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■
Mathematics
■
When I ask students at the end
of my courses what they have
learned about mathematics,
many of them say that they
have come to realize that
mathematics is much more
connected than they had
thought. They saw more
connections both among
concepts (such as fractions,
decimals, and percents) and
among problems that they had
solved. Do you see connections
between Pascal’s triangle and
other problems in this chapter?
Think and read on. . . .
If you didn’t, look back at
Investigation 1.3 and
Exploration 1.1. Those
sequences can be found within
Pascal’s triangle. Can you see
them?
5. Each number (not equal to 1) in the triangle is the sum of the two numbers
to its top right and its top left. Do you understand what I just wrote? Try it
out, and then read on. . . .
If you don’t understand what I just said, look at this visual representation:
a
b
c
Choose any number c (not equal to 1) in the triangle. If we let a and b represent the two numbers immediately above c, then the value of c is equal to the
value of a b!
6. Add the numbers in each row in the triangle. What do you get? Do it now
and then read on. . . .
The sums of the rows make the following sequence: 1, 2, 4, 8, 16. This
sequence can be described as the powers of 2. Can you guess the sum of the
numbers in the 10th row? Try it and then read on. . . .
In this case a table is helpful. See Table 1.9.
In other words, the sum of the numbers in the nth row can be obtained by
raising 2 to the n 1 power. If you have a calculator with an xy function, you
can determine 29 by doing the following: 2 xy 9 ; 512 will appear. More Explorations with Pascal’s triangle can be found in Exploration 1.1. If you type “Pascal’s
triangle” into a search engine, you can explore it even further.
TABLE 1.9
Row
Sum
1
1
2
2
3
4
4
8
5
16
n
2n1
The Power of Patterns
Some of the greatest mathematicians in history made their mark by discovering and articulating patterns. The wonderful thing about patterns, though, is
that they make mathematics more accessible to everyone. This was recognized
by the authors of Everybody Counts,8 who wrote,
Virtually all young children like mathematics. They do mathematics naturally,
discovering patterns and making conjectures based on observation. Natural curiosity is a powerful teacher, especially for mathematics. Unfortunately, as children
become socialized by school and society, they begin to view mathematics as a rigid
system of externally dictated rules governed by standards of accuracy, speed, and
memory. Their view of mathematics shifts gradually from enthusiasm to apprehension, from confidence to fear. Eventually, most students leave mathematics under
duress, convinced that only geniuses can learn it.
We explored several famous, powerful patterns in this section. You will
see many of these patterns again, both in this book and beyond it. But more
important, I hope you will see that stepping back from a problem can reveal
patterns within it. Sometimes these patterns can help you devise a plan for
solving the problem; at other times they emerge as you look back on your
work.
8National
Research Council, Everybody Counts: A Report to the Nation on the Future of Mathematics
Education (Washington, DC: National Academy Press, 1989), pp. 53–54.
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Section 1.4 / Representation
SECTION
23
1.4 REPRESENTATION
WHAT DO YOU THINK?
• Representation was not
one of the standards in the
1989 NCTM Curriculum
Standards. Why do you
think it was made a
separate standard in the
2000 Standards?
• Why do you think I chose
to put the Representation
standard (Standard 10)
after Patterns instead of at
the end of the chapter?
Standard 10: Representation
Instructional programs from prekindergarten through grade 12
should enable all students to
• create and use representations to organize, record, and communicate
mathematical ideas;
[When you can represent a situation or idea in multiple ways—
tables, graphs, symbols, diagrams, words—your understanding of
that situation or idea is much stronger than when you can represent
the situation/idea in only one way.]
• select, apply, and translate among mathematical representations to
solve problems;
[This relates to the toolbox metaphor and also to the notion of flexibility, as opposed to the mechanical application of rote procedures.]
• use representations to model and interpret physical, social, and mathematical phenomena.
[We will talk more about modeling later.]
