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I forgot the combination!
I’m ready to cry.
How many combinations
will I have to try?
Figure This! This combination lock uses three
numbers from 0 to 39. It opens when these
numbers are dialed in a particular order:
right, left, right. How many possible combinations are there?
Hint: Think about how many choices you have before
dialing each number.
Counting the number of possible arrangements in a given
situation is an important mathematical skill. Banks, phone
companies, and security systems use codes based on such
arrangements for Personal Identification Numbers (PIN).
© 2004 National Council of Teachers of Mathematics, www.figurethis.org/.
Used with permission.
Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource • 1
2
3
1
Combination
1
1-1-1
2
1-1-2
3
1-1-3
1
1-2-1
2
1-2-2
3
1-2-3
1
1-3-1
2
1-3-2
3
1-3-3
Start
If you start with 1, you get nine different combinations. If you start with
2 instead of 1, you also get 9 different combinations. If you start with 3,
you get 9 more possibilities for a total of 9 + 9 + 9, or 27 different combinations. Think about this in terms of choices. You can choose any of three
numbers as a possible first number in the combination, follow that with a
choice of any of the three numbers as the second number in the combination, and finally choose any of the three numbers for the final number in
the combination for a total of 3 x 3 x 3, or 27 different combinations. If the
lock used the numbers 1, 2, 3, and 4, you would have to choose from four
numbers, three different times. This would give you 4 x 4 x 4, or 64 choices
for the combination. The lock in the challenge requires that you choose from
40 different numbers, three different times. Therefore, there are 40 x 40 x 40,
or 64,000 different combinations.
Try This:
• Experiment with a combination lock. Do you have to dial each number exactly?
Things to Think About:
• Why do you think that telephone companies have to add new area
codes for certain regions?
• How many phone numbers do you think are served by each area code?
• Which kind of lock will allow more possible combinations: a keypad lock
or a lock with a dial?
• Find all the locks in your house. How are they alike? How are they
different?
Did You Know That?
• Mathematicians use the word combination differently than it is used in
a "combination lock." In a mathematical combination, the order in which
an item occurs is not important.
• Some old safes can be opened using more than one combination.
• A disc lock consists of a sequence of discs numbered on their outer
edges. To open the lock, you turn the discs (usually three) to the
appropriate numbers.
• Many apartment buildings, businesses, and airports—even restrooms—
use some form of keypad lock.
Resources:
Websites:
• www.howstuffworks.com/inside-lock.htm
• www.thelockman.com/cl1.htm
Answers to Additional Challenges:
Additional Challenges:
1. Suppose the combination for a particular brand of lock allows each number to appear only once. If the lock uses three numbers from 0 to 39
(as in the challenge), how many combinations are possible?
40 x 39 x 38 = 59,280.
1
3rd Number
3
2nd Number
(2.)
1st Number
3 = 27.
In the challenge, the lock uses the numbers 0 to 39. Start with an easier
problem using 1, 2, and 3. You can count the possibilities by drawing a
tree diagram. A portion of this tree diagram is shown below.
3. Some keypad locks have 10 keys numbered 0 to 9. You open the lock
using 3 keystrokes. A keystroke can consist of pressing 1 key or of
pressing 2 keys at the same time. How many combinations are possible
on this lock?
3
Complete Solution:
2. Suppose the combination for a bicycle lock is 10-24-32. With this lock,
however, the numbers on either side work as well as the actual number.
For example, the combinations 9-25-33 and 11-23-33 will also open the
lock. How many different combinations will open this particular lock?
(3.)
Try an easier problem. If the lock uses only the numbers 1, 2, and 3, how
many different three-number combinations are possible? Write out the
possible combinations in an ordered list and look for a pattern. What if the
lock uses the numbers 1, 2, 3, and 4?
55 = 166,375.
Get Started:
(1.)
2 • Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource
50
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Do dogs age
faster
?
than people?
?
?
Figure This! Ratio’s dad is 35 years
old. When will his dad and his newborn Irish setter puppy be the same
age in human years?
Hint: The growth rate is different the first
year. After the first year, how many human
years does a dog age each year?
Information is often converted from one
system of measurement to another.
Statisticians, scientists, and engineers all
use conversion formulas in their work.
© 2004 National Council of Teachers of Mathematics, www.figurethis.org/.
Used with permission.
Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource • 1
Get Started:
Add a row to the table that shows Ratio’s dad’s age with each passing year.
