
object. Students will notice that one of the six
sides of the net is blank. The goal of this
assignment is to determine what shape, letters,
and numbers belong on the blank face. The shapes,
letters, and numbers on the other faces of the net
should serve as clues. Each student or group should
give a written explanation of what they would put
on the blank face.
NCTM standard(s)Geometry-Identify, compare, and analyze attributes of two and three dimensional
shapes and develop vocabulary to describe the attributes.
Problem Solving- Build new mathematical knowledge through problem solving
Extension activity: www.illuminations.nctm.org/activitydetail.aspx?id=84
Problem #2 – “Cow ears and legs”
While we were driving down the road on our
summer vacation we noticed a large group of cows
in the field. As you know a cow has 4 legs and 2
ears. We love to count!! We started counting all
the legs and all the ears, but we lost count of the
number of cows in the field. Can you pitch
in and help us? If the difference between the
number of legs and the number of ears was 40,
then how many cows did we actually see in the
field? (Hint: make a chart)
sample chart for correct answer:
Cows
legs minus
ears =
1
2
3
18
19
20
4
8
12
72
76
80
-
difference
2
4
6
36
38
40
2
4
6
36
38
40
NCTM Standard(s)Data Analysis and Probability Standard for Grades 3–5•
represent data using tables and graphs such as line plots, bar graphs, and line graphs;
Problem #3 – “Number Puzzles”
1. 123456789 = 100 Insert mathematical symbols
between the digits on the left side to make the equation
correct.
2. 123456789 = 100 Can you make the equation true by
inserting only three plus or minus signs?
3. Use five 3's to make 37.
4. Use five 4's to make 55.
5. Use four 9's to make 20.
6. Use six identical digits to make 100.
Possible answers-many more possible
1. 1+2+3+4+5+6+7+(8X9) = 100
2.[123-(4+5)-(6+7)+8]-9=100
3. [(333/3)/3=37
4.(44/4) + 44 = 55
5. (99/9) +9 = 20
6. Number six can work with any number:
(999-99)/9 = 100
(888-88)/8 = 100
(777-77)/7 = 100
(666-66)/6 = 100
(555-55)/5 = 100
(444-44)/4 = 100
NCTM Standard(s):
Number and Operations Standard for Grades 3–5
-recognize equivalent representations for the same number and generate them by
decomposing and composing numbers;
-understand and use properties of operations, such as the distributivity of multiplication
over addition.
Problem #4 – “Mixed up houses…”
Five houses of different colors stand in a row. Each is owned by a person
with a different nationality, hobby, pet, and favorite drink:
the English person lives in the red house,
the Spaniard owns dogs,
coffee is drunk in the green house,
the Ukrainian drinks tea,
the green house is directly to the right of the white one,
the stamp collector owns snails,
the antique collector lives in the yellow house,
the person in the middle house drinks milk,
the Norwegian lives in the first house,
the person who sings lives next to the person with the fox,
the person who gardens drinks juice,
the antique collector lives next to the person with the horse,
the Japanese person's hobby is cooking, and
the Norwegian lives next to the blue house.
What I would like to know is: Who drinks water, who owns the zebra, and
how in the world did you figure this out?
Answer-
yellow
Norwegian
water
fox
antiques
blue
Ukranian
tea
horse
singer
red
English
milk
snails
stamps
white
Spaniard
juice
dog
gardner
green
Japanese
coffee
zebra
cooking
NCTM Standard(s):
Data Analysis and Probability Standard for Grades 3–5
- represent data using tables and graphs such as line plots, bar graphs, and line graphs;
- propose and justify conclusions and predictions that are based on data and design
studies to further investigate the conclusions or predictions
Problem #5 – “Adjacent numbers in a circle”
The diagram shows the numbers 1 through 10 (in order) at the
tips of 5 diameters. Only once does the sum of two adjacent
numbers equal the sum of the opposite two numbers:
Now…
Rearrange the numbers so that all sums are equal. You can expect
more than one solution to this problem. How many basic solutions
are there? (Do not include simple rotations of the same numbers for example 1,2,3,4,5,6,7,8,9,10 would be the same as
2,3,4,5,6,7,8,9,10,1.)
