palette of
problems
David Rock and Mary K. Porter
1. A carpenter goes to a hardware store to buy a specific
number of pounds of nails for his business. The first store
he enters sells his favorite-sized nails at 9 cents per pound.
At this price, the carpenter realizes that he would be short
$7.15 if he tried to buy the nails. The carpenter goes to a
competing hardware store and finds that the nails sell for
6 cents a pound. He realizes that after paying for the nails,
he would have $2.45 left over. How many pounds of nails
is he trying to buy?
5. While John is driving his car, he notices that the odometer reads 13931 miles. The mileage is a palindrome, a
number that reads the same forward as it does backward.
Exactly 2 hours later, John notices that the odometer
displays a different palindrome. What is the most likely
average speed at which the car has been traveling?
2. Based on the box of numbers below, determine the
missing value in the bottom-right box.
4
9
11
6
8
12
13
16
27
19
26
43
23
7
28
31
47
?
6. Find the smallest positive integer such that when it is
divided by each of the following integers 7, 8, 9, and 10, it
will produce a remainder that is 1 less than the divisor
(7, 8, 9, or 10).
3. You have 3 darts and are standing 2 meters from a
rapidly spinning sphere made of cork. The diameter of the
sphere is 1 meter. You throw all 3 darts individually. Each
one lands on the sphere. What is the probability that you
landed all three darts in the same hemisphere?
4. One-sixth of a steel beam of a bridge is in cement below
the bed of a river. Two-fifths of the beam is in the water,
and 78 feet of the beam are above the water. How long is
the steel beam?
400
Mathematics Teaching in the Middle School
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7. A single die is rolled six
times successively. Which
probability is greater and by
how many times greater:
a. The probability that
each different face
(number) will come up
exactly once, or
b. The probability that a 1
will occur on each roll
Vol. 13, No. 7, March 2008
Copyright © 2008 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, 4225 University Ave., Columbus, GA 31907, and Mary
K. Porter, [email protected], Saint Mary’s College, Notre Dame, IN 46556.
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 “Palette” editor, David Rock, at [email protected]. MTMS is also interested in students’ creative solutions to these problems. Send to “The Thinking of Students” editor, Edward S. Mooney, at [email protected]. Both
problems and solutions will be credited. For additional problems, see the NCTM publication, Menu Collection: Problems Adapted from
“Mathematics Teaching in the Middle School” (stock number 726).
8. Make the equation below a true statement by repositioning one and only one digit in the equation. You cannot add or
delete any mathematical symbol or move the – or = signs.
14. At what point on the number line should we draw a
vertical line so that the vertical line cuts the region (shown
below) into two parts of equal area?
26 – 63 = 1
9. Lindsay wants to store some DVDs in a rectangular
box that is 15 inches tall and that has a base measuring
2 ft. × 3 ft. 7 in. If each DVD package measures 5 3/8 in.
× 7 1/2 in. × 1/2 in., how many DVD packages will fit in
the storage box?
10. Continue the pattern by listing the next five terms in
the sequence below:
17, 8, 25, 7, 32, 5, 37, 10, 47,
11, 58, 13, 71, 8, __, __, __, __, __
11. Using the same pattern as in the previous problem
but starting with 22 as the first term instead of 17, list the
next five terms after 22 to continue the pattern.
12. Find the values of the positive numbers a and b, given
that a is 8 times the size of b and the reciprocal of ab is 1
less than 3.
1
2
3
4
5
6
7
15. Jason has 6 different
pairs of pants, 10 different shirts, 2 different belts,
and 4 different neckties.
Assume that wearing a
tie and a belt is optional.
How many different outfits
can Jason make from his
wardrobe?
1 2 any
3 digits,
4 5 insert
6 7
16. Without rearranging
one multiplication symbol (×) into the following
3 1/4 expression to obtain the
greatest possible product:
13. Find the values of a and b so that 8 is both the mean
and median of this set of numbers:
12345
{a, b, 10, 7, 11}
(Solutions on pages 402–3)
Vol. 13, No. 7, March 2008
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Mathematics Teaching in the Middle School
401
solutions to palette
(Continued from pages 400–401)
(Alternative approaches to those suggested here are encouraged.)
1. 320 pounds. Let x represent the number of pounds of nails, and let y represent the amount of money the carpenter
has to spend. Then, 9x and 6x represent
the costs of nails in cents from the two
hardware stores. Write two equations:
reach this value, John will have to travel
110 miles. If it takes him 2 hours, then
his average speed is 55 mph. The next
palindrome, 14141, would mean that
the car traveled 210 miles at 105 mph,
which is probably unrealistic.
y – 9x = –715 and y – 6x = 245
6. 2519. The least common multiple
of 7, 8, 9, and 10 is 2520, or 2 × 2 ×
2 × 3 × 3 × 5 × 7. When 2520 is
divided by each of the given numbers,
the remainder is 0. Therefore, if 2519,
which is 1 less than 2520, is divided
by these numbers, the remainder will
be 1 less than the divisor.
Solving for y and equating the two,
we have:
9x – 715 = 6x + 245
3x = 960
x = 320
2. 76. For each row, the number in
the third column equals the number
in the first column plus the number in
the second column minus two; therefore, 31 + 47 – 2 = 76.
3. 1, or 100 percent. Any 3 points on
a sphere lie in the same hemisphere.
The problem does not ask you to
identify the hemisphere before you
toss the first dart. After the 3 darts
have landed, it is guaranteed that you
will be able to divide the sphere into
two hemispheres so that all 3 darts
will be in the same hemisphere.
