[November Calendar]Author(s): Jean McGivney-Burelle and Janet A. WhiteSource: The Mathematics Teacher, Vol. 101, No. 4, Mathematical Discourse (NOVEMBER 2007),pp. 280-284Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20876112 .

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One line has a slope of m and a ^-intercept

of 2. A different line has a slope of 2 and a ^-intercept of m. At what coordinates, in terms of m, must the lines intersect?

4

The intersection of a unit cube and a plane passing through its center is a regular hexa gon. What is the area of the regular hexagon?

8

For some fixed constant(s) b, the statement sin x = cos (x + b) is

an

identity that is true for all x. Find all possible values of b in radi

ans, 0<b< 2k.

Jim has 18 hours to fly to Moscow and

then get to the train station to catch a train

to Siberia. It takes 30 minutes to load the plane and take off, 5 minutes to taxi to the

gate, 1 hour to get through customs, and 25 minutes to get to the train. The flight distance to Moscow is 12,000 km. What is the slowest speed the plane could average

to get Jim to the train on time?

1

Ka'rin has a banana. She eats 2/3 of it and then gives the rest to her younger brother, Jamal. Jamal eats 2/3 of what he was given and then gives the rest to his baby sister,

Kia. Kia eats half of what she was given

and then throws the rest away. What per

centage of the banana was thrown away?

A bicycle lock has a four-digit combination made by turning 4 wheels. Each wheel has the digits 0 through 9 in order; 0 reappears after 9. Unfortunately, the lock is broken; each time you turn 1 of the 4 wheels, an adjacent wheel also turns in the same direc tion. The correct combination is 2000. From

which one or more of the following starting

numbers is it possible to open the lock?

0000 1999 2001 3456

6543

7777 8161

8181

A school administration has instituted a rule that students' shorts may be no short er than 1/5 of their height. Sam's shorts were only 6 inches

long, and the adminis

tration said that these shorts were 3/7 of the minimum length. How tall is Sam?

NCTM

NATIONAL COUNCIL OF

TEACHERS OF MATHEMATICS

Dr. Small is 36 inches tall, and Ms. Tall is 96 inches tall. If Dr. Small shrinks 2 inches per year and Ms. Tall grows 2/3 of an inch

per year, how tall will Ms. Tall be when Dr. Small disappears altogether?

2

A mischievous tennis class decided to rebel by throwing tennis balls over a fence. One sixth of the students threw 5 balls each, 1/2 threw 4 balls each, 1 student threw 6 balls, and the rest threw 2 balls each. Of the balls

thrown, 75% went over the fence. When the students had to collect them later, they

found all 66 balls that went over the fence. How many students are in the class? ^ The product of the first n terms in a se quence is always equal to n. What is the

2007th term in this sequence?

10

How many base-two numbers are 4 (or is

it 100?) digits long?

Five students?Anita, Nalla, Tia, Thomas, and Igor?participated in an algebra contest and scored, not necessarily in order, 2, 5, 6,

8, and 9. Arrange the student names from lowest to highest scorer, using these clues: (1) Anita got more than 6 points. (2) Nalla got

an odd score. (3) Tia did not get the highest

score. (4) Thomas's score was a prime

number. (5) Igor's score was the

second lowest.

In rectangle ABCD, point E is on side AB so that AE = 10 and EB = 5. What frac tion of the area of the rectangle is inside

triangle AEC?

7

The sum of the first n terms in a sequence is always 1/n. Find the product of the first

2007 terms.

11

An 8.5 x 11-inch sheet of paper is cut in half lengthwise, while an identical sheet of paper is torn in half widthwise. Do the resulting half sheets have the same area,

the same perimeter, or both?

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It takes one man one day to dig a 2 m x

2 m x 2 m hole. How long does it take

3 men working at the same rate to dig a

4mx4mx4m

hole?

16

Find the

constant

sum for the magic hexa gon and fill in

the numbers

so that every

column or

diagonal

has that sum.

20

If grapefruit are piled in a pyramid with 1 grapefruit in the top layer, 4 in the second

layer (from the top), 9 in the third layer, and 16 in the

fourth layer, how many grapefruit will be needed to make a pile

with 10 layers?

24

In a recent survey, 40% of households con tained two or more people. Of those homes containing only 1 person, 25% contained a male. What is the

percentage of all houses

that contain

exactly

1

female and no males?

28

From a rectangular sheet of paper, Evan makes a small open rectangular prism by

cutting four

congruent

squares from each corner and taping

the sides together. The

final prism will have a 5 cm x 4 cm base; its

volume

is 60 cm3. What is the area of

the original piece

of paper?