Large Dog’s Age in Years
0
1
2
3
Equivalent Human Age
0 12 19 26
4
5
6
4. The following table shows the relationship between actual age and
”human age” for cats.
7
Ratio’s Dad’s Age in Years 35 36 37 38
Cat’s Age in Years
0
Equivalent Human Age
0 16 24 28
1
2
3
4
5
6
The formula for the human age of a cat is H = 16 + 4c, for c > 2.
When a cat is 36 in human years, how old is it in cat years?
Complete Solution:
• After the first year, the ”human age” of a large dog such as an Irish setter
increases by 7 each year. Use this information to complete the table.
When the dog is 5 years old, its human age will be 40. In 5 years, Ratio’s
dad also will be 40.
Large Dog’s Age in Years
0
1
2
3
4
5
Equivalent Human Age
0 12 19 26 33 40
Ratio’s Dad’s Age in Years
35 36 37 38 39 40
Try This:
• Find the ages of some neighbors and friends, along with the ages of their
dogs or cats. How many of them are ”older” than their pets? [Note: The
age table given in the Challenge is for large dogs. See the Additional
Challenges for age tables for small dogs and for cats.]
• Estimate the age (in months) of a large dog whose ”human age” is about
the same as your own.
Things to Think About:
• Given the actual age of a small dog, how could you find the actual age
of a cat with the same ”human age?”
• How many actual years does it take for a small dog to reach
”retirement age?”
• In the Challenge, you related the years in an animal’s life to a human
age. How could you relate the years in a human’s life to an animal’s
age? For example, if you were 13, what age would you be in cat years?
• How did biologists decide that a 1-year-old small dog is comparable to a
15-year-old human?
• Why do puppies grow faster in their first year of life than in later years?
Is this also true for human babies?
• Do larger animals typically live longer than smaller ones?
• Which animals live the longest?
Additional Challenges:
(Answers located in back of booklet)
1. Find a formula that shows the relationship between the human age H of
a large dog and the dog’s actual age d.
Did You Know That?
• An Asian elephant has an average life span of about 40 years. In 1998, the
average life span of a US resident was about 77. The average life span of
an opossum is 1 year.
2. The table below shows the relationship between actual age and ”human
age” for small dogs.
• One familiar conversion formula describes the relationship between
degrees Celsius and degrees Fahrenheit: F = 1.8C + 32.
Small Dog’s Age in Years
0
1
2
3
4
Equivalent Human Age
0 15 25 30 35
5
6
a. Based on this information, what type of dogs appear to age faster:
large or small?
b. When Ratio’s dad was 35, if he had adopted a small newborn dog,
when would the two have been the same age in human years?
c. If a small dog and a large dog were born on the same day, when will
the small dog be 10 human years younger than the large dog?
3. How would a formula for the human age of a small dog differ from the
one for large dogs?
• For an age older than 1, the graph of equivalent human age versus a large
dog’s age in years lies along a straight line.
• According to The Guinness Book of Records, the world’s oldest dog was
an Australian cattle-dog named Bluey. He died at the age of 29 years
and 5 months in 1939. The world’s oldest cat, a sphinx, was Granpa
Rexs Allen, adopted from the Humane Society of Travis County (Texas),
who died in 1998 at the age of 34 years, 2 months.
Resources:
Books:
• Burrill, G., M. Clifford, and R. Scheaffer. Exploring Symbols: Data Driven
Mathematics. White Plains, NY: Dale Seymour Publications, 1987.
• Roizen, M. Real Age: Are You As Young As You Could Be? Scranton, PA:
Harper Collins, 2000.
2 • Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource
C
ES
55
X
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TI NG B
How
do you
a
print
ribbon
on a box
?
Figure This! Which of the patterns can be folded
into a box with an orange ribbon printed continuously
all the way around it?
Hint: Think about folding each pattern to make a
box. Do the ends of the orange ribbon meet?
Two-dimensional patterns are often used to design
three-dimensional objects. Visualizing three-dimensional objects from two-dimensional patterns is
important for architects, artists, designers, model
manufacturers, and doctors.
© 2004 National Council of Teachers of Mathematics, www.figurethis.org/.
Used with permission.
Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource • 1
Get Started:
Get Started:
Draw each figure on a sheet of paper, then cut out the pattern. What
Draw
each
figure
onfold
a sheet
paper,into
then
cut out the pattern. What
happens
when
you
each of
pattern
a box?
happens when you fold each pattern into a box?