(Principles and Standards, p. 67)
At the simplest level, a representation is a picture—a picture of your
image of a problem. Representations can take many forms: diagrams, graphs,
tables, sketches, equations, words, etc. One of the key ideas about representations is that most problems and most mathematical concepts can be represented
in different ways. For example, there were several ways to represent the pigsand-chickens problem from Investigation 1.1. Students commonly ask me
which representation is “best,” but this is just not a useful question. A more useful question is whether this representation fits the purposes before us. Representing the problem with 24 circles (representing 24 bodies) and then putting
legs on the bodies is a very appropriate representation for this problem with
children. This notion of multiple representations is also related to the standard
on Connections, which will be discussed in Section 1.7. For example, some of
my students do represent the problem with 24 circles. In this case, the next step
is to connect this representation to other representations that are more useful
with larger numbers, such as the table and the formula.
Just as many problems can be represented in multiple ways, this is also true
for most important mathematical ideas. Let me use language to illustrate the
idea of multiple representations. Think of the word hot. What does it mean?
Write down the different meanings of hot before reading on. . . .
When I looked up this word in the dictionary, I found 14 different meanings. Following are phrases that illustrate some of these meanings: a hot forehead, hot peppers, a hot temper, hot for travel, a hot topic, a hot suspect, hot on
the trail, I’m not so hot at math, a hot sports car.9
Now consider the word fraction. Let me be more specific: In how many
different ways can you represent 3/4? Do this and then turn to page 50 in the
textbook for a sneak preview of this important idea.
9American
Heritage Dictionary of the English Language (Boston: Houghton Mifflin, 1992), p. 874.
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Just as a literate person knows the multiple meanings of many words, a
mathematically literate person knows multiple representations of important
mathematical ideas.
I chose the next investigation because it illustrates several important
aspects of representations.
Coin Combinations
I N V E S T I G AT I O N
1.5
Language
Teachers traditionally use a
phrase like “we will cover this
much material in this course.”
I urge the word uncover. Think
of an archaeological team that
has discovered an ancient city.
As they uncover the ruins
carefully, the city emerges
right before their eyes. So too
when we look at different
solution paths and different
representations of a problem,
new ideas and patterns
emerge right before our eyes.
You will find this and similar problems all over elementary school
mathematics, and for good reason. As you will soon see, there is rich
opportunity for mathematical development with these problems. The
problem: How many different ways can you make 25 cents using
dimes, nickels, and pennies? Solve this problem on your own before
reading on. . . .
DISCUSSION
Do you see patterns in your work? Did you work systematically or just randomly? If you don’t see patterns and/or you generated the combinations randomly, please go back and take a few minutes to do the problem over. This time
think how you might generate the combinations by being systematic, and then
look at your table. Patterns will help you to be more confident that you have all
the combinations.
One of the neat things about this investigation is that there are multiple ways
to represent the problem, some of which make it easier to see patterns. Furthermore, as shown with the pigs and chickens, there is more than one way to solve
the problem, even if you are systematic. Let us uncover this problem now. First,
look at the solution on the left, in which the combinations were generated randomly after the first two, which were combinations involving all of the same
coin. Now look at the solution on the right. What do you think the person said to
himself or herself when generating this table? Think before reading on. . . .
N
25P
D
D
N
N
D
D
D
N
N
D N
N
N
D
D
20P
N
N
N
15P
N
N
N
N
5P
N
N
N
10P
10P
N
N
15P
5P
N
N
5P
D
D
D
D
D
D
N
N
N
N
N
25P
D
D
N
N
N
15P
N
N
N
N
20P
N
5P
N
N
10P
N
N
N
15P
N
5P
N
N
10P
N
5P
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Section 1.4 / Representation
25
After the quarter, the person thought, how many ways would involve two
dimes, then how many ways would involve one dime, then no dimes. Now
describe patterns you see in this solution before reading on. . . .
Dimes
Nickels
Pennies
2
1
0
2
0
5
1
3
0
1
2
5
1
1
10
1
0
15
0
5
0
0
4
5
0
3
10
0
2
15
0
1
20
0
0
25
There are many patterns; we will examine just a couple here. In the righthand columns, the nickels form two triangles. Do you see them? Look at the
3rd, 4th, and 5th rows—you can draw a right triangle that encloses the nickels,
with the right angle in the top left corner. That is, there are 3 nickels in the 4th
row, 2 in the 5th row, and 1 in the 6th row. Then you skip a row, with no nickels, and then you have a similar, larger triangle. Another pattern is that the pennies go in a diagonal direction, increasing by 5 each time.