^

In the quadrilateral

ABCD, mZA = 120?, mZD is two-thirds of mZC, and mZB =

90?. Find rnZC

21

What is the surface area of the solid

figure?

1 cm

4 cm

Triangle PQR

is equilateral

with QR = 30 units. A is the

foot

of the perpendicular from Q to PR

and

B is the midpoint of

QA. What

is the length of PB?

29

A secondhand tire shop sells only one size

of retreaded tires, 0.75 m in diameter, at a

cost of $240 for

four

tires. To expand busi ness, the shop owner decides to sell tires

that are 1.2 m in diameter. If he wants prices to be proportional to the square of the tire diameters, then how much will it cost for a set of 18 of the larger tires?

18

Two gear wheels, A and B, are in contact. Wheel A has

36 teeth,

and wheel B has 24 teeth. How many revolutions must the smaller wheel make before the larger

wheel completes one revolution?

Triangle ABC is an isosceles right triangle with BC = AB = 2.

Circular

arcs of radius 2 centered at C and A meet the hypotenuse at

D and E, respectively.

What is the area of the shaded region?

26

A polyhedron is formed by connecting the midpoints of

adjacent

edges of a cube. If the cube has an

edge length of 4 cm, what

is the surface area of the polyhedron?

30

M and N are the

midpoints

of the sides of a square. What is the ratio of the area of

AAMN to the area of the complete square?

19

The points A, ?, C, D, and E are located on a straight line, in order, in accordance with the following

conditions. What is the

distance from B to C?

The distance from A to E is 20 cm. The distance from A to D is 15 cm. The distance from B to E is 10 cm. C is halfway between B and D.

23

Fred picked four numbers out of a hat. The average of the four numbers is 9. If three of the numbers are

5, 9 and 12, what is the

fourth number?

27

? National Council of Teachers of

Mathematics,

1906 Association Drive, Reston, VA 20191-1502

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Looking for more "Calendar" problems? Visit www.nctm.org/publications/ mtcalendar for a collection of previ ously published problems?sortable by topic?from the Mathematics Teacher.

Edited by Jean McGivney-Burelle, burelle?

hartford.edu, University of Hartford, West

Hartford, CT 06117, and Janet A. White,

jwhite@millersville.edu, Millersville Univer

sity, Millersville, PA 17551-0302.

Problems 1-12 were adapted from Gunn

Mathematics Competition, 2000. Problems

14-15 were submitted by William Jamski, In diana University Southeast. Problems 16-30 were adapted from The Canadian School Math Page: Word Problems for Kids, www.

stfx.ca/special/mathproblems/.

The Editorial Panel of the Mathematics Teacher is considering sets of problems sub

mitted by individuals, classes of prospective teachers, and mathematics clubs for publica tion in the monthly "Calendar." Send prob lems to the "Calendar" editors. Remember to

include a complete solution for each problem submitted.

Other sources of problems in calendar form

available from NCTM include Calendar Prob

lems from the Mathematics Teacher (a book

featuring more than 400 problems, orga nized by topic; stock number 12509, $22.95), and the 100 Problem Poster (stock number 13207, $9.00). Individual members receive a

20 percent discount off this price. A catalog of educational materials is available at www.

nctm.org.?Eds.

1. 750 km/hr. Of the 18 hours Jim has to

get to the Moscow train station, 2 hours total are spent not flying, which means the plane flies for a total of 16 hours. We divide 12,000 km by 16 hr. to find that the plane must travel an average of 750 km/hr. to arrive in Moscow on time. Note: Most commercial airliners cruise at an average rate of 500-900 km/hr.

2.108 inches, or 9 feet. Since Dr. Small is 36 inches tall and shrinks 2 inches a year, it will take Dr. Small 18 years to disap pear altogether. Meanwhile, Ms. Tall, cur

rently 96 inches tall, will grow (2/3) x 18 = 12 inches to reach 108 inches, or 9 feet.

3. Thomas, Igor, Tia, Anita, Nalla. Clue 5

implies Igor scored 5; clue 4, then, implies Thomas scored 2; clue 2 implies Nalla scored 9; clue 1 implies Anita scored 8; therefore Tia must have scored 6.

