palette of
problems
David Rock and Mary K. Porter
1. For every 3 minutes that Sarah reads a book, her parents allow her to spend 5 minutes on the computer. If she
wants to watch a 1 1/2-hour webcast, how long will she
need to read?
5. Approximately 56 M&M’s are in a 1.69-ounce bag. On
the basis of this information, approximately how many
M&M’s should be in a 12.69-ounce bag? How many
M&M’s should be in a 3-pound bag?
2. At David’s high school, the
basketball coach uses the
seniority system to choose
jersey numbers for each
player. The oldest player
receives the number 1, the
next oldest player receives
the number 3, followed by 7,
13, and 21. What jersey number
will the 6th and 7th players receive
if this pattern of selecting numbers
is continued?
6. A swimsuit company will embroider a star whose
number of points correspond with the number of gold
medals that a swimmer has won. A swimmer who has
won 1 gold medal will sport a star with 5 points. The
pattern of star points are shown below.
a. How many points will be on the star if a swimmer has
won 4 gold medals?
b. How many points will be on the star if a swimmer has
won 24 gold medals?
c. Create an expression for someone who has won x gold
medals.
3. The school’s softball coach decides to use the same
method of selecting jersey numbers as the basketball
coach from the previous problem. The team needs at least
12 players (10 players in the field and 2 substitutes) for
each game. Will there be enough two-digit jerseys for the
coach to outfit a team of 12 players? How many two-digit
jerseys can a coach assign using this system?
1 Gold Medal
4. Emma wants to bring candy to class on her birthday
and thinks that M&M’s® will be the easiest candy to
share. She notices three different-sized bags of M&M’s: a
1.69-ounce bag for $0.75; a 12.69-ounce bag for $3.25;
and 3-pound bag for $8.99. Which bag is the best buy?
Explain your reasoning.
266
Mathematics Teaching in the Middle School
●
2 Gold Medals
3 Gold Medals
7. A toy company makes stuffed Dalmatian puppies, which
have 4 brown spots for every 7 black spots.
a. How many brown spots will be on a puppy with 21
black spots?
b. How many black spots will be on a puppy with 28
brown spots?
Vol. 15, No. 5, December 2009/January 2010
Copyright © 2009 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Prepared by David Rock, [email protected], Columbus State University, Georgia, and Mary K. Porter, [email protected],
Saint Mary’s College, Notre Dame, Indiana. MTMS readers are encouraged to submit single problems or groups of problems by individuals, student groups, or mathematics clubs to be considered for publication. Send to the editor, David Rock, at [email protected].
Published solutions problems will be credited.
Problem 1 was submitted by Robin Vredenburg, Stanwood, Michigan; problem 2, by David Harris, Allendale, Michigan; problem 4, by
Kathy McCarthy, Manchester, Michigan; and problems 6 and 7, by Leah Wischmeyer, Byron Center, Michigan.
8. If the following patterns of numbers continue, determine
the next three numbers in each sequence:
a. 1, 2, 4, 6, 10, 12, ___, ___, ___
b. 1, 0, 2, 1, 3, 2, 4, 3, 5, ___, ___, ___
9. If it takes a man 1 hour to dig a hole 3 meters long,
3 meters wide, and 3 meters deep, how long would it take
him to dig a hole 6 meters long, 6 meters wide, and
6 meters deep? Assume that he digs at the same rate of
speed.
10. A state high school baseball tournament is single
elimination. If 600 high school teams are participating,
how many games will be played before a state champion
is declared?
11. Each year in the Chinese calendar is named after
1 of 12 animals in rotation. The year 1963 was a
Year of the Rabbit. How many years in the twenty-first
century will be a Year of the Rabbit? What is the first
Year of the Rabbit and last Year of the Rabbit in the
twenty-first century?
13. Jackson weighs 68
pounds. A Ping-Pong™ ball
weighs 1/10 of an ounce.
How many Ping-Pong balls
would equal John’s weight?
14. On November 19, 1863, President Abraham
Lincoln began his Gettysburg Address:
Four score and seven years ago our fathers brought forth
on this continent, a new nation, conceived in Liberty, and
dedicated to the proposition that all men are created equal.