Below is another representation of the same solution path. Which representation do you prefer? To you, which representation shows the patterns better? Interestingly, some of my students prefer the one above because they like
the letters, but others prefer the one below because seeing the numbers makes
more sense to them.
10
10
10
10
10
10
5
5
5
1 1 1 1 1
5 5
5 1 1 1 1 1
10
10
5
5
5
5
5
1
5
1
5
5
5
5
1
1
1
1
5
5
5
1
1
1
1
1
5
5
1
1
1
1
1
1
5
1
1
1
1
1
1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
At the left is yet another representation for the dimes, nickels, and pennies.
This one is more abstract in that you don’t see each individual coin. Because of
its abstractness, it will be harder for young children to make sense of than the
previous representations. However, again partly because of its abstractness,
certain patterns are easier to see. What patterns do you see in this table? Think
before reading on. . . .
Once again there are many patterns, and we will discuss just two.
The numbers of nickels form decreasing sequences: 1, 0, then 3, 2, 1, 0, then
5, 4, 3, 2, 1, 0. The lengths of the sequences are 2, 4, and 6. The numbers of pennies form increasing sequences: 0, 5 then 0, 5, 10, 15, then 0, 5, 10, 15, 20, 25. The
lengths of these sequences are 2, 4, and 6 also.
As we noted at the beginning of this discussion, there is more than one way
to be systematic. The solution at the left below and the table for that solution,
at the right below, illustrate another solution path. Before we look at patterns
in the table, take a minute to see if you can figure out the thinking process that
generated this solution. . . .
In this case, the person began with combinations with 0 pennies, then combinations with 5 pennies, then 10 pennies, and so on. Now look for patterns in
the solution path at the left and then in the table at the right. . . .
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N
D
D
N
N
N
D
N
N
N
N
5P
D
D
5P
N
N
D
5P
N
N
N
10P
N
D
10P
N
N
15P
D
15P
N
20P
N
N
25P
N
N
N
Pennies
Nickels
Dimes
0
1
2
0
3
1
0
5
0
5
0
2
5
2
1
5
4
0
10
1
1
10
3
0
15
0
1
15
2
0
20
1
0
25
0
0
There are 3 combinations with no pennies, 3 combinations with five pennies, 2 combinations with ten pennies, 2 combinations with fifteen pennies, 1
combination with twenty pennies, and 1 combination with twenty-five pennies. At the right, look at the nickels column. This time we have increasing sequences: 1, 3, 5, then 0, 2, 4, then 1, 3, then 0, 2, then 1, then 0. The sequences
rotate from all odd numbers to all even numbers to all odd numbers to all even
numbers to an odd number to an even number.
Summary
In the first half of Chapter 1, I have asked you to examine your beliefs and
attitudes toward mathematics, I have introduced the NCTM Curriculum Standards, and we have investigated the standards on problem-solving, representation, and patterns. Before moving on, many instructors find it helpful for
students to do some thinking and some problems that involve these themes. In
the second half of Chapter 1, we will explore the remaining process standards.
EXERCISES
1. At a bicycle store, there were a bunch of bicycles and tricycles. If there were 32 seats and 72 wheels, how many
bicycles and how many tricycles were there?
seven legs). She says to her son, “I count 97 heads and
436 feet. How many tribbles and how many chalkas are
out there?”
2. a. A farmer looks out into the barnyard and sees the
pigs and the chickens. He says to his daughter, “I
count 40 heads and 100 feet. How many pigs and
how many chickens are out there?”
4. At a benefit concert, 600 tickets were sold and $1500 was
raised. If there were $2 and $5 tickets, how many of each
were sold?
b. What if the farmer saw 169 heads and 398 feet? How
many pigs and how many chickens are out there?