Complete Solution:
Complete Solution:
One way to solve the problem is to assign letters to matching corners as
One
way
the problem
is tothe
assign
letters
matching
corners
shown
in to
thesolve
diagram
below. Fold
pattern
into to
a box
and see
if the as
shown
in the diagram below. Fold the pattern into a box and see if the
ribbon matches.
ribbon matches.
1.
1.
2.
2.
3.
3.
In pattern 2, the ends of the ribbon will not meet.
In pattern 2, the ends of the ribbon will not meet.
Pattern 3 is pictured below.
Pattern 3 is pictured below.
4. Pattern 4 is pictured below.
4. Pattern 4 is pictured below.
Try This:
Try This:
• Find a pattern, not in the challenge, that folds into a box so that a
• continuous
Find a pattern,
not isinformed
the challenge,
thatbox
folds
intotogether.
a box so that a
ribbon
when the
is put
continuous ribbon is formed when the box is put together.
• Determine how to wrap a package using the least amount of ribbon
• if
Determine
how to
a package using the least amount of ribbon
it has to touch
allwrap
six faces.
if it has to touch all six faces.
• Find the US Postal Service rules for wrapping and taping boxes for
• shipping.
Find the US Postal Service rules for wrapping and taping boxes for
shipping.
• Some products are sold in boxes with unusual shapes. Find some of
• these
Some boxes,
products
areunfold
sold inthem
boxes
with
unusual
shapes.
FindExamine
some of
then
and
examine
their
patterns.
these
boxes,
then
unfold
them
and
examine
their
patterns.
Examine
the patterns used to make them.
the patterns used to make them.
Additional Challenges:
Additional Challenges:
(Answers located in back of booklet)
(Answers located in back of booklet)
1. A rectangular box is tied with a ribbon so that the ribbon crosses the
1. A
rectangular
box is tied
with
a ribbon
that
ribbon
crosses
the
box
at the midpoints
of its
sides.
If theso
box
is 8the
inches
long,
6 inches
box
at
the
midpoints
of
its
sides.
If
the
box
is
8
inches
long,
6
inches
wide, and 5 inches high, how long is the ribbon?
wide, and 5 inches high, how long is the ribbon?
5 in.
5 in.
6 in.
6 in.
8 in.
8 in.
2. Complete each of the following diagrams so that it can be folded into
2. aComplete
eachlike
of the
diagrams
so that
it can
be folded
cube exactly
thisfollowing
one. (Each
of the three
lines
(white,
dottedinto
and
a
cubeshould
exactlywrap
like this
one. (Each
of the
lines (white, dotted and
black)
completely
around
thethree
cube.)
black) should wrap completely around the cube.)
3. What is the length of the shortest string that can stretch from point X
3. What is the length of the shortest string that can stretch from point X
to point Y along the outside of the box in the diagram below? (The box’s
to point Y along the outside of the box in the diagram below? (The box’s
dimensions are given in inches.)
dimensions are given in inches.)
4. Draw a two-dimensional pattern that, when folded, will make a square
4. Draw a two-dimensional pattern that, when folded, will make a square
pyramid as shown below.
pyramid as shown below.
Things to Think About:
Things to Think About:
• Which patterns that fold to make a box might a manufacturer use? Why?
• Which patterns that fold to make a box might a manufacturer use? Why?
• Why are parallel lines important when drawing two-dimensional pictures
• Why
are parallel
lines importantshapes?
when drawing two-dimensional pictures
to represent
three-dimensional
to represent three-dimensional shapes?
• How do painters create two-dimensional pictures that appear to have
• three
How do
painters create two-dimensional pictures that appear to have
dimensions?
three dimensions?
• What two-dimensional shape is used to make a quart milk carton?
• What two-dimensional shape is used to make a quart milk carton?
• The images you see at your local movie theater are stored on flat strips
• The
images
see at
your local
theater are stored
flat strips
of film.
Whyyou
do they
appear
to bemovie
three-dimensional
on theon
screen?
of film. Why do they appear to be three-dimensional on the screen?
• Why are parallelograms used in making cardboard tubes for rolls of
• Why
parallelograms used in making cardboard tubes for rolls of
paperare
towels?
paper towels?
Did You Know That?
Did You Know That?
• The shortest air route from New York to London passes over Iceland.
• It
The
shortest
air route
from
New
York toline.
London passes over Iceland.
is part
of a great
circle,
not
a straight
It is part of a great circle, not a straight line.