In 2009, how would President Barack Obama provide
a similar description using the words “score” and
“years ago”?
15. In what year will the United States of America
celebrate its tercentennial independence day?
16. You and a friend are playing tic-tac-toe. You win the
game. If you win in the shortest possible number of moves,
how many squares will be left unmarked by an X or an O?
12. A prime is a number whose only factors are 1 and
itself. A palindrome is a number that is read the same
forward and backward, such as 262. Find all two-digit
prime palindromes if any exist.
(Answers on page 303)
The solutions are appended to the online version of the
“Palette of Problems” at www.nctm.org/mtms.
Vol. 15, No. 5, December 2009/January 2010
●
Mathematics Teaching in the Middle School
267
solutions to palette
5x = (90 )( 3)
5 90
5x== 270
3 x
5xx==(54
90 )( 3)
5x = 270
(Continued from
pages 266–67)
1 .69
1
==54
xxxxxx
0 .75 x
(Alternative approaches to those suggested here are encouraged.)
1 .69x = ( 0 .75) • 1
xxxxx
1 .69
1
5 x=90
1 .69
= 0 .75
=
0 .75
xcosts about $0.44
ANSWERS
n2 – n + 1, in which n represents the
Therefore, this
3bag
x=≈x(00..44
1
.
69
x
5= (90
90 75) • 1
1. 54 minutes
player’s seniority level. For example,
per ounce. For5xthe
12.69-ounce
bag:
= )( 3)
=
.
1 .69
x
0
75
2. 31 and 43
the seventh player’s number is found as
3
x
5x = 270
12 .569
1.44)( 3)
3. No; the team cannot have more
follows: (7)2 – (7) + 1 = 49 – 7 + 1 = 43.
x ==(090
x =≈54
3
.
25
x
than 10 two-digit jerseys.
5x = 270
12 .69x = ( 3 .25) • 1
4. The 3-pound bag is the best buy
Player
Jersey Number
Add
.69x = 54
12
1
1 .69
since it costs about $0.19 per ounce.
.69=x=1= 3 .25
12
1
1
.
3
25
5. About 420 M&M’s are in the
0 .75 x x≈x0 .26
2
3
2
=( 0(13.75
12
.25) •)1• 1
1.69
.69
12.69-ounce bag; about 1591 M&M’s
1 .69
x x==
3
7
4
x==03.xcosts
.25 about $0.26
are in the 3-pound bag.
Therefore,12
0.69
.75
.69
1this
xbag
75
4
13
6
6. a. 11 star points for 4 gold medals;
per ounce. For
3-pound
1 .69the
(00.26
.75) • 1 bag, we
x=
≈
x ≈ 0 .44
5
21
8
48
1
b. 51 star points for 24 gold medals;
must convert
3
pounds
1 .69x ==0 .75into ounces
6
31
10
c. 2x + 3 star point for x gold medals
(16 oz. = 1 lb.;8 .399lbs. x= 48 oz.):
x ≈ 0 .44
7
43
12
12 .69
7. a. 12 brown spots on a puppy with
48=x 1= (8 .99) • 1
48 x1
21 black spots; b. 49 black spots on a
3 .25
48x== 8 .99
8.69
99
.69
puppy with 28 brown spots
3. Continue the table for 12 players.
1212
x ==( 31x.25) • 1
x ≈ 08 ..19
48
99) • 1
.
8. a. 16, 18, 22; b. 4, 6, 5
12 .36925
x x==3(.x25
12 .69
3.99
.25) • 1
9. 8 hours
48xx≈=0(8.26
Player
Jersey Number
Add
1
.
69
12
10. 599 games
12 .69x =
1
1
≈=30.25
19.69
11. 8 years; the first year will be 2011,
56x ≈ 0 .26
x
2
3
2
and the last year will be 2095
Therefore, the
3-pound
x
=
1
.
69
(
12
.bag
69)costs
• 56 about
3
7
4
1 .69and
12
.the
69 best buy.