4. (1, m + 2). The equations for the given lines are y = mx + 2 and y = 2x + m, so they intersect when mx +2 = 2x+m. Thus,

mx - 2x = m - 2, or x(m

- 2)

= m - 2, and

x-1 (provided m * 2). Theny = mx + 2,

so substituting 1 for x gives y = m(l) + 2 =

m + 2. Note: m*2because m-2 means

we do not have two distinct lines.

5. 5.56%. Jamal gets 1/3 of the banana, while Kia gets 1/3 of 1/3, or 1/9, of the banana. Half of what Kia received was thrown away, so (1/2) x (1/9) = 1/18, or

5.56%, of the banana was thrown away.

6. 24 students. Let z = number of stu dents. We have

0.75 5.- + 4.- + 1.6 + 2 6 2

Then

z z z-1

6 2 = 66.

? + 2z + 6 + 2z---z-2 = 88. 6 3

And

so

2=24.

7. 1/3. Let b represent the length of side AB and h represent the length of side BC. The base of triangle AEC is 2/3 the base of the rectangle ABCD. Further, triangle AEC and rectangle ABCD have the same height. Therefore, the area of

triangle AEC is (1/2) (2/3 b)h = 1/3 bh.

8. 3V3/4 square inches. The sides of the hexagon connect the midpoints of 6

edges of the cube and are therefore each V2/2 inches long. Since a hexagon can be divided into six equilateral triangles with

height V6/4, we can use the area formula for triangles to find the area of the hexa

gon, which is 3V3/4 square inches.

9.1999, 6543, and 8161. Each time two

adjacent wheels of a combination abed are

turned, the expression w-a-b + c-d

does not change or changes only by adding or subtracting a multiple of 10. Therefore, w must be 2,12, -8, or -18. Consequently,

1999, 6543, and 8161 are the solutions.

10. 2007/2006. If there is 1 term, the

product must be 1, so the first number must be 1. If there are 2 terms, the prod uct must be 2, so the second number must be 2: 1 2 = 2. For 3 terms, the

product must be 3, so the third number

282 MATHEMATICS TEACHER | Vol. 101, No. 4 November 2007

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must be 3/2: 1 2 (3/2) = 3. If there are n terms and the product must be n, then the nth term must be n/(n - 1) because

1-2 3 4

2 3

n-1

n-2 n-1 j \ = n.

To find the 2007th term, let n = 2007.

11. 1 2007

200612007! (2007!)2

The first term must equal 1. For the sec ond term, 1 + x = 1/2 -? x = -1/2. For the third term, 1-1/2+^=1/3-^

= -1/6.

Adding the fourth term, -1/12, allows us to identify the pattern in the sequence:

1 _ 1 _ 1 _1_

1' 2' 6' 12'-'

which is equivalent to

1_1_1_1_ l' 1-2' 2-3' 3-4'"'

We note that all terms after the first are

negative, and the 2007th term is

1

2006-2007

We require the product:

_l-(-l)2006_

1(1 2) (2 3) (3 4) (2006 2007) 1 2007

2006!2007! (2007!)2

12. 3^/2.

smx = cos ?x 2

= COS-X 2

k = COS| X

2

= COS 3k

Hence, b = 3n/2.

13. 5 feet 10 inches, or 70 inches. Since 6 inches is 3/7 of the allowed length, the allowed length equals 14 inches. If we

let h = Sam's height, then 14 inches =

(l/5)ft. Solving for h, we find that h = 70 inches.

14. 8. Base-two numbers are made with 0s and Is where each place value is a

power of 2. There are only eight pos sibilities: 1000, 1001, 1010, 1011, 1100, 1101,1110, and 1111.

Alternative solution. The number 8, written as 1000 in base two, requires 4

digits; 16, the next power of 2, requires 5 digits. There are 8 integers from 8 to

15, inclusive.

15. The area is the same; the perimeter is different. The half sheet of paper re

sulting from the lengthwise cut has an area of 4.25 x 11 = 46.75 square inches and a perimeter of 2(11) + 2(4.25) =

30.5 inches. The half sheet resulting from the widthwise cut has an area of 8.5 x 5.5 = 46.75 square inches and a

perimeter of 2(8.5) + 2(5.5) = 28 inches.

16. 2 2/3 days. Let x = rate of one man = (2 m)3/day

= 8 mVday. So 3x =

the rate of the three men = 3(8 mVday) =

24 mVday. Since the volume to be shov eled is a total of 64 m3, it would take the three men

64m3 2A = 2- days. 24 m3/day 3

17. 110 cm2. The volume of the rectangu lar box will be 5 x 4 x h, where h is the side of the square cut out of each corner.