• A CAT scan is based on the x-ray principle: as x-rays pass through the
• body
A CATthey
scanare
is absorbed
based on or
theweakened
x-ray principle:
as x-rays
pass
through
at differing
levels
creating
a the
body
they
are
absorbed
or
weakened
at
differing
levels
creating
a
matrix or profile of x-ray beams of different strength. This x-ray profile
matrix
or profile
of x-ray
beams
of different
strength.
This x-ray
is registered
on film,
thus
creating
an image.
(CAT stands
for profile
is
registered onAxial
film,Tomography.)
thus creating an image. (CAT stands for
Computerized
Computerized Axial Tomography.)
• Hospitals often use a continuous colored line on the floor to direct
• patients
Hospitalsfrom
oftenplace
use to
a continuous
line on the floor to direct
place in thecolored
building.
patients from place to place in the building.
• Some historical sites use lines on the pavement to indicate the path
• of
Some
historical
sitesOne
use example
lines on the
pavement
indicate
the path
self-guided
tours.
is the
FreedomtoTrail
in Boston,
of
self-guided tours. One example is the Freedom Trail in Boston,
Massachusetts.
Massachusetts.
Resources:
Resources:
Books:
Books:
• School Mathematics Project. Shape Review Book. Cambridge, England:
• Cambridge
School Mathematics
Shape Review Book. Cambridge, England:
UniversityProject.
Press, 1991.
Cambridge University Press, 1991.
• Winter, M., G. Lappan, E. Phillips, and W. Fitzgerald. Middle Grades
• Winter,
M., G.Project:
Lappan,Spatial
E. Phillips,
and W. Fitzgerald.
Middle
Grades
Mathematics
Visualization.
Menlo Park,
CA: AddisonMathematics
Project:
Spatial
Visualization.
Menlo
Park,
CA:
AddisonWesley Publishing Company, 1986.
Wesley Publishing Company, 1986.
Website:
Website:
• www.imaginis.com/ct-scan/how_ct.asp
• www.imaginis.com/ct-scan/how_ct.asp
2 • Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource
TC
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AY
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#
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Same birth month?
No Way!
Figure This! In any group of six people, what is the
probability that everyone was born in different months?
Hint: If there were only two people in a room, what
is the probability they were born in different months?
What if there were three people in the room?
The probability of an event is a measure of the
chance that it will occur. Insurance companies use
probability to set rates, weather forecasters use
probability to predict weather patterns, and doctors
use probability to determine treatments for disease.
© 2004 National Council of Teachers of Mathematics, www.figurethis.org/.
Used with permission.
Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource • 1
Get Started:
Since there are 12 months in a year, a person’s birth month has 12
different possibilities. For this problem, to make it simpler, assume the
probability of being born in any given month is 1/12. If there are two
people in the room, what is the probability that the second person’s
birthday occurred in a different month from the first person’s? If there are
three people in a room, what is the probability that all three people have
birthdays in different months?
Complete Solution:
With two people, the probability that the second person’s birth month
does not match the first person’s is 11/12.
A third person could be born in any of the other ten months and not match
the first two. The probability that the third person’s birth month will not
match either of the other two is 10/12. The probability that all three have
different birth months is:
Things to Think About:
• With 30 strangers in a room, the probability of having at least two
matching birthdays is about 71%.
• The probability that two people were born on the same day is different
from the probability that two people were born on a given day.
• How many US presidents have had the same birthday? How many
of them died on the same day?
• Think of an event that has a 100% probability of occurring.
Did You Know That?
• The probability that an event will occur and the probability that an event
will not occur are called “complementary” probabilities. The sum of
complementary probabilities is 1 or 100%.
• A probability of 0 means that the event cannot occur.
• The probability of any event is a percentage ranging from 0% to 100%.
Continuing this process, the probability that six people have different birth
months is:
Thus, the probability that at least two of the six people do not have the
same birth month is about 22%.
Resources:
Books:
• Jacobs, H. Mathematics, A Human Endeavor. San Francisco, CA:
W. H. Freeman and Co., 1970.
Websites:
• forum.swarthmore.edu/~isaac/problems/prob1.html
Try This:
• Choose a group of people, such as a classroom of students, a group of
friends, or the members of a club. Count the number of people in the
group and determine the probability that at least two of them were born
in the same month.
• Check several classes, or groups of people, of the same size. In how
many of the groups do two people share the same birthday? Use the
results to estimate the probability that two people in a group that size
share the same birthday.