Answers- many possible
We all worked on our own at home and between us generated 8 different
solutions:
Mike, Marina, Jim, George, Peter, Scott C., Claire : 6, 2, 8, 4, 10, 1, 7, 3, 9, 5
Scott P., Robby: 9, 2, 10, 1, 8, 4, 7, 5, 6, 3
Yiqiong: 10, 2, 8, 1, 9, 5, 7, 3, 6, 4
Eric: 10, 1, 8, 3, 6, 9, 2, 7, 4, 5
Gi-Soo: 10, 1, 9, 2, 8, 5, 6, 4, 7, 3
Katie:10, 1, 8, 2, 9, 5, 6, 3, 7, 4
Samantha: 10, 7, 6, 1, 4, 9, 8, 5, 2, 3
NCTM Standard(s):
Number and Operations Standard for Grades 3–5:- develop and use strategies to
estimate the results of whole-number computations and to judge the reasonableness of such
results
Problem #6 – “Desks in a classroom”
The desks in a classroom are lined up in straight rows.
Unless someone is sick, each desk is filled. Masaki is in
the second row from the front and the fourth row from
the back. She is also the third student from the left
end of the row and the fifth student from the right.
How many students are in the class?
AnswerROW 1 2 3 4 5 6 7
X X X X X X X
1
X X M X X X X
2
X X X X X X X
3
X X X X X X X
4
X X X X X X X
5
5 X 7 = 35
NCTM Standard(s):
Geometry Standard for Grades 3–5: describe location and movement using common
language and geometric vocabulary; make and use coordinate systems to specify locations
and to describe paths;
Problem #7 – “Toothpicks, ponies, and pens”
Carrie wanted to put her 6 toy ponies into a pen made of 13 toothpicks
which formed six equal spaces. It looked like the picture below.
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While she was making the pen, one of the toothpicks broke. Oh, no! she
exclaimed. Now one of my ponies won't have a pen.
Can you help Carrie make equal-sized pens for the six ponies with the
twelve remaining toothpicks?
AnswerMMMMMMMMM
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NCTM Standard(s):
Geometry Standard for Grades 3–5: build and draw geometric objects; create and
describe mental images of objects, patterns, and paths;
Problem #8 – “How many different paths?”
H
E
A
R
T
S
A
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T
S
E
A
R
T
S
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T
S
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How many paths are there to spell out the word ‘hearts’ ?
Answer- Use Pascal’s triangle to solve for this and any other word you might choose.
If you are looking at the word CRAZY, you would find sixteen possible paths (Find the
5th row, which means 2 to the 4th power and add them across; ie. 1+4+6+4+1=16).
Note: Each number represents a letter from the word
1=C, which is 2 to the 0 power
1,1=R, which is 2 to the 1st power
1,2,1=A, which is 2 to the 2nd power
1,3,3,1=Z, which is 2 to the 3rd power
1,4,6,4,1=Y, which is 2 to the 4th power
(2x2x2x2=16). See Pascal's Triangle forming?So how many could you find with the
word HEARTS? Using Pascal's Triangle should help you find the answer much easier.
Answer for hearts is 32
NCTM Standard(s):
Number and Operations Standard for Grades 3–5: develop and use strategies to estimate
the results of whole-number computations and to judge the reasonableness of such results;
Problem # 9 – “Different color maps”
Students will color maps with two, three, or four
colors. Each color will specify a specific region on the
map. Build toward the idea that four colors have been
proven to be all that one needs to color any map.
Remember- colors must always touch a different color.
Much of this lesson is trial and more trial. One fun rule:
have students draw their own map. They can draw any
map figure (they must end where they started) by never
taking their pencil off of the paper. They can cross over
themselves any number of times. This map will always be
two colors. Weird, huh? Give it a try.
NCTM Standard(s):
Problem Solving Standard for Grades 3–5: build new mathematical knowledge through
problem solving;