5. For a certain event, 812 tickets were sold, for a total of
$1912. If students paid $2 per ticket and nonstudents
paid $3 per ticket, how many student tickets were sold?
3. A Martian farmer looks out into the barnyard and sees
tribbles (which have four legs) and chalkas (which have
6. Make up your own problem like the ones in Exercises
1–5.
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Section 1.4 / Representation
7. If you have a bunch of 10¢ and 5¢ stamps, and you know
that there are 20 stamps and their total value is $1.50,
how many of each do you have?
8. This problem is adapted from the Figure This website,
which has 80 rich problems that upper elementary and
middle school children enjoy: Is there a combination of
15¢ and 33¢ stamps that can make $1.77? (Reprinted
with permission from Figure This! (http://www
.figurethis.org), copyright 2006 by the National Council
of Teachers of Mathematics.)
9. Let’s say you are building a patio in your back yard.
You have decided to make the patio 12 feet by 9 feet.
The local building supply store sells bricks that are 6
inches by 4 inches for 25¢ each. How much will the
patio cost?
10. At the beginning of the month, Jack and Jill had $642 in
a checking account. During the month, they wrote
checks for $22, $53, and $55. They made a withdrawal of
$50 from the automatic teller machine, and they made
deposits of $142 and $100. How much money did they
have at the end of the month? Solve this problem two
different ways.
11. Sarah’s diet allows 1500 calories per day. Thus far, she
has had a glass of milk (90 calories), pancakes (150 calories), an apple (75 calories), a salad (150 calories), salad
dressing (200 calories), and a piece of cake (350 calories).
Can she have a steak (250 calories) and a salad with
salad dressing for dinner? If not, what can she have?
Solve the problem two different ways.
12. Sally works 40 hours a week and makes $6.85 an hour,
but her kids are in child care for 32 hours a week and the
day care center charges her $15 per day. If you deduct
her child care expenses, how many dollars per hour does
she actually make?
miles per year, from the Arctic to the Antarctic and back.
If the bird flies an average of 25 miles per hour and an
average of 12 hours per day, how many days would it
take for a one-way flight?
18. A family is planning a three-week vacation for which
they will drive across the country. They have a van that
gets 18 miles per gallon, and they have a sedan that gets
32 miles per gallon. How much more will they pay for
gasoline if they take the van?
a. First describe the assumptions you need to make in
order to solve the problem.
b. Solve the problem and show your work.
c. What if the price of gas rose by 40¢ between the planning of the trip and the actual trip? How much more
would the gas cost for the trip?
19. Joni is the owner of Red Oak Furniture, which is doing
so well that the present staff is overworked. She has
two choices: continue to pay overtime or hire a new
person. She now has three employees, and they earn
time-and-a-half for all time over 40 hours worked in
a week. In the past three weeks, they have worked
the number of hours shown. Should she hire a new
employee?
Week 1
Week 2
Week 3
Gerald
47
55
42
Nancy
43
51
48
Jose
56
51
42
20. a. Using each of the numbers 1–9 exactly once, fill in the
blanks below:
13. Martha planted 14 rows of apple trees, and each row had
21 trees. If each tree yielded an average of 250 apples,
how many apples grew in her orchard?
14. A farmer needs to fence a rectangular piece of land.
She wants the length of the field to be 80 feet longer
than the width. If she has 1080 feet of fencing material, what should be the length and the width of the
field?
b. Find all solutions.
21. For each of the triangles below:
15. Since its beginning, the U.S. Mint has produced over
288.7 billion pennies.
(1) Describe as many patterns as you can find in the
triangle.
a. What if we lined these pennies up. How long would
the line be?
(2) Write the numbers that would be in the next row
of the triangle, and explain how you arrived at your
answer.
b. The mint currently makes about 30 million pennies a
day. How many is this per second? How many is this
per year?
a.
2
16. The Lewis and Clark expedition between 1804 and 1806
covered almost 8000 miles.
3
a. If you traveled along their route, how much would
you spend on gasoline?
b. How many days would it take you?