Additional Challenges:
(Answers located in back of booklet)
1. About how many people does it take to have a 99% probability that
at least two have the same birth month?
2. What is the probability that if you know the birthday of one person in
a room, a second person has the same birthday?
3. What is the least number of people required to make sure that at least
two have matching birthdays?
2 • Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource
77
VI
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O
Y
#
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Do movies
$
money
??
$
make
$
?
Figure This! The theater box office receipts for
the movie Ratio in Magicville for the past four weeks
were $15,000, $12,000, $12,000, and $10,000,
respectively. The theater owner pays the movie
distributor the following percentages of the
box-office receipts: 70% for each of the first two
weeks, 60% for the third week, and 50% for the
fourth week. Other operating expenses are $4500
per week. Did the theater make or lose money?
Hint: How much did the theater have to pay the
distributor the first week to show the movie?
Profit and loss calculations are essential to financing
and operating any business. Business owners, shippers,
accountants, and bankers all carefully examine income
and expenses when making financial decisions.
© 2004 National Council of Teachers of Mathematics, www.figurethis.org/.
Used with permission.
Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource • 1
2. For additional challenge 1, what would have to be the box office
receipts for the profit to be $500 for week 5 and $500 for week 6?
Get Started:
Make a table to keep track of the profit (or loss) each week.
Week
Box-Office
Receipts
1
$15,000
2
$12,000
3
$12,000
4
$10,000
Percentage
Amount to
to
Distributor
Distributor
Other
Expenses
Profit
(or Loss)
3. A theater’s typical profit for popcorn is around 85% of the price paid by
the patron. If you paid $4.00 for a bag of popcorn, about how much did
it cost the theater owner?
4. A theater charges $8.00 for admission. At the concession stand,
a 32-ounce soda which costs $0.30 to produce is sold for $3.00. During
the first four weeks of Ratio in Magicville, 40% of moviegoers bought
sodas. About how much did the theater earn on soda?
Things to Think About:
Total
Complete Solution:
For each week, subtract the distributor’s share and other expenses from
the theater box-office receipts. The result is the weekly profit (or loss).
See the table. After four weeks, the theater has a loss of $100.
Other
Expenses,
E
Profit or
(Loss)
R–A=E
$10,500
$4500
$0
$8400
$4500
–$900
60%
$7200
$4500
$300
50%
$5000
$4500
$500
Total
–$100
Percentage Amount,
p, to
A = Rp to
Distributor Distributor
Week
Box-Office
Receipts, R
1
$15,000
70%
2
$12,000
70%
3
$12,000
4
$10,000
Try This:
• Theater owners generally consider that it takes 10,000 ticket buyers to
support one movie screen. Based on this rule, determine if your town
has too many screens.
• Check your local newspaper for a report on box-office receipts, or consult an appropriate website such as
www.film.com/reviews/rev_box/index.jhtml.
• Identify the movie that has had the longest run at your local theaters.
• Do some research in publications such as Entertainment Weekly to
identify the box-office receipts made by your favorite movies.
Additional Challenges:
(Answers located in back of booklet)
1. Suppose the theater in the challenge extends the contract for Ratio in
Magicville. For the next two weeks, the distributor's percentages will
be 40% and 30%, respectively. If the theater hopes to earn a profit of
at least $1000 over the next two weeks, how much will have to be
taken in at the box office?
• Why have many movie theater chains filed for bankruptcy in recent
years?
• What is the most popular concession item sold at movies?
• Is it fair to compare box office receipts for movies such as Titanic and
Gone with the Wind?
• What kind of movies have the longest box-office runs?
Did You Know That?
• Theater owners generally do not like long movies (three hours or more)
because there’s time for only one nightly showing per screen. This
means fewer opportunities for concession sales and ticket sales.
• The bulbs for a movie projector cost about $750 each.
• Movie theaters make 95% of their money on weekends.
• Only 20% of movies are highly profitable.
• In 1999, the United States had about 37,000 movie screens.
• As of May, 2001, the current leading money-maker was Titanic, which
has grossed over $600.8 million.
Resources:
Newspaper:
• Dallas Morning News. Nov. 26, 2000.
Websites:
• www.howstuffworks.com/movie-distribution3.htm
• www.hollywood.com/nato
• faculty.washington.edu/baldasty/200.htm
• www.film.com
• www.imdb.com
• www.film.com/reviews/rev_box/index.jhtml
2 • Grade 9: Activity Centre Cards (Green) • Mathematics 7–9: Home–School Communication Resource