Thus 5 4 h = 60, and h = 3 cm. The

original piece of paper must then have had dimensions of (5 + 2 3) x (4 + 2 3), or 11 x 10 for an area of 110 cm2.

18. $2764.80. Let x = price for each big tire. The 0.75 m-diameter tires sell for

$60 each ($240 for four). This leads to

ML ML 60 x

'

and x = $153.60. Therefore, 18 tires will cost 18($153.60) = $2764.80.

20. Sum of 38.

21. 90?. Since the sum of the angles in the quadrilateral is 360?, mZA +

mZB + mZC + mZD = 360?. By substitution,

120? + 90? + mZC + - mZC = 360? 3

->-raZC = 150? 3

and therefore mZC = 90?.

22. 1.5. Each revolution of wheel B meets 24 teeth of wheel A. Therefore, (36

- 24) = 12 teeth of wheel A that are

not met during the first revolution of wheel B. This will then take 12/24 = 1/2 revolution to meet all the teeth, for a

total of 1 1/2 revolutions.

23. 2.5 cm. Since AE = 20 cm and BE =

10 cm, AB =10 cm. Similarly, since AE = 20 cm and AD = 15 cm, DE =

5 cm. Thus, BD = 5 cm, and BC = 2.5 cm.

10

A B C D E

15

24. 385. Each layer forms a square array of grapefruit: The first layer has 1 grape fruit, the second layer has 4, the third

layer has 9, the fourth layer has 16, and so forth. Thus we need to find the sum of 1 + 4 + 9 + 16 + 25 + + 81 + 100 = l2 +

22 + 32 + 42 + + 92 + 102 = 385. This

problem illustrates the fact that we call the sum of the first n squares the nth

"pyramidal number." Note: The formula for the rath pyramidal number is

Pz/r = i?(? + l)(2? + l). b

Vol. 101, No. 4 November 2007 | MATHEMATICS TEACHER 283

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25. 58 cm2. To find the total surface

area, we need to find the area of the

rectangles on the left and bottom of the figure. The front and back of the

figure will be congruent, and the right "side" and the "top" are made up of 6

rectangles (5 of which are congruent). This leads to finding the sum of each

part.

Left: 3 cm x 4 cm = 12 cm2 Bottom: 3 cm x 3 cm = 9 cm2 Front and back: 2 x (1 cm2 + 4 cm2 +

3 cm2) = 16 cm2

Right and top: 5 x (3 cm2) + (6 cm2) =

15cm2+6cm2 = 21cm2 The total area is 12 cm2 + 9 cm2+

16 cm2+21 cm2 = 58 cm2.

1 cm

3 cm

Alternative solution. We can view the solid as a rectangular prism with one base

facing us. Then the total surface area is

perimeter x height + 2 x area of base; or 14 cm x 3 cm + 2(1 + 4 + 3 cm2) = 58 cm2.

26. 4 - k ? 0.8584 units2.

The area of triangle ABC = -(2 2) =

2 units2. 2

Since triangle ABC is isosceles, raZA =

45? and

the area of sector ABE = the area of sector BDC

So the unshaded region is

2^j/r-2 = /r-2.

Thus, the shaded region is 2 -

(/r -

2) =

4-^-0.8584.

27. 10. Let x represent the fourth num ber. Then:

x + 5 + 9 + 12

4 x + 26

x

28. 45%. Since 40% of households con tain two or more people, 60% contain 1 person. Of those homes with exactly

= 9-?

= 36 = 10

1 person, 25% are male and 75% are female. Thus, the percentage of house holds with exactly 1 female and no males is 60% of 75% = (0.60)(0.75) =

0.45, or 45%.

29. ? V7? 19.84.

Since triangle APQ is a 30-60-90? tri

angle, AP= 15, and

QA = 15^3

30. 48 + 16V3 cm2. There are 14 total fac

es, 6 congruent squares, and 8 congruent triangles. Each square will have an area of

(2>/2)2=8cm2.

Each triangle will have an area of

^(2V2)2 = 273cm2.

So the total surface area is

6 8 + 8 2>/3 = 48 +16>/3 cm2.

The Albert Einstein

Distinguished Educator Fellowship

Program

K-12 Classroom Teachers

providing expertise to program managers and

policy makers

Paid Academic Year Fellowships in Washington, DC working with

Congress or in a Federal Agency Application available on-line

10/1/07 To apply please visit:

www.trianglecoalition.org/ein.htm

284 MATHEMATICS TEACHER | Vol. 101, No. 4 November 2007

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