4. 180 feet. Set up an algebraic equation
where x is the length of the steel beam.
(1/6)x + (2/5)x + 78 = x
Multiply both sides of the equation by
the common denominator 30.
5x + 12x + 2340 = 30x
17x + 2340 = 30x
2340 = 13x
x = 180
5. 55 mph. The next palindrome that
the odometer can display is 14041. To
402
7. The probability of rolling six different numbers is 720 times greater than
rolling six straight 1’s. The probability
of rolling a 1 is 1/6. The probability
of rolling 1 six consecutive times is
1/6 × 1/6 × 1/6 × 1/6 × 1/6 × 1/6,
or 1/46,656. The chance of rolling a
different number each time is much
greater. Consider the event of rolling six different numbers in a row.
The first roll is guaranteed to be a
new number. The probability of this
number showing is 6/6, or 1. The second roll can be any of five remaining
numbers. Therefore, the probability
is 5/6. The next roll can be any of the
other four numbers, with a probability
of 4/6, and so on. Therefore, the probability of rolling six different numbers
is 6/6 × 5/6 × 4/6 × 3/6 × 2/6 × 1/6 =
720/46,656, which is greater than rolling six consecutive 1’s. The probability
720/46,656 is 720 times greater.
in. × 2 = 15 in., so she can stack 2 of
the 7 1/2-inch edges along the 15inch height of the box. The remaining
dimension of the storage box is 2 feet,
or 24 inches, so she can fit 48 of the
1/2-inch edges along this 24-inch side.
Thus, Lindsay can fit 8 × 2 × 48 = 768
DVDs into the storage box.
10. 79, 16, 95, 14, 109. The second
term is found by adding the digits of
the first term: 1 + 7 = 8. The fourth
term is found by adding the digits of
the third term: 2 + 5 = 7. Similarly,
each of the remaining even-numbered
terms (sixth, eighth, tenth, and so on)
is found by adding the digits of the
previous term. To find the third term,
simply add the previous two terms:
17 + 8 = 25. The fifth term is the sum
of the previous two terms: 25 + 7 = 32.
Similarly, each of the remaining oddnumbered terms (seventh, ninth,
eleventh, and so on) is the sum of the
previous two terms. Thus, after the
terms 71 and 8, the next term is 71 +
8 = 79. The term after 79 is 7 + 9 =
16. The next term is 79 + 16 = 95.
The term after that is 9 + 5 = 14. The
next term after that is 95 + 14 = 109.
11. 4, 26, 8, 34, 7. Use the same pattern as in the previous problem to find
the terms after 22: 2 + 2 = 4, 22 + 4 =
26, 2 + 6 = 8, 26 + 8 = 34, 3 + 4 = 7.
8. Change 26 to 26, resulting in
26 – 63 = 1, which is 64 – 63 = 1.
12. a = 2, b = 1/4. We know that 1/ab =
3 – 1, or 1/ab = 2, so ab = 1/2. Since
a = 8b and ab = 1/2, by substitution,
1/2 = (8b)b = 8b2, so 1/16 = b2. Thus,
since b is positive, b = 1/4. Since a =
8b, then a = 8(1/4) = 2.
9. 768 DVDs. 5 3/8 inches is 43/8
inches, and 3 feet 7 inches is 43 inches,
so she can fit 8 of the 5 3/8-inch edges
along the longest side of the box; 7 1/2
13. 4 and 8. The set contains five
numbers (an odd number of elements) so the median, 8, will be the
middle number. Since 8 is not already
Mathematics Teaching in the Middle School
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Vol. 13, No. 7, March 2008
an element of the set, we know that
one of the numbers, a or b, must be 8.
Without loss of generality, let’s say b =
8. Since 8 is also the mean of this set,
we know that
Thus,
a + 8 + 10 + 7 + 11
8.
a + 8 + 10 + 7 +=11
= 8.
5
5
two portions (shown here) are not of
equal size. The shaded part only has
an area of four square units. Thus, we
need to
1 move
2 the
3 vertical
4 5 line
6 to7the
left, so that the shaded portion on the
right-hand side will have an area of
five square units.
15. 900. Jason has 6 options for pants,
10 options for shirts, 3 options for
belts (the third option is not wearing
a belt), and 5 options for neckties (the
fifth option is not wearing a necktie). Thus, he has this many different
combinations of outfits: 6 × 10 × 3 ×
5 = 900.
a + 36
8,
a +=36
= 8,
5
5
so a + 36 = 40, and therefore, a = 4.
Hence, our two numbers are 4 and 8.
14. Draw the vertical line at the
number 3.25 on the number line. The
entire region has an area of ten square
units (10 squares), so when we divide
the region into two equal regions,
each half will have an area of five
square units. If we drew the vertical
line at 3.5 on the number line, the
units. So 4w = 1, where w is the width
of the additional part. The width will
need to be 1/4 unit. Thus, we need
to move the vertical line 1/4 unit to
the left of where it is in the diagram
above, which is 3.25.
1
2
3
4
5
6
7
3 1/4
We must move our vertical line to the
left to gain an extra square unit on
the right-hand side. The height of the
additional part we want to shade is 4
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16. 1234 × 5 = 6170. There are only
four different positions for the multiplication symbol: 1 × 2345 = 2345,
12 × 345 = 4140, 123 × 45 = 5535,
and 1234 × 5 = 6170. The last one
gives the greatest product, 6170. l
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