12. 11 is the only two-digit prime
$0.19 per ounce
is
48
1
=
1 .69=x = 710 .64
4
13
6
56 x x
palindrome
8 .99
5
21
8
=56
148
.69
.69
13. 10,880 Ping-Pong balls
5. Since there
are
M&M’s
48
(8(112
.99
) • 1) • 56in a
x x==
6
31
10
1
.
69
48
14. 11 score and 13 years
1.69-ounce148
bag,
a proportion
to
.64
=8=710
.99
8.69
x x=use
.x99
7
43
12
56x = (8x.of
15. 2076
calculate the 48
number
in
) •1
99M&M’s
x ≈x 0=.(19
1 .69
) • 56
8
57
14
16. 4 squares
the other bags.
For a.48
12.69-ounce
bag:
99
148
.69x = 848
9
73
16
1 .69x== 2688
0x.19
56x ≈12
1 .69
.69
SOLUTIONS
10
91
18
1 .69x== ( 48) • 56
1. Since a 1 1/2 hour webcast is
11
111
20
x
56
1.69
.69
2688
x==(12
1
.
69
12
69) • 56
90 minutes, set up a proportion to
1
x
..69
12
133
22
7= 21
determine the number of minutes
= x.64
1 .6956
x = 710
x .69) • 56
she needs to read. Use 5 minutes of
Or use the expression to determine
1 .69x4= (12
7
x
=
(
21
) • 4 M&M’s.
computer time forxxxxx
every 3 minutes of
the numbers for the
10th, 11th, and
x ≈ 420.497,
or 7xabout
420,
xxxxx
1 .69
= 710
21 .64
1
.
69
48
reading, and solve:
12th players:
For a 3-pound,7or
bag:
x==48-ounce,
84
4= x x
56
12
102 – 10
1 = 91
7x x=x=(=48
(48
21
•4
5 +90
5 90
1 .69
1 .69
) •)56
=
2
=
=
11 – 11
+
1
=
111
3 x
7x x==2688
84
3 x
x
1 .6956
122 –512
1 =)(133
3)
x =+(90
5x = (90 )( 3)
1 .69x = (12
48) • 56
5x cost
= 270
5x = 270
4. Determine the
per ounce for
1 .69x = 2688
each bag by setting
54 a proportion
x = up
x = 54
21 1591, M&M’s.
for each. For the 1.69-ounce bag:
x ≈ 1590.53, or7 about
=
4
x
2. Making a table will show a pattern.
1 .69 1
1
.
69
1
7
21
7 x =listing
(See at right.) Beginning
with the first
6. Create a table
=
=
=( 21) • 4the number
0
.
75
x
x next number,
4= 84star
x points (see on
jersey, add 20to.75
get the
of gold medals
7 xand
1
.
69
=
(
.
)
x
0
75
•1
1
.
69
=
(
.
)
x
0
75
1
•
7
x
=
(
21)with
• 4 5 points
followed by adding 4, then 6, then 8,
the next page).x A
star
= 12
1 .69x = 0 .75
= 0The
.75 expression
1 .69
then 10, and
so xon.
7 x = 84
for determiningxthe
jersey
number
is
≈
.
x
0
44
≈ 0 .44
x = 12
Mathematics Teaching in the Middle School
12 .69 1
=
3 .25 x
●
Vol. 15, No. 5, December 2009/January 2010
12 .69 1
=
3 .25 x
3 .25 x
12 .69x = ( 3 .25) • 1
12 .69x = 3 .25
x ≈ 0 .26
No. of
No. of
Gold Medals
Star Points
48 1
=
1
5
8 .99 x
2
48x = (8 .99) •71
3
48x = 8 .99 9
4
x ≈ 0 .19 11
24
2x + 3 = 2(24) + 3 = 51
x
2x + 3
1 .69 12 .69
=
56
x
represents 1 gold medal. The number
1 .69x = (12 .69) • 56
of points increases by 2 with each gold
.64 can be
.69x = 710
medal. The 1number
of points
determined by the expression 2x + 3, in
which x is the1 .number
69 48 of gold medals.
=
56
x
7. For every17.69
black
the puppy
48) • 56
x = (spots,
has 4 brown1spots.