17. The record for the longest migration is held by the arctic
tern, which flies a round trip that can be as long as 20,000
4
5
b.
1
6
8
10
1
4
1
1
9
1
12 16
15 20
1
25
1
1
3
5
1
5
1
7 13 7
9 25 25 9
1
1
1 11 41 63 41 11 1
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CHAPTER 1 / Foundations for Learning Mathematics
c.
d.
1
2
4
3
3
5
c. Along the “what if” theme, this pattern seems to break
at 7 142857. Continue to find and write products of
142857 and 8, 9, 10, etc. What patterns do you see now?
3
3
6
3
3
6
9
3
9
3
7 9 11 12
13 16 20 23 24
3 15 30 30 15 3
3 12 18 12 3
25 29 36 43 47 48
3 18 45 60 45 18 3
22. List the next term in each of the sequences below, and
verbally describe the pattern that enabled you to predict
the next term.
a. , , , , ,
b.
c. 1, 4, 7, 10,
d. 2, 6, 18, 54,
e. 1, 4, 9, 16,
f. 1, 2, 4, 7, 11,
g. 5, 1, 4, 2, 3, 3,
h. 4, 5, 8, 13, 20, 29, 40,
i. 1, 7, 21, 46, 85, 141,
j. 1, 0, 2, 1, 2, 2, 4,
k. 1, 3, 6, 10, 15, 21,
l. 100, 90, 81, 73, 66,
, , ,
a. 12 22 22 32
22 32 62 72
32 42 122 132
b. 13 23 32
13 23 33 62
13 23 33 43 102
c. 1 2 3
45678
9 10 11 12 13 14 15
30. a. In how many different ways can you make change for
a quarter?
m. 81, 72, 63, 54, 45,
23. What patterns do you see in this sequence of numbers
that are the squares of the first 16 numbers: 1, 4, 9, 16, 25,
36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, . . .
24. Continue the pattern in the three number sentences
below, and then describe as many patterns as you can.
331 8
5 5 1 24
b. In how many different ways can you make change for
50 cents?
c. In how many different ways can you make change for
one dollar without using pennies?
31. A special rubber ball is dropped from the top of a wall
that is 16 meters high. Each time the ball bounces, it rises
half as high as the distance it fell. The ball is caught
when it bounces 1 meter high. How many times did the
ball bounce?
32. a. On a digital clock, how many times in one day will
the numbers be consecutive digits, as they are, for
example, in 1:23?
b. On a digital clock, which digit will appear most often
in one day?
7 7 1 48
25. Examine the following phrase made famous by Fred
Flintstone: y a b b a d a b b a d o o. If this pattern is
repeated over and over, what letter will be in the 275th
position?
26. Determine the following products: 67 67, 667 667.
a. Describe the pattern you see.
b. Predict the pattern of 6667 6667.
c. This is an example of “what if” and “I wonder.” What
if the two numbers didn’t have the same number of
6s? For example, is there any connection between
what we have seen above and, for example, 667 67?
Explore this question and report your findings.
27. Determine the following products: 1 1089, 2 1089,
3 1089.
a. Describe the pattern(s) you see.
b. Predict the next several products.
c. Along the “what if” theme, will this pattern continue
when you reach 10 1089 and further?
28. Determine the following
2 142857, 3 142857.
29. Find a pattern in each of the following sets of equations
and predict the next equation.
products:
1 142857,
a. Describe the pattern(s) you see.
b. Do more computations until you can predict the next
product.
33. One of the most famous patterns in mathematics was
discovered by the Italian mathematician Leonardo of
Pisa (1170–1250), known to us as Fibonacci. He discovered this sequence, which we today call the Fibonacci sequence, while studying the birth rates of rabbits. He
posed the following question: Suppose that a pair of rabbits produce a pair of baby rabbits every month, and that
rabbits cannot reproduce until they are two months of
age. How many pairs of rabbits will you have after one
year (assuming, of course, that no rabbits die)? He found
that if he listed the number of pairs of rabbits he had
after each month, he had a very interesting sequence: 1,
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .
a. Add the first seven Fibonacci numbers. What do you
notice?
Add the first eight Fibonacci numbers. What do you
notice?
Try to describe this relationship both in words and
using notation.
b. Pick three consecutive terms. Multiply the first term
and the third term. How does this compare to the
square of the middle term? Describe this relationship
using notation.
34. Palindromes are numbers whose value is the same backward as forward. For example, 1331 is a palindrome.
Most children love explorations with palindromes
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Section 1.4 / Representation
35. Making palindromes from nonpalindromes is another
interesting exploration. For example, take any number,
reverse the digits, and add the two numbers together.
Then determine how many steps it takes until a palindrome is reached. For example, 38 83 121. Thus, we
can say that 38 is a 1-step palindrome. On the other
hand, 87 is a 4-step palindrome. Explore different 2-digit
numbers. Can you describe those numbers that will be
2-step palindromes? 3-step palindromes? etc.?
36. Let’s say there are 145 students in your class. If each student greeted and shook hands with every other student
in the class, how many handshakes would there be?
37. This variation of the handshakes problem is on video 17
of Teaching Math: A Video Library, K–4, which many colleges and schools have: “If there are 20 children in a
classroom and every child gives every child a valentine,
how many valentines are distributed that day?” Does
this problem have the same answer as the handshakes
problem? Why or why not?
38. This problem was posed in the November 2001 “Problem
Solvers” section of Teaching Children Mathematics:
In the Animal Football League, the teams can score only 3point field goals and 7-point touchdowns; no safeties or
extra-credit points are allowed. In one contest, the
Anteaters defeated the Bobcats 42 to 37. In how many different ways could each team have arrived at its final score?
a. Do the problem yourself before reading on.
b. Solve the problem using the approach described by
this fifth grader and shown in the table: “I found the
different ways that teams could arrive at the score by
making a chart and checking each possibility. After
subtracting the touchdown points from the total
score, the difference would have to be divisible by 3
to be a possible combination.”
Touchdowns
Field Goals
1.
7
3
2.
14
6
3.
21
9
4.
28
12
5.
35
15
6.
42
18
7.
21
8.
24
9.
27
10.
30
11.
33
12.
36
13.
39
14.
42
This problem has several extensions:
c. What scores are impossible to get?
d. What is the greatest score that is impossible to get?
Prove it.
39. This problem was posed in the April 2001 “Problem
Solvers” section of Teaching Children Mathematics: Lori
and Betty are pretending to have a garage sale. They
found five 30¢ price tags and five 40¢ price tags left over
from their mothers’ last sale. If they use these price tags
on their sale items, how many different amounts of
money could they make from their sale?
a. Solve the problem yourself.
b. The two tables below illustrate one child’s strategy.
Continue this strategy to determine the answer. Describe this strategy as if to someone who missed class
and doesn’t understand it. Critique the strategy. That
is, cite its strengths and weaknesses (or limitations).
30
30 30
30 30 30
30 30 30 30
30 30 30 30 30
30 40
30 30 40
30 30 30 40
30 30 30 30 40
30 30 30 30 30 40
c. The table below shows another student’s solution.
Combine this strategy with being systematic to find
all solutions.
Possibilities for 40¢
Possibilities for 30¢
because they find the patterns fun and the investigations
are so mathematically rich. How many palindromes can
you find between 100 and 999?
0
1
2
3
4
5
0
0
40
80
120
160
200
1
30
70
110
150
190
230
2
60
100
140
180
220
260
3
90
130
170
210
250
290
4
5
120
150
160
190
200
230
240
270
280
310
320
350
d. Use this table to find all the possibilities. Explain this
table so that someone who doesn’t understand it
could use it.
40. This problem was explored in the October 1998 issue of
Teaching Children Mathematics, pp. 105–107: South Main
School has 618 students. Each student will have his or
her picture taken next week by a photographer hired by
the school. The photographer uses rolls of film having
twenty-four exposures each. How many rolls of film will
the photographer have to buy?
Source: Reprinted with permission from Teaching Children
Mathematics, October 1988, copyright 1988 by the National
Council of Teachers of Mathematics.