Since
21
.69x = 2688 black
spots is 3 × 7, the puppy will have
3 × 4 = 12 brown spots, or set up a
proportion:
7 21
=
4 x
7 x = ( 21) • 4
7 x = 84
x = 12
A total of 28 brown spots is seven
times greater than 4 brown spots. The
puppy will have seven times, or 7 × 7
= 49, more black spots.
8. a. The numbers are all one fewer
than consecutive prime numbers, which
are 17, 19, 23. So, the next three numbers in the pattern are 16, 18, and 22.
b. The pattern begins with 1. The
next terms are determined by using
the pattern subtracting 1, then adding
2 for the next term, and then subtracting 1, etc. Therefore, the second term
is found by starting with the first term
and subtracting 1 to yield 0; for the
next term, add 2 to yield 2; for the
next term, subtract 1 to yield 1; for
the next, add 2 to yield 3. The pattern
continues so that the next three terms
are 4, 6, and 5.
9. The new hole is twice as wide,
twice as long, and twice as deep as the
original, so it will take 2 × 2 × 2, or 8,
times as long. You could also find the
volume of each hole. The volume of
the first hole is 3 × 3 × 3 = 27 cubic
meters. The volume of the second
hole is 6 × 6 × 6 = 216 cubic meters,
which is 216/27 = 8 times greater.
10. Since this tournament is single
elimination, each loss will send a team
home. With an odd number of teams
at a given stage, one team gets a “bye,”
which means that it passes through
to the next round. Of the 600 teams,
there is only 1 champion, which means
there must be 599 losses, or 599 games
to declare the winner. Another explanation is the following:
Round 1:600 teams in 300 games,
300 winners
Round 2:300 teams in 150 games,
150 winners
Round 3:150 teams in 75 games,
75 winners
Round 4:75 teams in 37 games (one
team gets a bye),
37 winners (with 38
teams remaining)
Round 5:38 teams in 19 games,
19 winners
Round 6:19 teams in 9 games (one
team gets a bye), 9 winners
(with 10 teams remaining)
Round 7:10 teams in 5 games,
5 winners
Round 8:5 teams in 2 games (one
team gets a bye), 2 winners
(with 3 teams remaining)
Round 9:3 teams in 1 game (one
team gets a bye), 1 winner
(with 2 teams remaining)
Round 10:2 teams play 1 game to
declare the champion
1975, 1987, 1999, 2011, 2023, 2035,
2047, 2059, 2071, 2083, 2095. Eight
Years of the Rabbit will occur in the
twenty-first century, beginning with
2011 and ending with 2095.
12. The two-digit palindromes are 11,
22, 33, 44, 55, 66, 77, 88, and 99. Of
these, only 11 is a prime.
13. Ten Ping-Pong balls equal 1 oz.;
16 oz. are in 1 lb. Therefore, 16 × 10
= 160 Ping-Pong balls weigh 1 lb.
Since John weighs 68 lbs., it will take
160 × 68 = 10,880 Ping-Pong balls to
equal John’s 68 lbs.
14. A score is 20 years. Therefore, 4
score equals 4 × 20 = 80 years; 4 score
and 7 would be 80 + 7 = 87 years.
Lincoln spoke in 1863, and was referring to 1776. In 2009, the year 1776 is
2009 – 1776 = 233 years ago. To find
the number of scores:
233 ÷ 20 = 11 R13
Therefore, the time period would be
11 score and 13 years.
15. Tercentennial represents the celebration of the 300th anniversary of
an event. So 1776 + 300 = 2076.
16. If you start first and win with your
first three Xs, the other player would
have placed only two Os on the board.
Therefore, five squares will have been
played and there will be four remaining, unmarked squares.
Therefore, 300 + 150 + 75 + 37 + 19 +
9 + 5 + 2 + 1 + 1 = 599 games.
11. If 1963 is a Year of the Rabbit,
and it occurs every 12 years, then
these will also be Years of the Rabbit:
Vol. 15, No. 5, December 2009/January 2010
●
Mathematics Teaching in the Middle School